From 335c4f54318adc08c45c92e25cf80fa37d2bbf62 Mon Sep 17 00:00:00 2001 From: MichelJuillard Date: Tue, 19 Dec 2023 22:18:53 +0100 Subject: [PATCH] reorganize documentation --- {doc => docs}/Project.toml | 0 .../development/algorighms_derivatives.aux | 6 + docs/build/development/derivative_UQME.aux | 2 + .../development/derivativs_eigenvalue.aux | 2 + .../development/derivativs_eigenvalue.pdf | Bin 0 -> 35396 bytes .../development/derivativs_eigenvalue.tex | 22 + docs/build/development/distributions.aux | 4 + docs/build/development/distributions.pdf | Bin 0 -> 132679 bytes .../development}/distributions.tex | 0 .../development/lre_solution_derivatives.aux | 6 + docs/build/orig/the-model-file.md.new | 10384 ++++++++++++++++ {doc => docs}/make.jl | 0 docs/src/development/Outputs.md | 17 + .../src/development/RecursiveBlocks.blocks.md | 286 + docs/src/{ => development}/RecursiveBlocks.md | 0 .../development/algorighms_derivatives.aux | 6 + docs/src/development/derivative_UQME.aux | 2 + .../src/development/derivativs_eigenvalue.aux | 2 + .../src/development/derivativs_eigenvalue.pdf | Bin 0 -> 35396 bytes .../src/development/derivativs_eigenvalue.tex | 22 + docs/src/development/distributions.aux | 4 + docs/src/development/distributions.pdf | Bin 0 -> 132679 bytes docs/src/development/distributions.tex | 94 + docs/src/{ => development}/example1.md | 0 .../development/lre_solution_derivatives.aux | 6 + .../src/{ => development}/perfectforesight.md | 0 {doc => docs}/src/index.md | 0 .../src/installation-and-configuration.md | 0 {doc => docs}/src/introduction.md | 0 {doc => docs}/src/macroprocessor.md | 0 .../model-file/deterministic-simulations.md | 0 {doc => docs}/src/model-file/estimation.md | 0 .../src/model-file/filtersmoother.md | 0 {doc => docs}/src/model-file/forecasting.md | 0 .../model-file/initial-terminal-conditions.md | 0 .../src/model-file/local-approxiation.md | 0 .../src/model-file/model-declaration.md | 0 .../src/model-file/optimal-policy.md | 0 {doc => docs}/src/model-file/shocks.md | 0 {doc => docs}/src/model-file/steady-state.md | 0 .../src/model-file/syntax-elements.md | 0 .../src/model-file/variable-declarations.md | 0 {doc => docs}/src/orig/bibliography.md | 0 .../src/orig/dynare-misc-commands.md | 0 {doc => docs}/src/orig/examples.md | 0 {doc => docs}/src/orig/index.md | 0 .../orig/installation-and-configuration.md | 0 {doc => docs}/src/orig/introduction.md | 0 {doc => docs}/src/orig/reporting.md | 0 {doc => docs}/src/orig/running-dynare.md | 0 .../src/orig/the-configuration-file.md | 0 {doc => docs}/src/orig/the-model-file.md | 0 docs/src/orig/the-model-file.md.new | 10384 ++++++++++++++++ {doc => docs}/src/orig/time-series.md | 0 {doc => docs}/src/running-dynare.md | 0 {doc => docs}/src/the-model-file.md | 0 56 files changed, 21249 insertions(+) rename {doc => docs}/Project.toml (100%) create mode 100644 docs/build/development/algorighms_derivatives.aux create mode 100644 docs/build/development/derivative_UQME.aux create mode 100644 docs/build/development/derivativs_eigenvalue.aux create mode 100644 docs/build/development/derivativs_eigenvalue.pdf create mode 100644 docs/build/development/derivativs_eigenvalue.tex create mode 100644 docs/build/development/distributions.aux create mode 100644 docs/build/development/distributions.pdf rename docs/{src => build/development}/distributions.tex (100%) create mode 100644 docs/build/development/lre_solution_derivatives.aux create mode 100644 docs/build/orig/the-model-file.md.new rename {doc => docs}/make.jl (100%) create mode 100644 docs/src/development/Outputs.md create mode 100644 docs/src/development/RecursiveBlocks.blocks.md rename docs/src/{ => development}/RecursiveBlocks.md (100%) create mode 100644 docs/src/development/algorighms_derivatives.aux create mode 100644 docs/src/development/derivative_UQME.aux create mode 100644 docs/src/development/derivativs_eigenvalue.aux create mode 100644 docs/src/development/derivativs_eigenvalue.pdf create mode 100644 docs/src/development/derivativs_eigenvalue.tex create mode 100644 docs/src/development/distributions.aux create mode 100644 docs/src/development/distributions.pdf create mode 100644 docs/src/development/distributions.tex rename docs/src/{ => development}/example1.md (100%) create mode 100644 docs/src/development/lre_solution_derivatives.aux rename docs/src/{ => development}/perfectforesight.md (100%) rename {doc => docs}/src/index.md (100%) rename {doc => docs}/src/installation-and-configuration.md (100%) rename {doc => docs}/src/introduction.md (100%) rename {doc => docs}/src/macroprocessor.md (100%) rename {doc => docs}/src/model-file/deterministic-simulations.md (100%) rename {doc => docs}/src/model-file/estimation.md (100%) rename {doc => docs}/src/model-file/filtersmoother.md (100%) rename {doc => docs}/src/model-file/forecasting.md (100%) rename {doc => docs}/src/model-file/initial-terminal-conditions.md (100%) rename {doc => docs}/src/model-file/local-approxiation.md (100%) rename {doc => docs}/src/model-file/model-declaration.md (100%) rename {doc => docs}/src/model-file/optimal-policy.md (100%) rename {doc => docs}/src/model-file/shocks.md (100%) rename {doc => docs}/src/model-file/steady-state.md (100%) rename {doc => docs}/src/model-file/syntax-elements.md (100%) rename {doc => docs}/src/model-file/variable-declarations.md (100%) rename {doc => docs}/src/orig/bibliography.md (100%) rename {doc => docs}/src/orig/dynare-misc-commands.md (100%) rename {doc => docs}/src/orig/examples.md (100%) rename {doc => docs}/src/orig/index.md (100%) rename {doc => docs}/src/orig/installation-and-configuration.md (100%) rename {doc => docs}/src/orig/introduction.md (100%) rename {doc => docs}/src/orig/reporting.md (100%) rename {doc => docs}/src/orig/running-dynare.md (100%) rename {doc => docs}/src/orig/the-configuration-file.md (100%) rename {doc => docs}/src/orig/the-model-file.md (100%) create mode 100644 docs/src/orig/the-model-file.md.new rename {doc => 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+\begin{document} + +Main results on distributions used in Dynare. We use the +parametrization used in Distrubtions.jl + +\section{Inverse gamma} +\begin{itemize} +\item AKA inverse gamma type 2 +\item shape: $\alpha$ +\item scale: $\theta$ +\end{itemize} +\begin{align*} +\alpha &= \frac{\nu}{2}\\ +\theta &= \frac{s}{2}\\ +X \sim IG_2(\alpha, \theta) &\Leftrightarrow Z = X^{-1} \sim G(\alpha, \theta)\\ +\mathbb{E}(X) &= \frac{\theta}{\alpha - 1}\\ +\mathbb{V}ar(X) &= \frac{1}{\alpha - 2}[\mathbb{E}(X)]^2 \mbox{ for + }\alpha > 2\\ + \alpha &= 2+\frac{[\mathbb{E}(X)]^2}{\mathbb{V}ar(X)}\\ + \theta &= (\alpha - 1)\mathbb{E}(X)\\ +f_{IG_2}(x, \alpha, \theta) &= + \frac{\theta^\alpha}{\Gamma(\alpha)}x^{-(\alpha+1)}e^{-\frac{\theta}{x}}\\ +\end{align*} + +\section{Inverse gamma type I} +\begin{align*} + X \sim IG_2(\alpha, \theta) &\Leftrightarrow Y = \sqrt{X} \sim IG_1(\alpha, \theta)\\ + & \Leftrightarrow Z = X^{-1} \sim G(\alpha, \theta)\\ +\mathbb{E}(Y) &= \sqrt{\theta}\frac{\Gamma(\alpha-\frac{1}{2})}{\Gamma(\alpha)} \mbox{ for }\alpha > \frac{1}{2} \\ +\mathbb{V}ar(Y) &= \frac{\theta}{\alpha - 1} - [\mathbb{E}(Y)]^2 \mbox{ for }\alpha > 1\\ +\mbox{mode }Y &= \sqrt{\frac{\theta}{\alpha + \frac{1}{2}}}\\ + f_{IG_1}(y, \alpha, \theta) &= f_{IG_2}(y^2, \alpha, \theta)|2y|\\ + &= 2\frac{\theta^\alpha}{\Gamma(\alpha)}y^{-(2\alpha+1)}e^{-\frac{\theta}{y^2}}\\ +\end{align*} +$\alpha$ solves +\[ +(\alpha - 1)\left(\mathbb{V}ar(Y) + [\mathbb{E}(Y)]^2\right)- +\left[\mathbb{E}(Y)\right]^2\frac{\Gamma(\alpha)}{\Gamma(\alpha-\frac{1}{2})} =0 +\] +and $\theta = (\alpha - 1)(\mathbb{V}ar(Y) + [\mathbb{E}(Y)]^2)$ + +\section*{Appendices} +\subsection*{Bauwens et (1999)} + +\begin{align*} +X \sim IG_2(s, \nu) &\Leftrightarrow Y = \sqrt{X} \sim IG_1(s, \nu)\\ +& \Leftrightarrow Z = X^{-1} \sim G(\frac{\nu}{2}, \frac{2}{s})\\ +\mathbb{E}(X) &= \frac{s}{\nu - 2}\\ +\mathbb{V}ar(X) &= \frac{2}{\nu - 4}[\mathbb{E}(X)]^2 \mbox{ for }\nu > 4\\ +\mathbb{E}(Y) &= \sqrt{\frac{s}{2}}\frac{\Gamma\left(\frac{\nu +-1}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)} \mbox{ for }\nu > 1 \\ +\mathbb{V}ar(Y) &= \frac{s}{\nu - 2} - [\mathbb{E}(Y)]^2 \mbox{ for }\nu > 2\\ +f_{IG_2}(x|\frac{\nu}{2}, \frac{2}{s}) &= \frac{1}{\Gamma(\frac{\nu}{2})\left(\frac{2}{s}\right)^{\frac{\nu}{2}}}x^{-\frac{1}{2}(\nu+2)}e^{-\frac{s}{2x}} \\ +f_{IG_2}(x|\alpha,\theta) &= + \frac{1}{\Gamma(\alpha)\theta^\alpha} + x^{-(\alpha+1)}e^{-\frac{1}{sx}} +\end{align*} + +\subsection*{Inverse gamma type I} +\subsubsection*{Mode derivation} +\begin{align*} + \max_x y &= + 2\frac{\theta^\alpha}{\Gamma(\alpha)}x^{-(2\alpha+1)}e^{-\frac{\theta}{x^2}}\\ + \frac{dy}{dx} &= -\left[(2\alpha+1)x^{-1}-2\theta x^{-3}\right]y\\ + &= 0\\ + x^\star &= \sqrt{\frac{\theta}{\alpha + \frac{1}{2}}} +\end{align*} +\subsubsection*{Obtaining $\alpha$ and $\theta$ from mean and + variance} +\begin{align*} + \theta &= + \left[\mathbb{E}(Y)\right]^2\left(\frac{\Gamma(\alpha)}{\Gamma(\alpha-\frac{1}{2})}\right)^2\\ + \theta &= (\alpha - 1)\left(\mathbb{V}ar(Y) + [\mathbb{E}(Y)]^2\right) +\end{align*} +Solve numerically +\[ +(\alpha - 1)\left(\mathbb{V}ar(Y) + [\mathbb{E}(Y)]^2\right)- +\left[\mathbb{E}(Y)\right]^2\left(\frac{\Gamma(\alpha)}{\Gamma(\alpha-\frac{1}{2})}\right)^2 =0 +\] +or taking the logarithm for numerical stabilitye: +\[ +\ln(\alpha - 1) + \ln\left(\mathbb{V}ar(Y) + [\mathbb{E}(Y)]^2\right)- +\ln\left[\mathbb{E}(Y)\right]^2 - 2*\ln\left(\frac{\Gamma(\alpha)}{\Gamma(\alpha-\frac{1}{2})}\right) =0 +\] + +\end{document} + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: diff --git a/docs/src/example1.md b/docs/src/development/example1.md similarity index 100% rename from docs/src/example1.md rename to docs/src/development/example1.md diff --git a/docs/src/development/lre_solution_derivatives.aux b/docs/src/development/lre_solution_derivatives.aux new file mode 100644 index 00000000..753dc682 --- /dev/null +++ b/docs/src/development/lre_solution_derivatives.aux @@ -0,0 +1,6 @@ +\relax 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to docs/src/orig/the-model-file.md diff --git a/docs/src/orig/the-model-file.md.new b/docs/src/orig/the-model-file.md.new new file mode 100644 index 00000000..a488f595 --- /dev/null +++ b/docs/src/orig/the-model-file.md.new @@ -0,0 +1,10384 @@ +::: {.default-domain} +dynare +::: + +The model file {#model-file} +============== + +Conventions {#conv} +----------- + +A model file contains a list of commands and of blocks. Each command and +each element of a block is terminated by a semicolon (;). Blocks are +terminated by `end;`. + +If Dynare encounters an unknown expression at the beginning of a line or +after a semicolon, it will parse the rest of that line as native MATLAB +code, even if there are more statements separated by semicolons present. +To prevent cryptic error messages, it is strongly recommended to always +only put one statement/command into each line and start a new line after +each semicolon.[^1] + +Lines of codes can be commented out line by line or as a block. +Single-line comments begin with `//` and stop at the end of the line. +Multiline comments are introduced by `/*` and terminated by `*/`. + +*Examples* + +> // This is a single line comment +> +> var x; // This is a comment about x +> +> /* This is another inline comment about alpha */ alpha = 0.3; +> +> /* +> This comment is spanning +> two lines. +> */ + +Note that these comment marks should not be used in native MATLAB code +regions where the [%]{.title-ref} should be preferred instead to +introduce a comment. In a `verbatim` block, see +`verbatim`{.interpreted-text role="ref"}, this would result in a crash +since `//` is not a valid MATLAB statement). + +Most Dynare commands have arguments and several accept options, +indicated in parentheses after the command keyword. Several options are +separated by commas. + +In the description of Dynare commands, the following conventions are +observed: + +- Optional arguments or options are indicated between square brackets: + '\[\]'; +- Repeated arguments are indicated by ellipses: "\..."; +- Mutually exclusive arguments are separated by vertical bars: '\|'; +- INTEGER indicates an integer number; +- INTEGER\_VECTOR indicates a vector of integer numbers separated by + spaces, enclosed by square brackets; +- DOUBLE indicates a double precision number. The following syntaxes + are valid: `1.1e3`, `1.1E3`, `1.1d3`, `1.1D3`. In some places, + infinite Values `Inf` and `-Inf` are also allowed; +- NUMERICAL\_VECTOR indicates a vector of numbers separated by spaces, + enclosed by square brackets; +- EXPRESSION indicates a mathematical expression valid outside the + model description (see `expr`{.interpreted-text role="ref"}); +- MODEL\_EXPRESSION (sometimes MODEL\_EXP) indicates a mathematical + expression valid in the model description (see + `expr`{.interpreted-text role="ref"} and + `model-decl`{.interpreted-text role="ref"}); +- MACRO\_EXPRESSION designates an expression of the macro processor + (see `macro-exp`{.interpreted-text role="ref"}); +- VARIABLE\_NAME (sometimes VAR\_NAME) indicates a variable name + starting with an alphabetical character and can't contain: + '()+-\*/\^=!;:@\#.' or accentuated characters; +- PARAMETER\_NAME (sometimes PARAM\_NAME) indicates a parameter name + starting with an alphabetical character and can't contain: + '()+-\*/\^=!;:@\#.' or accentuated characters; +- LATEX\_NAME (sometimes TEX\_NAME) indicates a valid LaTeX expression + in math mode (not including the dollar signs); +- FUNCTION\_NAME indicates a valid MATLAB function name; +- FILENAME indicates a filename valid in the underlying operating + system; it is necessary to put it between quotes when specifying the + extension or if the filename contains a non-alphanumeric character; +- QUOTED\_STRING indicates an arbitrary string enclosed between + (single) quotes. + +Variable declarations {#var-decl} +--------------------- + +While Dynare allows the user to choose their own variable names, there +are some restrictions to be kept in mind. First, variables and +parameters must not have the same name as Dynare commands or built-in +functions. In this respect, Dynare is not case-sensitive. For example, +do not use `Ln` or `Sigma_e` to name your variable. Not conforming to +this rule might yield hard-to-debug error messages or crashes. Second, +when employing user-defined steady state files it is recommended to +avoid using the name of MATLAB functions as this may cause conflicts. In +particular, when working with user-defined steady state files, do not +use correctly-spelled greek names like [alpha]{.title-ref}, because +there are MATLAB functions of the same name. Rather go for `alppha` or +`alph`. Lastly, please do not name a variable or parameter `i`. This may +interfere with the imaginary number i and the index in many loops. +Rather, name investment `invest`. Using `inv` is also not recommended as +it already denotes the inverse operator. Commands for declaring +variables and parameters are described below. + +::: {.command} +var VAR\_NAME \[\$TEX\_NAME\$\] +\[(long\_name=QUOTED\_STRINGNAME=QUOTED\_STRING)\]\...; +var(deflator=MODEL\_EXPR) VAR\_NAME (\... same options apply) var(log, +deflator=MODEL\_EXPR) VAR\_NAME (\... same options apply) +var(log\_deflator=MODEL\_EXPR) VAR\_NAME (\... same options apply) + +This required command declares the endogenous variables in the model. +See `conv`{.interpreted-text role="ref"} for the syntax of *VAR\_NAME* +and *MODEL\_EXPR*. Optionally it is possible to give a LaTeX name to the +variable or, if it is nonstationary, provide information regarding its +deflator. The variables in the list can be separated by spaces or by +commas. `var` commands can appear several times in the file and Dynare +will concatenate them. Dynare stores the list of declared parameters, in +the order of declaration, in a column cell array `M_.endo_names`. + +If the model is nonstationary and is to be written as such in the +`model` block, Dynare will need the trend deflator for the appropriate +endogenous variables in order to stationarize the model. The trend +deflator must be provided alongside the variables that follow this +trend. + +*Options* + +::: {.option} +log + +In addition to the endogenous variable(s) thus declared, this option +also triggers the creation of auxiliary variable(s) equal to the log of +the corresponding endogenous variable(s). For example, given a +`var(log) y` statement, two endogenous will be created (`y` and +`LOG_y`), and an auxiliary equation linking the two will also be added +(equal to `LOG_y = log(y)`). Moreover, every occurence of `y` in the +model will be replaced by `exp(LOG_y)`. This option is for example +useful when one wants to perform a loglinear approximation of some +variable(s) in the context of a first-order stochastic approximation; or +when one wants to ensure the variable(s) stay(s) in the definition +domain of the function defining the steady state or the dynamic +residuals when the nonlinear solver is used. +::: + +::: {.option} +deflator = MODEL\_EXPR + +The expression used to detrend an endogenous variable. All trend +variables, endogenous variables and parameters referenced in MODEL\_EXPR +must already have been declared by the `trend_var, log_trend_var, var` +and `parameters` commands. The deflator is assumed to be multiplicative; +for an additive deflator, use `log_deflator`. This option can be used +together with the `log` option (the latter must come first). +::: + +::: {.option} +log\_deflator = MODEL\_EXPR + +Same as `deflator`, except that the deflator is assumed to be additive +instead of multiplicative (or, to put it otherwise, the declared +variable is equal to the log of a variable with a multiplicative trend). +This option cannot be used together with the `log` option, because it +would not make much sense from an economic point of view (the +corresponding auxiliary variable would correspond to the log taken two +times on a variable with a multiplicative trend). +::: + +::: {#long-name} +::: {.option} +long\_name = QUOTED\_STRING + +This is the long version of the variable name. Its value is stored in +`M_.endo_names_long` (a column cell array, in the same order as +`M_.endo_names`). In case multiple `long_name` options are provided, the +last one will be used. Default: `VAR_NAME`. +::: +::: + +::: {#partitioning} +::: {.option} +NAME = QUOTED\_STRING + +This is used to create a partitioning of variables. It results in the +direct output in the `.m` file analogous to: +`M_.endo_partitions.NAME = QUOTED_STRING`;. +::: +::: + +*Example (variable partitioning)* + +> var c gnp cva (country=`US', state=`VA') +> cca (country=`US', state=`CA', long_name=`Consumption CA'); +> var(deflator=A) i b; +> var c $C$ (long_name=`Consumption'); +::: + +::: {.command} +varexo\_det VAR\_NAME \[\$TEX\_NAME\$\] +\[(long\_name=QUOTED\_STRING\|NAME=QUOTED\_STRING)\...\]; + +This optional command declares exogenous deterministic variables in a +stochastic model. See `conv`{.interpreted-text role="ref"} for the +syntax of VARIABLE\_NAME. Optionally it is possible to give a LaTeX name +to the variable. The variables in the list can be separated by spaces or +by commas. `varexo_det` commands can appear several times in the file +and Dynare will concatenate them. + +It is possible to mix deterministic and stochastic shocks to build +models where agents know from the start of the simulation about future +exogenous changes. In that case `stoch_simul` will compute the rational +expectation solution adding future information to the state space +(nothing is shown in the output of `stoch_simul`) and forecast will +compute a simulation conditional on initial conditions and future +information. + +Note that exogenous deterministic variables cannot appear with a lead or +a lag in the model. + +*Options* + +::: {.option} +long\_name = QUOTED\_STRING + +Like `long_name `{.interpreted-text role="ref"} but value +stored in `M_.exo_det_names_long`. +::: + +::: {.option} +NAME = QUOTED\_STRING + +Like `partitioning `{.interpreted-text role="ref"} but +QUOTED\_STRING stored in `M_.exo_det_partitions.NAME`. +::: + +*Example* + +> varexo m gov; +> varexo_det tau; +::: + +::: {.command} +var\_remove VAR\_NAME \| PARAM\_NAME\...; + +Removes the listed variables (or parameters) from the model. Removing a +variable that has already been used in a model equation or elsewhere +will lead to an error. +::: + +::: {.command} +predetermined\_variables VAR\_NAME\...; + +In Dynare, the default convention is that the timing of a variable +reflects when this variable is decided. The typical example is for +capital stock: since the capital stock used at current period is +actually decided at the previous period, then the capital stock entering +the production function is `k(-1)`, and the law of motion of capital +must be written: + + k = i + (1-delta)*k(-1) + +Put another way, for stock variables, the default in Dynare is to use a +"stock at the end of the period" concept, instead of a "stock at the +beginning of the period" convention. + +The `predetermined_variables` is used to change that convention. The +endogenous variables declared as predetermined variables are supposed to +be decided one period ahead of all other endogenous variables. For stock +variables, they are supposed to follow a "stock at the beginning of the +period" convention. + +Note that Dynare internally always uses the "stock at the end of the +period" concept, even when the model has been entered using the +`predetermined_variables` command. Thus, when plotting, computing or +simulating variables, Dynare will follow the convention to use variables +that are decided in the current period. For example, when generating +impulse response functions for capital, Dynare will plot `k`, which is +the capital stock decided upon by investment today (and which will be +used in tomorrow's production function). This is the reason that capital +is shown to be moving on impact, because it is `k` and not the +predetermined `k(-1)` that is displayed. It is important to remember +that this also affects simulated time series and output from smoother +routines for predetermined variables. Compared to non-predetermined +variables they might otherwise appear to be falsely shifted to the +future by one period. + +*Example* + +> The following two program snippets are strictly equivalent. +> +> Using default Dynare timing convention: +> +> var y, k, i; +> ... +> model; +> y = k(-1)^alpha; +> k = i + (1-delta)*k(-1); +> ... +> end; +> +> Using the alternative timing convention: +> +> var y, k, i; +> predetermined_variables k; +> ... +> model; +> y = k^alpha; +> k(+1) = i + (1-delta)*k; +> ... +> end; +::: + +::: {.command} +trend\_var (growth\_factor = MODEL\_EXPR) VAR\_NAME +\[\$LATEX\_NAME\$\]\...; + +This optional command declares the trend variables in the model. See +`conv`{.interpreted-text role="ref"} for the syntax of MODEL\_EXPR and +VAR\_NAME. Optionally it is possible to give a LaTeX name to the +variable. + +The variable is assumed to have a multiplicative growth trend. For an +additive growth trend, use `log_trend_var` instead. + +Trend variables are required if the user wants to be able to write a +nonstationary model in the `model` block. The `trend_var` command must +appear before the var command that references the trend variable. + +`trend_var` commands can appear several times in the file and Dynare +will concatenate them. + +If the model is nonstationary and is to be written as such in the +`model` block, Dynare will need the growth factor of every trend +variable in order to stationarize the model. The growth factor must be +provided within the declaration of the trend variable, using the +`growth_factor` keyword. All endogenous variables and parameters +referenced in MODEL\_EXPR must already have been declared by the var and +parameters commands. + +*Example* + +> trend_var (growth_factor=gA) A; +::: + +::: {.command} +model\_local\_variable VARIABLE\_NAME \[LATEX\_NAME\]\... ; + +This optional command declares a model local variable. See +`conv`{.interpreted-text role="ref"} for the syntax of VARIABLE\_NAME. +As you can create model local variables on the fly in the model block +(see `model-decl`{.interpreted-text role="ref"}), the interest of this +command is primarily to assign a LATEX\_NAME to the model local +variable. + +*Example* + +> model_local_variable GDP_US $GDPUS$; +::: + +### On-the-fly Model Variable Declaration {#on-the-fly-declaration} + +Endogenous variables, exogenous variables, and parameters can also be +declared inside the model block. You can do this in two different ways: +either via the equation tag or directly in an equation. + +To declare a variable on-the-fly in an equation tag, simply state the +type of variable to be declared (`endogenous`, `exogenous`, or +`parameter` followed by an equal sign and the variable name in single +quotes. Hence, to declare a variable `c` as endogenous in an equation +tag, you can type `[endogenous='c']`. + +To perform on-the-fly variable declaration in an equation, simply follow +the symbol name with a vertical line (`|`, pipe character) and either an +`e`, an `x`, or a `p`. For example, to declare a parameter named +`alphaa` in the model block, you could write `alphaa|p` directly in an +equation where it appears. Similarly, to declare an endogenous variable +`c` in the model block you could write `c|e`. Note that in-equation +on-the-fly variable declarations must be made on contemporaneous +variables. + +On-the-fly variable declarations do not have to appear in the first +place where this variable is encountered. + +*Example* + +> The following two snippets are equivalent: +> +> > model; +> > [endogenous='k',name='law of motion of capital'] +> > k(+1) = i|e + (1-delta|p)*k; +> > y|e = k^alpha|p; +> > ... +> > end; +> > delta = 0.025; +> > alpha = 0.36; +> > +> > var k, i, y; +> > parameters delta, alpha; +> > delta = 0.025; +> > alpha = 0.36; +> > ... +> > model; +> > [name='law of motion of capital'] +> > k(1) = i|e + (1-delta|p)*k; +> > y|e = k|e^alpha|p; +> > ... +> > end; + +Expressions {#expr} +----------- + +Dynare distinguishes between two types of mathematical expressions: +those that are used to describe the model, and those that are used +outside the model block (e.g. for initializing parameters or variables, +or as command options). In this manual, those two types of expressions +are respectively denoted by MODEL\_EXPRESSION and EXPRESSION. + +Unlike MATLAB or Octave expressions, Dynare expressions are necessarily +scalar ones: they cannot contain matrices or evaluate to matrices.[^2] + +Expressions can be constructed using integers (INTEGER), floating point +numbers (DOUBLE), parameter names (PARAMETER\_NAME), variable names +(VARIABLE\_NAME), operators and functions. + +The following special constants are also accepted in some contexts: + +::: {.constant} +inf + +Represents infinity. +::: + +::: {.constant} +nan + +"Not a number": represents an undefined or unrepresentable value. +::: + +### Parameters and variables + +Parameters and variables can be introduced in expressions by simply +typing their names. The semantics of parameters and variables is quite +different whether they are used inside or outside the model block. + +#### Inside the model + +Parameters used inside the model refer to the value given through +parameter initialization (see `param-init`{.interpreted-text +role="ref"}) or `homotopy_setup` when doing a simulation, or are the +estimated variables when doing an estimation. + +Variables used in a MODEL\_EXPRESSION denote current period values when +neither a lead or a lag is given. A lead or a lag can be given by +enclosing an integer between parenthesis just after the variable name: a +positive integer means a lead, a negative one means a lag. Leads or lags +of more than one period are allowed. For example, if `c` is an +endogenous variable, then `c(+1)` is the variable one period ahead, and +`c(-2)` is the variable two periods before. + +When specifying the leads and lags of endogenous variables, it is +important to respect the following convention: in Dynare, the timing of +a variable reflects when that variable is decided. A control variable +--- which by definition is decided in the current period --- must have +no lead. A predetermined variable --- which by definition has been +decided in a previous period --- must have a lag. A consequence of this +is that all stock variables must use the "stock at the end of the +period" convention. + +Leads and lags are primarily used for endogenous variables, but can be +used for exogenous variables. They have no effect on parameters and are +forbidden for local model variables (see Model declaration). + +#### Outside the model + +When used in an expression outside the model block, a parameter or a +variable simply refers to the last value given to that variable. More +precisely, for a parameter it refers to the value given in the +corresponding parameter initialization (see +`param-init`{.interpreted-text role="ref"}); for an endogenous or +exogenous variable, it refers to the value given in the most recent +`initval` or `endval` block. + +### Operators + +The following operators are allowed in both MODEL\_EXPRESSION and +EXPRESSION: + +- Binary arithmetic operators: `+`, `-`, `*`, `/`, `^` +- Unary arithmetic operators: `+`, `-` +- Binary comparison operators (which evaluate to either 0 or 1): `<`, + `>`, `<=`, `>=`, `==`, `!=` + +Note the binary comparison operators are differentiable everywhere +except on a line of the 2-dimensional real plane. However for +facilitating convergence of Newton-type methods, Dynare assumes that, at +the points of non-differentiability, the partial derivatives of these +operators with respect to both arguments is equal to 0 (since this is +the value of the partial derivatives everywhere else). + +The following special operators are accepted in MODEL\_EXPRESSION (but +not in EXPRESSION): + +::: {.operator} +STEADY\_STATE (MODEL\_EXPRESSION) + +This operator is used to take the value of the enclosed expression at +the steady state. A typical usage is in the Taylor rule, where you may +want to use the value of GDP at steady state to compute the output gap. + +Exogenous and exogenous deterministic variables may not appear in +MODEL\_EXPRESSION. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +The concept of a steady state is ambiguous in a perfect foresight +context with permament and potentially anticipated shocks occuring. +Dynare will use the contents of `oo_.steady_state` as its reference for +calls to the `STEADY_STATE()`-operator. In the presence of `endval`, +this implies that the terminal state provided by the user is used. This +may be a steady state computed by Dynare (if `endval` is followed by +`steady`) or simply the terminal state provided by the user (if `endval` +is not followed by `steady`). Put differently, Dynare will not +automatically compute the steady state conditional on the specificed +value of the exogenous variables in the respective periods. +::: +::: + +::: {.operator} +EXPECTATION (INTEGER) (MODEL\_EXPRESSION) + +This operator is used to take the expectation of some expression using a +different information set than the information available at current +period. For example, `EXPECTATION(-1)(x(+1))` is equal to the expected +value of variable x at next period, using the information set available +at the previous period. See `aux-variables`{.interpreted-text +role="ref"} for an explanation of how this operator is handled +internally and how this affects the output. +::: + +### Functions + +#### Built-in functions + +The following standard functions are supported internally for both +MODEL\_EXPRESSION and EXPRESSION: + +::: {.function} +exp(x) + +Natural exponential. +::: + +::: {.function} +log(x) +::: + +::: {.function} +ln(x) + +Natural logarithm. +::: + +::: {.function} +log10(x) + +Base 10 logarithm. +::: + +::: {.function} +sqrt(x) + +Square root. +::: + +::: {.function} +cbrt(x) + +Cube root. +::: + +::: {.function} +sign(x) + +Signum function, defined as: + +> $$\begin{aligned} +> \textrm{sign}(x) = +> \begin{cases} +> -1 &\quad\text{if }x<0\\ +> 0 &\quad\text{if }x=0\\ +> 1 &\quad\text{if }x>0 +> \end{cases} +> \end{aligned}$$ + +Note that this function is not continuous, hence not differentiable, at +$x=0$. However, for facilitating convergence of Newton-type methods, +Dynare assumes that the derivative at $x=0$ is equal to $0$. This +assumption comes from the observation that both the right- and +left-derivatives at this point exist and are equal to $0$, so we can +remove the singularity by postulating that the derivative at $x=0$ is +$0$. +::: + +::: {.function} +abs(x) + +Absolute value. + +Note that this continuous function is not differentiable at $x=0$. +However, for facilitating convergence of Newton-type methods, Dynare +assumes that the derivative at $x=0$ is equal to $0$ (even if the +derivative does not exist). The rational for this mathematically +unfounded definition, rely on the observation that the derivative of +$\mathrm{abs}(x)$ is equal to $\mathrm{sign}(x)$ for any $x\neq 0$ in +$\mathbb R$ and from the convention for the value of $\mathrm{sign}(x)$ +at $x=0$). +::: + +::: {.function} +sin(x) +::: + +::: {.function} +cos(x) +::: + +::: {.function} +tan(x) +::: + +::: {.function} +asin(x) +::: + +::: {.function} +acos(x) +::: + +::: {.function} +atan(x) + +Trigonometric functions. +::: + +::: {.function} +sinh(x) +::: + +::: {.function} +cosh(x) +::: + +::: {.function} +tanh(x) +::: + +::: {.function} +asinh(x) +::: + +::: {.function} +acosh(x) +::: + +::: {.function} +atanh(x) + +Hyperbolic functions. +::: + +::: {.function} +max(a, b) +::: + +::: {.function} +min(a, b) + +Maximum and minimum of two reals. + +Note that these functions are differentiable everywhere except on a line +of the 2-dimensional real plane defined by $a=b$. However for +facilitating convergence of Newton-type methods, Dynare assumes that, at +the points of non-differentiability, the partial derivative of these +functions with respect to the first (resp. the second) argument is equal +to $1$ (resp. to $0$) (i.e. the derivatives at the kink are equal to the +derivatives observed on the half-plane where the function is equal to +its first argument). +::: + +::: {.function} +normcdf(x) normcdf(x, mu, sigma) + +Gaussian cumulative density function, with mean *mu* and standard +deviation *sigma*. Note that `normcdf(x)` is equivalent to +`normcdf(x,0,1)`. +::: + +::: {.function} +normpdf(x) normpdf(x, mu, sigma) + +Gaussian probability density function, with mean *mu* and standard +deviation *sigma*. Note that `normpdf(x)` is equivalent to +`normpdf(x,0,1)`. +::: + +::: {.function} +erf(x) + +Gauss error function. +::: + +::: {.function} +erfc(x) + +Complementary error function, *i.e.* +$\mathrm{erfc}(x) = 1-\mathrm{erf}(x)$. +::: + +#### External functions + +Any other user-defined (or built-in) MATLAB or Octave function may be +used in both a MODEL\_EXPRESSION and an EXPRESSION, provided that this +function has a scalar argument as a return value. + +To use an external function in a MODEL\_EXPRESSION, one must declare the +function using the `external_function` statement. This is not required +for external functions used in an EXPRESSION outside of a `model` block +or `steady_state_model` block. + +::: {.command} +external\_function (OPTIONS\...); + +This command declares the external functions used in the model block. It +is required for every unique function used in the model block. + +`external_function` commands can appear several times in the file and +must come before the model block. + +*Options* + +::: {.option} +name = NAME + +The name of the function, which must also be the name of the Julia file +implementing it. This option is mandatory. +::: + +::: {.option} +nargs = INTEGER + +The number of arguments of the function. If this option is not provided, +Dynare assumes `nargs = 1`. +::: + +::: {.option} +first\_deriv\_provided \[= NAME\] + +If NAME is provided, this tells Dynare that the Jacobian is provided as +the only output of the Julia file given as the option argument. If NAME +is not provided, this tells Dynare that the Julia file specified by the +argument passed to NAME returns the Jacobian as its second output +argument. When this option is not provided, Dynare will use finite +difference approximations for computing the derivatives of the function, +whenever needed. +::: + +::: {.option} +second\_deriv\_provided \[= NAME\] + +If NAME is provided, this tells Dynare that the Hessian is provided as +the only output of the Julia file given as the option argument. If NAME +is not provided, this tells Dynare that the Julia file specified by the +argument passed to NAME returns the Hessian as its third output +argument. NB: This option can only be used if the `first_deriv_provided` +option is used in the same `external_function` command. When this option +is not provided, Dynare will use finite difference approximations for +computing the Hessian derivatives of the function, whenever needed. +::: + +*Example* + +> external_function(name = funcname); +> external_function(name = otherfuncname, nargs = 2, first_deriv_provided, second_deriv_provided); +> external_function(name = yetotherfuncname, nargs = 3, first_deriv_provided = funcname_deriv); +::: + +### A few words of warning in stochastic context + +The use of the following functions and operators is strongly discouraged +in a stochastic context: `max`, `min`, `abs`, `sign`, `<`, `>`, `<=`, +`>=`, `==`, `!=`. + +The reason is that the local approximation used by `stoch_simul` or +`estimation` will by nature ignore the non-linearities introduced by +these functions if the steady state is away from the kink. And, if the +steady state is exactly at the kink, then the approximation will be +bogus because the derivative of these functions at the kink is bogus (as +explained in the respective documentations of these functions and +operators). + +Note that `extended_path` is not affected by this problem, because it +does not rely on a local approximation of the mode. + +Parameter initialization {#param-init} +------------------------ + +When using Dynare for computing simulations, it is necessary to +calibrate the parameters of the model. This is done through parameter +initialization. + +The syntax is the following: + + PARAMETER_NAME = EXPRESSION; + +Here is an example of calibration: + + parameters alpha, beta; + + beta = 0.99; + alpha = 0.36; + A = 1-alpha*beta; + +Internally, the parameter values are stored in `M_.params`: + +::: {.matvar} +[M]().params + +Contains the values of model parameters. The parameters are in the order +that was used in the `parameters` command, hence ordered as in +`M_.param_names`. +::: + +The parameter names are stored in `M_.param_names`: + +::: {.matvar} +[M]().param\_names + +Cell array containing the names of the model parameters. +::: + +::: {.matcomm} +get\_param\_by\_name (\'PARAMETER\_NAME\'); + +Given the name of a parameter, returns its calibrated value as it is +stored in `M_.params`. +::: + +::: {.matcomm} +set\_param\_value (\'PARAMETER\_NAME\', MATLAB\_EXPRESSION); + +Sets the calibrated value of a parameter to the provided expression. +This does essentially the same as the parameter initialization syntax +described above, except that it accepts arbitrary MATLAB/Octave +expressions, and that it works from MATLAB/Octave scripts. +::: + +Model declaration {#model-decl} +----------------- + +The model is declared inside a `model` block: + +::: {.block} +model ; model (OPTIONS\...); + +The equations of the model are written in a block delimited by `model` +and `end` keywords. + +There must be as many equations as there are endogenous variables in the +model, except when computing the unconstrained optimal policy with +`ramsey_model`, `ramsey_policy` or `discretionary_policy`. + +The syntax of equations must follow the conventions for +MODEL\_EXPRESSION as described in `expr`{.interpreted-text role="ref"}. +Each equation must be terminated by a semicolon (';'). A normal equation +looks like: + +> MODEL\_EXPRESSION = MODEL\_EXPRESSION; + +When the equations are written in homogenous form, it is possible to +omit the '=0' part and write only the left hand side of the equation. A +homogenous equation looks like: + +> MODEL\_EXPRESSION; + +Inside the model block, Dynare allows the creation of *model-local +variables*, which constitute a simple way to share a common expression +between several equations. The syntax consists of a pound sign (\#) +followed by the name of the new model local variable (which must **not** +be declared as in `var-decl`{.interpreted-text role="ref"}, but may have +been declared by `model_local_variable`{.interpreted-text role="comm"}), +an equal sign, and the expression for which this new variable will +stand. Later on, every time this variable appears in the model, Dynare +will substitute it by the expression assigned to the variable. Note that +the scope of this variable is restricted to the model block; it cannot +be used outside. To assign a LaTeX name to the model local variable, use +the declaration syntax outlined by +`model_local_variable`{.interpreted-text role="comm"}. A model local +variable declaration looks like: + +> \#VARIABLE\_NAME = MODEL\_EXPRESSION; + +It is possible to tag equations written in the model block. A tag can +serve different purposes by allowing the user to attach arbitrary +informations to each equation and to recover them at runtime. For +instance, it is possible to name the equations with a `name`-tag, using +a syntax like: + + model; + + [name = 'Budget constraint']; + c + k = k^theta*A; + + end; + +Here, `name` is the keyword indicating that the tag names the equation. +If an equation of the model is tagged with a name, the `resid` command +will display the name of the equations (which may be more informative +than the equation numbers) in addition to the equation number. Several +tags for one equation can be separated using a comma: + + model; + + [name='Taylor rule',mcp = 'r > -1.94478'] + r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e; + + end; + +More information on tags is available at +. + +There can be several `model` blocks, in which case they are simply +concatenated. The set of effective options is also the concatenation of +the options declared in all the blocks, but in that case you may rather +want to use the `model_options`{.interpreted-text role="comm"} command. + +*Options* + +::: {.option} +linear + +Declares the model as being linear. It spares oneself from having to +declare initial values for computing the steady state of a stationary +linear model. This option can't be used with non-linear models, it will +NOT trigger linearization of the model. +::: + +::: {.option} +no\_static + +Don't create the static model file. This can be useful for models which +don't have a steady state. +::: + +::: {.option} +differentiate\_forward\_vars differentiate\_forward\_vars = ( +VARIABLE\_NAME \[VARIABLE\_NAME \...\] ) + +Tells Dynare to create a new auxiliary variable for each endogenous +variable that appears with a lead, such that the new variable is the +time differentiate of the original one. More precisely, if the model +contains `x(+1)`, then a variable `AUX_DIFF_VAR` will be created such +that `AUX_DIFF_VAR=x-x(-1)`, and `x(+1)` will be replaced with +`x+AUX_DIFF_VAR(+1)`. + +The transformation is applied to all endogenous variables with a lead if +the option is given without a list of variables. If there is a list, the +transformation is restricted to endogenous with a lead that also appear +in the list. + +This option can useful for some deterministic simulations where +convergence is hard to obtain. Bad values for terminal conditions in the +case of very persistent dynamics or permanent shocks can hinder correct +solutions or any convergence. The new differentiated variables have +obvious zero terminal conditions (if the terminal condition is a steady +state) and this in many cases helps convergence of simulations. +::: + +::: {.option} +parallel\_local\_files = ( FILENAME \[, FILENAME\]\... ) + +Declares a list of extra files that should be transferred to follower +nodes when doing a parallel computation (see +`paral-conf`{.interpreted-text role="ref"}). +::: + +::: {.option} +balanced\_growth\_test\_tol = DOUBLE + +Tolerance used for determining whether cross-derivatives are zero in the +test for balanced growth path (the latter is documented on +). Default: +`1e-6` +::: + +*Example* (Elementary RBC model) + +> var c k; +> varexo x; +> parameters aa alph bet delt gam; +> +> model; +> c = - k + aa*x*k(-1)^alph + (1-delt)*k(-1); +> c^(-gam) = (aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam)/(1+bet); +> end; + +*Example* (Use of model local variables) + +> The following program: +> +> model; +> # gamma = 1 - 1/sigma; +> u1 = c1^gamma/gamma; +> u2 = c2^gamma/gamma; +> end; +> +> \...is formally equivalent to: +> +> model; +> u1 = c1^(1-1/sigma)/(1-1/sigma); +> u2 = c2^(1-1/sigma)/(1-1/sigma); +> end; + +*Example* (A linear model) + +> model(linear); +> x = a*x(-1)+b*y(+1)+e_x; +> y = d*y(-1)+e_y; +> end; +::: + +::: {.command} +model\_options (OPTIONS\...); + +This command accepts the same options as the `model`{.interpreted-text +role="bck"} block. + +The purpose of this statement is to specify the options that apply to +the whole model, when there are several `model` blocks, so as to restore +the symmetry between those blocks (since otherwise one `model` block +would typically bear the options, while the other ones would typically +have no option). +::: + +::: {.command} +model\_remove (TAGS\...); + +This command removes equations that appeared in a previous +`model`{.interpreted-text role="bck"} block. + +The equations must be specified by a list of tag values, separated by +commas. Each element of the list is either a simple quoted string, in +which case it designates an equation by its `name` tag; or a tag name +(without quotes), followed by an equal sign, then by the tag value +(within quotes). + +Each removed equation must either have an `endogenous` tag, or have a +left hand side containing a single endogenous variable. The +corresponding endogenous variable will be either turned into an +exogenous (if it is still used in somewhere in the model at that point), +otherwise it will be removed from the model. + +*Example* + +> var c k dummy1 dummy2; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1) + dummy1; +> c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> [ name = 'eq:dummy1', endogenous = 'dummy1' ] +> c*k = dummy1; +> [ foo = 'eq:dummy2' ] +> log(dummy2) = k + 2; +> end; +> +> model_remove('eq:dummy1', foo = 'eq:dummy2'); +> +> In the above example, the last two equations will be removed, `dummy1` +> will be turned into an exogenous, and `dummy2` will be removed. +::: + +::: {.block} +model\_replace (TAGS\...); + +This block replaces several equations in the model. It removes the +equations given by the tags list (with the same syntax as in +`model_remove`{.interpreted-text role="comm"}), and it adds equations +given within the block (with the same syntax as +`model`{.interpreted-text role="bck"}). + +No variable is removed or has its type changed in the process. + +*Example* + +> var c k; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); +> [ name = 'dummy' ] +> c*k = 1; +> end; +> +> model_replace('dummy'); +> c^(-gam) = (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> end; +> +> In the above example, the dummy equation is replaced by a proper Euler +> equation. +::: + +Dynare has the ability to output the original list of model equations to +a LaTeX file, using the `write_latex_original_model` command, the list +of transformed model equations using the +`write_latex_dynamic_model command`, and the list of static model +equations using the `write_latex_static_model` command. + +::: {.command} +write\_latex\_original\_model (OPTIONS); + +This command creates two LaTeX files: one containing the model as +defined in the model block and one containing the LaTeX document header +information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/original.tex`, which includes a file called +`FILENAME/latex/original_content.tex` (also created by Dynare) +containing the list of all the original model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Time subscripts (`t`, `t+1`, `t-1`, \...) will be appended to the +variable names, as LaTeX subscripts. + +Compiling the TeX file requires the following LaTeX packages: +`geometry, fullpage, breqn`. + +*Options* + +::: {.option} +write\_equation\_tags + +Write the equation tags in the LaTeX output. The equation tags will be +interpreted with LaTeX markups. +::: +::: + +::: {.command} +write\_latex\_dynamic\_model ; write\_latex\_dynamic\_model (OPTIONS); + +This command creates two LaTeX files: one containing the dynamic model +and one containing the LaTeX document header information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/dynamic.tex`, which includes a file called +`FILENAME/latex/dynamic_content.tex` (also created by Dynare) containing +the list of all the dynamic model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Time subscripts (`t`, `t+1`, `t-1`, \...) will be appended to the +variable names, as LaTeX subscripts. + +Note that the model written in the TeX file will differ from the model +declared by the user in the following dimensions: + +> - The timing convention of predetermined variables (see +> `predetermined_variables`{.interpreted-text role="comm"}) will +> have been changed to the default Dynare timing convention; in +> other words, variables declared as predetermined will be lagged on +> period back, +> - The `EXPECTATION` operators will have been removed, replaced by +> auxiliary variables and new equations (as explained in the +> documentation of +> `EXPECTATION `{.interpreted-text +> role="op"}), +> - Endogenous variables with leads or lags greater or equal than two +> will have been removed, replaced by new auxiliary variables and +> equations, +> - Exogenous variables with leads or lags will also have been +> replaced by new auxiliary variables and equations. + +For the required LaTeX packages, see +`write_latex_original_model`{.interpreted-text role="comm"}. + +*Options* + +::: {.option} +write\_equation\_tags + +See `write_equation_tags`{.interpreted-text role="opt"} +::: +::: + +::: {.command} +write\_latex\_static\_model (OPTIONS); + +This command creates two LaTeX files: one containing the static model +and one containing the LaTeX document header information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/static.tex`, which includes a file called +`FILENAME/latex/static_content.tex` (also created by Dynare) containing +the list of all the steady state model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Note that the model written in the TeX file will differ from the model +declared by the user in the some dimensions (see +`write_latex_dynamic_model`{.interpreted-text role="comm"} for details). + +Also note that this command will not output the contents of the optional +`steady_state_model` block (see `steady_state_model`{.interpreted-text +role="bck"}); it will rather output a static version (i.e. without leads +and lags) of the dynamic `model` declared in the model block. To write +the LaTeX contents of the `steady_state_model` see +`write_latex_steady_state_model`{.interpreted-text role="comm"}. + +For the required LaTeX packages, see +`write_latex_original_model`{.interpreted-text role="comm"}. + +*Options* + +::: {.option} +write\_equation\_tags + +See `write_equation_tags`{.interpreted-text role="opt"}. +::: +::: + +::: {.command} +write\_latex\_steady\_state\_model + +This command creates two LaTeX files: one containing the steady state +model and one containing the LaTeX document header information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/steady_state.tex`, which includes a file called +`FILENAME/latex/steady_state_content.tex` (also created by Dynare) +containing the list of all the steady state model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Note that the model written in the `.tex` file will differ from the +model declared by the user in some dimensions (see +`write_latex_dynamic_model`{.interpreted-text role="comm"} for details). + +For the required LaTeX packages, see +`write_latex_original_model`{.interpreted-text role="comm"}. +::: + +Auxiliary variables {#aux-variables} +------------------- + +The model which is solved internally by Dynare is not exactly the model +declared by the user. In some cases, Dynare will introduce auxiliary +endogenous variables---along with corresponding auxiliary +equations---which will appear in the final output. + +The main transformation concerns leads and lags. Dynare will perform a +transformation of the model so that there is only one lead and one lag +on endogenous variables and no leads/lags on exogenous variables. + +This transformation is achieved by the creation of auxiliary variables +and corresponding equations. For example, if `x(+2)` exists in the +model, Dynare will create one auxiliary variable +`AUX_ENDO_LEAD = x(+1)`, and replace `x(+2)` by `AUX_ENDO_LEAD(+1)`. + +A similar transformation is done for lags greater than 2 on endogenous +(auxiliary variables will have a name beginning with `AUX_ENDO_LAG`), +and for exogenous with leads and lags (auxiliary variables will have a +name beginning with `AUX_EXO_LEAD` or `AUX_EXO_LAG` respectively). + +Another transformation is done for the `EXPECTATION` operator. For each +occurrence of this operator, Dynare creates an auxiliary variable +defined by a new equation, and replaces the expectation operator by a +reference to the new auxiliary variable. For example, the expression +`EXPECTATION(-1)(x(+1))` is replaced by `AUX_EXPECT_LAG_1(-1)`, and the +new auxiliary variable is declared as `AUX_EXPECT_LAG_1 = x(+2)`. + +Auxiliary variables are also introduced by the preprocessor for the +`ramsey_model` and `ramsey_policy` commands. In this case, they are used +to represent the Lagrange multipliers when first order conditions of the +Ramsey problem are computed. The new variables take the form `MULT_i`, +where *i* represents the constraint with which the multiplier is +associated (counted from the order of declaration in the model block). + +Auxiliary variables are also introduced by the +`differentiate_forward_vars` option of the model block. The new +variables take the form `AUX_DIFF_FWRD_i`, and are equal to `x-x(-1)` +for some endogenous variable `x`. + +Finally, auxiliary variables will arise in the context of employing the +`diff`-operator. + +Once created, all auxiliary variables are included in the set of +endogenous variables. The output of decision rules (see below) is such +that auxiliary variable names are replaced by the original variables +they refer to. + +The number of endogenous variables before the creation of auxiliary +variables is stored in `M_.orig_endo_nbr`, and the number of endogenous +variables after the creation of auxiliary variables is stored in +`M_.endo_nbr`. + +See +for more technical details on auxiliary variables. + +Initial and terminal conditions {#init-term-cond} +------------------------------- + +For most simulation exercises, it is necessary to provide initial (and +possibly terminal) conditions. It is also necessary to provide initial +guess values for non-linear solvers. This section describes the +statements used for those purposes. + +In many contexts (deterministic or stochastic), it is necessary to +compute the steady state of a non-linear model: `initval` then specifies +numerical initial values for the non-linear solver. The command `resid` +can be used to compute the equation residuals for the given initial +values. + +Used in perfect foresight mode, the types of forward-looking models for +which Dynare was designed require both initial and terminal conditions. +Most often these initial and terminal conditions are static equilibria, +but not necessarily. + +One typical application is to consider an economy at the equilibrium at +time 0, trigger a shock in first period, and study the trajectory of +return to the initial equilibrium. To do that, one needs `initval` and +`shocks` (see `shocks-exo`{.interpreted-text role="ref"}). + +Another one is to study how an economy, starting from arbitrary initial +conditions at time 0 converges towards equilibrium. In this case models, +the command `histval` permits to specify different historical initial +values for variables with lags for the periods before the beginning of +the simulation. Due to the design of Dynare, in this case `initval` is +used to specify the terminal conditions. + +::: {.block} +initval ; initval(OPTIONS\...); + +The `initval` block has two main purposes: providing guess values for +non-linear solvers in the context of perfect foresight simulations and +providing guess values for steady state computations in both perfect +foresight and stochastic simulations. Depending on the presence of +`histval` and `endval` blocks it is also used for declaring the initial +and terminal conditions in a perfect foresight simulation exercise. +Because of this interaction of the meaning of an `initval` block with +the presence of `histval` and `endval` blocks in perfect foresight +simulations, it is strongly recommended to check that the constructed +`oo_.endo_simul` and `oo_.exo_simul` variables contain the desired +values after running `perfect_foresight_setup` and before running +`perfect_foresight_solver`. In the presence of leads and lags, these +subfields of the results structure will store the historical values for +the lags in the first column/row and the terminal values for the leads +in the last column/row. + +The `initval` block is terminated by `end;` and contains lines of the +form: + +> VARIABLE\_NAME = EXPRESSION; + +*In a deterministic (i.e. perfect foresight) model* + +First, both the `oo_.endo_simul` and `oo_.exo_simul` variables storing +the endogenous and exogenous variables will be filled with the values +provided by this block. If there are no other blocks present, it will +therefore provide the initial and terminal conditions for all the +endogenous and exogenous variables, because it will also fill the last +column/row of these matrices. For the intermediate simulation periods it +thereby provides the starting values for the solver. In the presence of +a `histval` block (and therefore absence of an `endval` block), this +`histval` block will provide/overwrite the historical values for the +state variables (lags) by setting the first column/row of +`oo_.endo_simul` and `oo_.exo_simul`. This implies that the `initval` +block in the presence of `histval` only sets the terminal values for the +variables with leads and provides initial values for the perfect +foresight solver. + +Because of these various functions of `initval` it is often necessary to +provide values for all the endogenous variables in an `initval` block. +Initial and terminal conditions are strictly necessary for lagged/leaded +variables, while feasible starting values are required for the solver. +It is important to be aware that if some variables, endogenous or +exogenous, are not mentioned in the `initval` block, a zero value is +assumed. It is particularly important to keep this in mind when +specifying exogenous variables using `varexo` that are not allowed to +take on the value of zero, like e.g. TFP. + +Note that if the `initval` block is immediately followed by a `steady` +command, its semantics are slightly changed. The `steady` command will +compute the steady state of the model for all the endogenous variables, +assuming that exogenous variables are kept constant at the value +declared in the `initval` block. These steady state values conditional +on the declared exogenous variables are then written into +`oo_.endo_simul` and take up the potential roles as historical and +terminal conditions as well as starting values for the solver. An +`initval` block followed by `steady` is therefore formally equivalent to +an `initval` block with the specified values for the exogenous +variables, and the endogenous variables set to the associated steady +state values conditional on the exogenous variables. + +*In a stochastic model* + +The main purpose of `initval` is to provide initial guess values for the +non-linear solver in the steady state computation. Note that if the +`initval` block is not followed by `steady`, the steady state +computation will still be triggered by subsequent commands +(`stoch_simul`, `estimation`\...). + +As such, `initval` allows specifying the initial instrument value for +steady state finding when providing an analytical conditional steady +state file for `ramsey_model`-computations. + +It is not necessary to declare 0 as initial value for exogenous +stochastic variables, since it is the only possible value. + +The subsequently computed steady state (not the initial values, use +histval for this) will be used as the initial condition at all the +periods preceeding the first simulation period for the three possible +types of simulations in stochastic mode: + +> - `stoch_simul`{.interpreted-text role="comm"}, if the `periods` +> option is specified. +> - `forecast`{.interpreted-text role="comm"} as the initial point at +> which the forecasts are computed. +> - `conditional_forecast`{.interpreted-text role="comm"} as the +> initial point at which the conditional forecasts are computed. + +To start simulations at a particular set of starting values that are not +a computed steady state, use `histval`{.interpreted-text role="bck"}. + +*Options* + +::: {.option} +all\_values\_required + +Issues an error and stops processing the .mod file if there is at least +one endogenous or exogenous variable that has not been set in the +initval block. +::: + +*Example* + +: initval; + c = 1.2; + k = 12; + x = 1; + end; + + steady; +::: + +::: {.block} +endval ; endval (OPTIONS\...); + +This block is terminated by `end;` and contains lines of the form: + +> VARIABLE\_NAME = EXPRESSION; + +The `endval` block makes only sense in a deterministic model and cannot +be used together with `histval`. Similar to the `initval` command, it +will fill both the `oo_.endo_simul` and `oo_.exo_simul` variables +storing the endogenous and exogenous variables with the values provided +by this block. If no `initval` block is present, it will fill the whole +matrices, therefore providing the initial and terminal conditions for +all the endogenous and exogenous variables, because it will also fill +the first and last column/row of these matrices. Due to also filling the +intermediate simulation periods it will provide the starting values for +the solver as well. + +If an `initval` block is present, `initval` will provide the historical +values for the variables (if there are states/lags), while `endval` will +fill the remainder of the matrices, thereby still providing *i*) the +terminal conditions for variables entering the model with a lead and +*ii*) the initial guess values for all endogenous variables at all the +simulation dates for the perfect foresight solver. + +Note that if some variables, endogenous or exogenous, are NOT mentioned +in the `endval` block, the value assumed is that of the last `initval` +block or `steady` command (if present). Therefore, in contrast to +`initval`, omitted variables are not automatically assumed to be 0 in +this case. Again, it is strongly recommended to check the constructed +`oo_.endo_simul` and `oo_.exo_simul` variables after running +`perfect_foresight_setup` and before running `perfect_foresight_solver` +to see whether the desired outcome has been achieved. + +Like `initval`, if the `endval` block is immediately followed by a +`steady` command, its semantics are slightly changed. The `steady` +command will compute the steady state of the model for all the +endogenous variables, assuming that exogenous variables are kept +constant to the value declared in the `endval` block. These steady state +values conditional on the declared exogenous variables are then written +into `oo_.endo_simul` and therefore take up the potential roles as +historical and terminal conditions as well as starting values for the +solver. An `endval` block followed by `steady` is therefore formally +equivalent to an `endval` block with the specified values for the +exogenous variables, and the endogenous variables set to the associated +steady state values. + +*Options* + +::: {.option} +all\_values\_required + +See `all_values_required`{.interpreted-text role="opt"}. +::: + +*Example* + +> var c k; +> varexo x; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); +> c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> end; +> +> initval; +> c = 1.2; +> k = 12; +> x = 1; +> end; +> +> steady; +> +> endval; +> c = 2; +> k = 20; +> x = 2; +> end; +> +> steady; +> +> perfect_foresight_setup(periods=200); +> perfect_foresight_solver; +> +> In this example, the problem is finding the optimal path for +> consumption and capital for the periods $t=1$ to $T=200$, given the +> path of the exogenous technology level `x`. `c` is a forward-looking +> variable and the exogenous variable `x` appears with a lead in the +> expected return of physical capital, while `k` is a purely +> backward-looking (state) variable. +> +> The initial equilibrium is computed by `steady` conditional on `x=1`, +> and the terminal one conditional on `x=2`. The `initval` block sets +> the initial condition for `k` (since it is the only backward-looking +> variable), while the `endval` block sets the terminal condition for +> `c` (since it is the only forward-looking endogenous variable). The +> starting values for the perfect foresight solver are given by the +> `endval` block. See below for more details. + +*Example* + +> var c k; +> varexo x; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); +> c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> end; +> +> initval; +> k = 12; +> end; +> +> endval; +> c = 2; +> x = 1.1; +> end; +> +> perfect_foresight_setup(periods=200); +> perfect_foresight_solver; +> +> In this example, there is no [steady]{.title-ref} command, hence the +> conditions are exactly those specified in the [initval]{.title-ref} +> and [endval]{.title-ref} blocks. We need terminal conditions for `c` +> and `x`, since both appear with a lead, and an initial condition for +> `k`, since it appears with a lag. +> +> Setting `x=1.1` in the `endval` block without a `shocks` block implies +> that technology is at $1.1$ in $t=1$ and stays there forever, because +> `endval` is filling all entries of `oo_.endo_simul` and +> `oo_.exo_simul` except for the very first one, which stores the +> initial conditions and was set to $0$ by the `initval` block when not +> explicitly specifying a value for it. +> +> Because the law of motion for capital is backward-looking, we need an +> initial condition for `k` at time $0$. Due to the presence of +> `endval`, this cannot be done via a `histval` block, but rather must +> be specified in the `initval` block. Similarly, because the Euler +> equation is forward-looking, we need a terminal condition for `c` at +> $t=201$, which is specified in the `endval` block. +> +> As can be seen, it is not necessary to specify `c` and `x` in the +> `initval` block and `k` in the `endval` block, because they have no +> impact on the results. Due to the optimization problem in the first +> period being to choose `c,k` at $t=1$ given the predetermined capital +> stock `k` inherited from $t=0$ as well as the current and future +> values for technology `x`, the values for `c` and `x` at time $t=0$ +> play no role. The same applies to the choice of `c,k` at time $t=200$, +> which does not depend on `k` at $t=201$. As the Euler equation shows, +> that choice only depends on current capital as well as future +> consumption `c` and technology `x`, but not on future capital `k`. The +> intuitive reason is that those variables are the consequence of +> optimization problems taking place in at periods $t=0$ and $t=201$, +> respectively, which are not modeled here. + +*Example* + +> initval; +> c = 1.2; +> k = 12; +> x = 1; +> end; +> +> endval; +> c = 2; +> k = 20; +> x = 1.1; +> end; +> +> In this example, initial conditions for the forward-looking variables +> `x` and `c` are provided, together with a terminal condition for the +> backward-looking variable `k`. As shown in the previous example, these +> values will not affect the simulation results. Dynare simply takes +> them as given and basically assumes that there were realizations of +> exogenous variables and states that make those choices equilibrium +> values (basically initial/terminal conditions at the unspecified time +> periods $t<0$ and $t>201$). +> +> The above example suggests another way of looking at the use of +> `steady` after `initval` and `endval`. Instead of saying that the +> implicit unspecified conditions before and after the simulation range +> have to fit the initial/terminal conditions of the endogenous +> variables in those blocks, steady specifies that those conditions at +> $t<0$ and $t>201$ are equal to being at the steady state given the +> exogenous variables in the `initval` and `endval` blocks. The +> endogenous variables at $t=0$ and $t=201$ are then set to the +> corresponding steady state equilibrium values. +> +> The fact that `c` at $t=0$ and `k` at $t=201$ specified in `initval` +> and `endval` are taken as given has an important implication for +> plotting the simulated vector for the endogenous variables, i.e. the +> rows of `oo_.endo_simul`: this vector will also contain the initial +> and terminal conditions and thus is 202 periods long in the example. +> When you specify arbitrary values for the initial and terminal +> conditions for forward- and backward-looking variables, respectively, +> these values can be very far away from the endogenously determined +> values at $t=1$ and $t=200$. While the values at $t=0$ and $t=201$ are +> unrelated to the dynamics for $0 strange-looking large jumps. In the example above, consumption will +> display a large jump from $t=0$ to $t=1$ and capital will jump from +> $t=200$ to $t=201$ when using `rplot`{.interpreted-text role="comm"} +> or manually plotting `oo_.endo_val`. +::: + +::: {.block} +histval ; histval (OPTIONS\...); + +*In a deterministic perfect foresight context* + +In models with lags on more than one period, the `histval` block permits +to specify different historical initial values for different periods of +the state variables. In this case, the `initval` block takes over the +role of specifying terminal conditions and starting values for the +solver. Note that the `histval` block does not take non-state variables. + +This block is terminated by `end;` and contains lines of the form: + +> VARIABLE\_NAME(INTEGER) = EXPRESSION; + +EXPRESSION is any valid expression returning a numerical value and can +contain already initialized variable names. + +By convention in Dynare, period 1 is the first period of the simulation. +Going backward in time, the first period before the start of the +simulation is period 0, then period -1, and so on. + +State variables not initialized in the `histval` block are assumed to +have a value of zero at period 0 and before. Note that `histval` cannot +be followed by `steady`. + +*Example* + +> model; +> x=1.5*x(-1)-0.6*x(-2)+epsilon; +> log(c)=0.5*x+0.5*log(c(+1)); +> end; +> +> histval; +> x(0)=-1; +> x(-1)=0.2; +> end; +> +> initval; +> c=1; +> x=1; +> end; +> +> In this example, `histval` is used to set the historical conditions +> for the two lags of the endogenous variable `x`, stored in the first +> column of `oo_.endo_simul`. The `initval` block is used to set the +> terminal condition for the forward looking variable `c`, stored in the +> last column of `oo_.endo_simul`. Moreover, the `initval` block defines +> the starting values for the perfect foresight solver for both +> endogenous variables `c` and `x`. + +*In a stochastic simulation context* + +In the context of stochastic simulations, `histval` allows setting the +starting point of those simulations in the state space. As for the case +of perfect foresight simulations, all not explicitly specified variables +are set to 0. Moreover, as only states enter the recursive policy +functions, all values specified for control variables will be ignored. +This can be used + +> - In `stoch_simul`{.interpreted-text role="comm"}, if the `periods` +> option is specified. Note that this only affects the starting +> point for the simulation, but not for the impulse response +> functions. When using the `loglinear `{.interpreted-text +> role="ref"} option, the `histval` block nevertheless takes the +> unlogged starting values. +> - In `forecast`{.interpreted-text role="comm"} as the initial point +> at which the forecasts are computed. When using the `loglinear +> `{.interpreted-text role="ref"} option, the `histval` block +> nevertheless takes the unlogged starting values. +> - In `conditional_forecast`{.interpreted-text role="comm"} for a +> calibrated model as the initial point at which the conditional +> forecasts are computed. When using the +> `loglinear `{.interpreted-text role="ref"} option, the +> histval-block nevertheless takes the unlogged starting values. +> - In `Ramsey policy `{.interpreted-text role="comm"}, +> where it also specifies the values of the endogenous states +> (including lagged exogenous) at which the objective function of +> the planner is computed. Note that the initial values of the +> Lagrange multipliers associated with the planner's problem cannot +> be set (see `evaluate_planner_objective`{.interpreted-text +> role="comm"}). + +*Options* + +::: {.option} +all\_values\_required + +See `all_values_required`{.interpreted-text role="opt"}. +::: + +*Example* + +> var x y; +> varexo e; +> +> model; +> x = y(-1)^alpha*y(-2)^(1-alpha)+e; +> +> end; +> +> initval; +> x = 1; +> y = 1; +> e = 0.5; +> end; +> +> steady; +> +> histval; +> y(0) = 1.1; +> y(-1) = 0.9; +> end; +> +> stoch_simul(periods=100); +::: + +::: {.command} +resid ; + +This command will display the residuals of the static equations of the +model, using the values given for the endogenous in the last `initval` +or `endval` block (or the steady state file if you provided one, see +`st-st`{.interpreted-text role="ref"}). + +*Options* + +::: {.option} +non\_zero + +Only display non-zero residuals. +::: +::: + +::: {.command} +initval\_file (OPTIONS\...); + +In a deterministic setup, this command is used to specify a path for all +endogenous and exogenous variables. The length of these paths must be +equal to the number of simulation periods, plus the number of leads and +the number of lags of the model (for example, with 50 simulation +periods, in a model with 2 lags and 1 lead, the paths must have a length +of 53). Note that these paths cover two different things: + +> - The constraints of the problem, which are given by the path for +> exogenous and the initial and terminal values for endogenous +> - The initial guess for the non-linear solver, which is given by the +> path for endogenous variables for the simulation periods +> (excluding initial and terminal conditions) + +In perfect foresight and stochastic contexts, `steady` uses the first +observation loaded by `initval_file` as guess value to solve for the +steady state of the model. This first observation is determined by the +`first_obs` option when it is used. + +Don't mix `initval_file` with `initval` statements. However, after +`initval_file`, you can modify the historical initial values with +`histval` or `histval_file` statement. + +There can be several `initval_file` statements in a model file. Each +statement resets `oo_.initval_series`. + +*Options* + +::: {.option} +datafile = FILENAME filename = FILENAME (deprecated) + +The name of the file containing the data. It must be included in quotes +if the filename contains a path or an extension. The command accepts the +following file formats: + +- M-file (extension `.m`): for each endogenous and exogenous variable, + the file must contain a row or column vector of the same name. +- MAT-file (extension `.mat`): same as for M-files. +- Excel file (extension `.xls` or `.xlsx`): for each endogenous and + exogenous variable, the file must contain a column of the same name. + NB: Octave only supports the `.xlsx` file extension and must have + the [io](https://octave.sourceforge.io/io/) package installed + (easily done via octave by typing '`pkg install -forge io`'). The + first column may contain the date +::: + +> of each observation. +> +> : - CSV files (extension `.csv`): for each endogenous and +> exogenous variable, the file must contain a column of the same +> name. The first column may contain the date of each +> observation. +> +::: {.option} +first\_obs = {INTEGER \| DATE} + +The observation number or the date (see +::: + +`dates-members`{.interpreted-text role="ref"}) of the first observation +to be used in the file + +::: {.option} +first\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates `{.interpreted-text role="ref"}) at which the +simulation (or the forecast) is starting. This option avoids to have to +compute the maximum number of lags in the model. The observation +corresponding to the first period of simulation doesn't need to exist in +the file as the only dates necessary for initialization are before that +date. + +::: {.option} +last\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates `{.interpreted-text role="ref"}) at which the +simulation (or the forecast) is ending. This option avoids to have to +compute the maximum number of leads in the model. + +::: {.option} +last\_obs = {INTEGER \| DATE} + +The observaton number or the date (see +::: + +`dates-members`{.interpreted-text role="ref"}) of the last observation +to be used in the file. + +::: {.option} +nobs = INTEGER +::: + +The number of observations to be used in the file (starting with first +of `first_obs` observation). + +::: {.option} +series = DSERIES NAME + +The name of a DSERIES containing the data (see +`dseries-members`{.interpreted-text role="ref"}) +::: + +*Example 1* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv` +(nothing guarantees that these vales are the steady state of the model). +The guess value for the trajectories are also taken from the file. The +file must contain at least 203 observations of variables `c`, `x` and +`e`. If there are more than 203 observations available in the file, the +first 203 are used by `perfect_foresight_setup(periods=200)`. Note that +the values for the auxiliary variable corresponding to `x(-2)` are +automatically computed by `initval_file`. + +*Example 2* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs=10); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv` +starting with the 10th observation in the file. There must be at least +212 observations in the file. + +*Example 3* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> ds = dseries(mydata.csv); lds = log(ds); +> +> initval\_file(series=lds, +> +> : first\_obs=2010Q1); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from dseries `lds`. All +observations are loaded starting with the 1st quarter of 2010 until the +end of the file. There must be data available at least until 2050Q3. + +*Example 4* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_simulation\_period=2010Q1); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv`. The +observations in the file must have dates. All observations are loaded +from the 3rd quarter of 2009 until the end of the file. There must be +data available in the file at least until 2050Q1. + +*Example 5* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : last\_obs = 212); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv`. The +first 212 observations are loaded and the first 203 observations will be +used by `perfect_foresight_setup(periods=200)`. + +*Example 6* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs = 10, +> +> > nobs = 203); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv`. +Observations 10 to 212 are loaded. + +*Example 7* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs = 10); +> +> steady; +> +> The values of the 10th observation of `mydata.csv` are used + +as guess value to compute the steady state. The exogenous variables are +set to values found in the file or zero if these variables aren\'t +present. +::: + +::: {.command} +histval\_file (OPTIONS\...); + +This command is equivalent to `histval`, except that it reads its input +from a file, and is typically used in conjunction with +`smoother2histval`. + +> *Options* + +::: {.option} +datafile = FILENAME filename = FILENAME (deprecated) + +The name of the file containing the data. The command accepts +::: + +the following file formats: + +> - M-file (extension `.m`): for each endogenous and exogenous +> variable, the file must contain a row or column vector of the same +> name. +> - MAT-file (extension `.mat`): same as for M-files. +> - Excel file (extension `.xls` or `.xlsx`): for each endogenous and +> exogenous variable, the file must contain a column of the same +> name. NB: Octave only supports the `.xlsx` file extension and must +> have the [io](https://octave.sourceforge.io/io/) package installed +> (easily done via octave by typing '`pkg install -forge io`'). The +> first column may contain the date of each observation. +> - CSV files (extension `.csv`): for each endogenous and exogenous +> variable, the file must contain a column of the same name. The +> first column may contain the date of each observation. + +::: {.option} +first\_obs = {INTEGER \| DATE} + +The observation number or the date (see +`dates-members`{.interpreted-text role="ref"}) of +::: + +the first observation to be used in the file + +::: {.option} +first\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates-members`{.interpreted-text role="ref"}) at which the simulation +(or the forecast) is starting. This option avoids to have to compute the +maximum number of lags in the model. The observation corresponding to +the first period of simulation doesn't need to exist in the file as the +only dates necessary for initialization are before that date. + +::: {.option} +last\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates `{.interpreted-text role="ref"}) at which the +simulation (or the forecast) is ending. This option avoids to have to +compute the maximum number of leads in the model. + +::: {.option} +last\_obs = {INTEGER \| DATE} + +The observation number or the date (see +`dates-members`{.interpreted-text role="ref"}) of the +::: + +last observation to be used in the file. + +::: {.option} +nobs = INTEGER +::: + +The number of observations to be used in the file (starting with first +of `first_obs` observation). + +::: {.option} +series = DSERIES NAME + +The name of a DSERIES containing the data (see +`dseries-members`{.interpreted-text role="ref"}) +::: + +*Example 1* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> steady\_state\_model; x = 0; c = exp(c\*x/(1 - d)); end; +> +> histval\_file(datafile=mydata.csv); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from the two +first rows of file `mydata.csv`. + +*Example 2* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : first\_obs=10); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from rows 10 +and 11 of file `mydata.csv`. + +*Example 3* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : first\_obs=2010Q1); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from +observations 2010Q1 and 2010Q2 of file `mydata.csv`. + +*Example 4* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : first\_simulation\_period=2010Q1) +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from +observations 2009Q3 and 2009Q4 of file `mydata.csv`. + +*Example 5* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : last\_obs = 4); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from the two +first rows of file `mydata.csv`. + +*Example 6* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs = 10, +> +> > nobs = 4); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from rows 10 +and 11 of file `mydata.csv`. + +*Example 7* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs=10); +> +> > histval\_file(datafile=myotherdata.csv); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +Historical initial values for the simulation are taken from the two +first rows of file `myotherdata.csv`. + +> Terminal values and guess values for the simulation are taken + +from file `mydata.csv` starting with the 12th observation in the file. +There must be at least 212 observations in the file. +::: + +Shocks on exogenous variables {#shocks-exo} +----------------------------- + +In a deterministic context, when one wants to study the transition of +one equilibrium position to another, it is equivalent to analyze the +consequences of a permanent shock and this in done in Dynare through the +proper use of `initval` and `endval`. + +Another typical experiment is to study the effects of a temporary shock +after which the system goes back to the original equilibrium (if the +model is stable\...). A temporary shock is a temporary change of value +of one or several exogenous variables in the model. Temporary shocks are +specified with the command `shocks`. + +In a stochastic framework, the exogenous variables take random values in +each period. In Dynare, these random values follow a normal distribution +with zero mean, but it belongs to the user to specify the variability of +these shocks. The non-zero elements of the matrix of variance-covariance +of the shocks can be entered with the `shocks` command. Or, the entire +matrix can be directly entered with `Sigma_e` (this use is however +deprecated). + +If the variance of an exogenous variable is set to zero, this variable +will appear in the report on policy and transition functions, but isn't +used in the computation of moments and of Impulse Response Functions. +Setting a variance to zero is an easy way of removing an exogenous +shock. + +Note that, by default, if there are several `shocks` or `mshocks` blocks +in the same `.mod` file, then they are cumulative: all the shocks +declared in all the blocks are considered; however, if a `shocks` or +`mshocks` block is declared with the `overwrite` option, then it +replaces all the previous `shocks` and `mshocks` blocks. + +::: {.block} +shocks ; shocks(overwrite); + +See above for the meaning of the `overwrite` option. + +*In deterministic context* + +For deterministic simulations, the `shocks` block specifies temporary +changes in the value of exogenous variables. For permanent shocks, use +an `endval` block. + +The block should contain one or more occurrences of the following group +of three lines: + + var VARIABLE_NAME; + periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; + values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...; + +It is possible to specify shocks which last several periods and which +can vary over time. The `periods` keyword accepts a list of several +dates or date ranges, which must be matched by as many shock values in +the `values` keyword. Note that a range in the `periods` keyword can be +matched by only one value in the `values` keyword. If `values` +represents a scalar, the same value applies to the whole range. If +`values` represents a vector, it must have as many elements as there are +periods in the range. + +Note that shock values are not restricted to numerical constants: +arbitrary expressions are also allowed, but you have to enclose them +inside parentheses. + +The feasible range of `periods` is from 0 to the number of `periods` +specified in `perfect_foresight_setup`. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +Note that the first endogenous simulation period is period 1. Thus, a +shock value specified for the initial period 0 may conflict with (i.e. +may overwrite or be overwritten by) values for the initial period +specified with `initval` or `endval` (depending on the exact context). +Users should always verify the correct setting of `oo_.exo_simul` after +`perfect_foresight_setup`. +::: + +*Example* (with scalar values) + + shocks; + + var e; + periods 1; + values 0.5; + var u; + periods 4:5; + values 0; + var v; + periods 4:5 6 7:9; + values 1 1.1 0.9; + var w; + periods 1 2; + values (1+p) (exp(z)); + + end; + +*Example* (with vector values) + + xx = [1.2; 1.3; 1]; + + shocks; + var e; + periods 1:3; + values (xx); + end; + +*In stochastic context* + +For stochastic simulations, the `shocks` block specifies the non zero +elements of the covariance matrix of the shocks of exogenous variables. + +You can use the following types of entries in the block: + +- Specification of the standard error of an exogenous variable. + + var VARIABLE_NAME; stderr EXPRESSION; + +- Specification of the variance of an exogenous variable. + + var VARIABLE_NAME = EXPRESSION; + +- Specification the covariance of two exogenous variables. + + var VARIABLE_NAME, VARIABLE_NAME = EXPRESSION; + +- Specification of the correlation of two exogenous variables. + + corr VARIABLE_NAME, VARIABLE_NAME = EXPRESSION; + +In an estimation context, it is also possible to specify variances and +covariances on endogenous variables: in that case, these values are +interpreted as the calibration of the measurement errors on these +variables. This requires the `varobs` command to be specified before the +`shocks` block. + +*Example* + + shocks; + var e = 0.000081; + var u; stderr 0.009; + corr e, u = 0.8; + var v, w = 2; + end; + +*In stochastic optimal policy context* + +When computing conditional welfare in a `ramsey_model` or +`discretionary_policy` context, welfare is conditional on the state +values inherited by planner when making choices in the first period. The +information set of the first period includes the respective exogenous +shock realizations. Thus, their known value can be specified using the +perfect foresight syntax. Note that i) all other values specified for +periods than period 1 will be ignored and ii) the value of lagged shocks +(e.g. in the case of news shocks) is specified with `histval`. + +*Example* + + shocks; + var u; stderr 0.008; + var u; + periods 1; + values 1; + end; + +*Mixing deterministic and stochastic shocks* + +It is possible to mix deterministic and stochastic shocks to build +models where agents know from the start of the simulation about future +exogenous changes. In that case `stoch_simul` will compute the rational +expectation solution adding future information to the state space +(nothing is shown in the output of `stoch_simul`) and `forecast` will +compute a simulation conditional on initial conditions and future +information. + +*Example* + + varexo_det tau; + varexo e; + ... + shocks; + var e; stderr 0.01; + var tau; + periods 1:9; + values -0.15; + end; + + stoch_simul(irf=0); + + forecast; +::: + + + +::: {.matcomm} +get\_shock\_stderr\_by\_name (\'EXOGENOUS\_NAME\'); + +Given the name of an exogenous variable, returns its standard deviation, +as set by a previous `shocks` block. +::: + +::: {.matcomm} +set\_shock\_stderr\_value (\'EXOGENOUS\_NAME\', MATLAB\_EXPRESSION); + +Sets the standard deviation of an exgonous variable. This does +essentially the same as setting the standard error via a `shocks` block, +except that it accepts arbitrary Julia expressions, and that it +works from Julia scripts. +::: + +Steady state {#st-st} +------------ + +There are two ways of computing the steady state (i.e. the static +equilibrium) of a model. The first way is to let Dynare compute the +steady state using a nonlinear Newton-type solver; this should work for +most models, and is relatively simple to use. The second way is to give +more guidance to Dynare, using your knowledge of the model, by providing +it with a method to compute the steady state, either using a +[steady\_state\_model]{.title-ref} block or writing matlab routine. + +### Finding the steady state with Dynare nonlinear solver + +::: {.command} +steady ; steady (OPTIONS\...); + +This command computes the steady state of a model using a nonlinear +Newton-type solver and displays it. When a steady state file is used +`steady` displays the steady state and checks that it is a solution of +the static model. + +More precisely, it computes the equilibrium value of the endogenous +variables for the value of the exogenous variables specified in the +previous `initval` or `endval` block. + +`steady` uses an iterative procedure and takes as initial guess the +value of the endogenous variables set in the previous `initval` or +`endval` block. + +For complicated models, finding good numerical initial values for the +endogenous variables is the trickiest part of finding the equilibrium of +that model. Often, it is better to start with a smaller model and add +new variables one by one. + +*Options* + +::: {#steady_maxit} +::: {.option} +maxit = INTEGER + +Determines the maximum number of iterations used in the non-linear +solver. The default value of `maxit` is 50. +::: +::: + +::: {#steady_tolf} +::: {.option} +tolf = DOUBLE + +Convergence criterion for termination based on the function value. +Iteration will cease when the residuals are smaller than `tolf`. +Default: `eps^(1/3)` +::: +::: + +::: {.option} +tolx = DOUBLE + +Convergence criterion for termination based on the step tolerance along. +Iteration will cease when the attempted step size is smaller than +`tolx`. Default: `eps^(2/3)` +::: + + +::: {.option} +homotopy\_mode = INTEGER + +Use a homotopy (or divide-and-conquer) technique to solve for the steady +state. If you use this option, you must specify a `homotopy_setup` +block. This option can take three possible values: + +> `1` +> +> > In this mode, all the parameters are changed simultaneously, and the +> > distance between the boundaries for each parameter is divided in as +> > many intervals as there are steps (as defined by the +> > `homotopy_steps` option); the problem is solved as many times as +> > there are steps. +> +> `2` +> +> > Same as mode `1`, except that only one parameter is changed at a +> > time; the problem is solved as many times as steps times number of +> > parameters. +> +> `3` +> +> > Dynare tries first the most extreme values. If it fails to compute +> > the steady state, the interval between initial and desired values is +> > divided by two for all parameters. Every time that it is impossible +> > to find a steady state, the previous interval is divided by two. +> > When it succeeds to find a steady state, the previous interval is +> > multiplied by two. In that last case `homotopy_steps` contains the +> > maximum number of computations attempted before giving up. +::: + +::: {.option} +homotopy\_steps = INTEGER + +Defines the number of steps when performing a homotopy. See +`homotopy_mode` option for more details. +::: + +::: {.option} +homotopy\_force\_continue = INTEGER + +This option controls what happens when homotopy fails. + +> `0` +> +> > `steady` fails with an error message +> +> `1` +> +> > `steady` keeps the values of the last homotopy step that was +> > successful and continues. **BE CAREFUL**: parameters and/or +> > exogenous variables are NOT at the value expected by the user + +Default is `0`. +::: + +::: {.option} +nocheck + +Don't check the steady state values when they are provided explicitly +either by a steady state file or a `steady_state_model` block. This is +useful for models with unit roots as, in this case, the steady state is +not unique or doesn't exist. +::: + + +*Example* + +See `init-term-cond`{.interpreted-text role="ref"}. +::: + +After computation, the steady state is available in the following +variable: + +::: {.matvar} +[oo]().steady\_state + +Contains the computed steady state. Endogenous variables are ordered in +the order of declaration used in the `var` command (which is also the +order used in `M_.endo_names`). +::: + +::: {.matcomm} +get\_mean (\'ENDOGENOUS\_NAME\' \[, \'ENDOGENOUS\_NAME\'\]\... ); + +Returns the steady of state of the given endogenous variable(s), as it +is stored in `oo_.steady_state`. Note that, if the steady state has not +yet been computed with `steady`, it will first try to compute it. +::: + +::: {.block} +homotopy\_setup ; + +This block is used to declare initial and final values when using a +homotopy method. It is used in conjunction with the option +`homotopy_mode` of the steady command. + +The idea of homotopy (also called divide-and-conquer by some authors) is +to subdivide the problem of finding the steady state into smaller +problems. It assumes that you know how to compute the steady state for a +given set of parameters, and it helps you finding the steady state for +another set of parameters, by incrementally moving from one to another +set of parameters. + +The purpose of the `homotopy_setup` block is to declare the final (and +possibly also the initial) values for the parameters or exogenous that +will be changed during the homotopy. It should contain lines of the +form: + + VARIABLE_NAME, EXPRESSION, EXPRESSION; + +This syntax specifies the initial and final values of a given +parameter/exogenous. + +There is an alternative syntax: + + VARIABLE_NAME, EXPRESSION; + +Here only the final value is specified for a given parameter/exogenous; +the initial value is taken from the preceeding `initval` block. + +A necessary condition for a successful homotopy is that Dynare must be +able to solve the steady state for the initial parameters/exogenous +without additional help (using the guess values given in the `initval` +block). + +If the homotopy fails, a possible solution is to increase the number of +steps (given in `homotopy_steps` option of `steady`). + +*Example* + +In the following example, Dynare will first compute the steady state for +the initial values (`gam=0.5` and `x=1`), and then subdivide the problem +into 50 smaller problems to find the steady state for the final values +(`gam=2` and `x=2`): + + var c k; + varexo x; + + parameters alph gam delt bet aa; + alph=0.5; + delt=0.02; + aa=0.5; + bet=0.05; + + model; + c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); + c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); + end; + + initval; + x = 1; + k = ((delt+bet)/(aa*x*alph))^(1/(alph-1)); + c = aa*x*k^alph-delt*k; + end; + + homotopy_setup; + gam, 0.5, 2; + x, 2; + end; + + steady(homotopy_mode = 1, homotopy_steps = 50); +::: + +### Providing the steady state to Dynare + +If you know how to compute the steady state for your model, you can +provide a MATLAB/Octave function doing the computation instead of using +`steady`. Again, there are two options for doing that: + +> - The easiest way is to write a `steady_state_model` block, which is +> described below in more details. See also `fs2000.mod` in the +> `examples` directory for an example. The steady state file +> generated by Dynare will be called `+FILENAME/steadystate.m.` +> - You can write the corresponding MATLAB function by hand. If your +> MOD-file is called `FILENAME.mod`, the steady state file must be +> called `FILENAME_steadystate.m`. See `NK_baseline_steadystate.m` +> in the examples directory for an example. This option gives a bit +> more flexibility (loops and conditional structures can be used), +> at the expense of a heavier programming burden and a lesser +> efficiency. + +Note that both files allow to update parameters in each call of the +function. This allows for example to calibrate a model to a labor supply +of 0.2 in steady state by setting the labor disutility parameter to a +corresponding value (see `NK_baseline_steadystate.m` in the `examples` +directory). They can also be used in estimation where some parameter may +be a function of an estimated parameter and needs to be updated for +every parameter draw. For example, one might want to set the capital +utilization cost parameter as a function of the discount rate to ensure +that capacity utilization is 1 in steady state. Treating both parameters +as independent or not updating one as a function of the other would lead +to wrong results. But this also means that care is required. Do not +accidentally overwrite your parameters with new values as it will lead +to wrong results. + +::: {.block} +steady\_state\_model ; + +When the analytical solution of the model is known, this command can be +used to help Dynare find the steady state in a more efficient and +reliable way, especially during estimation where the steady state has to +be recomputed for every point in the parameter space. + +Each line of this block consists of a variable (either an endogenous, a +temporary variable or a parameter) which is assigned an expression +(which can contain parameters, exogenous at the steady state, or any +endogenous or temporary variable already declared above). Each line +therefore looks like: + + VARIABLE_NAME = EXPRESSION; + +Note that it is also possible to assign several variables at the same +time, if the main function in the right hand side is a MATLAB/Octave +function returning several arguments: + + [ VARIABLE_NAME, VARIABLE_NAME... ] = EXPRESSION; + +Dynare will automatically generate a steady state file (of the form +`+FILENAME/steadystate.m`) using the information provided in this block. + +*Steady state file for deterministic models* + +The `steady_state_model` block also works with deterministic models. An +`initval` block and, when necessary, an `endval` block, is used to set +the value of the exogenous variables. Each `initval` or `endval` block +must be followed by `steady` to execute the function created by +`steady_state_model` and set the initial, respectively terminal, steady +state. + +*Example* + +> var m P c e W R k d n l gy_obs gp_obs y dA; +> varexo e_a e_m; +> +> parameters alp bet gam mst rho psi del; +> +> ... +> // parameter calibration, (dynamic) model declaration, shock calibration... +> ... +> +> steady_state_model; +> dA = exp(gam); +> gst = 1/dA; // A temporary variable +> m = mst; +> +> // Three other temporary variables +> khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1)); +> xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1); +> nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp ); +> +> n = xist/(nust+xist); +> P = xist + nust; +> k = khst*n; +> +> l = psi*mst*n/( (1-psi)*(1-n) ); +> c = mst/P; +> d = l - mst + 1; +> y = k^alp*n^(1-alp)*gst^alp; +> R = mst/bet; +> +> // You can use MATLAB functions which return several arguments +> [W, e] = my_function(l, n); +> +> gp_obs = m/dA; +> gy_obs = dA; +> end; +> +> steady; +::: + +### Replace some equations during steady state computations {#eq-tag-ss} + +When there is no steady state file, Dynare computes the steady state by +solving the static model, i.e. the model from the `.mod` file from which +leads and lags have been removed. + +In some specific cases, one may want to have more control over the way +this static model is created. Dynare therefore offers the possibility to +explicitly give the form of equations that should be in the static +model. + +More precisely, if an equation is prepended by a `[static]` tag, then it +will appear in the static model used for steady state computation, but +that equation will not be used for other computations. For every +equation tagged in this way, you must tag another equation with +`[dynamic]`: that equation will not be used for steady state +computation, but will be used for other computations. + +This functionality can be useful on models with a unit root, where there +is an infinity of steady states. An equation (tagged `[dynamic]`) would +give the law of motion of the nonstationary variable (like a random +walk). To pin down one specific steady state, an equation tagged +`[static]` would affect a constant value to the nonstationary variable. +Another situation where the `[static]` tag can be useful is when one has +only a partial closed form solution for the steady state. + +*Example* + +This is a trivial example with two endogenous variables. The second +equation takes a different form in the static model: + + var c k; + varexo x; + ... + model; + c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); + [dynamic] c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); + [static] k = ((delt+bet)/(x*aa*alph))^(1/(alph-1)); + end; + +Getting information about the model +----------------------------------- + +::: {.command} +check ; check (OPTIONS\...); + +Computes the eigenvalues of the model linearized around the values +specified by the last `initval`, `endval` or `steady` statement. +Generally, the eigenvalues are only meaningful if the linearization is +done around a steady state of the model. It is a device for local +analysis in the neighborhood of this steady state. + +A necessary condition for the uniqueness of a stable equilibrium in the +neighborhood of the steady state is that there are as many eigenvalues +larger than one in modulus as there are forward looking variables in the +system. An additional rank condition requires that the square submatrix +of the right Schur vectors corresponding to the forward looking +variables (jumpers) and to the explosive eigenvalues must have full +rank. + +Note that the outcome may be different from what would be suggested by +`sum(abs(oo_.dr.eigval))` when eigenvalues are very close to +`qz_criterium `{.interpreted-text role="opt"}. + +*Options* + +::: {.option} +solve\_algo = INTEGER + +See `solve_algo `{.interpreted-text role="ref"}, for the +possible values and their meaning. +::: + +::: {.option} +qz\_zero\_threshold = DOUBLE + +Value used to test if a generalized eigenvalue is $0/0$ in the +generalized Schur decomposition (in which case the model does not admit +a unique solution). Default: `1e-6`. +::: + +*Output* + +`check` returns the eigenvalues in the global variable `oo_.dr.eigval`. +::: + +::: {.matvar} +[oo]().dr.eigval + +Contains the eigenvalues of the model, as computed by the `check` +command. +::: + +::: {.command} +model\_diagnostics ; + +This command performs various sanity checks on the model, and prints a +message if a problem is detected (missing variables at current period, +invalid steady state, singular Jacobian of static model). +::: + +::: {.command} +model\_info ; model\_info (OPTIONS\...); + +This command provides information about the model. + +When used outside the context of the `block` option of the `model` +block, it will provide a list of predetermined state variables, +forward-looking variables, and purely static variables. + + +Deterministic simulation {#det-simul} +------------------------ + +### Perfect foresight + +When the framework is deterministic, Dynare can be used for models with +the assumption of perfect foresight. Typically, the system is supposed +to be in a state of equilibrium before a period `1` when the news of a +contemporaneous or of a future shock is learned by the agents in the +model. The purpose of the simulation is to describe the reaction in +anticipation of, then in reaction to the shock, until the system returns +to the old or to a new state of equilibrium. In most models, this return +to equilibrium is only an asymptotic phenomenon, which one must +approximate by an horizon of simulation far enough in the future. +Another exercise for which Dynare is well suited is to study the +transition path to a new equilibrium following a permanent shock. For +deterministic simulations, the numerical problem consists of solving a +nonlinear system of simultaneous equations in `n` endogenous variables +in `T` periods. Dynare offers several algorithms for solving this +problem, which can be chosen via the `stack_solve_algo` option. By +default (`stack_solve_algo=0`), Dynare uses a Newton-type method to +solve the simultaneous equation system. Because the resulting Jacobian +is in the order of `n` by `T` and hence will be very large for long +simulations with many variables, Dynare makes use of the sparse matrix +capacities of MATLAB/Octave. A slower but potentially less memory +consuming alternative (`stack_solve_algo=1`) is based on a Newton-type +algorithm first proposed by *Laffargue (1990)* and *Boucekkine (1995)*, +which avoids ever storing the full Jacobian. The details of the +algorithm can be found in *Juillard (1996)*. The third type of +algorithms makes use of block decomposition techniques +(divide-and-conquer methods) that exploit the structure of the model. +The principle is to identify recursive and simultaneous blocks in the +model structure and use this information to aid the solution process. +These solution algorithms can provide a significant speed-up on large +models. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +Be careful when employing auxiliary variables in the context of perfect +foresight computations. The same model may work for stochastic +simulations, but fail for perfect foresight simulations. The issue +arises when an equation suddenly only contains variables dated `t+1` (or +`t-1` for that matter). In this case, the derivative in the last (first) +period with respect to all variables will be 0, rendering the stacked +Jacobian singular. + +*Example* + +> Consider the following specification of an Euler equation with log +> utility: +> +> Lambda = beta*C(-1)/C; +> Lambda(+1)*R(+1)= 1; +> +> Clearly, the derivative of the second equation with respect to all +> endogenous variables at time `t` is zero, causing +> `perfect_foresight_solver` to generally fail. This is due to the use +> of the Lagrange multiplier `Lambda` as an auxiliary variable. Instead, +> employing the identical +> +> beta*C/C(+1)*R(+1)= 1; +> +> will work. +::: + +::: {.command} +perfect\_foresight\_setup ; perfect\_foresight\_setup (OPTIONS\...); + +Prepares a perfect foresight simulation, by extracting the information +in the `initval`, `endval` and `shocks` blocks and converting them into +simulation paths for exogenous and endogenous variables. + +This command must always be called before running the simulation with +`perfect_foresight_solver`. + +*Options* + +::: {.option} +periods = INTEGER + +Number of periods of the simulation. +::: + +::: {.option} +datafile = FILENAME + +Used to specify path for all endogenous and exogenous variables. +Strictly equivalent to `initval_file`{.interpreted-text role="comm"}. +::: + +*Output* + +The paths for the exogenous variables are stored into `oo_.exo_simul`. + +The initial and terminal conditions for the endogenous variables and the +initial guess for the path of endogenous variables are stored into +`oo_.endo_simul`. +::: + +::: {.command} +perfect\_foresight\_solver ; perfect\_foresight\_solver (OPTIONS\...); + +Computes the perfect foresight (or deterministic) simulation of the +model. + +Note that `perfect_foresight_setup` must be called before this command, +in order to setup the environment for the simulation. + +*Options* + +::: {.option} +maxit = INTEGER + +Determines the maximum number of iterations used in the non-linear +solver. The default value of `maxit` is `50`. +::: + +::: {.option} +tolf = DOUBLE + +Convergence criterion for termination based on the function value. +Iteration will cease when it proves impossible to improve the function +value by more than `tolf`. Default: `1e-5` +::: + +::: {.option} +tolx = DOUBLE + +Convergence criterion for termination based on the change in the +function argument. Iteration will cease when the solver attempts to take +a step that is smaller than `tolx`. Default: `1e-5` +::: + +::: {.option} +noprint + +Don't print anything. Useful for loops. +::: + +::: {.option} +print + +Print results (opposite of `noprint`). +::: + + +::: {.option} +lmmcp + +Solves the perfect foresight model with a Levenberg-Marquardt mixed +complementarity problem (LMMCP) solver (*Kanzow and Petra (2004)*), +which allows to consider inequality constraints on the endogenous +variables (such as a ZLB on the nominal interest rate or a model with +irreversible investment). This option is equivalent to +`stack_solve_algo=7` **and** `solve_algo=10`. Using the LMMCP solver +requires a particular model setup as the goal is to get rid of any +min/max operators and complementary slackness conditions that might +introduce a singularity into the Jacobian. This is done by attaching an +equation tag (see `model-decl`{.interpreted-text role="ref"}) with the +`mcp` keyword to affected equations. This tag states that the equation +to which the tag is attached has to hold unless the expression within +the tag is binding. For instance, a ZLB on the nominal interest rate +would be specified as follows in the model block: + + model; + ... + [mcp = 'r > -1.94478'] + r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e; + ... + end; + +where `1.94478` is the steady state level of the nominal interest rate +and `r` is the nominal interest rate in deviation from the steady state. +This construct implies that the Taylor rule is operative, unless the +implied interest rate `r<=-1.94478`, in which case the `r` is fixed at +`-1.94478` (thereby being equivalent to a complementary slackness +condition). By restricting the value of `r` coming out of this equation, +the `mcp` tag also avoids using `max(r,-1.94478)` for other occurrences +of `r` in the rest of the model. It is important to keep in mind that, +because the `mcp` tag effectively replaces a complementary slackness +condition, it cannot be simply attached to any equation. Rather, it must +be attached to the correct affected equation as otherwise the solver +will solve a different problem than originally intended. Also, since the +problem to be solved is nonlinear, the sign of the residuals of the +dynamic equation matters. In the previous example, for the nominal +interest rate rule, if the LHS and RHS are reversed the sign of the +residuals (the difference between the LHS and the RHS) will change and +it may happen that solver fails to identify the solution path. More +generally, convergence of the nonlinear solver is not guaranteed when +using mathematically equivalent representations of the same equation. + +Note that in the current implementation, the content of the `mcp` +equation tag is not parsed by the preprocessor. The inequalities must +therefore be as simple as possible: an endogenous variable, followed by +a relational operator, followed by a number (not a variable, parameter +or expression). +::: + +::: {.option} +endogenous\_terminal\_period + +The number of periods is not constant across Newton iterations when +solving the perfect foresight model. The size of the nonlinear system of +equations is reduced by removing the portion of the paths (and +associated equations) for which the solution has already been identified +(up to the tolerance parameter). This strategy can be interpreted as a +mix of the shooting and relaxation approaches. Note that round off +errors are more important with this mixed strategy (user should check +the reported value of the maximum absolute error). Only available with +option `stack_solve_algo==0`. +::: + +::: {.option} +linear\_approximation + +Solves the linearized version of the perfect foresight model. The model +must be stationary. Only available with option `stack_solve_algo==0` or +`stack_solve_algo==7`. +::: + +*Output* + +The simulated endogenous variables are available in global matrix +`oo_.endo_simul`. +::: + + +This variable stores the path of exogenous variables during a simulation +(computed by `perfect_foresight_solver`, `simul`, `stoch_simul` or +`extended_path`). The variables are arranged in columns, in order of +declaration (as in `M_.exo_names`). Periods are in rows. Note that this +convention regarding columns and rows is the opposite of the convention +for `oo_.endo_simul`! +::: + + +Stochastic solution and simulation {#stoch-sol} +---------------------------------- + +In a stochastic context, Dynare computes one or several simulations +corresponding to a random draw of the shocks. + +The main algorithm for solving stochastic models relies on a Taylor +approximation, up to third order, of the expectation functions (see +*Judd (1996)*, *Collard and Juillard (2001a, 2001b)*, and *Schmitt-Grohé +and Uríbe (2004)*). The details of the Dynare implementation of the +first order solution are given in *Villemot (2011)*. Such a solution is +computed using the `stoch_simul` command. + +As an alternative, it is possible to compute a simulation to a +stochastic model using the *extended path* method presented by *Fair and +Taylor (1983)*. This method is especially useful when there are strong +nonlinearities or binding constraints. Such a solution is computed using +the `extended_path` command. + +### Computing the stochastic solution {#stoch-sol-simul} + +::: {.command} +stoch\_simul \[VARIABLE\_NAME\...\]; stoch\_simul (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +Solves a stochastic (i.e. rational expectations) model, using +perturbation techniques. + +More precisely, `stoch_simul` computes a Taylor approximation of the +model around the deterministic steady state and solves of the the +decision and transition functions for the approximated model. Using +this, it computes impulse response functions and various descriptive +statistics (moments, variance decomposition, correlation and +autocorrelation coefficients). For correlated shocks, the variance +decomposition is computed as in the VAR literature through a Cholesky +decomposition of the covariance matrix of the exogenous variables. When +the shocks are correlated, the variance decomposition depends upon the +order of the variables in the `varexo` command. + +The Taylor approximation is computed around the steady state (see +`st-st`{.interpreted-text role="ref"}). + +The IRFs are computed as the difference between the trajectory of a +variable following a shock at the beginning of period `1` and its steady +state value. More details on the computation of IRFs can be found at +. + +Variance decomposition, correlation, autocorrelation are only displayed +for variables with strictly positive variance. Impulse response +functions are only plotted for variables with response larger than +$10^{-10}$. + +Variance decomposition is computed relative to the sum of the +contribution of each shock. Normally, this is of course equal to +aggregate variance, but if a model generates very large variances, it +may happen that, due to numerical error, the two differ by a significant +amount. Dynare issues a warning if the maximum relative difference +between the sum of the contribution of each shock and aggregate variance +is larger than `0.01%`. + +The covariance matrix of the shocks is specified with the `shocks` +command (see `shocks-exo`{.interpreted-text role="ref"}). + +When a list of `VARIABLE_NAME` is specified, results are displayed only +for these variables. + +The `stoch_simul` command with a first order approximation can benefit +from the block decomposition of the model (see `block`{.interpreted-text +role="opt"}). + +*Options* + +::: {.option} +ar = INTEGER + +Order of autocorrelation coefficients to compute and to print. Default: +`5`. +::: + +::: {.option} +drop = INTEGER + +Number of points (burnin) dropped at the beginning of simulation before +computing the summary statistics. Note that this option does not affect +the simulated series stored in `oo_.endo_simul` and the workspace. Here, +no periods are dropped. Default: `100`. +::: + +::: {.option} +hp\_filter = DOUBLE + +Uses HP filter with $\lambda =$ `DOUBLE` before computing moments. If +theoretical moments are requested, the spectrum of the model solution is +filtered following the approach outlined in Uhlig (2001). Default: no +filter. +::: + +::: {.option} +one\_sided\_hp\_filter = DOUBLE + +Uses the one-sided HP filter with $\lambda =$ `DOUBLE` described in +*Stock and Watson (1999)* before computing moments. This option is only +available with simulated moments. Default: no filter. +::: + +::: {.option} +bandpass\_filter + +Uses a bandpass filter with the default passband before computing +moments. If theoretical moments are requested, the spectrum of the model +solution is filtered using an ideal bandpass filter. If empirical +moments are requested, the *Baxter and King (1999)* filter is used. +Default: no filter. +::: + +::: {.option} +bandpass\_filter = \[HIGHEST\_PERIODICITY LOWEST\_PERIODICITY\] + +Uses a bandpass filter before computing moments. The passband is set to +a periodicity of to LOWEST\_PERIODICITY, e.g. $6$ to $32$ quarters if +the model frequency is quarterly. Default: `[6,32]`. +::: + +::: {.option} +filtered\_theoretical\_moments\_grid = INTEGER + +When computing filtered theoretical moments (with either option +`hp_filter` or option `bandpass_filter`), this option governs the number +of points in the grid for the discrete Inverse Fast Fourier Transform. +It may be necessary to increase it for highly autocorrelated processes. +Default: `512`. +::: + +::: {.option} +irf = INTEGER + +Number of periods on which to compute the IRFs. Setting `irf=0` +suppresses the plotting of IRFs. Default: `40`. +::: + +::: {.option} +irf\_shocks = ( VARIABLE\_NAME \[\[,\] VARIABLE\_NAME \...\] ) + +The exogenous variables for which to compute IRFs. Default: all. +::: + +::: {.option} +relative\_irf + +Requests the computation of normalized IRFs. At first order, the normal +shock vector of size one standard deviation is divided by the standard +deviation of the current shock and multiplied by 100. The impulse +responses are hence the responses to a unit shock of size 1 (as opposed +to the regular shock size of one standard deviation), multiplied by 100. +Thus, for a loglinearized model where the variables are measured in +percent, the IRFs have the interpretation of the percent responses to a +100 percent shock. For example, a response of 400 of output to a TFP +shock shows that output increases by 400 percent after a 100 percent TFP +shock (you will see that TFP increases by 100 on impact). Given +linearity at `order=1`, it is straightforward to rescale the IRFs stored +in `oo_.irfs` to any desired size. At higher order, the interpretation +is different. The `relative_irf` option then triggers the generation of +IRFs as the response to a 0.01 unit shock (corresponding to 1 percent +for shocks measured in percent) and no multiplication with 100 is +performed. That is, the normal shock vector of size one standard +deviation is divided by the standard deviation of the current shock and +divided by 100. For example, a response of 0.04 of log output (thus +measured in percent of the steady state output level) to a TFP shock +also measured in percent then shows that output increases by 4 percent +after a 1 percent TFP shock (you will see that TFP increases by 0.01 on +impact). +::: + +::: {.option} +irf\_plot\_threshold = DOUBLE + +Threshold size for plotting IRFs. All IRFs for a particular variable +with a maximum absolute deviation from the steady state smaller than +this value are not displayed. Default: `1e-10`. +::: + +::: {.option} +nocorr + +Don't print the correlation matrix (printing them is the default). +::: + +::: {.option} +nodecomposition + +Don't compute (and don't print) unconditional variance decomposition. +::: + +::: {.option} +nofunctions + +Don't print the coefficients of the approximated solution (printing them +is the default). +::: + +::: {.option} +nomoments + +Don't print moments of the endogenous variables (printing them is the +default). +::: + +::: {.option} +nograph + +Do not create graphs (which implies that they are not saved to the disk +nor displayed). If this option is not used, graphs will be saved to disk +(to the format specified by `graph_format` option, except if +`graph_format=none`) and displayed to screen (unless `nodisplay` option +is used). +::: + +::: {.option} +graph + +Re-enables the generation of graphs previously shut off with `nograph`. +::: + +::: {.option} +nodisplay + +Do not display the graphs, but still save them to disk (unless `nograph` +is used). +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +Specify the file format(s) for graphs saved to disk. Possible values are +`eps` (the default), `pdf`, `fig` and `none` (under Octave, `fig` is +unavailable). If the file format is set equal to `none`, the graphs are +displayed but not saved to the disk. +::: + +::: {.option} +noprint + +See `noprint`{.interpreted-text role="opt"}. +::: + +::: {.option} +print + +See `print`{.interpreted-text role="opt"}. +::: + +::: {.option} +order = INTEGER + +Order of Taylor approximation. Note that for third order and above, the +`k_order_solver` option is implied and only empirical moments are +available (you must provide a value for `periods` option). Default: `2` +(except after an `estimation` command, in which case the default is the +value used for the estimation). +::: + +::: {.option} +k\_order\_solver + +Use a k-order solver (implemented in C++) instead of the default Dynare +solver. This option is not yet compatible with the `bytecode` option +(see `model-decl`{.interpreted-text role="ref"}). Default: disabled for +order 1 and 2, enabled for order 3 and above. +::: + +::: {.option} +periods = INTEGER + +If different from zero, empirical moments will be computed instead of +theoretical moments. The value of the option specifies the number of +periods to use in the simulations. Values of the initval block, possibly +recomputed by `steady`, will be used as starting point for the +simulation. The simulated endogenous variables are made available to the +user in a vector for each variable and in the global matrix +`oo_.endo_simul` (see `oo_.endo_simul`{.interpreted-text role="mvar"}). +The simulated exogenous variables are made available in `oo_.exo_simul` +(see `oo_.exo_simul`{.interpreted-text role="mvar"}). Default: `0`. +::: + +::: {.option} +qz\_criterium = DOUBLE + +Value used to split stable from unstable eigenvalues in reordering the +Generalized Schur decomposition used for solving first order problems. +Default: `1.000001` (except when estimating with `lik_init` option equal +to `1`: the default is `0.999999` in that case; see +`estim`{.interpreted-text role="ref"}). +::: + +::: {.option} +qz\_zero\_threshold = DOUBLE + +See `qz_zero_threshold `{.interpreted-text +role="opt"}. +::: + +::: {.option} +replic = INTEGER + +Number of simulated series used to compute the IRFs. Default: `1` if +`order=1`, and `50` otherwise. +::: + +::: {.option} +simul\_replic = INTEGER + +Number of series to simulate when empirical moments are requested (i.e. +`periods` $>$ 0). Note that if this option is greater than 1, the +additional series will not be used for computing the empirical moments +but will simply be saved in binary form to the file `FILENAME_simul` in +the `FILENAME/Output`-folder. Default: `1`. +::: + +::: {.option} +solve\_algo = INTEGER + +See `solve_algo `{.interpreted-text role="ref"}, for the +possible values and their meaning. +::: + + +::: {.option} +conditional\_variance\_decomposition = INTEGER +conditional\_variance\_decomposition = \[INTEGER1:INTEGER2\] +conditional\_variance\_decomposition = \[INTEGER1 INTEGER2 \...\] + +Computes a conditional variance decomposition for the specified +period(s). The periods must be strictly positive. Conditional variances +are given by $var(y_{t+k}\vert t)$. For period 1, the conditional +variance decomposition provides the decomposition of the effects of +shocks upon impact. + +The results are stored in `oo_.conditional_variance_decomposition` (see +`oo_.conditional_variance_decomposition`{.interpreted-text +role="mvar"}). In the presence of measurement error, the +`oo_.conditional_variance_decomposition` field will contain the variance +contribution after measurement error has been taken out, i.e. the +decomposition will be conducted of the actual as opposed to the measured +variables. The variance decomposition of the measured variables will be +stored in `oo_.conditional_variance_decomposition_ME` (see +`oo_.conditional_variance_decomposition_ME`{.interpreted-text +role="mvar"}). The variance decomposition is only conducted, if +theoretical moments are requested, *i.e.* using the `periods=0`-option. +Only available at `order<3` and without +`pruning''. In case of`order=2`, Dynare provides a second-order accurate approximation to the true second moments based on the linear terms of the second-order solution (see *Kim, Kim, Schaumburg and Sims (2008)*). Note that the unconditional variance decomposition *i.e.* at horizon infinity) is automatically conducted if theoretical moments are requested and if`nodecomposition\`[ +is not set (see :mvar:\`oo\_.variance\_decomposition]{.title-ref}). +::: + +::: {.option} +pruning + +Discard higher order terms when iteratively computing simulations of the +solution. At second order, Dynare uses the algorithm of *Kim, Kim, +Schaumburg and Sims (2008)*, while at third order its generalization by +*Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018)* is used. Not +available above third order. When specified, theoretical moments are +based on the pruned state space, i.e. the computation of second moments +uses all terms as in *Andreasen, Fernández-Villaverde and Rubio-Ramírez +(2018), page 10* as opposed to simply providing a second-order accurate +result based on the linear solution as in *Kim, Kim, Schaumburg and Sims +(2008)*. +::: + + +::: {.option} +sylvester = OPTION + +Determines the algorithm used to solve the Sylvester equation for block +decomposed model. Possible values for OPTION are: + +> `default` +> +> > Uses the default solver for Sylvester equations (`gensylv`) based on +> > Ondra Kamenik's algorithm (see +> > [here](https://www.dynare.org/assets/dynare++/sylvester.pdf) for +> > more information). +> +> `fixed_point` +> +> > Uses a fixed point algorithm to solve the Sylvester equation +> > (`gensylv_fp`). This method is faster than the default one for large +> > scale models. + +Default value is `default`. +::: + +::: {.option} +sylvester\_fixed\_point\_tol = DOUBLE + +The convergence criterion used in the fixed point Sylvester solver. Its +default value is `1e-12`. +::: + +::: {.option} +dr = OPTION + +Determines the method used to compute the decision rule. Possible values +for OPTION are: + +> `default` +> +> > Uses the default method to compute the decision rule based on the +> > generalized Schur decomposition (see *Villemot (2011)* for more +> > information). +> +> `cycle_reduction` +> +> > Uses the cycle reduction algorithm to solve the polynomial equation +> > for retrieving the coefficients associated to the endogenous +> > variables in the decision rule. This method is faster than the +> > default one for large scale models. +> + +Default value is `default`. +::: + +::: {.option} +dr\_cycle\_reduction\_tol = DOUBLE + +The convergence criterion used in the cycle reduction algorithm. Its +default value is `1e-7`. +::: + +::: {.option} +tex + +Requests the printing of results and graphs in TeX tables and graphics +that can be later directly included in LaTeX files. +::: + +::: {.option} +dr\_display\_tol = DOUBLE + +Tolerance for the suppression of small terms in the display of decision +rules. Rows where all terms are smaller than `dr_display_tol` are not +displayed. Default value: `1e-6`. +::: + +::: {.option} +contemporaneous\_correlation + +Saves the contemporaneous correlation between the endogenous variables +in `oo_.contemporaneous_correlation`. Requires the `nocorr` option not +to be set. +::: + +::: {.option} +spectral\_density + +Triggers the computation and display of the theoretical spectral density +of the (filtered) model variables. Results are stored in +`oo_.SpectralDensity`, defined below. Default: do not request spectral +density estimates. +::: + +::: {.option} +hp\_ngrid = INTEGER + +Deprecated option. It has the same effect as +`filtered_theoretical_moments_grid `{.interpreted-text +role="opt"}. +::: + +*Output* + +This command sets `oo_.dr`, `oo_.mean`, `oo_.var`, `oo_.var_list`, and +`oo_.autocorr`, which are described below. + +If the `periods` option is present, sets `oo_.skewness`, `oo_.kurtosis`, +and `oo_.endo_simul` (see `oo_.endo_simul`{.interpreted-text +role="mvar"}), and also saves the simulated variables in MATLAB/Octave +vectors of the global workspace with the same name as the endogenous +variables. + +If option `irf` is different from zero, sets `oo_.irfs` (see below) and +also saves the IRFs in MATLAB/Octave vectors of the global workspace +(this latter way of accessing the IRFs is deprecated and will disappear +in a future version). + +If the option `contemporaneous_correlation` is different from `0`, sets +`oo_.contemporaneous_correlation`, which is described below. + +*Example* + +> shocks; +> var e; +> stderr 0.0348; +> end; +> +> stoch_simul; +> +> Performs the simulation of the 2nd-order approximation of a model with +> a single stochastic shock `e`, with a standard error of `0.0348`. + +*Example* + +> stoch_simul(irf=60) y k; +> +> Performs the simulation of a model and displays impulse response +> functions on 60 periods for variables `y` and `k`. +::: + +::: {.matvar} +[oo]().mean + +After a run of `stoch_simul`, contains the mean of the endogenous +variables. Contains theoretical mean if the `periods` option is not +present, and simulated mean otherwise. The variables are arranged in +declaration order. +::: + +::: {.matvar} +[oo]().var + +After a run of `stoch_simul`, contains the variance-covariance of the +endogenous variables. Contains theoretical variance if the `periods` +option is not present and simulated variance otherwise. Only available +for `order<4`. At `order=2` it will be be a second-order accurate +approximation (i.e. ignoring terms of order 3 and 4 that would arise +when using the full second-order policy function). At `order=3`, +theoretical moments are only available with `pruning`. The variables are +arranged in declaration order. +::: + +::: {.matvar} +[oo]().var\_list + +The list of variables for which results are displayed. +::: + +::: {.matvar} +[oo]().skewness + +After a run of `stoch_simul` contains the skewness (standardized third +moment) of the simulated variables if the `periods` option is present. +The variables are arranged in declaration order. +::: + +::: {.matvar} +[oo]().kurtosis + +After a run of `stoch_simul` contains the excess kurtosis (standardized +fourth moment) of the simulated variables if the `periods` option is +present. The variables are arranged in declaration order. +::: + +::: {.matvar} +[oo]().autocorr + +After a run of `stoch_simul`, contains a cell array of the +autocorrelation matrices of the endogenous variables. The element number +of the matrix in the cell array corresponds to the order of +autocorrelation. The option ar specifies the number of autocorrelation +matrices available. Contains theoretical autocorrelations if the +`periods` option is not present and simulated autocorrelations +otherwise. Only available for `order<4`. At `order=2` it will be be a +second-order accurate approximation. At `order=3`, theoretical moments +are only available with `pruning`. The field is only created if +stationary variables are present. + +The element `oo_.autocorr{i}(k,l)` is equal to the correlation between +$y^k_t$ and $y^l_{t-i}$, where $y^k$ (resp. $y^l$) is the $k$-th (resp. +$l$-th) endogenous variable in the declaration order. + +Note that if theoretical moments have been requested, `oo_.autocorr{i}` +is the same than `oo_.gamma_y{i+1}`. +::: + +::: {.matvar} +[oo]().gamma\_y + +After a run of `stoch_simul`, if theoretical moments have been requested +(i.e. if the `periods` option is not present), this variable contains a +cell array with the following values (where `ar` is the value of the +option of the same name): + +> `oo_.gamma{1}` +> +> > Variance/covariance matrix. +> +> `oo_.gamma{i+1}` (for i=1:ar) +> +> > Autocorrelation function. See `oo_.autocorr`{.interpreted-text +> > role="mvar"} for more details. **Beware**, this is the +> > autocorrelation function, not the autocovariance function. +> +> `oo_.gamma{ar+2}` +> +> > Unconditional variance decomposition, see +> > `oo_.variance_decomposition`{.interpreted-text role="mvar"}. +> +> `oo_.gamma{ar+3}` +> +> > If a second order approximation has been requested, contains the +> > vector of the mean correction terms. +> > +> > Only available at `order<4`. In case `order=2`, the theoretical +> > second moments are a second order accurate approximation of the true +> > second moments. See conditional\_variance\_decomposition. At +> > `order=3`, theoretical moments are only available with `pruning`. +::: + +::: {.matvar} +[oo]().variance\_decomposition + +After a run of `stoch_simul` when requesting theoretical moments +(`periods=0`), contains a matrix with the result of the unconditional +variance decomposition (i.e. at horizon infinity). The first dimension +corresponds to the endogenous variables (in the order of declaration +after the command or in `M_.endo_names`) and the second dimension +corresponds to exogenous variables (in the order of declaration). +Numbers are in percent and sum up to 100 across columns. In the presence +of measurement error, the field will contain the variance contribution +after measurement error has been taken out, *i.e.* the decomposition +will be conducted of the actual as opposed to the measured variables. +::: + +::: {.matvar} +[oo]().variance\_decomposition\_ME + +Field set after a run of `stoch_simul` when requesting theoretical +moments (`periods=0`) if measurement error is present. It is similar to +`oo_.variance_decomposition`{.interpreted-text role="mvar"}, but the +decomposition will be conducted of the measured variables. The field +contains a matrix with the result of the unconditional variance +decomposition (*i.e.* at horizon infinity). The first dimension +corresponds to the observed endoogenous variables (in the order of +declaration after the command) and the second dimension corresponds to +exogenous variables (in the order of declaration), with the last column +corresponding to the contribution of measurement error. Numbers are in +percent and sum up to 100 across columns. +::: + +::: {.matvar} +[oo]().conditional\_variance\_decomposition + +After a run of `stoch_simul` with the +`conditional_variance_decomposition` option, contains a +three-dimensional array with the result of the decomposition. The first +dimension corresponds to the endogenous variables (in the order of +declaration after the command or in `M_.endo_names` if not specified), +the second dimension corresponds to the forecast horizons (as declared +with the option), and the third dimension corresponds to the exogenous +variables (in the order of declaration). In the presence of measurement +error, the field will contain the variance contribution after +measurement error has been taken out, *i.e.* the decomposition will be +conductedof the actual as opposed to the measured variables. +::: + +::: {.matvar} +[oo]().conditional\_variance\_decomposition\_ME + +Field set after a run of `stoch_simul` with the +`conditional_variance_decomposition` option if measurement error is +present. It is similar to +`oo_.conditional_variance_decomposition`{.interpreted-text role="mvar"}, +but the decomposition will be conducted of the measured variables. It +contains a three-dimensional array with the result of the decomposition. +The first dimension corresponds to the endogenous variables (in the +order of declaration after the command or in `M_.endo_names` if not +specified), the second dimension corresponds to the forecast horizons +(as declared with the option), and the third dimension corresponds to +the exogenous variables (in the order of declaration), with the last +column corresponding to the contribution of the measurement error. +::: + +::: {.matvar} +[oo]().contemporaneous\_correlation + +After a run of `stoch_simul` with the +`contemporaneous_correlation option`, contains theoretical +contemporaneous correlations if the `periods` option is not present, and +simulated contemporaneous correlations otherwise. Only available for +`order<4`. At `order=2` it will be be a second-order accurate +approximation. At `order=3`, theoretical moments are only available with +`pruning`. The variables are arranged in declaration order. +::: + +::: {.matvar} +[oo]().SpectralDensity + +After a run of `stoch_simul` with option `spectral_density`, contains +the spectral density of the model variables. There will be a `nvars` by +`nfrequencies` subfield `freqs` storing the respective frequency grid +points ranging from $0$ to $2\pi$ and a same sized subfield `density` +storing the corresponding density. +::: + +::: {.matvar} +[oo]().irfs + +After a run of `stoch_simul` with option `irf` different from zero, +contains the impulse responses, with the following naming convention: +[VARIABLE\_NAME\_SHOCK\_NAME]{.title-ref}. + +> For example, `oo_.irfs.gnp_ea` contains the effect on `gnp` of a +> one-standard deviation shock on `ea`. +::: + +::: {.matcomm} +get\_irf (\'EXOGENOUS\_NAME\' \[, \'ENDOGENOUS\_NAME\'\]\... ); + +Given the name of an exogenous variables, returns the IRFs for the +requested endogenous variable(s), as they are stored in `oo_.irfs`. +::: + +The approximated solution of a model takes the form of a set of decision +rules or transition equations expressing the current value of the +endogenous variables of the model as function of the previous state of +the model and shocks observed at the beginning of the period. The +decision rules are stored in the structure `oo_.dr` which is described +below. + +::: {.matvar} +[oo]().dr + +Structure storing the decision rules. The subfields for different orders +of approximation are explained below. +::: + +::: {.command} +extended\_path ; extended\_path (OPTIONS\...); + +Simulates a stochastic (i.e. rational expectations) model, using the +extended path method presented by *Fair and Taylor (1983)*. Time series +for the endogenous variables are generated by assuming that the agents +believe that there will no more shocks in the following periods. + +This function first computes a random path for the exogenous variables +(stored in `oo_.exo_simul`, see `oo_.exo_simul`{.interpreted-text +role="mvar"}) and then computes the corresponding path for endogenous +variables, taking the steady state as starting point. The result of the +simulation is stored in `oo_.endo_simul` (see +`oo_.endo_simul`{.interpreted-text role="mvar"}). Note that this +simulation approach does not solve for the policy and transition +equations but for paths for the endogenous variables. + +*Options* + +::: {.option} +periods = INTEGER + +The number of periods for which the simulation is to be computed. No +default value, mandatory option. +::: + +::: {.option} +solver\_periods = INTEGER + +The number of periods used to compute the solution of the perfect +foresight at every iteration of the algorithm. Default: `200`. +::: + +::: {.option} +order = INTEGER + +If order is greater than `0` Dynare uses a gaussian quadrature to take +into account the effects of future uncertainty. If `order` $=S$ then the +time series for the endogenous variables are generated by assuming that +the agents believe that there will no more shocks after period $t+S$. +This is an experimental feature and can be quite slow. A non-zero value +is not compatible with either the `bytecode` or the `block` option of +the `model` block. Default: `0`. +::: + +::: {.option} +hybrid + +Use the constant of the second order perturbation reduced form to +correct the paths generated by the (stochastic) extended path algorithm. +::: + +::: {.option} +lmmcp + +Solves the perfect foresight model with a Levenberg-Marquardt mixed +complementarity problem (LMMCP) solver (*Kanzow and Petra (2004)*), +which allows to consider inequality constraints on the endogenous +variables (such as a ZLB on the nominal interest rate or a model with +irreversible investment). For specifying the necessary `mcp`-tag, see +`lmmcp`{.interpreted-text role="opt"}. +::: +::: + +### Typology and ordering of variables + +Dynare distinguishes four types of endogenous variables: + +*Purely backward (or purely predetermined) variables* + +> Those that appear only at current and past period in the model, but +> not at future period (i.e. at $t$ and $t-1$ but not $t+1$). The number +> of such variables is equal to `M_.npred`. + +*Purely forward variables* + +> Those that appear only at current and future period in the model, but +> not at past period (i.e. at $t$ and $t+1$ but not $t-1$). The number +> of such variables is stored in `M_.nfwrd`. + +*Mixed variables* + +> Those that appear at current, past and future period in the model +> (i.e. at $t$, $t+1$ and $t-1$). The number of such variables is stored +> in `M_.nboth`. + +*Static variables* + +> Those that appear only at current, not past and future period in the +> model (i.e. only at $t$, not at $t+1$ or $t-1$). The number of such +> variables is stored in `M_.nstatic`. + +Note that all endogenous variables fall into one of these four +categories, since after the creation of auxiliary variables (see +`aux-variables`{.interpreted-text role="ref"}), all endogenous have at +most one lead and one lag. We therefore have the following identity: + +> ``` {.sourceCode .matlab} +> M_.npred + M_.both + M_.nfwrd + M_.nstatic = M_.endo_nbr +> ``` + +::: {.matvar} +[M]().state\_var + +Vector of numerical indices identifying the state variables in the +vector of declared variables. `M_.endo_names(M_.state_var)` therefore +yields the name of all variables that are states in the model +declaration, i.e. that show up with a lag. +::: + +Internally, Dynare uses two orderings of the endogenous variables: the +order of declaration (which is reflected in `M_.endo_names`), and an +order based on the four types described above, which we will call the +DR-order ("DR" stands for decision rules). Most of the time, the +declaration order is used, but for elements of the decision rules, the +DR-order is used. + +The DR-order is the following: static variables appear first, then +purely backward variables, then mixed variables, and finally purely +forward variables. Inside each category, variables are arranged +according to the declaration order. + +::: {.matvar} +[oo]().dr.order\_var + +This variables maps DR-order to declaration order. +::: + +::: {.matvar} +[oo]().dr.inv\_order\_var + +This variable contains the inverse map. +::: + +In other words, the k-th variable in the DR-order corresponds to the +endogenous variable numbered `oo_.dr.order_var(k)` in declaration order. +Conversely, k-th declared variable is numbered `oo_.dr.inv_order_var(k)` +in DR-order. + +Finally, the state variables of the model are the purely backward +variables and the mixed variables. They are ordered in DR-order when +they appear in decision rules elements. There are +`M_.nspred = M_.npred + M_.nboth` such variables. Similarly, one has +`M_.nsfwrd = M_.nfwrd + M_.nboth`, and +`M_.ndynamic = M_.nfwrd + M_.nboth + M_.npred`. + +### First-order approximation + +The approximation has the stylized form: + +$$y_t = y^s + A y^h_{t-1} + B u_t$$ + +where $y^s$ is the steady state value of $y$ and $y^h_t=y_t-y^s$. + +::: {.matvar} +oo.dr.state\_var + +Vector of numerical indices identifying the state variables in the +vector of declared variables, *given the current parameter values* for +which the decision rules have been computed. It may differ from +`M_.state_var` in case a state variable drops from the model given the +current parameterization, because it only gets 0 coefficients in the +decision rules. See `M_.state_var`{.interpreted-text role="mvar"}. +::: + +The coefficients of the decision rules are stored as follows: + +- $y^s$ is stored in `oo_.dr.ys`. The vector rows correspond to all + endogenous in the declaration order. +- $A$ is stored in `oo_.dr.ghx`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to state + variables in DR-order, as given by `oo_.dr.state_var`. (N.B.: if the + `block` option to the `model` block has been specified, then rows + are in declaration order, and columns are ordered according to + `oo_.dr.state_var` which may differ from DR-order.) +- $B$ is stored `oo_.dr.ghu`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to exogenous + variables in declaration order. (N.B.: if the `block` option to the + `model` block has been specified, then rows are in declaration + order.) + +Of course, the shown form of the approximation is only stylized, because +it neglects the required different ordering in $y^s$ and $y^h_t$. The +precise form of the approximation that shows the way Dynare deals with +differences between declaration and DR-order, is + +$$y_t(\mathrm{oo\_.dr.order\_var}) = +y^s(\mathrm{oo\_.dr.order\_var}) + A \cdot +y_{t-1}(\mathrm{oo\_.dr.order\_var(k2)}) - +y^s(\mathrm{oo\_.dr.order\_var(k2)}) + B\cdot u_t$$ + +where $\mathrm{k2}$ selects the state variables, $y_t$ and $y^s$ are in +declaration order and the coefficient matrices are in DR-order. +Effectively, all variables on the right hand side are brought into DR +order for computations and then assigned to $y_t$ in declaration order. + +### Second-order approximation + +The approximation has the form: + +$$y_t = y^s + 0.5 \Delta^2 + A y^h_{t-1} + B u_t + 0.5 C (y^h_{t-1}\otimes y^h_{t-1}) + 0.5 D (u_t \otimes u_t) + E (y^h_{t-1} \otimes u_t)$$ + +where $y^s$ is the steady state value of $y$, $y^h_t=y_t-y^s$, and +$\Delta^2$ is the shift effect of the variance of future shocks. For the +reordering required due to differences in declaration and DR order, see +the first order approximation. + +The coefficients of the decision rules are stored in the variables +described for first order approximation, plus the following variables: + +- $\Delta^2$ is stored in `oo_.dr.ghs2`. The vector rows correspond to + all endogenous in DR-order. +- $C$ is stored in `oo_.dr.ghxx`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of the vector of state variables in DR-order. +- $D$ is stored in `oo_.dr.ghuu`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of exogenous variables in declaration order. +- $E$ is stored in `oo_.dr.ghxu`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of the vector of state variables (in DR-order) by + the vector of exogenous variables (in declaration order). + +### Third-order approximation + +The approximation has the form: + +$$y_t = y^s + G_0 + G_1 z_t + G_2 (z_t \otimes z_t) + G_3 (z_t \otimes z_t \otimes z_t)$$ + +where $y^s$ is the steady state value of $y$, and $z_t$ is a vector +consisting of the deviation from the steady state of the state variables +(in DR-order) at date $t-1$ followed by the exogenous variables at date +$t$ (in declaration order). The vector $z_t$ is therefore of size $n_z$ += `M_.nspred` + `M_.exo_nbr`. + +The coefficients of the decision rules are stored as follows: + +- $y^s$ is stored in `oo_.dr.ys`. The vector rows correspond to all + endogenous in the declaration order. +- $G_0$ is stored in `oo_.dr.g_0`. The vector rows correspond to all + endogenous in DR-order. +- $G_1$ is stored in `oo_.dr.g_1`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to state + variables in DR-order, followed by exogenous in declaration order. +- $G_2$ is stored in `oo_.dr.g_2`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of state variables (in DR-order), followed by + exogenous (in declaration order). Note that the Kronecker product is + stored in a folded way, i.e. symmetric elements are stored only + once, which implies that the matrix has $n_z(n_z+1)/2$ columns. More + precisely, each column of this matrix corresponds to a pair + $(i_1, i_2)$ where each index represents an element of $z_t$ and is + therefore between $1$ and $n_z$. Only non-decreasing pairs are + stored, i.e. those for which $i_1 + \leq i_2$. The columns are arranged in the lexicographical order of + non-decreasing pairs. Also note that for those pairs where + $i_1 \neq i_2$, since the element is stored only once but appears + two times in the unfolded $G_2$ matrix, it must be multiplied by 2 + when computing the decision rules. +- $G_3$ is stored in `oo_.dr.g_3`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the third + Kronecker power of state variables (in DR-order), followed by + exogenous (in declaration order). Note that the third Kronecker + power is stored in a folded way, i.e. symmetric elements are stored + only once, which implies that the matrix has $n_z(n_z+1)(n_z+2)/6$ + columns. More precisely, each column of this matrix corresponds to a + tuple $(i_1, i_2, i_3)$ where each index represents an element of + $z_t$ and is therefore between $1$ and $n_z$. Only non-decreasing + tuples are stored, i.e. those for which $i_1 \leq i_2 \leq i_3$. The + columns are arranged in the lexicographical order of non-decreasing + tuples. Also note that for tuples that have three distinct indices + (i.e. $i_1 \neq i_2$ and $i_1 \neq i_3$ and $i_2 + \neq i_3$), since these elements are stored only once but appears + six times in the unfolded $G_3$ matrix, they must be multiplied by 6 + when computing the decision rules. Similarly, for those tuples that + have two equal indices (i.e. of the form $(a,a,b)$ or $(a,b,a)$ or + $(b,a,a)$), since these elements are stored only once but appears + three times in the unfolded $G_3$ matrix, they must be multiplied by + 3 when computing the decision rules. + +### Higher-order approximation + +Higher-order approximations are simply a generalization of what is done +at order 3. + +The steady state is stored in `oo_.dr.ys` and the constant correction is +stored in `oo_.dr.g_0`. The coefficient for orders 1, 2, 3, 4... are +respectively stored in `oo_.dr.g_0`, `oo_.dr.g_1`, `oo_.dr.g_2`, +`oo_.dr.g_3`, `oo_.dr.g_4`... The columns of those matrices correspond +to multidimensional indices of state variables, in such a way that +symmetric elements are never repeated (for more details, see the +description of `oo_.dr.g_3` in the third-order case). + + +Estimation based on likelihood {#estim} +------------------------------ + +Provided that you have observations on some endogenous variables, it is +possible to use Dynare to estimate some or all parameters. Both maximum +likelihood (as in *Ireland (2004)*) and Bayesian techniques (as in +*Fernández-Villaverde and Rubio-Ramírez (2004)*, *Rabanal and +Rubio-Ramirez (2003)*, *Schorfheide (2000)* or *Smets and Wouters +(2003)*) are available. Using Bayesian methods, it is possible to +estimate DSGE models, VAR models, or a combination of the two techniques +called DSGE-VAR. + +Note that in order to avoid stochastic singularity, you must have at +least as many shocks or measurement errors in your model as you have +observed variables. + +The estimation using a first order approximation can benefit from the +block decomposition of the model (see `block`{.interpreted-text +role="opt"}). + +::: {#varobs} +::: {.command} +varobs VARIABLE\_NAME\...; + +This command lists the name of observed endogenous variables for the +estimation procedure. These variables must be available in the data file +(see `estimation_cmd `{.interpreted-text role="ref"}). + +Alternatively, this command is also used in conjunction with the +`partial_information` option of `stoch_simul`, for declaring the set of +observed variables when solving the model under partial information. + +Only one instance of `varobs` is allowed in a model file. If one needs +to declare observed variables in a loop, the macro processor can be used +as shown in the second example below. + +*Example* + +> varobs C y rr; +> +> Declares endogenous variables `C`, `y` and `rr` as observed variables. + +*Example* (with a macro processor loop) + +> varobs +> @#for co in countries +> GDP_@{co} +> @#endfor +> ; +::: +::: + +::: {.block} +observation\_trends ; + +This block specifies linear trends for observed variables as functions +of model parameters. In case the `loglinear` option is used, this +corresponds to a linear trend in the logged observables, i.e. an +exponential trend in the level of the observables. + +Each line inside of the block should be of the form: + + VARIABLE_NAME(EXPRESSION); + +In most cases, variables shouldn't be centered when `observation_trends` +is used. + +*Example* + +> observation_trends; +> Y (eta); +> P (mu/eta); +> end; +::: + +::: {.block} +estimated\_params ; estimated\_params (overwrite) ; + +This block lists all parameters to be estimated and specifies bounds and +priors as necessary. + +Each line corresponds to an estimated parameter. + +In a maximum likelihood or a method of moments estimation, each line +follows this syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME + , INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ]; + +In a Bayesian MCMC or a penalized method of moments estimation, each +line follows this syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT + [, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE, + PRIOR_MEAN, PRIOR_STANDARD_ERROR [, PRIOR_3RD_PARAMETER [, + PRIOR_4TH_PARAMETER [, SCALE_PARAMETER ] ] ]; + +The first part of the line consists of one of the four following +alternatives: + +- `stderr VARIABLE_NAME` + + Indicates that the standard error of the exogenous variable + VARIABLE\_NAME, or of the observation error/measurement errors + associated with endogenous observed variable VARIABLE\_NAME, is to + be estimated. + +- `corr VARIABLE_NAME1, VARIABLE_NAME2` + + Indicates that the correlation between the exogenous variables + VARIABLE\_NAME1 and VARIABLE\_NAME2, or the correlation of the + observation errors/measurement errors associated with endogenous + observed variables VARIABLE\_NAME1 and VARIABLE\_NAME2, is to be + estimated. Note that correlations set by previous shocks-blocks or + estimation-commands are kept at their value set prior to estimation + if they are not estimated again subsequently. Thus, the treatment is + the same as in the case of deep parameters set during model + calibration and not estimated. + +- `PARAMETER_NAME` + + The name of a model parameter to be estimated + +- `DSGE_PRIOR_WEIGHT` + + Special name for the weigh of the DSGE model in DSGE-VAR model. + +The rest of the line consists of the following fields, some of them +being optional: + +::: {.option} +INITIAL\_VALUE + +Specifies a starting value for the posterior mode optimizer or the +maximum likelihood estimation. If unset, defaults to the prior mean. +::: + +::: {.option} +LOWER\_BOUND + +Specifies a lower bound for the parameter value in maximum likelihood +estimation. In a Bayesian estimation context, sets a lower bound only +effective while maximizing the posterior kernel. This lower bound does +not modify the shape of the prior density, and is only aimed at helping +the optimizer in identifying the posterior mode (no consequences for the +MCMC). For some prior densities (namely inverse gamma, gamma, uniform, +beta or Weibull) it is possible to shift the support of the prior +distributions to the left or the right using +`prior_3rd_parameter `{.interpreted-text +role="opt"}. In this case the prior density is effectively modified +(note that the truncated Gaussian density is not implemented in Dynare). +If unset, defaults to minus infinity (ML) or the natural lower bound of +the prior (Bayesian estimation). +::: + +::: {.option} +UPPER\_BOUND + +Same as `lower_bound`, but specifying an upper bound instead. +::: + +::: {.option} +PRIOR\_SHAPE + +A keyword specifying the shape of the prior density. The possible values +are: `beta_pdf`, `gamma_pdf`, `normal_pdf`, `uniform_pdf`, +`inv_gamma_pdf`, `inv_gamma1_pdf`, `inv_gamma2_pdf` and `weibull_pdf`. +Note that `inv_gamma_pdf` is equivalent to `inv_gamma1_pdf`. +::: + +::: {.option} +PRIOR\_MEAN + +The mean of the prior distribution. +::: + +::: {.option} +PRIOR\_STANDARD\_ERROR + +The standard error of the prior distribution. +::: + +::: {.option} +PRIOR\_3RD\_PARAMETER + +A third parameter of the prior used for generalized beta distribution, +generalized gamma, generalized Weibull and for the uniform distribution. +Default: `0`. +::: + +::: {.option} +PRIOR\_4TH\_PARAMETER + +A fourth parameter of the prior used for generalized beta distribution +and for the uniform distribution. Default: `1`. +::: + +::: {.option} +SCALE\_PARAMETER + +A parameter specific scale parameter for the jumping distribution's +covariance matrix of the Metropolis-Hasting algorithm. +::: + +Note that INITIAL\_VALUE, LOWER\_BOUND, UPPER\_BOUND, PRIOR\_MEAN, +PRIOR\_STANDARD\_ERROR, PRIOR\_3RD\_PARAMETER, PRIOR\_4TH\_PARAMETER and +SCALE\_PARAMETER can be any valid EXPRESSION. Some of them can be empty, +in which Dynare will select a default value depending on the context and +the prior shape. + +In case of the uniform distribution, it can be specified either by +providing an upper and a lower bound using +`PRIOR_3RD_PARAMETER`{.interpreted-text role="opt"} and +`PRIOR_4TH_PARAMETER`{.interpreted-text role="opt"} or via mean and +standard deviation using `PRIOR_MEAN`{.interpreted-text role="opt"}, +`PRIOR_STANDARD_ERROR`{.interpreted-text role="opt"}. The other two will +automatically be filled out. Note that providing both sets of +hyperparameters will yield an error message. + +As one uses options more towards the end of the list, all previous +options must be filled: for example, if you want to specify +SCALE\_PARAMETER, you must specify `PRIOR_3RD_PARAMETER` and +`PRIOR_4TH_PARAMETER`. Use empty values, if these parameters don't +apply. + +*Example* + +> corr eps_1, eps_2, 0.5, , , beta_pdf, 0, 0.3, -1, 1; +> +> Sets a generalized beta prior for the correlation between `eps_1` and +> `eps_2` with mean `0` and variance `0.3`. By setting +> `PRIOR_3RD_PARAMETER` to `-1` and `PRIOR_4TH_PARAMETER` to `1` the +> standard beta distribution with support `[0,1]` is changed to a +> generalized beta with support `[-1,1]`. Note that LOWER\_BOUND and +> UPPER\_BOUND are left empty and thus default to `-1` and `1`, +> respectively. The initial value is set to `0.5`. + +*Example* + +> corr eps_1, eps_2, 0.5, -0.5, 1, beta_pdf, 0, 0.3, -1, 1; +> +> Sets the same generalized beta distribution as before, but now +> truncates this distribution to `[-0.5,1]` through the use of +> LOWER\_BOUND and UPPER\_BOUND. + +*Parameter transformation* + +Sometimes, it is desirable to estimate a transformation of a parameter +appearing in the model, rather than the parameter itself. It is of +course possible to replace the original parameter by a function of the +estimated parameter everywhere is the model, but it is often +unpractical. + +In such a case, it is possible to declare the parameter to be estimated +in the parameters statement and to define the transformation, using a +pound sign (\#) expression (see `model-decl`{.interpreted-text +role="ref"}). + +*Example* + +> parameters bet; +> +> model; +> # sig = 1/bet; +> c = sig*c(+1)*mpk; +> end; +> +> estimated_params; +> bet, normal_pdf, 1, 0.05; +> end; + +It is possible to have several `estimated_params` blocks. By default, +subsequent blocks are concatenated with the previous ones; this can be +useful when building models in a modular fashion (see also +`estimated_params_remove`{.interpreted-text role="bck"} for that use +case). However, if an `estimated_params` block has the `overwrite` +option, its contents becomes the new list of estimated parameters, +cancelling previous blocks; this can be useful when doing several +estimations in a single `.mod` file. +::: + +::: {.block} +estimated\_params\_init ; estimated\_params\_init (OPTIONS\...); + +This block declares numerical initial values for the optimizer when +these ones are different from the prior mean. It should be specified +after the `estimated_params` block as otherwise the specified starting +values are overwritten by the latter. + +Each line has the following syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, INITIAL_VALUE; + +*Options* + +::: {.option} +use\_calibration + +For not specifically initialized parameters, use the deep parameters and +the elements of the covariance matrix specified in the `shocks` block +from calibration as starting values for estimation. For components of +the `shocks` block that were not explicitly specified during calibration +or which violate the prior, the prior mean is used. +::: + +See `estimated_params`{.interpreted-text role="bck"}, for the meaning +and syntax of the various components. +::: + +::: {.block} +estimated\_params\_bounds ; + +This block declares lower and upper bounds for parameters in maximum +likelihood estimation. + +Each line has the following syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND; + +See `estimated_params`{.interpreted-text role="bck"}, for the meaning +and syntax of the various components. +::: + +::: {.block} +estimated\_params\_remove ; + +This block partially undoes the effect of a previous +`estimated_params`{.interpreted-text role="bck"} block, by removing some +parameters from the estimation. + +Each line has the following syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME; +::: + +::: {#estim-comm} +::: {.command} +estimation \[VARIABLE\_NAME\...\]; estimation (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +This command runs Bayesian or maximum likelihood estimation. + +The following information will be displayed by the command: + +- Results from posterior optimization (also for maximum likelihood) +- Marginal log data density +- Posterior mean and highest posterior density interval (shortest + credible set) from posterior simulation +- Convergence diagnostic table when only one MCM chain is used or + Metropolis-Hastings convergence graphs documented in + *Pfeifer (2014)* in case of multiple MCM chains +- Table with numerical inefficiency factors of the MCMC +- Graphs with prior, posterior, and mode +- Graphs of smoothed shocks, smoothed observation errors, smoothed and + historical variables + +Note that the posterior moments, smoothed variables, k-step ahead +filtered variables and forecasts (when requested) will only be computed +on the variables listed after the `estimation` command. Alternatively, +one can choose to compute these quantities on all endogenous or on all +observed variables (see `consider_all_endogenous`, +`consider_all_endogenous_and_auxiliary`, and `consider_only_observed` +options below). If no variable is listed after the estimation command, +then Dynare will interactively ask which variable set to use. + +Also, during the MCMC (Bayesian estimation with `mh_replic` $>0$) a +(graphical or text) waiting bar is displayed showing the progress of the +Monte-Carlo and the current value of the acceptance ratio. Note that if +the `load_mh_file` option is used (see below) the reported acceptance +ratio does not take into account the draws from the previous MCMC. In +the literature there is a general agreement for saying that the +acceptance ratio should be close to one third or one quarter. If this +not the case, you can stop the MCMC (`Ctrl-C`) and change the value of +option `mh_jscale` (see below). + +Note that by default Dynare generates random numbers using the algorithm +`mt199937ar` (i.e. Mersenne Twister method) with a seed set equal to +`0`. Consequently the MCMCs in Dynare are deterministic: one will get +exactly the same results across different Dynare runs (*ceteris +paribus*). For instance, the posterior moments or posterior densities +will be exactly the same. This behaviour allows to easily identify the +consequences of a change on the model, the priors or the estimation +options. But one may also want to check that across multiple runs, with +different sequences of proposals, the returned results are almost +identical. This should be true if the number of iterations (i.e. the +value of `mh_replic`) is important enough to ensure the convergence of +the MCMC to its ergodic distribution. In this case the default behaviour +of the random number generators in not wanted, and the user should set +the seed according to the system clock before the estimation command +using the following command: + + set_dynare_seed('clock'); + +so that the sequence of proposals will be different across different +runs. + +Finally, Dynare does not always properly distinguish between maximum +likelihood and Bayesian estimation in its field names. While there is an +important conceptual distinction between frequentist confidence +intervals and Bayesian highest posterior density intervals (HPDI) as +well as between posterior density and likelilhood, Dynare sometimes uses +the Bayesian terms as a stand-in in its display of maximum likelihood +results. An example is the storage of the output of the +`forecast`-option of `estimation` with ML, which will use +`HPDinf/HPDsup` to denote the confidence interval. + +*Algorithms* + +The Monte Carlo Markov Chain (MCMC) diagnostics are generated by the +estimation command if +`mh_replic `{.interpreted-text role="opt"} is +larger than 2000 and if option `nodiagnostic`{.interpreted-text +role="opt"} is not used. If +`mh_nblocks `{.interpreted-text role="opt"} is +equal to one, the convergence diagnostics of *Geweke (1992,1999)* is +computed. It uses a chi-square test to compare the means of the first +and last draws specified by `geweke_interval +`{.interpreted-text role="opt"} after +discarding the burn-in of `mh_drop `{.interpreted-text +role="opt"}. The test is computed using variance estimates under the +assumption of no serial correlation as well as using tapering windows +specified in `taper_steps +`{.interpreted-text role="opt"}. +If `mh_nblocks +`{.interpreted-text role="opt"} is larger than 1, +the convergence diagnostics of *Brooks and Gelman (1998)* are used +instead. As described in section 3 of *Brooks and Gelman (1998)* the +univariate convergence diagnostics are based on comparing pooled and +within MCMC moments (Dynare displays the second and third order moments, +and the length of the Highest Probability Density interval covering 80% +of the posterior distribution). Due to computational reasons, the +multivariate convergence diagnostic does not follow *Brooks and Gelman +(1998)* strictly, but rather applies their idea for univariate +convergence diagnostics to the range of the posterior likelihood +function instead of the individual parameters. The posterior kernel is +used to aggregate the parameters into a scalar statistic whose +convergence is then checked using the *Brooks and Gelman (1998)* +univariate convergence diagnostic. + +The inefficiency factors are computed as in *Giordano et al.(2011)* +based on Parzen windows as in e.g. *Andrews (1991)*. + +*Options* + +::: {#dataf} +::: {.option} +datafile = FILENAME + +The datafile: a `.m` file, a `.mat` file, a `.csv` file, or a +`.xls/.xlsx` file (under Octave, the +[io](https://octave.sourceforge.io/io/) package from Octave-Forge is +required for the `.csv` and `.xlsx` formats and the `.xls` file +extension is not supported). Note that the base name (i.e. without +extension) of the datafile has to be different from the base name of the +model file. If there are several files named FILENAME, but with +different file endings, the file name must be included in quoted strings +and provide the file ending like: + + estimation(datafile='../fsdat_simul.mat',...); +::: +::: + +::: {.option} +dirname = FILENAME + +Directory in which to store `estimation` output. To pass a subdirectory +of a directory, you must quote the argument. Default: ``. +::: + +::: {.option} +xls\_sheet = QUOTED\_STRING + +The name of the sheet with the data in an Excel file. +::: + +::: {.option} +xls\_range = RANGE + +The range with the data in an Excel file. For example, +`xls_range=B2:D200`. +::: + +::: {.option} +nobs = INTEGER + +The number of observations following `first_obs `{.interpreted-text role="opt"} to be used. +Default: all observations in the file after `first_obs`. +::: + +::: {.option} +nobs = \[INTEGER1:INTEGER2\] + +Runs a recursive estimation and forecast for samples of size ranging of +`INTEGER1` to `INTEGER2`. Option `forecast` must also be specified. The +forecasts are stored in the `RecursiveForecast` field of the results +structure (see +`RecursiveForecast `{.interpreted-text +role="mvar"}). The respective results structures `oo_` are saved in +`oo_recursive_` (see `oo_recursive_`{.interpreted-text role="mvar"}) and +are indexed with the respective sample length. +::: + +::: {.option} +first\_obs = INTEGER + +The number of the first observation to be used. In case of estimating a +DSGE-VAR, `first_obs` needs to be larger than the number of lags. +Default: `1`. +::: + +::: {.option} +first\_obs = \[INTEGER1:INTEGER2\] + +Runs a rolling window estimation and forecast for samples of fixed size +`nobs` starting with the first observation ranging from `INTEGER1` to +`INTEGER2`. Option `forecast` must also be specified. This option is +incompatible with requesting recursive forecasts using an expanding +window (see `nobs +`{.interpreted-text role="opt"}). The +respective results structures `oo_` are saved in `oo_recursive_` (see +`oo_recursive_`{.interpreted-text role="mvar"}) and are indexed with the +respective first observation of the rolling window. +::: + +::: {.option} +prefilter = INTEGER + +A value of 1 means that the estimation procedure will demean each data +series by its empirical mean. If the `loglinear +`{.interpreted-text role="ref"} option without the +`logdata`{.interpreted-text role="opt"} option is requested, the data +will first be logged and then demeaned. Default: `0`, i.e. no +prefiltering. +::: + +::: {.option} +presample = INTEGER + +The number of observations after `first_obs `{.interpreted-text role="opt"} to be skipped before +evaluating the likelihood. These presample observations do not enter the +likelihood, but are used as a training sample for starting the Kalman +filter iterations. This option is incompatible with estimating a +DSGE-VAR. Default: `0`. +::: + +::: {#logl} +::: {.option} +loglinear + +Computes a log-linear approximation of the model instead of a linear +approximation. As always in the context of estimation, the data must +correspond to the definition of the variables used in the model (see +*Pfeifer (2013)* for more details on how to correctly specify +observation equations linking model variables and the data). If you +specify the loglinear option, Dynare will take the logarithm of both +your model variables and of your data as it assumes the data to +correspond to the original non-logged model variables. The displayed +posterior results like impulse responses, smoothed variables, and +moments will be for the logged variables, not the original un-logged +ones. Default: computes a linear approximation. +::: +::: + +::: {.option} +logdata + +Dynare applies the $log$ transformation to the provided data if a +log-linearization of the model is requested +(`loglinear`{.interpreted-text role="opt"}) unless `logdata` option is +used. This option is necessary if the user provides data already in +logs, otherwise the $log$ transformation will be applied twice (this may +result in complex data). +::: + +::: {.option} +plot\_priors = INTEGER + +Control the plotting of priors. + +> `0` +> +> > No prior plot. +> +> `1` +> +> > Prior density for each estimated parameter is plotted. It is +> > important to check that the actual shape of prior densities matches +> > what you have in mind. Ill-chosen values for the prior standard +> > density can result in absurd prior densities. + +Default value is `1`. +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +posterior\_nograph + +Suppresses the generation of graphs associated with Bayesian IRFs +(`bayesian_irf`{.interpreted-text role="opt"}), posterior smoothed +objects (`smoother`{.interpreted-text role="opt"}), and posterior +forecasts (`forecast`{.interpreted-text role="opt"}). +::: + +::: {.option} +posterior\_graph + +Re-enables the generation of graphs previously shut off with +`posterior_nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See +`graph_format `{.interpreted-text +role="opt"}. +::: + +::: {.option} +no\_init\_estimation\_check\_first\_obs + +Do not check for stochastic singularity in first period. If used, +[ESTIMATION CHECKS]{.title-ref} does not return an error if the check +fails only in first observation. This should only be used when observing +stock variables (e.g. capital) in first period, on top of their +associated flow (e.g. investment). Using this option may lead to a crash +or provide undesired/wrong results for badly specified problems (e.g. +the additional variable observed in first period is not predetermined). + +For advanced use only. +::: + +::: {.option} +lik\_init = INTEGER + +Type of initialization of Kalman filter: + +> `1` +> +> > For stationary models, the initial matrix of variance of the error +> > of forecast is set equal to the unconditional variance of the state +> > variables. +> +> `2` +> +> > For nonstationary models: a wide prior is used with an initial +> > matrix of variance of the error of forecast diagonal with 10 on the +> > diagonal (follows the suggestion of *Harvey and Phillips(1979)*). +> +> `3` +> +> > For nonstationary models: use a diffuse filter (use rather the +> > `diffuse_filter` option). +> +> `4` +> +> > The filter is initialized with the fixed point of the Riccati +> > equation. +> +> `5` +> +> > Use i) option 2 for the non-stationary elements by setting their +> > initial variance in the forecast error matrix to 10 on the diagonal +> > and all covariances to 0 and ii) option 1 for the stationary +> > elements. + +Default value is 1. For advanced use only. +::: + +::: {.option} +lik\_algo = INTEGER + +For internal use and testing only. +::: + +::: {.option} +conf\_sig = DOUBLE + +Level of significance of the confidence interval used for classical +forecasting after estimation. Default: 0.9. +::: + +::: {.option} +mh\_conf\_sig = DOUBLE + +Confidence/HPD interval used for the computation of prior and posterior +statistics like: parameter distributions, prior/posterior moments, +conditional variance decomposition, impulse response functions, Bayesian +forecasting. Default: `0.9`. +::: + +::: {.option} +mh\_replic = INTEGER + +Number of replications for each chain of the Metropolis-Hastings +algorithm. The number of draws should be sufficient to achieve +convergence of the MCMC and to meaningfully compute posterior objects. +Default: `20000`. +::: + +::: {.option} +sub\_draws = INTEGER + +Number of draws from the MCMC that are used to compute posterior +distribution of various objects (smoothed variable, smoothed shocks, +forecast, moments, IRF). The draws used to compute these posterior +moments are sampled uniformly in the estimated empirical posterior +distribution (i.e. draws of the MCMC). `sub_draws` should be smaller +than the total number of MCMC draws available. Default: +`min(posterior_max_subsample_draws, (Total number of draws)*(number of chains) )`. +::: + +::: {.option} +posterior\_max\_subsample\_draws = INTEGER + +Maximum number of draws from the MCMC used to compute posterior +distribution of various objects (smoothed variable, smoothed shocks, +forecast, moments, IRF), if not overriden by option `sub_draws`. +Default: `1200`. +::: + +::: {.option} +mh\_nblocks = INTEGER + +Number of parallel chains for Metropolis-Hastings algorithm. Default: +`2`. +::: + +::: {.option} +mh\_drop = DOUBLE + +The fraction of initially generated parameter vectors to be dropped as a +burn-in before using posterior simulations. Default: `0.5`. +::: + +::: {.option} +mh\_jscale = DOUBLE + +The scale parameter of the jumping distribution's covariance matrix +(Metropolis-Hastings or TaRB-algorithm). The default value is rarely +satisfactory. This option must be tuned to obtain, ideally, an +acceptance ratio of 25%-33%. Basically, the idea is to increase the +variance of the jumping distribution if the acceptance ratio is too +high, and decrease the same variance if the acceptance ratio is too low. +In some situations it may help to consider parameter-specific values for +this scale parameter. This can be done in the +`estimated_params`{.interpreted-text role="bck"} block. + +Note that `mode_compute=6` will tune the scale parameter to achieve an +acceptance rate of `AcceptanceRateTarget`{.interpreted-text +role="ref"}. The resulting scale parameter will be saved into a file +named `MODEL_FILENAME_mh_scale.mat` in the `FILENAME/Output`-folder. +This file can be loaded in subsequent runs via the +`posterior_sampler_options` option +`scale_file `{.interpreted-text role="ref"}. Both +`mode_compute=6` and `scale_file` will overwrite any value specified in +`estimated_params` with the tuned value. Default: `0.2`. + +Note also that for the Random Walk Metropolis Hastings algorithm, it is +possible to use option `mh_tune_jscale +`{.interpreted-text role="opt"}, to +automatically tune the value of `mh_jscale`. In this case, the +`mh_jscale` option must not be used. +::: + +::: {.option} +mh\_init\_scale = DOUBLE + +The scale to be used for drawing the initial value of the +Metropolis-Hastings chain. Generally, the starting points should be +overdispersed for the *Brooks and Gelman (1998)* convergence diagnostics +to be meaningful. Default: `2*mh_jscale.` + +It is important to keep in mind that `mh_init_scale` is set at the +beginning of Dynare execution, i.e. the default will not take into +account potential changes in `mh_jscale` introduced by either +`mode_compute=6` or the `posterior_sampler_options` option `scale_file +`{.interpreted-text role="ref"}. If `mh_init_scale` is too +wide during initalization of the posterior sampler so that 100 tested +draws are inadmissible (e.g. Blanchard-Kahn conditions are always +violated), Dynare will request user input of a new `mh_init_scale` value +with which the next 100 draws will be drawn and tested. If the +`nointeractive`{.interpreted-text role="opt"} option has been invoked, +the program will instead automatically decrease `mh_init_scale` by 10 +percent after 100 futile draws and try another 100 draws. This iterative +procedure will take place at most 10 times, at which point Dynare will +abort with an error message. +::: + +::: {.option} +mh\_tune\_jscale \[= DOUBLE\] + +Automatically tunes the scale parameter of the jumping distribution\'s +covariance matrix (Metropolis-Hastings), so that the overall acceptance +ratio is close to the desired level. Default value is `0.33`. It is not +possible to match exactly the desired acceptance ratio because of the +stochastic nature of the algorithm (the proposals and the initial +conditions of the markov chains if `mh_nblocks>1`). This option is only +available for the Random Walk Metropolis Hastings algorithm. Must not be +used in conjunction with `mh_jscale = DOUBLE`{.interpreted-text +role="opt"}. +::: + +::: {.option} +mh\_tune\_guess = DOUBLE + +Specifies the initial value for the `mh_tune_jscale +`{.interpreted-text role="opt"} option. +Default: `0.2`. Must not be set if +`mh_tune_jscale `{.interpreted-text +role="opt"} is not used. +::: + +::: {.option} +mh\_recover + +Attempts to recover a Metropolis-Hastings simulation that crashed +prematurely, starting with the last available saved `mh`-file. Shouldn't +be used together with `load_mh_file` or a different `mh_replic` than in +the crashed run. Since Dynare 4.5 the proposal density from the previous +run will automatically be loaded. In older versions, to assure a neat +continuation of the chain with the same proposal density, you should +provide the `mode_file` used in the previous run or the same +user-defined `mcmc_jumping_covariance` when using this option. Note that +under Octave, a neat continuation of the crashed chain with the +respective last random number generator state is currently not +supported. +::: + +::: {.option} +mh\_posterior\_mode\_estimation + +Skip optimizer-based mode-finding and instead compute the mode based on +a run of a MCMC. The MCMC will start at the prior mode and use the prior +variances to compute the inverse Hessian. +::: + +::: {.option} +mode\_file = FILENAME + +Name of the file containing previous value for the mode. When computing +the mode, Dynare stores the mode (`xparam1`) and the hessian (`hh`, only +if `cova_compute=1`) in a file called `MODEL_FILENAME_mode.mat` in the +`FILENAME/Output`-folder. After a successful run of the estimation +command, the `mode_file` will be disabled to prevent other function +calls from implicitly using an updated mode-file. Thus, if the `.mod` +file contains subsequent `estimation` commands, the `mode_file` option, +if desired, needs to be specified again. +::: + + +::: {.option} +silent\_optimizer + +Instructs Dynare to run mode computing/optimization silently without +displaying results or saving files in between. Useful when running +loops. +::: + +::: {.option} +mcmc\_jumping\_covariance = OPTION + +Tells Dynare which covariance to use for the proposal density of the +MCMC sampler. OPTION can be one of the following: + +> `hessian` +> +> > Uses the Hessian matrix computed at the mode. +> +> `prior_variance` +> +> > Uses the prior variances. No infinite prior variances are allowed in +> > this case. +> +> `identity_matrix` +> +> > Uses an identity matrix. +> +> `FILENAME` +> +> > Loads an arbitrary user-specified covariance matrix from +> > `FILENAME.mat`. The covariance matrix must be saved in a variable +> > named `jumping_covariance`, must be square, positive definite, and +> > have the same dimension as the number of estimated parameters. + +Note that the covariance matrices are still scaled with +`mh_jscale `{.interpreted-text role="opt"}. Default +value is `hessian`. +::: + +::: {.option} +mode\_check + +Tells Dynare to plot the posterior density for values around the +computed mode for each estimated parameter in turn. This is helpful to +diagnose problems with the optimizer. Note that for `order>1` the +likelihood function resulting from the particle filter is not +differentiable anymore due to the resampling step. For this reason, the +`mode_check` plot may look wiggly. +::: + +::: {.option} +mode\_check\_neighbourhood\_size = DOUBLE + +Used in conjunction with option `mode_check`, gives the width of the +window around the posterior mode to be displayed on the diagnostic +plots. This width is expressed in percentage deviation. The `Inf` value +is allowed, and will trigger a plot over the entire domain (see also +`mode_check_symmetric_plots`). Default:`0.5`. +::: + +::: {.option} +mode\_check\_symmetric\_plots = INTEGER + +Used in conjunction with option `mode_check`, if set to `1`, tells +Dynare to ensure that the check plots are symmetric around the posterior +mode. A value of `0` allows to have asymmetric plots, which can be +useful if the posterior mode is close to a domain boundary, or in +conjunction with `mode_check_neighbourhood_size = Inf` when the domain +in not the entire real line. Default: `1`. +::: + +::: {.option} +mode\_check\_number\_of\_points = INTEGER + +Number of points around the posterior mode where the posterior kernel is +evaluated (for each parameter). Default is `20`. +::: + +::: {.option} +prior\_trunc = DOUBLE + +Probability of extreme values of the prior density that is ignored when +computing bounds for the parameters. Default: `1e-32`. +::: + +::: {.option} +huge\_number = DOUBLE + +Value for replacing infinite values in the definition of (prior) bounds +when finite values are required for computational reasons. Default: +`1e7`. +::: + +::: {.option} +load\_mh\_file + +Tells Dynare to add to previous Metropolis-Hastings simulations instead +of starting from scratch. Since Dynare 4.5 the proposal density from the +previous run will automatically be loaded. In older versions, to assure +a neat continuation of the chain with the same proposal density, you +should provide the `mode_file` used in the previous run or the same +user-defined `mcmc_jumping_covariance` when using this option. Shouldn't +be used together with `mh_recover`. Note that under Octave, a neat +continuation of the chain with the last random number generator state of +the already present draws is currently not supported. +::: + +::: {.option} +load\_results\_after\_load\_mh + +This option is available when loading a previous MCMC run without adding +additional draws, i.e. when `load_mh_file` is specified with +`mh_replic=0`. It tells Dynare to load the previously computed +convergence diagnostics, marginal data density, and posterior statistics +from an existing `_results` file instead of recomputing them. +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc + +This option allows to pick initial values for new MCMC from a previous +one, where the model specification, the number of estimated parameters, +(some) prior might have changed (so a situation where `load_mh_file` +would not work). If an additional parameter is estimated, it is +automatically initialized from prior\_draw. Note that, if this option is +used to skip the optimization step, you should use a sampling method +which does not require a proposal density, like slice. Otherwise, +optimization should always be done beforehand or a mode file with an +appropriate posterior covariance matrix should be used. +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc\_directory = FILENAME + +If `mh_initialize_from_previous_mcmc` is set, users must provide here +the path to the standard FNAME folder from where to load prior +definitions and last MCMC values to be used to initialize the new MCMC. + +Example: if previous project directory is `/my_previous_dir` and FNAME +is `mymodel`, users should set the option as + +`mh_initialize_from_previous_mcmc_directory = '/my_previous_dir/mymodel'` + +Dynare will then look for the last record file into + +`/my_previous_dir/mymodel/metropolis/mymodel_mh_history_.mat` + +and for the prior definition file into + +`/my_previous_dir/mymodel/prior/definition.mat` +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc\_record = FILENAME + +If `mh_initialize_from_previous_mcmc` is set, and whenever the standard +file or directory tree is not applicable to load initial values, users +may directly provide here the path to the record file from which to load +values to be used to initialize the new MCMC. +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc\_prior = FILENAME + +If `mh_initialize_from_previous_mcmc` is set, and whenever the standard +file or directory tree is not applicable to load initial values, users +may directly provide here the path to the prior definition file, to get +info in the priors used in previous MCMC. +::: + +::: {.option} +optim = (NAME, VALUE, \...) + +A list of NAME and VALUE pairs. Can be used to set options for the +optimization routines. The set of available options depends on the +selected optimization routine (i.e. on the value of option +`mode_compute `{.interpreted-text role="opt"}): + +> `1, 3, 7, 12, 13` +> +> > Available options are given in the documentation of the MATLAB +> > Optimization Toolbox or in Octave's documentation. +> +> `2` +> +> > Available options are: +> > +> > > `'initial_step_length'` +> > > +> > > > Initial step length. Default: `1`. +> > > +> > > `'initial_temperature'` +> > > +> > > > Initial temperature. Default: `15`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of function evaluations. Default: `100000`. +> > > +> > > `'neps'` +> > > +> > > > Number of final function values used to decide upon termination. +> > > > Default: `10`. +> > > +> > > `'ns'` +> > > +> > > > Number of cycles. Default: `10`. +> > > +> > > `'nt'` +> > > +> > > > Number of iterations before temperature reduction. Default: +> > > > `10`. +> > > +> > > `'step_length_c'` +> > > +> > > > Step length adjustment. Default: `0.1`. +> > > +> > > `'TolFun'` +> > > +> > > > Stopping criteria. Default: `1e-8`. +> > > +> > > `'rt'` +> > > +> > > > Temperature reduction factor. Default: `0.1`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization, ranging from +> > > > `0` (silent) to `3` (each function evaluation). Default: `1` +> +> `4` +> +> > Available options are: +> > +> > > `'InitialInverseHessian'` +> > > +> > > > Initial approximation for the inverse of the Hessian matrix of +> > > > the posterior kernel (or likelihood). Obviously this +> > > > approximation has to be a square, positive definite and +> > > > symmetric matrix. Default: `'1e-4*eye(nx)'`, where nx is the +> > > > number of parameters to be estimated. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `1000`. +> > > +> > > `'NumgradAlgorithm'` +> > > +> > > > Possible values are `2`, `3` and `5`, respectively, +> > > > corresponding to the two, three and five points formula used to +> > > > compute the gradient of the objective function (see *Abramowitz +> > > > and Stegun (1964)*). Values `13` and `15` are more experimental. +> > > > If perturbations on the right and the left increase the value of +> > > > the objective function (we minimize this function) then we force +> > > > the corresponding element of the gradient to be zero. The idea +> > > > is to temporarily reduce the size of the optimization problem. +> > > > Default: `2`. +> > > +> > > `'NumgradEpsilon'` +> > > +> > > > Size of the perturbation used to compute numerically the +> > > > gradient of the objective function. Default: `1e-6`. +> > > +> > > `'TolFun'` +> > > +> > > > Stopping criteria. Default: `1e-7`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> > > +> > > `'SaveFiles'` +> > > +> > > > Controls saving of intermediate results during optimization. Set +> > > > to `0` to shut off saving. Default: `1`. +> +> `5` +> +> > Available options are: +> > +> > `'Hessian'` +> > +> > > Triggers three types of Hessian computations. `0`: outer product +> > > gradient; `1`: default Dynare Hessian routine; `2`: 'mixed' outer +> > > product gradient, where diagonal elements are obtained using +> > > second order derivation formula and outer product is used for +> > > correlation structure. Both {0} and {2} options require univariate +> > > filters, to ensure using maximum number of individual densities +> > > and a positive definite Hessian. Both {0} and {2} are quicker than +> > > default Dynare numeric Hessian, but provide decent starting values +> > > for Metropolis for large models (option {2} being more accurate +> > > than {0}). Default: `1`. +> > +> > `'MaxIter'` +> > +> > > Maximum number of iterations. Default: `1000`. +> > +> > `'TolFun'` +> > +> > > Stopping criteria. Default: `1e-5` for numerical derivatives, +> > > `1e-7` for analytic derivatives. +> > +> > `'verbosity'` +> > +> > > Controls verbosity of display during optimization. Set to `0` to +> > > set to silent. Default: `1`. +> > +> > `'SaveFiles'` +> > +> > > Controls saving of intermediate results during optimization. Set +> > > to `0` to shut off saving. Default: `1`. +> +> `6` +> +> > Available options are: +> > +> > > ::: {#art} +> > > `'AcceptanceRateTarget'` +> > > ::: +> > > +> > > > A real number between zero and one. The scale parameter of the +> > > > jumping distribution is adjusted so that the effective +> > > > acceptance rate matches the value of option +> > > > `'AcceptanceRateTarget'`. Default: `1.0/3.0`. +> > > +> > > `'InitialCovarianceMatrix'` +> > > +> > > > Initial covariance matrix of the jumping distribution. Default +> > > > is `'previous'` if option `mode_file` is used, `'prior'` +> > > > otherwise. +> > > +> > > `'nclimb-mh'` +> > > +> > > > Number of iterations in the last MCMC (climbing mode). Default: +> > > > `200000`. +> > > +> > > `'ncov-mh'` +> > > +> > > > Number of iterations used for updating the covariance matrix of +> > > > the jumping distribution. Default: `20000`. +> > > +> > > `'nscale-mh'` +> > > +> > > > Maximum number of iterations used for adjusting the scale +> > > > parameter of the jumping distribution. Default: `200000`. +> > > +> > > `'NumberOfMh'` +> > > +> > > > Number of MCMC run sequentially. Default: `3`. +> +> `8` +> +> > Available options are: +> > +> > > `'InitialSimplexSize'` +> > > +> > > > Initial size of the simplex, expressed as percentage deviation +> > > > from the provided initial guess in each direction. Default: +> > > > `.05`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `5000`. +> > > +> > > `'MaxFunEvals'` +> > > +> > > > Maximum number of objective function evaluations. No default. +> > > +> > > `'MaxFunvEvalFactor'` +> > > +> > > > Set `MaxFunvEvals` equal to `MaxFunvEvalFactor` times the number +> > > > of estimated parameters. Default: `500`. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-4`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-4`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> +> `9` +> +> > Available options are: +> > +> > > `'CMAESResume'` +> > > +> > > > Resume previous run. Requires the `variablescmaes.mat` from the +> > > > last run. Set to `1` to enable. Default: `0`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. +> > > +> > > `'MaxFunEvals'` +> > > +> > > > Maximum number of objective function evaluations. Default: +> > > > `Inf`. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-7`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-7`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> > > +> > > `'SaveFiles'` +> > > +> > > > Controls saving of intermediate results during optimization. Set +> > > > to `0` to shut off saving. Default: `1`. +> +> `10` +> +> > Available options are: +> > +> > > `'EndTemperature'` +> > > +> > > > Terminal condition w.r.t the temperature. When the temperature +> > > > reaches `EndTemperature`, the temperature is set to zero and the +> > > > algorithm falls back into a standard simplex algorithm. Default: +> > > > `0.1`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `5000`. +> > > +> > > `'MaxFunvEvals'` +> > > +> > > > Maximum number of objective function evaluations. No default. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-4`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-4`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> +> `101` +> +> > Available options are: +> > +> > > `'LBGradientStep'` +> > > +> > > > Lower bound for the stepsize used for the difference +> > > > approximation of gradients. Default: `1e-11`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `15000` +> > > +> > > `'SpaceDilation'` +> > > +> > > > Coefficient of space dilation. Default: `2.5`. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-6`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-6`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> +> `102` +> +> > Available options are given in the documentation of the MATLAB +> > Global Optimization Toolbox. + +*Example* + +> To change the defaults of `csminwel` (`mode_compute=4`): +> +> estimation(..., mode_compute=4,optim=('NumgradAlgorithm',3,'TolFun',1e-5),...); +::: + +::: {.option} +nodiagnostic + +Does not compute the convergence diagnostics for Metropolis-Hastings. +Default: diagnostics are computed and displayed. +::: + +::: {.option} +bayesian\_irf + +Triggers the computation of the posterior distribution of IRFs. The +length of the IRFs are controlled by the `irf` option. Results are +stored in `oo_.PosteriorIRF.dsge` (see below for a description of this +variable). +::: + +::: {.option} +relative\_irf + +See `relative_irf`{.interpreted-text role="opt"}. +::: + + +::: {.option} +posterior\_sampling\_method = NAME + +Selects the sampler used to sample from the posterior distribution +during Bayesian estimation. Default:`’random_walk_metropolis_hastings’`. + +> `'random_walk_metropolis_hastings'` +> +> > Instructs Dynare to use the Random-Walk Metropolis-Hastings. In this +> > algorithm, the proposal density is recentered to the previous draw +> > in every step. +> +> `'tailored_random_block_metropolis_hastings'` +> +> > Instructs Dynare to use the Tailored randomized block (TaRB) +> > Metropolis-Hastings algorithm proposed by *Chib and Ramamurthy +> > (2010)* instead of the standard Random-Walk Metropolis-Hastings. In +> > this algorithm, at each iteration the estimated parameters are +> > randomly assigned to different blocks. For each of these blocks a +> > mode-finding step is conducted. The inverse Hessian at this mode is +> > then used as the covariance of the proposal density for a +> > Random-Walk Metropolis-Hastings step. If the numerical Hessian is +> > not positive definite, the generalized Cholesky decomposition of +> > *Schnabel and Eskow (1990)* is used, but without pivoting. The +> > TaRB-MH algorithm massively reduces the autocorrelation in the MH +> > draws and thus reduces the number of draws required to +> > representatively sample from the posterior. However, this comes at a +> > computational cost as the algorithm takes more time to run. +> +> `'independent_metropolis_hastings'` +> +> > Use the Independent Metropolis-Hastings algorithm where the proposal +> > distribution - in contrast to the Random Walk Metropolis-Hastings +> > algorithm - does not depend on the state of the chain. +> +> `'slice'` +> +> > Instructs Dynare to use the Slice sampler of *Planas, Ratto, and +> > Rossi (2015)*. Note that `'slice'` is incompatible with +> > `prior_trunc=0`. +::: + +::: {.option} +posterior\_sampler\_options = (NAME, VALUE, \...) + +A list of NAME and VALUE pairs. Can be used to set options for the +posterior sampling methods. The set of available options depends on the +selected posterior sampling routine (i.e. on the value of option +`posterior_sampling_method +`{.interpreted-text role="opt"}): + +> `'random_walk_metropolis_hastings'` +> +> > Available options are: +> +> ::: {#prop_distrib} +> `'proposal_distribution'` +> ::: +> +> > Specifies the statistical distribution used for the proposal +> > density. +> +> `'rand_multivariate_normal'` +> +> > Use a multivariate normal distribution. This is the default. +> +> `'rand_multivariate_student'` +> +> > Use a multivariate student distribution. +> +> `'student_degrees_of_freedom'` +> +> > Specifies the degrees of freedom to be used with the multivariate +> > student distribution. Default: `3`. +> +> ::: {#usemhcov} +> `'use_mh_covariance_matrix'` +> ::: +> +> > Indicates to use the covariance matrix of the draws from a previous +> > MCMC run to define the covariance of the proposal distribution. +> > Requires the `load_mh_file`{.interpreted-text role="opt"} option to +> > be specified. Default: `0`. +> +> ::: {#scale-file} +> `'scale_file'` +> ::: +> +> > Provides the name of a `_mh_scale.mat` file storing the tuned scale +> > factor from a previous run of `mode_compute=6`. +> +> ::: {#savetmp} +> `'save_tmp_file'` +> ::: +> +> > Save the MCMC draws into a `_mh_tmp_blck` file at the refresh rate +> > of the status bar instead of just saving the draws when the current +> > `_mh*_blck` file is full. Default: `0` +> +> `'independent_metropolis_hastings'` +> +> > Takes the same options as in the case of +> > `random_walk_metropolis_hastings`. +> +> `'slice'` +> +> `'rotated'` +> +> > Triggers rotated slice iterations using a covariance matrix from +> > initial burn-in iterations. Requires either +> > `use_mh_covariance_matrix` or `slice_initialize_with_mode`. Default: +> > `0`. +> +> `'mode_files'` +> +> > For multimodal posteriors, provide the name of a file containing a +> > `nparam` by `nmodes` variable called `xparams` storing the different +> > modes. This array must have one column vector per mode and the +> > estimated parameters along the row dimension. With this info, the +> > code will automatically trigger the `rotated` and `mode` options. +> > Default: `[]`. +> +> `'slice_initialize_with_mode'` +> +> > The default for slice is to set `mode_compute=0` and start the +> > chain(s) from a random location in the prior space. This option +> > first runs the mode-finder and then starts the chain from the mode. +> > Together with `rotated`, it will use the inverse Hessian from the +> > mode to perform rotated slice iterations. Default: `0`. +> +> `'initial_step_size'` +> +> > Sets the initial size of the interval in the stepping-out procedure +> > as fraction of the prior support, i.e. the size will be +> > `initial_step_size * (UB-LB)`. `initial_step_size` must be a real +> > number in the interval `[0,1]`. Default: `0.8`. +> +> `'use_mh_covariance_matrix'` +> +> > See `use_mh_covariance_matrix `{.interpreted-text +> > role="ref"}. Must be used with `'rotated'`. Default: `0`. +> +> `'save_tmp_file'` +> +> > See `save_tmp_file `{.interpreted-text role="ref"}. +> > Default: `1`. +> +> `'tailored_random_block_metropolis_hastings'` +> +> `'proposal_distribution'` +> +> > Specifies the statistical distribution used for the proposal +> > density. See +> > `proposal_distribution `{.interpreted-text +> > role="ref"}. +> +> `new_block_probability = DOUBLE` +> +> > Specifies the probability of the next parameter belonging to a new +> > block when the random blocking in the TaRB Metropolis-Hastings +> > algorithm is conducted. The higher this number, the smaller is the +> > average block size and the more random blocks are formed during each +> > parameter sweep. Default: `0.25`. +> +> `mode_compute = INTEGER` +> +> > Specifies the mode-finder run in every iteration for every block of +> > the TaRB Metropolis-Hastings algorithm. See +> > `mode_compute > INTEGER | FUNCTION_NAME>`{.interpreted-text role="opt"}. Default: +> > `4`. +> +> `optim = (NAME, VALUE,...)` +> +> > Specifies the options for the mode-finder used in the TaRB +> > Metropolis-Hastings algorithm. See `optim +> > `{.interpreted-text role="opt"}. +> +> `'scale_file'` +> +> > See `scale_file `{.interpreted-text role="ref"}.. +> +> `'save_tmp_file'` +> +> > See `save_tmp_file `{.interpreted-text role="ref"}. +> > Default: `1`. +::: + +::: {.option} +moments\_varendo + +Triggers the computation of the posterior distribution of the +theoretical moments of the endogenous variables. Results are stored in +`oo_.PosteriorTheoreticalMoments` (see +`oo_.PosteriorTheoreticalMoments`{.interpreted-text role="mvar"}). The +number of lags in the autocorrelation function is controlled by the `ar` +option. +::: + +::: {.option} +contemporaneous\_correlation + +See `contemporaneous_correlation`{.interpreted-text role="opt"}. Results +are stored in `oo_.PosteriorTheoreticalMoments`. Note that the `nocorr` +option has no effect. +::: + +::: {.option} +no\_posterior\_kernel\_density + +Shuts off the computation of the kernel density estimator for the +posterior objects (see `density `{.interpreted-text role="ref"} +field). +::: + +::: {.option} +conditional\_variance\_decomposition = INTEGER +conditional\_variance\_decomposition = \[INTEGER1:INTEGER2\] +conditional\_variance\_decomposition = \[INTEGER1 INTEGER2 \...\] + +Computes the posterior distribution of the conditional variance +decomposition for the specified period(s). The periods must be strictly +positive. Conditional variances are given by $var(y_{t+k}\vert t)$. For +period 1, the conditional variance decomposition provides the +decomposition of the effects of shocks upon impact. The results are +stored in +`oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecomposition`.. +Note that this option requires the option `moments_varendo` to be +specified. In the presence of measurement error, the field will contain +the variance contribution after measurement error has been taken out, +*i.e.* the decomposition will be conducted of the actual as opposed to +the measured variables. The variance decomposition of the measured +variables will be stored in +`oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecompositionME`. +::: + +::: {.option} +filtered\_vars + +Triggers the computation of the posterior distribution of filtered +endogenous variables/one-step ahead forecasts, i.e. $E_{t}{y_{t+1}}$. +Results are stored in `oo_.FilteredVariables` (see below for a +description of this variable) +::: + +::: {.option} +smoother + +Triggers the computation of the posterior distribution of smoothed +endogenous variables and shocks, i.e. the expected value of variables +and shocks given the information available in all observations up to the +final date ($E_{T}{y_t}$). Results are stored in +`oo_.SmoothedVariables`, `oo_.SmoothedShocks` and +`oo_.SmoothedMeasurementErrors`. Also triggers the computation of +`oo_.UpdatedVariables`, which contains the estimation of the expected +value of variables given the information available at the current date +($E_{t}{y_t}$). See below for a description of all these variables. +::: + +::: {.option} +smoother\_redux + +Triggers a faster computation of the smoothed endogenous variables and +shocks for large models. It runs the smoother only for the state +variables (i.e. with the same representation used for likelihood +computations) and computes the remaining variables ex-post. Static +unobserved objects (filtered, smoothed, updated, k-step ahead) are +recovered, but there are exceptions to a full recovery, depending on how +static unobserved variables depend on the restricted state space +adopted. For example, lagged shocks which are ONLY used to recover +NON-observed static variables will not be recovered). For such +exceptions, only the following output is provided: + +> `FilteredVariablesKStepAhead`: will be fully recovered +> +> `SmoothedVariables`, `FilteredVariables`, `UpdatedVariables`: recovered for all periods beyond period `d+1`, +> +> : where `d` denotes the number of diffuse filtering steps. +> +> `FilteredVariablesKStepAheadVariances`, `Variance`, and +> `State_uncertainty` cannot be recovered, and ZERO is provided as +> output. + +If you need variances for those variables, either do not set the option, +or declare the variable as observed, using NaNs as data points. +::: + +::: {.option} +forecast = INTEGER + +Computes the posterior distribution of a forecast on INTEGER periods +after the end of the sample used in estimation. If no +Metropolis-Hastings is computed, the result is stored in variable +`oo_.forecast` and corresponds to the forecast at the posterior mode. If +a Metropolis-Hastings is computed, the distribution of forecasts is +stored in variables `oo_.PointForecast` and `oo_.MeanForecast`. See +`fore`{.interpreted-text role="ref"}, for a description of these +variables. +::: + +::: {.option} +tex + +See `tex`{.interpreted-text role="opt"}. +::: + +::: {.option} +kalman\_algo = INTEGER + +`0` + +> Automatically use the Multivariate Kalman Filter for stationary models +> and the Multivariate Diffuse Kalman Filter for non-stationary models. + +`1` + +> Use the Multivariate Kalman Filter. + +`2` + +> Use the Univariate Kalman Filter. + +`3` + +> Use the Multivariate Diffuse Kalman Filter. + +`4` + +> Use the Univariate Diffuse Kalman Filter. +::: + +> Default value is `0`. In case of missing observations of single or all +> series, Dynare treats those missing values as unobserved states and +> uses the Kalman filter to infer their value (see e.g. *Durbin and +> Koopman (2012)*, Ch. 4.10) This procedure has the advantage of being +> capable of dealing with observations where the forecast error variance +> matrix becomes singular for some variable(s). If this happens, the +> respective observation enters with a weight of zero in the +> log-likelihood, i.e. this observation for the respective variable(s) +> is dropped from the likelihood computations (for details see *Durbin +> and Koopman (2012)*, Ch. 6.4 and 7.2.5 and *Koopman and Durbin +> (2000)*). If the use of a multivariate Kalman filter is specified and +> a singularity is encountered, Dynare by default automatically switches +> to the univariate Kalman filter for this parameter draw. This behavior +> can be changed via the +> `use_univariate_filters_if_singularity_is_detected +> `{.interpreted-text +> role="opt"} option. + +::: {.option} +fast\_kalman\_filter + +Select the fast Kalman filter using Chandrasekhar recursions as +described by `Herbst (2015)`. This setting is only used with +`kalman_algo=1` or `kalman_algo=3`. In case of using the diffuse Kalman +filter (`kalman_algo=3/lik_init=3`), the observables must be stationary. +This option is not yet compatible with +`analytic_derivation`{.interpreted-text role="opt"}. +::: + +::: {.option} +kalman\_tol = DOUBLE + +Numerical tolerance for determining the singularity of the covariance +matrix of the prediction errors during the Kalman filter (minimum +allowed reciprocal of the matrix condition number). Default value is +`1e-10`. +::: + +::: {.option} +diffuse\_kalman\_tol = DOUBLE + +Numerical tolerance for determining the singularity of the covariance +matrix of the prediction errors ($F_{\infty}$) and the rank of the +covariance matrix of the non-stationary state variables ($P_{\infty}$) +during the Diffuse Kalman filter. Default value is `1e-6`. +::: + +::: {.option} +filter\_covariance + +Saves the series of one step ahead error of forecast covariance +matrices. With Metropolis, they are saved in +`oo_.FilterCovariance`{.interpreted-text role="mvar"}, otherwise in +`oo_.Smoother.Variance`{.interpreted-text role="mvar"}. Saves also +k-step ahead error of forecast covariance matrices if +`filter_step_ahead` is set. +::: + +::: {.option} +filter\_step\_ahead = \[INTEGER1:INTEGER2\] filter\_step\_ahead = +\[INTEGER1 INTEGER2 \...\] + +Triggers the computation k-step ahead filtered values, i.e. +$E_{t}{y_{t+k}}$. Stores results in `oo_.FilteredVariablesKStepAhead`. +Also stores 1-step ahead values in `oo_.FilteredVariables`. +`oo_.FilteredVariablesKStepAheadVariances` is stored if +`filter_covariance`. +::: + +::: {.option} +filter\_decomposition + +Triggers the computation of the shock decomposition of the above k-step +ahead filtered values. Stores results in +`oo_.FilteredVariablesShockDecomposition`. +::: + +::: {.option} +smoothed\_state\_uncertainty + +Triggers the computation of the variance of smoothed estimates, i.e. +$var_T(y_t)$. Stores results in `oo_.Smoother.State_uncertainty`. +::: + +::: {.option} +diffuse\_filter + +Uses the diffuse Kalman filter (as described in *Durbin and Koopman +(2012)* and *Koopman and Durbin (2003)* for the multivariate and +*Koopman and Durbin (2000)* for the univariate filter) to estimate +models with non-stationary observed variables. This option will also +reset the `qz_criterium` to count unit root variables towards the stable +variables. Trying to estimate a model with unit roots will otherwise +result in a Blanchard-Kahn error. + +When `diffuse_filter` is used the `lik_init` option of `estimation` has +no effect. + +When there are nonstationary exogenous variables in a model, there is no +unique deterministic steady state. For instance, if productivity is a +pure random walk: + +> $$a_t = a_{t-1} + e_t$$ + +any value of $\bar a$ of $a$ is a deterministic steady state for +productivity. Consequently, the model admits an infinity of steady +states. In this situation, the user must help Dynare in selecting one +steady state, except if zero is a trivial model's steady state, which +happens when the `linear` option is used in the model declaration. The +user can either provide the steady state to Dynare using a +`steady_state_model` block (or writing a steady state file) if a closed +form solution is available, see `steady_state_model`{.interpreted-text +role="bck"}, or specify some constraints on the steady state, see +`equation_tag_for_conditional_steady_state `{.interpreted-text +role="ref"}, so that Dynare computes the steady state conditionally on +some predefined levels for the non stationary variables. In both cases, +the idea is to use dummy values for the steady state level of the +exogenous non stationary variables. + +Note that the nonstationary variables in the model must be integrated +processes (their first difference or k-difference must be stationary). +::: + + +::: {.option} +selected\_variables\_only + +Only run the classical smoother on the variables listed just after the +`estimation` command. This option is incompatible with requesting +classical frequentist forecasts and will be overridden in this case. +When using Bayesian estimation, the smoother is by default only run on +the declared endogenous variables. Default: run the smoother on all the +declared endogenous variables. +::: + +::: {.option} +cova\_compute = INTEGER + +When `0`, the covariance matrix of estimated parameters is not computed +after the computation of posterior mode (or maximum likelihood). This +increases speed of computation in large models during development, when +this information is not always necessary. Of course, it will break all +successive computations that would require this covariance matrix. +Otherwise, if this option is equal to `1`, the covariance matrix is +computed and stored in variable `hh` of `MODEL_FILENAME_mode.mat`. +Default is `1`. +::: + +::: {.option} +solve\_algo = INTEGER + +See `solve_algo `{.interpreted-text role="ref"}. +::: + +::: {.option} +order = INTEGER + +Order of approximation around the deterministic steady state. When +greater than 1, the likelihood is evaluated with a particle or nonlinear +filter (see *Fernández-Villaverde and Rubio-Ramírez (2005)*). Default is +`1`, i.e. the likelihood of the linearized model is evaluated using a +standard Kalman filter. +::: + +::: {.option} +irf = INTEGER + +See `irf `{.interpreted-text role="opt"}. Only used if +`bayesian_irf`{.interpreted-text role="opt"} is passed. +::: + +::: {.option} +irf\_shocks = ( VARIABLE\_NAME \[\[,\] VARIABLE\_NAME \...\] ) + +See `irf_shocks `{.interpreted-text role="opt"}. Only used if +`bayesian_irf`{.interpreted-text role="opt"} is passed. +::: + +::: {.option} +irf\_plot\_threshold = DOUBLE + +See `irf_plot_threshold `{.interpreted-text role="opt"}. Only used if +`bayesian_irf`{.interpreted-text role="opt"} is passed. +::: + +::: {.option} +aim\_solver + +See `aim_solver`{.interpreted-text role="opt"}. +::: + +::: {.option} +sylvester = OPTION + +See `sylvester `{.interpreted-text role="opt"}. +::: + +::: {.option} +sylvester\_fixed\_point\_tol = DOUBLE + +See `sylvester_fixed_point_tol `{.interpreted-text role="opt"} . +::: + +::: {.option} +lyapunov = OPTION + +Determines the algorithm used to solve the Lyapunov equation to +initialized the variance-covariance matrix of the Kalman filter using +the steady-state value of state variables. Possible values for OPTION +are: + +> `default` +> +> > Uses the default solver for Lyapunov equations based on +> > Bartels-Stewart algorithm. +> +> `fixed_point` +> +> > Uses a fixed point algorithm to solve the Lyapunov equation. This +> > method is faster than the `default` one for large scale models, but +> > it could require a large amount of iterations. +> +> `doubling` +> +> > Uses a doubling algorithm to solve the Lyapunov equation +> > (`disclyap_fast`). This method is faster than the two previous one +> > for large scale models. +> +> `square_root_solver` +> +> > Uses a square-root solver for Lyapunov equations (`dlyapchol`). This +> > method is fast for large scale models (available under MATLAB if the +> > Control System Toolbox is installed; available under Octave if the +> > [control](https://octave.sourceforge.io/control/) package from +> > Octave-Forge is installed) + +Default value is `default`. +::: + +::: {.option} +lyapunov\_fixed\_point\_tol = DOUBLE + +This is the convergence criterion used in the fixed point Lyapunov +solver. Its default value is `1e-10`. +::: + +::: {.option} +lyapunov\_doubling\_tol = DOUBLE + +This is the convergence criterion used in the doubling algorithm to +solve the Lyapunov equation. Its default value is `1e-16`. +::: + +::: {.option} +use\_penalized\_objective\_for\_hessian + +Use the penalized objective instead of the objective function to compute +numerically the hessian matrix at the mode. The penalties decrease the +value of the posterior density (or likelihood) when, for some +perturbations, Dynare is not able to solve the model (issues with steady +state existence, Blanchard and Kahn conditions, \...). In pratice, the +penalized and original objectives will only differ if the posterior mode +is found to be near a region where the model is ill-behaved. By default +the original objective function is used. +::: + +::: {.option} +analytic\_derivation + +Triggers estimation with analytic gradient at `order=1`. The final +hessian at the mode is also computed analytically. Only works for +stationary models without missing observations, i.e. for +`kalman_algo<3`. Optimizers that rely on analytic gradients are +`mode_compute=1,3,4,5,101`. +::: + +::: {.option} +ar = INTEGER + +See `ar `{.interpreted-text role="opt"}. Only useful in +conjunction with option `moments_varendo`. +::: + +::: {.option} +endogenous\_prior + +Use endogenous priors as in *Christiano, Trabandt and Walentin (2011)*. +The procedure is motivated by sequential Bayesian learning. Starting +from independent initial priors on the parameters, specified in the +`estimated_params` block, the standard deviations observed in a +\"pre-sample\", taken to be the actual sample, are used to update the +initial priors. Thus, the product of the initial priors and the +pre-sample likelihood of the standard deviations of the observables is +used as the new prior (for more information, see the technical appendix +of *Christiano, Trabandt and Walentin (2011)*). This procedure helps in +cases where the regular posterior estimates, which minimize in-sample +forecast errors, result in a large overprediction of model variable +variances (a statistic that is not explicitly targeted, but often of +particular interest to researchers). +::: + +::: {.option} +use\_univariate\_filters\_if\_singularity\_is\_detected = INTEGER + +Decide whether Dynare should automatically switch to univariate filter +if a singularity is encountered in the likelihood computation (this is +the behaviour if the option is equal to `1`). Alternatively, if the +option is equal to `0`, Dynare will not automatically change the filter, +but rather use a penalty value for the likelihood when such a +singularity is encountered. Default: `1`. +::: + +::: {.option} +keep\_kalman\_algo\_if\_singularity\_is\_detected + +With the default `use_univariate_filters_if_singularity_is_detected=1 +`{.interpreted-text +role="opt"}, Dynare will switch to the univariate Kalman filter when it +encounters a singular forecast error variance matrix during Kalman +filtering. Upon encountering such a singularity for the first time, all +subsequent parameter draws and computations will automatically rely on +univariate filter, i.e. Dynare will never try the multivariate filter +again. Use the `keep_kalman_algo_if_singularity_is_detected` option to +have the `use_univariate_filters_if_singularity_is_detected` only affect +the behavior for the current draw/computation. +::: + +::: {.option} +rescale\_prediction\_error\_covariance + +Rescales the prediction error covariance in the Kalman filter to avoid +badly scaled matrix and reduce the probability of a switch to univariate +Kalman filters (which are slower). By default no rescaling is done. +::: + +::: {.option} +qz\_zero\_threshold = DOUBLE + +See `qz_zero_threshold `{.interpreted-text +role="opt"}. +::: + +::: {.option} +taper\_steps = \[INTEGER1 INTEGER2 \...\] + +Percent tapering used for the spectral window in the *Geweke +(1992,1999)* convergence diagnostics (requires +`mh_nblocks=1 `{.interpreted-text role="opt"}). +The tapering is used to take the serial correlation of the posterior +draws into account. Default: `[4 8 15]`. +::: + +::: {.option} +geweke\_interval = \[DOUBLE DOUBLE\] + +Percentage of MCMC draws at the beginning and end of the MCMC chain +taken to compute the *Geweke (1992,1999)* convergence diagnostics +(requires `mh_nblocks=1 `{.interpreted-text role="opt"}) after discarding the first +`mh_drop = DOUBLE +`{.interpreted-text role="opt"} percent of draws as a burnin. +Default: \[0.2 0.5\]. +::: + +::: {.option} +raftery\_lewis\_diagnostics + +Triggers the computation of the *Raftery and Lewis (1992)* convergence +diagnostics. The goal is deliver the number of draws required to +estimate a particular quantile of the CDF `q` with precision `r` with a +probability `s`. Typically, one wants to estimate the `q=0.025` +percentile (corresponding to a 95 percent HPDI) with a precision of 0.5 +percent (`r=0.005`) with 95 percent certainty (`s=0.95`). The defaults +can be changed via `raftery_lewis_qrs +`{.interpreted-text +role="opt"}. Based on the theory of first order Markov Chains, the +diagnostics will provide a required burn-in (`M`), the number of draws +after the burnin (`N`) as well as a thinning factor that would deliver a +first order chain (`k`). The last line of the table will also deliver +the maximum over all parameters for the respective values. +::: + +::: {.option} +raftery\_lewis\_qrs = \[DOUBLE DOUBLE DOUBLE\] + +Sets the quantile of the CDF `q` that is estimated with precision `r` +with a probability `s` in the *Raftery and Lewis (1992)* convergence +diagnostics. Default: `[0.025 0.005 0.95]`. +::: + +::: {.option} +consider\_all\_endogenous + +Compute the posterior moments, smoothed variables, k-step ahead filtered +variables and forecasts (when requested) on all the endogenous +variables. This is equivalent to manually listing all the endogenous +variables after the `estimation` command. +::: + +::: {.option} +consider\_all\_endogenous\_and\_auxiliary + +Compute the posterior moments, smoothed variables, k-step ahead filtered +variables and forecasts (when requested) on all the endogenous variables +and the auxiliary variables introduced by the preprocessor. This option +is useful when e.g. running `smoother2histval` on the results of the +Kalman smoother. +::: + +::: {.option} +consider\_only\_observed + +Compute the posterior moments, smoothed variables, k-step ahead filtered +variables and forecasts (when requested) on all the observed variables. +This is equivalent to manually listing all the observed variables after +the `estimation` command. +::: + + +*"Endogenous" prior restrictions* + +It is also possible to impose implicit "endogenous" priors about IRFs +and moments on the model during estimation. For example, one can specify +that all valid parameter draws for the model must generate fiscal +multipliers that are bigger than 1 by specifying how the IRF to a +government spending shock must look like. The prior restrictions can be +imposed via `irf_calibration` and `moment_calibration` blocks (see +`irf-momcal`{.interpreted-text role="ref"}). The way it works internally +is that any parameter draw that is inconsistent with the "calibration" +provided in these blocks is discarded, i.e. assigned a prior density of +0. When specifying these blocks, it is important to keep in mind that +one won't be able to easily do `model_comparison` in this case, because +the prior density will not integrate to 1. + +*Output* + +After running estimation, the parameters `M_.params` and the variance +matrix `M_.Sigma_e` of the shocks are set to the mode for maximum +likelihood estimation or posterior mode computation without Metropolis +iterations. After estimation with Metropolis iterations (option +`mh_replic > 0` or option `load_mh_file` set) the parameters `M_.params` +and the variance matrix `M_.Sigma_e` of the shocks are set to the +posterior mean. + +Depending on the options, `estimation` stores results in various fields +of the `oo_` structure, described below. In the following variables, we +will adopt the following shortcuts for specific field names: + +> `MOMENT_NAME` +> +> > This field can take the following values: +> > +> > `HPDinf` +> > +> > > Lower bound of a 90% HPD interval.[^4] +> > +> > `HPDsup` +> > +> > > Upper bound of a 90% HPD interval. +> > +> > `HPDinf_ME` +> > +> > > Lower bound of a 90% HPD interval[^5] for observables when taking +> > > measurement error into account (see e.g. *Christoffel et al. +> > > (2010*), p.17). +> > +> > `HPDsup_ME` +> > +> > > Upper bound of a 90% HPD interval for observables when taking +> > > measurement error into account. +> > +> > `Mean` +> > +> > > Mean of the posterior distribution. +> > +> > `Median` +> > +> > > Median of the posterior distribution. +> > +> > `Std` +> > +> > > Standard deviation of the posterior distribution. +> > +> > `Variance` +> > +> > > Variance of the posterior distribution. +> > +> > `deciles` +> > +> > > Deciles of the distribution. +> > +> > ::: {#dens} +> > `density` +> > ::: +> > +> > > Non parametric estimate of the posterior density following the +> > > approach outlined in *Skoeld and Roberts (2003)*. First and second +> > > columns are respectively abscissa and ordinate coordinates. +> +> `ESTIMATED_OBJECT` +> +> > This field can take the following values: +> > +> > `measurement_errors_corr` +> > +> > > Correlation between two measurement errors. +> > +> > `measurement_errors_std` +> > +> > > Standard deviation of measurement errors. +> > +> > `parameters` +> > +> > > Parameters. +> > +> > `shocks_corr` +> > +> > > Correlation between two structural shocks. +> > +> > `shocks_std` +> > +> > > Standard deviation of structural shocks. + +::: {.matvar} +[oo]().MarginalDensity.LaplaceApproximation + +Variable set by the `estimation` command. Stores the marginal data +density based on the Laplace Approximation. +::: + +::: {.matvar} +[oo]().MarginalDensity.ModifiedHarmonicMean + +Variable set by the `estimation command`, if it is used with +`mh_replic > 0` or `load_mh_file` option. Stores the marginal data +density based on *Geweke (1999)* Modified Harmonic Mean estimator. +::: + +::: {.matvar} +[oo]().posterior.optimization + +Variable set by the `estimation` command if mode-finding is used. Stores +the results at the mode. Fields are of the form: + + oo_.posterior.optimization.OBJECT + +where OBJECT is one of the following: + +> `mode` +> +> > Parameter vector at the mode. +> +> `Variance` +> +> > Inverse Hessian matrix at the mode or MCMC jumping covariance matrix +> > when used with the `MCMC_jumping_covariance > = OPTION>`{.interpreted-text role="opt"} option. +> +> `log_density` +> +> > Log likelihood (ML)/log posterior density (Bayesian) at the mode +> > when used with `mode_compute>0`. +::: + +::: {.matvar} +[oo]().posterior.metropolis + +Variable set by the `estimation` command if `mh_replic>0` is used. +Fields are of the form: + + oo_.posterior.metropolis.OBJECT + +where OBJECT is one of the following: + +> `mean` +> +> > Mean parameter vector from the MCMC. +> +> `Variance` +> +> > Covariance matrix of the parameter draws in the MCMC. +::: + +::: {.matvar} +[oo]().FilteredVariables + +Variable set by the `estimation` command, if it is used with the +`filtered_vars` option. + +After an estimation without Metropolis, fields are of the form: + + oo_.FilteredVariables.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.FilteredVariables.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().FilteredVariablesKStepAhead + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead` option. The k-steps are stored along the rows while +the columns indicate the respective variables. The third dimension of +the array provides the observation for which the forecast has been made. +For example, if `filter_step_ahead=[1 2 4]` and `nobs=200`, the element +(3,5,204) stores the four period ahead filtered value of variable 5 +computed at time t=200 for time t=204. The periods at the beginning and +end of the sample for which no forecasts can be made, e.g. entries +(1,5,1) and (1,5,204) in the example, are set to zero. Note that in case +of Bayesian estimation the variables will be ordered in the order of +declaration after the estimation command (or in general declaration +order if no variables are specified here). In case of running the +classical smoother, the variables will always be ordered in general +declaration order. If the `selected_variables_only`{.interpreted-text +role="opt"} option is specified with the classical smoother, +non-requested variables will be simply left out in this order. +::: + +::: {.matvar} +[oo]().FilteredVariablesKStepAheadVariances + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead option`. It is a 4 dimensional array where the +k-steps are stored along the first dimension, while the fourth dimension +of the array provides the observation for which the forecast has been +made. The second and third dimension provide the respective variables. +For example, if `filter_step_ahead=[1 2 4]` and `nobs=200`, the element +(3,4,5,204) stores the four period ahead forecast error covariance +between variable 4 and variable 5, computed at time t=200 for time +t=204. Padding with zeros and variable ordering is analogous to +`oo_.FilteredVariablesKStepAhead`. +::: + +::: {.matvar} +[oo]().Filtered\_Variables\_X\_step\_ahead + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead option` in the context of Bayesian estimation. Fields +are of the form: + + oo_.Filtered_Variables_X_step_ahead.VARIABLE_NAME + +The n-th entry stores the k-step ahead filtered variable computed at +time n for time n+k. +::: + +::: {.matvar} +[oo]().FilteredVariablesShockDecomposition + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead` option. The k-steps are stored along the rows while +the columns indicate the respective variables. The third dimension +corresponds to the shocks in declaration order. The fourth dimension of +the array provides the observation for which the forecast has been made. +For example, if `filter_step_ahead=[1 2 4]` and `nobs=200`, the element +(3,5,2,204) stores the contribution of the second shock to the four +period ahead filtered value of variable 5 (in deviations from the mean) +computed at time t=200 for time t=204. The periods at the beginning and +end of the sample for which no forecasts can be made, e.g. entries +(1,5,1) and (1,5,204) in the example, are set to zero. Padding with +zeros and variable ordering is analogous to +`oo_.FilteredVariablesKStepAhead`. +::: + +::: {.matvar} +[oo]().PosteriorIRF.dsge + +Variable set by the `estimation` command, if it is used with the +`bayesian_irf` option. Fields are of the form: + + oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME +::: + +::: {.matvar} +[oo]().SmoothedMeasurementErrors + +Variable set by the `estimation` command, if it is used with the +`smoother` option. Fields are of the form: + + oo_.SmoothedMeasurementErrors.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().SmoothedShocks + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. + +After an estimation without Metropolis, or if computed by +`calib_smoother`, fields are of the form: + + oo_.SmoothedShocks.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.SmoothedShocks.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().SmoothedVariables + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. + +After an estimation without Metropolis, or if computed by +`calib_smoother`, fields are of the form: + + oo_.SmoothedVariables.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.SmoothedVariables.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matcomm} +get\_smooth (\'VARIABLE\_NAME\' \[, \'VARIABLE\_NAME\'\]\...); + +Returns the smoothed values of the given endogenous or exogenous +variable(s), as they are stored in the `oo_.SmoothedVariables` and +`oo_.SmoothedShocks` variables. +::: + +::: {.matvar} +[oo]().UpdatedVariables + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. Contains the estimation of +the expected value of variables given the information available at the +current date. + +After an estimation without Metropolis, or if computed by +`calib_smoother`, fields are of the form: + + oo_.UpdatedVariables.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.UpdatedVariables.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matcomm} +get\_update (\'VARIABLE\_NAME\' \[, \'VARIABLE\_NAME\'\]\...); + +Returns the updated values of the given variable(s), as they are stored +in the `oo_.UpdatedVariables` variable. +::: + +::: {.matvar} +[oo]().FilterCovariance + +Three-dimensional array set by the `estimation` command if used with the +`smoother` and Metropolis, if the `filter_covariance` option has been +requested. Contains the series of one-step ahead forecast error +covariance matrices from the Kalman smoother. The `M_.endo_nbr` times +`M_.endo_nbr` times `T+1` array contains the variables in declaration +order along the first two dimensions. The third dimension of the array +provides the observation for which the forecast has been made. Fields +are of the form: + + oo_.FilterCovariance.MOMENT_NAME + +Note that density estimation is not supported. +::: + +::: {.matvar} +[oo]().Smoother.Variance + +Three-dimensional array set by the `estimation` command (if used with +the `smoother`) without Metropolis, or by the `calib_smoother` command, +if the `filter_covariance` option has been requested. Contains the +series of one-step ahead forecast error covariance matrices from the +Kalman smoother. The `M_.endo_nbr` times `M_.endo_nbr` times `T+1` array +contains the variables in declaration order along the first two +dimensions. The third dimension of the array provides the observation +for which the forecast has been made. +::: + +::: {.matvar} +[oo]().Smoother.State\_uncertainty + +Three-dimensional array set by the `estimation` command (if used with +the `smoother` option) without Metropolis, or by the `calib_smoother` +command, if the `smoothed_state_uncertainty` option has been requested. +Contains the series of covariance matrices for the state estimate given +the full data from the Kalman smoother. The `M_.endo_nbr` times +`M_.endo_nbr` times `T` array contains the variables in declaration +order along the first two dimensions. The third dimension of the array +provides the observation for which the smoothed estimate has been made. +::: + +::: {.matvar} +[oo]().Smoother.SteadyState + +Variable set by the `estimation` command (if used with the `smoother`) +without Metropolis, or by the `calib_smoother` command. Contains the +steady state component of the endogenous variables used in the smoother +in order of variable declaration. +::: + +::: {.matvar} +[oo]().Smoother.TrendCoeffs + +Variable set by the `estimation` command (if used with the `smoother`) +without Metropolis, or by the `calib_smoother` command. Contains the +trend coefficients of the observed variables used in the smoother in +order of declaration of the observed variables. +::: + +::: {.matvar} +[oo]().Smoother.Trend + +Variable set by the `estimation command` (if used with the `smoother` +option), or by the `calib_smoother` command. Contains the trend +component of the variables used in the smoother. + +Fields are of the form: + + oo_.Smoother.Trend.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().Smoother.Constant + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. Contains the constant part +of the endogenous variables used in the smoother, accounting e.g. for +the data mean when using the prefilter option. + +Fields are of the form: + + oo_.Smoother.Constant.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().Smoother.loglinear + +Indicator keeping track of whether the smoother was run with the +`loglinear `{.interpreted-text role="ref"} option and thus whether +stored smoothed objects are in logs. +::: + +::: {.matvar} +[oo]().PosteriorTheoreticalMoments + +Variable set by the `estimation` command, if it is used with the +`moments_varendo` option. Fields are of the form: + + oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_NAME.VARIABLE_NAME + +where *THEORETICAL\_MOMENT* is one of the following: + +> `covariance` +> +> > Variance-covariance of endogenous variables. +> +> `contemporaneous_correlation` +> +> > Contemporaneous correlation of endogenous variables when the +> > `contemporaneous_correlation`{.interpreted-text role="opt"} option +> > is specified. +> +> `correlation` +> +> > Auto- and cross-correlation of endogenous variables. Fields are +> > vectors with correlations from 1 up to order `options_.ar`. +> +> ::: {#VarianceDecomposition} +> `VarianceDecomposition` +> ::: +> +> > Decomposition of variance (unconditional variance, i.e. at horizon +> > infinity).[^6] +> +> `VarianceDecompositionME` +> +> > Same as [VarianceDecomposition](), but contains the decomposition of +> > the measured as opposed to the actual variable. The joint +> > contribution of the measurement error will be saved in a field named +> > `ME`. +> +> ::: {#ConditionalVarianceDecomposition} +> `ConditionalVarianceDecomposition` +> ::: +> +> > Only if the `conditional_variance_decomposition` option has been +> > specified. In the presence of measurement error, the field will +> > contain the variance contribution after measurement error has been +> > taken out, i.e. the decomposition will be conducted of the actual as +> > opposed to the measured variables. +> +> `ConditionalVarianceDecompositionME` +> +> > Only if the `conditional_variance_decomposition` option has been +> > specified. Same as [ConditionalVarianceDecomposition](), but +> > contains the decomposition of the measured as opposed to the actual +> > variable. The joint contribution of the measurement error will be +> > saved in a field names `ME`. +::: + +::: {.matvar} +[oo]().posterior\_density + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_density.PARAMETER_NAME +::: + +::: {.matvar} +[oo]().posterior\_hpdinf + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_hpdsup + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_mean + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_mode + +Variable set by the `estimation` command during mode-finding. Fields are +of the form: + + oo_.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_std\_at\_mode + +Variable set by the `estimation` command during mode-finding. It is +based on the inverse Hessian at `oo_.posterior_mode`. Fields are of the +form: + + oo_.posterior_std_at_mode.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_std + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_var + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_var.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_median + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_median.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +*Example* + +Here are some examples of generated variables: + + oo_.posterior_mode.parameters.alp + oo_.posterior_mean.shocks_std.ex + oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso + +::: {.matvar} +[oo]().dsge\_var.posterior\_mode + +Structure set by the `dsge_var` option of the `estimation` command after +mode\_compute. + +The following fields are saved: + +> `PHI_tilde` +> +> > Stacked posterior DSGE-BVAR autoregressive matrices at the mode +> > (equation (28) of *Del Negro and Schorfheide (2004)*). +> +> `SIGMA_u_tilde` +> +> > Posterior covariance matrix of the DSGE-BVAR at the mode (equation +> > (29) of *Del Negro and Schorfheide (2004)*). +> +> `iXX` +> +> > Posterior population moments in the DSGE-BVAR at the mode ( +> > $inv(\lambda T \Gamma_{XX}^*+ X'X)$). +> +> `prior` +> +> > Structure storing the DSGE-BVAR prior. +> +> `PHI_star` +> +> > Stacked prior DSGE-BVAR autoregressive matrices at the mode +> > (equation (22) of *Del Negro and Schorfheide (2004)*). +> +> `SIGMA_star` +> +> > Prior covariance matrix of the DSGE-BVAR at the mode (equation (23) +> > of *Del Negro and Schorfheide (2004)*). +> +> `ArtificialSampleSize` +> +> > Size of the artifical prior sample ( $inv(\lambda T)$). +> +> `DF` +> +> > Prior degrees of freedom ( $inv(\lambda T-k-n)$). +> +> `iGXX_star` +> +> > Inverse of the theoretical prior "covariance" between X and X +> > ($\Gamma_{xx}^*$ in *Del Negro and Schorfheide (2004)*). +::: + +::: {.matvar} +[oo]().RecursiveForecast + +Variable set by the `forecast` option of the `estimation` command when +used with the nobs = \[INTEGER1:INTEGER2\] option (see +`nobs `{.interpreted-text role="opt"}). + +Fields are of the form: + + oo_.RecursiveForecast.FORECAST_OBJECT.VARIABLE_NAME + +where `FORECAST_OBJECT` is one of the following[^7] : + +`Mean` + +> Mean of the posterior forecast distribution. + +`HPDinf/HPDsup` + +> Upper/lower bound of the 90% HPD interval taking into account only +> parameter uncertainty (corresponding to +> `oo_.MeanForecast`{.interpreted-text role="mvar"}). + +`HPDTotalinf/HPDTotalsup`. + +> Upper/lower bound of the 90% HPD interval taking into account both +> parameter and future shock uncertainty (corresponding to +> `oo_.PointForecast`{.interpreted-text role="mvar"}) + +`VARIABLE_NAME` contains a matrix of the following size: number of time +periods for which forecasts are requested using the +`nobs = [INTEGER1:INTEGER2]` option times the number of forecast +horizons requested by the forecast option. i.e., the row indicates the +period at which the forecast is performed and the column the respective +k-step ahead forecast. The starting periods are sorted in ascending +order, not in declaration order. +::: + +::: {.matvar} +[oo]().convergence.geweke + +Variable set by the convergence diagnostics of the `estimation` command +when used with `mh_nblocks=1` option (see +`mh_nblocks `{.interpreted-text role="opt"}). + +Fields are of the form: + + oo_.convergence.geweke.VARIABLE_NAME.DIAGNOSTIC_OBJECT + +where *DIAGNOSTIC\_OBJECT* is one of the following: + +`posteriormean` + +> Mean of the posterior parameter distribution. + +`posteriorstd` + +> Standard deviation of the posterior parameter distribution. + +`nse_iid` + +> Numerical standard error (NSE) under the assumption of iid draws. + +`rne_iid` + +> Relative numerical efficiency (RNE) under the assumption of iid draws. + +`nse_x` + +> Numerical standard error (NSE) when using an x% taper. + +`rne_x` + +> Relative numerical efficiency (RNE) when using an x% taper. + +`pooled_mean` + +> Mean of the parameter when pooling the beginning and end parts of the +> chain specified in `geweke_interval +> `{.interpreted-text role="opt"} and +> weighting them with their relative precision. It is a vector +> containing the results under the iid assumption followed by the ones +> using the `taper_steps` option (see `taper_steps = [INTEGER1 INTEGER2 ...]>`{.interpreted-text role="opt"}). + +`pooled_nse` + +> NSE of the parameter when pooling the beginning and end parts of the +> chain and weighting them with their relative precision. See +> `pooled_mean`. + +`prob_chi2_test` + +> p-value of a chi-squared test for equality of means in the beginning +> and the end of the MCMC chain. See `pooled_mean`. A value above 0.05 +> indicates that the null hypothesis of equal means and thus convergence +> cannot be rejected at the 5 percent level. Differing values along the +> `taper_steps` signal the presence of significant autocorrelation in +> draws. In this case, the estimates using a higher tapering are usually +> more reliable. +::: +::: +::: + +::: {.command} +unit\_root\_vars VARIABLE\_NAME\...; + +This command is deprecated. Use `estimation` option `diffuse_filter` +instead for estimating a model with non-stationary observed variables or +`steady` option `nocheck` to prevent `steady` to check the steady state +returned by your steady state file. +::: + +Dynare also has the ability to estimate Bayesian VARs: + +::: {.command} +bvar\_density ; + +Computes the marginal density of an estimated BVAR model, using +Minnesota priors. + +See `bvar-a-la-sims.pdf`, which comes with Dynare distribution, for more +information on this command. +::: + + +Shock Decomposition +------------------- + +::: {.command} +shock\_decomposition \[VARIABLE\_NAME\]\...; shock\_decomposition +(OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command computes the historical shock decomposition for a given +sample based on the Kalman smoother, i.e. it decomposes the historical +deviations of the endogenous variables from their respective steady +state values into the contribution coming from the various shocks. The +`variable_names` provided govern for which variables the decomposition +is plotted. + +Note that this command must come after either `estimation` (in case of +an estimated model) or `stoch_simul` (in case of a calibrated model). + +*Options* + +::: {.option} +parameter\_set = OPTION + +Specify the parameter set to use for running the smoother. Possible +values for OPTION are: + +> - `calibration` +> - `prior_mode` +> - `prior_mean` +> - `posterior_mode` +> - `posterior_mean` +> - `posterior_median` +> - `mle_mode` + +Note that the parameter set used in subsequent commands like +`stoch_simul` will be set to the specified `parameter_set`. Default +value: `posterior_mean` if Metropolis has been run, `mle_mode` if MLE +has been run. +::: + +::: {.option} +datafile = FILENAME + +See `datafile `{.interpreted-text role="ref"}. Useful when +computing the shock decomposition on a calibrated model. +::: + +::: {.option} +first\_obs = INTEGER + +See `first_obs `{.interpreted-text role="opt"}. +::: + +::: {.option} +nobs = INTEGER + +See `nobs `{.interpreted-text role="opt"}. +::: + +::: {.option} +prefilter = INTEGER + +See `prefilter `{.interpreted-text role="opt"}. +::: + +::: {.option} +loglinear + +See `loglinear `{.interpreted-text role="opt"}. +::: + +::: {.option} +diffuse\_kalman\_tol = DOUBLE + +See `diffuse_kalman_tol `{.interpreted-text +role="opt"}. +::: + +::: {.option} +diffuse\_filter + +See `diffuse_filter `{.interpreted-text role="opt"}. +::: + +::: {.option} +xls\_sheet = QUOTED\_STRING + +See `xls_sheet `{.interpreted-text +role="opt"}. +::: + +::: {.option} +xls\_range = RANGE + +See `xls_range `{.interpreted-text role="opt"}. +::: + +::: {.option} +use\_shock\_groups \[= NAME\] + +Uses shock grouping defined by the string instead of individual shocks +in the decomposition. The groups of shocks are defined in the +`shock_groups`{.interpreted-text role="bck"} block. If no group name is +given, `default` is assumed. +::: + +::: {.option} +colormap = VARIABLE\_NAME + +Controls the `colormap` used for the shocks decomposition graphs. +VARIABLE\_NAME must be the name of a MATLAB/Octave variable that has +been declared beforehand and whose value will be passed to the +MATLAB/Octave `colormap` function (see the MATLAB/Octave manual for the +list of acceptable values). +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. Suppresses the display and +creation only within the `shock_decomposition` command, but does not +affect other commands. See `plot_shock_decomposition`{.interpreted-text +role="comm"} for plotting graphs. +::: + +::: {.option} +init\_state = BOOLEAN + +If equal to 0, the shock decomposition is computed conditional on the +smoothed state variables in period `0`, i.e. the smoothed shocks +starting in period 1 are used. If equal to `1`, the shock decomposition +is computed conditional on the smoothed state variables in period 1. +Default: `0`. +::: + +::: {.option} +with\_epilogue + +If set, then also compute the decomposition for variables declared in +the `epilogue` block (see `epilogue`{.interpreted-text role="ref"}). +::: + +*Output* + +::: {.matvar} +[oo]().shock\_decomposition + +The results are stored in the field `oo_.shock_decomposition`, which is +a three dimensional array. The first dimension contains the +`M_.endo_nbr` endogenous variables. The second dimension stores in the +first `M_.exo_nbr` columns the contribution of the respective shocks. +Column `M_.exo_nbr+1` stores the contribution of the initial conditions, +while column `M_.exo_nbr+2` stores the smoothed value of the respective +endogenous variable in deviations from their steady state, i.e. the mean +and trends are subtracted. The third dimension stores the time periods. +Both the variables and shocks are stored in the order of declaration, +i.e. `M_.endo_names` and `M_.exo_names`, respectively. +::: +::: + +::: {.block} +shock\_groups ; shock\_groups(OPTIONS\...); + +Shocks can be regrouped for the purpose of shock decomposition. The +composition of the shock groups is written in a block delimited by +`shock_groups` and `end`. + +Each line defines a group of shocks as a list of exogenous variables: + + SHOCK_GROUP_NAME = VARIABLE_1 [[,] VARIABLE_2 [,]...]; + 'SHOCK GROUP NAME' = VARIABLE_1 [[,] VARIABLE_2 [,]...]; + +*Options* + +::: {.option} +name = NAME + +Specifies a name for the following definition of shock groups. It is +possible to use several `shock_groups` blocks in a model file, each +grouping being identified by a different name. This name must in turn be +used in the `shock_decomposition` command. If no name is given, +`default` is used. +::: + +*Example* + +> varexo e_a, e_b, e_c, e_d; +> ... +> +> shock_groups(name=group1); +> supply = e_a, e_b; +> 'aggregate demand' = e_c, e_d; +> end; +> +> shock_decomposition(use_shock_groups=group1); +> +> This example defines a shock grouping with the name `group1`, +> containing a set of supply and demand shocks and conducts the shock +> decomposition for these two groups. +::: + +::: {.command} +realtime\_shock\_decomposition \[VARIABLE\_NAME\]\...; +realtime\_shock\_decomposition (OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command computes the realtime historical shock decomposition for a +given sample based on the Kalman smoother. For each period +$T=[\texttt{presample},\ldots,\texttt{nobs}]$, it recursively computes +three objects: + +> - Real-time historical shock decomposition $Y(t\vert T)$ for +> $t=[1,\ldots,T]$, i.e. without observing data in +> $[T+1,\ldots,\texttt{nobs}]$. This results in a standard shock +> decomposition being computed for each additional datapoint +> becoming available after `presample`. +> - Forecast shock decomposition $Y(T+k\vert T)$ for +> $k=[1,\ldots,forecast]$, i.e. the $k$-step ahead forecast made for +> every $T$ is decomposed in its shock contributions. +> - Real-time conditional shock decomposition of the difference +> between the real-time historical shock decomposition and the +> forecast shock decomposition. If `vintage INTEGER>`{.interpreted-text role="opt"} is equal to `0`, it +> computes the effect of shocks realizing in period $T$, i.e. +> decomposes $Y(T\vert T)-Y(T\vert T-1)$. Put differently, it +> conducts a $1$-period ahead shock decomposition from $T-1$ to $T$, +> by decomposing the update step of the Kalman filter. If +> `vintage>0` and smaller than `nobs`, the decomposition is +> conducted of the forecast revision +> $Y(T+k\vert T+k)-Y(T+k\vert T)$. + +Like `shock_decomposition`{.interpreted-text role="comm"} it decomposes +the historical deviations of the endogenous variables from their +respective steady state values into the contribution coming from the +various shocks. The `variable_names` provided govern for which variables +the decomposition is plotted. + +Note that this command must come after either `estimation` (in case of +an estimated model) or `stoch_simul` (in case of a calibrated model). + +*Options* + +::: {.option} +parameter\_set = OPTION + +See `parameter_set `{.interpreted-text +role="opt"} for possible values. +::: + +::: {.option} +datafile = FILENAME + +See `datafile `{.interpreted-text role="ref"}. +::: + +::: {.option} +first\_obs = INTEGER + +See `first_obs `{.interpreted-text role="opt"}. +::: + +::: {.option} +nobs = INTEGER + +See `nobs `{.interpreted-text role="opt"}. +::: + +::: {.option} +use\_shock\_groups \[= NAME\] + +See `use_shock_groups `{.interpreted-text +role="opt"}. +::: + +::: {.option} +colormap = VARIABLE\_NAME + +See `colormap `{.interpreted-text role="opt"}. +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. Only shock decompositions +are computed and stored in `oo_.realtime_shock_decomposition`, +`oo_.conditional_shock_decomposition` and +`oo_.realtime_forecast_shock_decomposition` but no plot is made (See +`plot_shock_decomposition`{.interpreted-text role="comm"}). +::: + +::: {.option} +presample = INTEGER + +Data point above which recursive realtime shock decompositions are +computed, *i.e.* for $T=[\texttt{presample+1} \ldots \texttt{nobs}]$. +::: + +::: {.option} +forecast = INTEGER + +Compute shock decompositions up to $T+k$ periods, i.e. get shock +contributions to k-step ahead forecasts. +::: + +::: {.option} +save\_realtime = INTEGER\_VECTOR + +Choose for which vintages to save the full realtime shock decomposition. +Default: `0`. +::: + +::: {.option} +fast\_realtime = INTEGER fast\_realtime = \[INTEGER1:INTEGER2\] +fast\_realtime = \[INTEGER1 INTEGER2 \...\] + +Runs the smoother only for the data vintages provided by the specified +integer (vector). +::: + +::: {.option} +with\_epilogue + +See `with_epilogue`{.interpreted-text role="opt"}. +::: + +*Output* + +::: {.matvar} +[oo]().realtime\_shock\_decomposition + +Structure storing the results of realtime historical decompositions. +Fields are three-dimensional arrays with the first two dimension equal +to the ones of `oo_.shock_decomposition`{.interpreted-text role="mvar"}. +The third dimension stores the time periods and is therefore of size +`T+forecast`. Fields are of the form: + + oo_.realtime_shock_decomposition.OBJECT + +where OBJECT is one of the following: + +> `pool` +> +> > Stores the pooled decomposition, i.e. for every real-time shock +> > decomposition terminal period +> > $T=[\texttt{presample},\ldots,\texttt{nobs}]$ it collects the last +> > period's decomposition $Y(T\vert T)$ (see also +> > `plot_shock_decomposition`{.interpreted-text role="comm"}). The +> > third dimension of the array will have size `nobs+forecast`. +> +> `time_*` +> +> > Stores the vintages of realtime historical shock decompositions if +> > `save_realtime` is used. For example, if `save_realtime=[5]` and +> > `forecast=8`, the third dimension will be of size `13`. +::: + +::: {.matvar} +[oo]().realtime\_conditional\_shock\_decomposition + +Structure storing the results of real-time conditional decompositions. +Fields are of the form: + + oo_.realtime_conditional_shock_decomposition.OBJECT + +where OBJECT is one of the following: + +> `pool` +> +> > Stores the pooled real-time conditional shock decomposition, i.e. +> > collects the decompositions of $Y(T\vert T)-Y(T\vert T-1)$ for the +> > terminal periods $T=[\texttt{presample},\ldots,\texttt{nobs}]$. The +> > third dimension is of size `nobs`. +> +> `time_*` +> +> > Store the vintages of $k$-step conditional forecast shock +> > decompositions $Y(t\vert T+k)$, for $t=[T \ldots T+k]$. See `vintage +> > `{.interpreted-text role="opt"}. The third +> > dimension is of size `1+forecast`. +::: + +::: {.matvar} +[oo]().realtime\_forecast\_shock\_decomposition + +Structure storing the results of realtime forecast decompositions. +Fields are of the form: + + oo_.realtime_forecast_shock_decomposition.OBJECT + +where `OBJECT` is one of the following: + +> `pool` +> +> > Stores the pooled real-time forecast decomposition of the $1$-step +> > ahead effect of shocks on the $1$-step ahead prediction, i.e. +> > $Y(T\vert +> > T-1)$. +> +> `time_*` +> +> > Stores the vintages of $k$-step out-of-sample forecast shock +> > decompositions, i.e. $Y(t\vert +> > T)$, for $t=[T \ldots T+k]$. See `vintage +> > `{.interpreted-text role="opt"}. +::: +::: + +::: {.command} +plot\_shock\_decomposition \[VARIABLE\_NAME\]\...; +plot\_shock\_decomposition (OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command plots the historical shock decomposition already computed +by `shock_decomposition` or `realtime_shock_decomposition`. For that +reason, it must come after one of these commands. The `variable_names` +provided govern which variables the decomposition is plotted for. + +Further note that, unlike the majority of Dynare commands, the options +specified below are overwritten with their defaults before every call to +`plot_shock_decomposition`. Hence, if you want to reuse an option in a +subsequent call to `plot_shock_decomposition`, you must pass it to the +command again. + +*Options* + +::: {.option} +use\_shock\_groups \[= NAME\] + +See `use_shock_groups `{.interpreted-text +role="opt"}. +::: + +::: {.option} +colormap = VARIABLE\_NAME + +See `colormap `{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See `graph_format `{.interpreted-text +role="opt"}. +::: + +::: {.option} +detail\_plot + +Plots shock contributions using subplots, one per shock (or group of +shocks). Default: not activated +::: + +::: {.option} +interactive + +Under MATLAB, add uimenus for detailed group plots. Default: not +activated +::: + +::: {.option} +screen\_shocks + +For large models (i.e. for models with more than 16 shocks), plots only +the shocks that have the largest historical contribution for chosen +selected `variable_names`. Historical contribution is ranked by the mean +absolute value of all historical contributions. +::: + +::: {.option} +steadystate + +If passed, the the $y$-axis value of the zero line in the shock +decomposition plot is translated to the steady state level. Default: not +activated +::: + +::: {.option} +type = qoq \| yoy \| aoa + +For quarterly data, valid arguments are: `qoq` for quarter-on-quarter +plots, `yoy` for year-on-year plots of growth rates, `aoa` for +annualized variables, i.e. the value in the last quarter for each year +is plotted. Default value: empty, i.e. standard period-on-period plots +(`qoq` for quarterly data). +::: + +::: {.option} +fig\_name = STRING + +Specifies a user-defined keyword to be appended to the default figure +name set by `plot_shock_decomposition`. This can avoid to overwrite +plots in case of sequential calls to `plot_shock_decomposition`. +::: + +::: {.option} +write\_xls + +Saves shock decompositions to Excel-file in the main directory, named +`FILENAME_shock_decomposition_TYPE_FIG_NAME.xls`. This option requires +your system to be configured to be able to write Excel files.[^8] +::: + +::: {.option} +realtime = INTEGER + +Which kind of shock decomposition to plot. INTEGER can take the +following values: + +> - `0`: standard historical shock decomposition. See +> `shock_decomposition`{.interpreted-text role="comm"}. +> - `1`: realtime historical shock decomposition. See +> `realtime_shock_decomposition`{.interpreted-text role="comm"}. +> - `2`: conditional realtime shock decomposition. See +> `realtime_shock_decomposition`{.interpreted-text role="comm"}. +> - `3`: realtime forecast shock decomposition. See +> `realtime_shock_decomposition`{.interpreted-text role="comm"}. + +If no vintage is requested, i.e. `vintage=0` then the pooled objects +from `realtime_shock_decomposition`{.interpreted-text role="comm"} will +be plotted and the respective vintage otherwise. Default: `0`. +::: + +::: {.option} +vintage = INTEGER + +Selects a particular data vintage in $[presample,\ldots,nobs]$ for which +to plot the results from +`realtime_shock_decomposition`{.interpreted-text role="comm"} selected +via the `realtime `{.interpreted-text role="opt"} +option. If the standard historical shock decomposition is selected +(`realtime=0`), `vintage` will have no effect. If `vintage=0` the pooled +objects from `realtime_shock_decomposition`{.interpreted-text +role="comm"} will be plotted. If `vintage>0`, it plots the shock +decompositions for vintage $T=\texttt{vintage}$ under the following +scenarios: + +> - `realtime=1`: the full vintage shock decomposition $Y(t\vert T)$ +> for $t=[1,\ldots,T]$ +> - `realtime=2`: the conditional forecast shock decomposition from +> $T$, i.e. plots $Y(T+j\vert T+j)$ and the shock contributions +> needed to get to the data $Y(T+j)$ conditional on $T=$ vintage, +> with $j=[0,\ldots,\texttt{forecast}]$. +> - `realtime=3`: plots unconditional forecast shock decomposition +> from $T$, i.e. $Y(T+j\vert +> T)$, where $T=\texttt{vintage}$ and +> $j=[0,\ldots,\texttt{forecast}]$. + +Default: `0`. +::: + +::: {.option} +plot\_init\_date = DATE + +If passed, plots decomposition using `plot_init_date` as initial period. +Default: first observation in estimation +::: + +::: {.option} +plot\_end\_date = DATE + +If passed, plots decomposition using `plot_end_date` as last period. +Default: last observation in estimation +::: + +::: {.option} +diff + +If passed, plot the decomposition of the first difference of the list of +variables. If used in combination with `flip`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: + +::: {.option} +flip + +If passed, plot the decomposition of the opposite of the list of +variables. If used in combination with `diff`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: + +::: {.option} +max\_nrows + +Maximum number of rows in the subplot layout of detailed shock +decomposition graphs. Note that columns are always 3. Default: 6 +::: + +::: {.option} +with\_epilogue + +See `with_epilogue`{.interpreted-text role="opt"}. +::: + +::: {.option} +init2shocks init2shocks = NAME + +Use the information contained in an `init2shocks`{.interpreted-text +role="bck"} block, in order to attribute initial conditions to shocks. +The name of the block can be explicitly given, otherwise it defaults to +the `default` block. +::: +::: + +::: {.block} +init2shocks ; init2shocks (OPTIONS\...); + +This blocks gives the possibility of attributing the initial condition +of endogenous variables to the contribution of exogenous variables in +the shock decomposition. + +For example, in an AR(1) process, the contribution of the initial +condition on the process variable can naturally be assigned to the +innovation of the process. + +Each line of the block should have the syntax: + + VARIABLE_1 [,] VARIABLE_2; + +Where VARIABLE\_1 is an endogenous variable whose initial condition will +be attributed to the exogenous VARIABLE\_2. + +The information contained in this block is used by the +`plot_shock_decomposition`{.interpreted-text role="comm"} command when +given the `init2shocks` option. + +*Options* + +::: {.option} +name = NAME + +Specifies a name for the block, that can be referenced from +`plot_shock_decomposition`, so that several such blocks can coexist in a +single model file. If the name is unspecified, it defaults to `default`. +::: + +*Example* + +> var y y_s R pie dq pie_s de A y_obs pie_obs R_obs; +> varexo e_R e_q e_ys e_pies e_A; +> ... +> +> model; +> dq = rho_q*dq(-1)+e_q; +> A = rho_A*A(-1)+e_A; +> ... +> end; +> +> ... +> +> init2shocks; +> dq e_q; +> A e_A; +> end; +> +> shock_decomposition(nograph); +> +> plot_shock_decomposition(init2shocks) y_obs R_obs pie_obs dq de; +> +> In this example, the initial conditions of `dq` and `A` will be +> respectively attributed to `e_q` and `e_A`. +::: + +::: {.command} +initial\_condition\_decomposition \[VARIABLE\_NAME\]\...; +initial\_condition\_decomposition (OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command computes and plots the decomposition of the effect of +smoothed initial conditions of state variables. The `variable_names` +provided govern which variables the decomposition is plotted for. + +Further note that, unlike the majority of Dynare commands, the options +specified below are overwritten with their defaults before every call to +`initial_condition_decomposition`. Hence, if you want to reuse an option +in a subsequent call to `initial_condition_decomposition`, you must pass +it to the command again. + +*Options* + +::: {.option} +colormap = VARIABLE\_NAME + +See `colormap `{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See `graph_format `{.interpreted-text +role="opt"}. +::: + +::: {.option} +detail\_plot + +Plots shock contributions using subplots, one per shock (or group of +shocks). Default: not activated +::: + +::: {.option} +steadystate + +If passed, the the $y$-axis value of the zero line in the shock +decomposition plot is translated to the steady state level. Default: not +activated +::: + +::: {.option} +type = qoq \| yoy \| aoa + +For quarterly data, valid arguments are: `qoq` for quarter-on-quarter +plots, `yoy` for year-on-year plots of growth rates, `aoa` for +annualized variables, i.e. the value in the last quarter for each year +is plotted. Default value: empty, i.e. standard period-on-period plots +(`qoq` for quarterly data). +::: + +::: {.option} +fig\_name = STRING + +Specifies a user-defined keyword to be appended to the default figure +name set by `plot_shock_decomposition`. This can avoid to overwrite +plots in case of sequential calls to `plot_shock_decomposition`. +::: + +::: {.option} +write\_xls + +Saves shock decompositions to Excel-file in the main directory, named +`FILENAME_shock_decomposition_TYPE_FIG_NAME_initval.xls`. This option +requires your system to be configured to be able to write Excel +files.[^9] +::: + +::: {.option} +plot\_init\_date = DATE + +If passed, plots decomposition using `plot_init_date` as initial period. +Default: first observation in estimation +::: + +::: {.option} +plot\_end\_date = DATE + +If passed, plots decomposition using `plot_end_date` as last period. +Default: last observation in estimation +::: + +::: {.option} +diff + +If passed, plot the decomposition of the first difference of the list of +variables. If used in combination with `flip`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: + +::: {.option} +flip + +If passed, plot the decomposition of the opposite of the list of +variables. If used in combination with `diff`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: +::: + +::: {.command} +squeeze\_shock\_decomposition \[VARIABLE\_NAME\]\...; + +For large models, the size of the information stored by shock +decompositions (especially various settings of realtime decompositions) +may become huge. This command allows to squeeze this information in two +possible ways: + +> - Automatic (default): only the variables for which plotting has +> been explicitly required with `plot_shock_decomposition` will have +> their decomposition left in `oo_` after this command is run; +> - If a list of variables is passed to the command, then only those +> variables will have their decomposition left in `oo_` after this +> command is run. +::: + +Calibrated Smoother +------------------- + +Dynare can also run the smoother on a calibrated model: + +::: {.command} +calib\_smoother \[VARIABLE\_NAME\]\...; calib\_smoother (OPTIONS\...) +\[VARIABLE\_NAME\]\...; + +This command computes the smoothed variables (and possible the filtered +variables) on a calibrated model. + +A datafile must be provided, and the observable variables declared with +`varobs`. The smoother is based on a first-order approximation of the +model. + +By default, the command computes the smoothed variables and shocks and +stores the results in `oo_.SmoothedVariables` and `oo_.SmoothedShocks`. +It also fills `oo_.UpdatedVariables`. + +*Options* + +::: {.option} +datafile = FILENAME + +See `datafile `{.interpreted-text role="ref"}. +::: + +::: {.option} +filtered\_vars + +Triggers the computation of filtered variables. See +`filtered_vars`{.interpreted-text role="opt"}, for more details. +::: + +::: {.option} +filter\_step\_ahead = \[INTEGER1:INTEGER2\] + +See +`filter_step_ahead `{.interpreted-text +role="opt"}. +::: + +::: {.option} +prefilter = INTEGER + +See `prefilter `{.interpreted-text role="opt"}. +::: + +::: {.option} +parameter\_set = OPTION + +See `parameter_set `{.interpreted-text +role="opt"} for possible values. Default: `calibration`. +::: + +::: {.option} +loglinear + +See `loglinear `{.interpreted-text role="ref"}. +::: + +::: {.option} +first\_obs = INTEGER + +See `first_obs `{.interpreted-text role="opt"}. +::: + +::: {.option} +filter\_decomposition + +See `filter_decomposition`{.interpreted-text role="opt"}. +::: + +::: {.option} +filter\_covariance + +See `filter_covariance`{.interpreted-text role="opt"}. +::: + +::: {.option} +smoother\_redux + +See `smoother_redux`{.interpreted-text role="opt"}. +::: + +::: {.option} +kalman\_algo = INTEGER + +See `kalman_algo `{.interpreted-text role="opt"}. +::: + +::: {.option} +diffuse\_filter = INTEGER + +See `diffuse_filter`{.interpreted-text role="opt"}. +::: + +::: {.option} +diffuse\_kalman\_tol = DOUBLE + +See `diffuse_kalman_tol `{.interpreted-text +role="opt"}. +::: + +::: {.option} +xls\_sheet = QUOTED\_STRING + +See `xls_sheet `{.interpreted-text +role="opt"}. +::: + +::: {.option} +xls\_range = RANGE + +See `xls_range `{.interpreted-text role="opt"}. +::: +::: + +Forecasting {#fore} +----------- + +On a calibrated model, forecasting is done using the `forecast` command. +On an estimated model, use the `forecast` option of `estimation` +command. + +It is also possible to compute forecasts on a calibrated or estimated +model for a given constrained path of the future endogenous variables. +This is done, from the reduced form representation of the DSGE model, by +finding the structural shocks that are needed to match the restricted +paths. Use `conditional_forecast`{.interpreted-text role="comm"}, +`conditional_forecast_paths`{.interpreted-text role="bck"} and +`plot_conditional_forecast`{.interpreted-text role="comm"} for that +purpose. + +Finally, it is possible to do forecasting with a Bayesian VAR using the +`bvar_forecast`{.interpreted-text role="comm"} command. + +::: {.command} +forecast \[VARIABLE\_NAME\...\]; forecast (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +This command computes a simulation of a stochastic model from an +arbitrary initial point. + +When the model also contains deterministic exogenous shocks, the +simulation is computed conditionally to the agents knowing the future +values of the deterministic exogenous variables. + +`forecast` must be called after `stoch_simul`. + +`forecast` plots the trajectory of endogenous variables. When a list of +variable names follows the command, only those variables are plotted. A +90% confidence interval is plotted around the mean trajectory. Use +option `conf_sig` to change the level of the confidence interval. + +*Options* + +::: {.option} +periods = INTEGER + +Number of periods of the forecast. Default: `5`. +::: + +::: {#confsig} +::: {.option} +conf\_sig = DOUBLE + +Level of significance for confidence interval. Default: `0.90`. +::: +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See `graph_format = FORMAT`{.interpreted-text role="opt"}. +::: + +*Initial Values* + +`forecast` computes the forecast taking as initial values the values +specified in `histval` (see `histval`{.interpreted-text role="bck"}). +When no `histval` block is present, the initial values are the one +stated in `initval`. When `initval` is followed by command `steady`, the +initial values are the steady state (see `steady`{.interpreted-text +role="comm"}). + +*Output* + +The results are stored in `oo_.forecast`, which is described below. + +*Example* + +> varexo_det tau; +> +> varexo e; +> ... +> shocks; +> var e; stderr 0.01; +> var tau; +> periods 1:9; +> values -0.15; +> end; +> +> stoch_simul(irf=0); +> +> forecast; + +::: {.matvar} +[oo]().forecast + +Variable set by the `forecast` command, or by the `estimation` command +if used with the `forecast` option and ML or if no Metropolis-Hastings +has been computed (in that case, the forecast is computed for the +posterior mode). Fields are of the form: + + oo_.forecast.FORECAST_MOMENT.VARIABLE_NAME + +where `FORECAST_MOMENT` is one of the following: + +> `HPDinf` +> +> > Lower bound of a 90% HPD interval[^10] of forecast due to parameter +> > uncertainty, but ignoring the effect of measurement error on +> > observed variables. In case of ML, it stores the lower bound of the +> > confidence interval. +> +> `HPDsup` +> +> > Upper bound of a 90% HPD forecast interval due to parameter +> > uncertainty, but ignoring the effect of measurement error on +> > observed variables. In case of ML, it stores the upper bound of the +> > confidence interval. +> +> `HPDinf_ME` +> +> > Lower bound of a 90% HPD interval[^11] of forecast for observed +> > variables due to parameter uncertainty and measurement error. In +> > case of ML, it stores the lower bound of the confidence interval. +> +> `HPDsup_ME` +> +> > Upper bound of a 90% HPD interval of forecast for observed variables +> > due to parameter uncertainty and measurement error. In case of ML, +> > it stores the upper bound of the confidence interval. +> +> `Mean` +> +> > Mean of the posterior distribution of forecasts. +::: + +::: {.matvar} +[oo]().PointForecast + +Set by the `estimation` command, if it is used with the `forecast` +option and if either `mh_replic > 0` or the `load_mh_file` option are +used. + +Contains the distribution of forecasts taking into account the +uncertainty about both parameters and shocks. + +Fields are of the form: + + oo_.PointForecast.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().MeanForecast + +Set by the `estimation` command, if it is used with the `forecast` +option and if either `mh_replic > 0` or `load_mh_file` option are used. + +Contains the distribution of forecasts where the uncertainty about +shocks is averaged out. The distribution of forecasts therefore only +represents the uncertainty about parameters. + +Fields are of the form: + + oo_.MeanForecast.MOMENT_NAME.VARIABLE_NAME +::: +::: + +::: {.command} +conditional\_forecast (OPTIONS\...); + +This command computes forecasts on an estimated or calibrated model for +a given constrained path of some future endogenous variables. This is +done using the reduced form first order state-space representation of +the DSGE model by finding the structural shocks that are needed to match +the restricted paths. Consider the augmented state space representation +that stacks both predetermined and non-predetermined variables into a +vector $y_{t}$: + +> $$y_t=Ty_{t-1}+R\varepsilon_t$$ + +Both $y_t$ and $\varepsilon_t$ are split up into controlled and +uncontrolled ones, and we assume without loss of generality that the +constrained endogenous variables and the controlled shocks come first : + +> $$\begin{aligned} +> \begin{pmatrix} +> y_{c,t}\\ +> y_{u,t} +> \end{pmatrix} +> = +> \begin{pmatrix} +> T_{c,c} & T_{c,u}\\ +> T_{u,c} & T_{u,u} +> \end{pmatrix} +> \begin{pmatrix} +> y_{c,t-1}\\ +> y_{u,t-1} +> \end{pmatrix} +> + +> \begin{pmatrix} +> R_{c,c} & R_{c,u}\\ +> R_{u,c} & R_{u,u} +> \end{pmatrix} +> \begin{pmatrix} +> \varepsilon_{c,t}\\ +> \varepsilon_{u,t} +> \end{pmatrix} +> \end{aligned}$$ + +where matrices $T$ and $R$ are partitioned consistently with the vectors +of endogenous variables and innovations. Provided that matrix $R_{c,c}$ +is square and full rank (a necessary condition is that the number of +free endogenous variables matches the number of free innovations), given +$y_{c,t}$, $\varepsilon_{u,t}$ and $y_{t-1}$ the first block of +equations can be solved for $\varepsilon_{c,t}$: + +> $$\varepsilon_{c,t} = R_{c,c}^{-1}\bigl( y_{c,t} - T_{c,c}y_{c,t} - T_{c,u}y_{u,t} - R_{c,u}\varepsilon_{u,t}\bigr)$$ + +and $y_{u,t}$ can be updated by evaluating the second block of +equations: + +> $$y_{u,t} = T_{u,c}y_{c,t-1} + T_{u,u}y_{u,t-1} + R_{u,c}\varepsilon_{c,t} + R_{u,u}\varepsilon_{u,t}$$ + +By iterating over these two blocks of equations, we can build a forecast +for all the endogenous variables in the system conditional on paths for +a subset of the endogenous variables. If the distribution of the free +innovations $\varepsilon_{u,t}$ is provided (*i.e.* some of them have +positive variances) this exercise is replicated (the number of +replication is controlled by the option `replic`{.interpreted-text +role="opt"} described below) by drawing different sequences of free +innovations. The result is a predictive distribution for the +uncontrolled endogenous variables, $y_{u,t}$, that Dynare will use to +report confidence bands around the point conditional forecast. + +A few things need to be noted. First, the controlled exogenous variables +are set to zero for the uncontrolled periods. This implies that there is +no forecast uncertainty arising from these exogenous variables in +uncontrolled periods. Second, by making use of the first order state +space solution, even if a higher-order approximation was performed, the +conditional forecasts will be based on a first order approximation. +Since the controlled exogenous variables are identified on the basis of +the reduced form model (*i.e.* after solving for the expectations), they +are unforeseen shocks from the perspective of the agents in the model. +That is, agents expect the endogenous variables to return to their +respective steady state levels but are surprised in each period by the +realisation of shocks keeping the endogenous variables along a +predefined (unexpected) path. Fourth, if the structural innovations are +correlated, because the calibrated or estimated covariance matrix has +non zero off diagonal elements, the results of the conditional forecasts +will depend on the ordering of the innovations (as declared after +`varexo`). As in VAR models, a Cholesky decomposition is used to +factorise the covariance matrix and identify orthogonal impulses. It is +preferable to declare the correlations in the model block (explicitly +imposing the identification restrictions), unless you are satisfied with +the implicit identification restrictions implied by the Cholesky +decomposition. + +This command has to be called after `estimation` or `stoch_simul`. + +Use `conditional_forecast_paths`{.interpreted-text role="bck"} block to +give the list of constrained endogenous, and their constrained future +path. Option `controlled_varexo` is used to specify the structural +shocks which will be matched to generate the constrained path. + +Use `plot_conditional_forecast`{.interpreted-text role="comm"} to graph +the results. + +*Options* + +::: {.option} +parameter\_set = OPTION + +See `parameter_set `{.interpreted-text +role="opt"} for possible values. No default value, mandatory option. +::: + +::: {.option} +controlled\_varexo = (VARIABLE\_NAME\...) + +Specify the exogenous variables to use as control variables. No default +value, mandatory option. +::: + +::: {.option} +periods = INTEGER + +Number of periods of the forecast. Default: `40`. `periods` cannot be +smaller than the number of constrained periods. +::: + +::: {.option} +replic = INTEGER + +Number of simulations used to compute the conditional forecast +uncertainty. Default: `5000`. +::: + +::: {.option} +conf\_sig = DOUBLE + +Level of significance for confidence interval. Default: `0.80`. +::: + +*Output* + +The results are stored in `oo_.conditional_forecast`, which is described +below. + +*Example* + +> var y a; +> varexo e u; +> ... +> estimation(...); +> +> conditional_forecast_paths; +> var y; +> periods 1:3, 4:5; +> values 2, 5; +> var a; +> periods 1:5; +> values 3; +> end; +> +> conditional_forecast(parameter_set = calibration, controlled_varexo = (e, u), replic = 3000); +> +> plot_conditional_forecast(periods = 10) a y; + +::: {.matvar} +[oo]().conditional\_forecast.cond + +Variable set by the `conditional_forecast` command. It stores the +conditional forecasts. Fields are `periods+1` by `1` vectors storing the +steady state (time 0) and the subsequent `periods` forecasts periods. +Fields are of the form: + + oo_.conditional_forecast.cond.FORECAST_MOMENT.VARIABLE_NAME + +where FORECAST\_MOMENT is one of the following: + +> `Mean` +> +> > Mean of the conditional forecast distribution. +> +> `ci` +> +> > Confidence interval of the conditional forecast distribution. The +> > size corresponds to `conf_sig`. +::: + +::: {.matvar} +[oo]().conditional\_forecast.uncond + +Variable set by the `conditional_forecast` command. It stores the +unconditional forecasts. Fields are of the form: + + oo_.conditional_forecast.uncond.FORECAST_MOMENT.VARIABLE_NAME +::: + +::: {.matvar} +forecasts.instruments + +Variable set by the `conditional_forecast command`. Stores the names of +the exogenous instruments. +::: + +::: {.matvar} +[oo]().conditional\_forecast.controlled\_variables + +Variable set by the `conditional_forecast` command. Stores the position +of the constrained endogenous variables in declaration order. +::: + +::: {.matvar} +[oo]().conditional\_forecast.controlled\_exo\_variables + +Variable set by the `conditional_forecast` command. Stores the values of +the controlled exogenous variables underlying the conditional forecasts +to achieve the constrained endogenous variables. Fields are +`[number of constrained periods]` by `1` vectors and are of the form: + + oo_.conditional_forecast.controlled_exo_variables.FORECAST_MOMENT.SHOCK_NAME +::: + +::: {.matvar} +[oo]().conditional\_forecast.graphs + +Variable set by the `conditional_forecast` command. Stores the +information for generating the conditional forecast plots. +::: +::: + +::: {.block} +conditional\_forecast\_paths ; + +Describes the path of constrained endogenous, before calling +`conditional_forecast`. The syntax is similar to deterministic shocks in +`shocks`, see `conditional_forecast` for an example. + +The syntax of the block is the same as for the deterministic shocks in +the `shocks` blocks (see `shocks-exo`{.interpreted-text role="ref"}). +Note that you need to specify the full path for all constrained +endogenous variables between the first and last specified period. If an +intermediate period is not specified, a value of 0 is assumed. That is, +if you specify only values for periods 1 and 3, the values for period 2 +will be 0. Currently, it is not possible to have uncontrolled +intermediate periods. + +It is however possible to have different number of controlled periods +for different variables. In that case, the order of declaration of +endogenous controlled variables and of `controlled_varexo` matters: if +the second endogenous variable is controlled for less periods than the +first one, the second `controlled_varexo` isn\'t set for the last +periods. + +In case of the presence of `observation_trends`, the specified +controlled path for these variables needs to include the trend +component. When using the `loglinear `{.interpreted-text +role="ref"} option, it is necessary to specify the logarithm of the +controlled variables. +::: + +::: {.block} +filter\_initial\_state ; + +This block specifies the initial values of the endogenous states at the +beginning of the Kalman filter recursions. That is, if the Kalman filter +recursion starts with time t=1 being the first observation, this block +provides the state estimate at time 0 given information at time 0, +$E_0(x_0)$. If nothing is specified, the initial condition is assumed to +be at the steady state (which is the unconditional mean for a stationary +model). + +This block is terminated by `end;`. + +Each line inside of the block should be of the form: + + VARIABLE_NAME(INTEGER)=EXPRESSION; + +`EXPRESSION` is any valid expression returning a numerical value and can +contain parameter values. This allows specifying relationships that will +be honored during estimation. `INTEGER` refers to the lag with which a +variable appears. By convention in Dynare, period 1 is the first period. +Going backwards in time, the first period before the start of the +simulation is period 0, then period -1, and so on. Note that the +`filter_initial_state` block does not take non-state variables. + +*Example* + +> filter_initial_state; +> k(0)= ((1/bet-(1-del))/alp)^(1/(alp-1))*l_ss; +> P(0)=2.5258; +> m(0)= mst; +> end; +::: + +::: {.command} +plot\_conditional\_forecast \[VARIABLE\_NAME\...\]; +plot\_conditional\_forecast (periods = INTEGER) \[VARIABLE\_NAME\...\]; + +Plots the conditional (plain lines) and unconditional (dashed lines) +forecasts. + +To be used after `conditional_forecast`. + +*Options* + +::: {.option} +periods = INTEGER + +Number of periods to be plotted. Default: equal to periods in +`conditional_forecast`. The number of periods declared in +`plot_conditional_forecast` cannot be greater than the one declared in +`conditional_forecast`. +::: +::: + +::: {.command} +bvar\_forecast ; + +This command computes (out-of-sample) forecasts for an estimated BVAR +model, using Minnesota priors. + +See `bvar-a-la-sims.pdf`, which comes with Dynare distribution, for more +information on this command. +::: + +If the model contains strong non-linearities or if some perfectly +expected shocks are considered, the forecasts and the conditional +forecasts can be computed using an extended path method. The forecast +scenario describing the shocks and/or the constrained paths on some +endogenous variables should be build. The first step is the forecast +scenario initialization using the function `init_plan`: + +::: {.matcomm} +HANDLE = init\_plan (DATES); + +Creates a new forecast scenario for a forecast period (indicated as a +dates class, see `dates class members +`{.interpreted-text role="ref"}). This function return a +handle on the new forecast scenario. +::: + +The forecast scenario can contain some simple shocks on the exogenous +variables. This shocks are described using the function `basic_plan`: + +::: {.matcomm} +HANDLE = basic\_plan (HANDLE, \`VAR\_NAME\', \`SHOCK\_TYPE\', DATES, +MATLAB VECTOR OF DOUBLE \| \[DOUBLE \| EXPR \[DOUBLE \| EXPR\] \] ); + +Adds to the forecast scenario a shock on the exogenous variable +indicated between quotes in the second argument. The shock type has to +be specified in the third argument between quotes: 'surprise' in case of +an unexpected shock or 'perfect\_foresight' for a perfectly anticipated +shock. The fourth argument indicates the period of the shock using a +dates class (see `dates class +members `{.interpreted-text role="ref"}). The last +argument is the shock path indicated as a MATLAB vector of double. This +function return the handle of the updated forecast scenario. +::: + +The forecast scenario can also contain a constrained path on an +endogenous variable. The values of the related exogenous variable +compatible with the constrained path are in this case computed. In other +words, a conditional forecast is performed. This kind of shock is +described with the function `flip_plan`: + +::: {.matcomm} +HANDLE = flip\_plan (HANDLE, \`VAR\_NAME\', \`VAR\_NAME\', +\`SHOCK\_TYPE\', DATES, MATLAB VECTOR OF DOUBLE \| \[DOUBLE \| EXPR +\[DOUBLE \| EXPR\] \] ); + +Adds to the forecast scenario a constrained path on the endogenous +variable specified between quotes in the second argument. The associated +exogenous variable provided in the third argument between quotes, is +considered as an endogenous variable and its values compatible with the +constrained path on the endogenous variable will be computed. The nature +of the expectation on the constrained path has to be specified in the +fourth argument between quotes: 'surprise' in case of an unexpected path +or 'perfect\_foresight' for a perfectly anticipated path. The fifth +argument indicates the period where the path of the endogenous variable +is constrained using a dates class (see `dates class +members `{.interpreted-text role="ref"}). The last +argument contains the constrained path as a MATLAB vector of double. +This function return the handle of the updated forecast scenario. +::: + +Once the forecast scenario if fully described, the forecast is computed +with the command `det_cond_forecast`: + +::: {.matcomm} +DSERIES = det\_cond\_forecast (HANDLE\[, DSERIES \[, DATES\]\]); + +Computes the forecast or the conditional forecast using an extended path +method for the given forecast scenario (first argument). The past values +of the endogenous and exogenous variables provided with a dseries class +(see `dseries class +members `{.interpreted-text role="ref"}) can be +indicated in the second argument. By default, the past values of the +variables are equal to their steady-state values. The initial date of +the forecast can be provided in the third argument. By default, the +forecast will start at the first date indicated in the +`init_plan command`. This function returns a dset containing the +historical and forecast values for the endogenous and exogenous +variables. +::: + +*Example* + +> % conditional forecast using extended path method +> % with perfect foresight on r path +> +> var y r; +> varexo e u; +> ... +> smoothed = dseries('smoothed_variables.csv'); +> +> fplan = init_plan(2013Q4:2029Q4); +> fplan = flip_plan(fplan, 'y', 'u', 'surprise', 2013Q4:2014Q4, [1 1.1 1.2 1.1 ]); +> fplan = flip_plan(fplan, 'r', 'e', 'perfect_foresight', 2013Q4:2014Q4, [2 1.9 1.9 1.9 ]); +> +> dset_forecast = det_cond_forecast(fplan, smoothed); +> +> plot(dset_forecast.{'y','u'}); +> plot(dset_forecast.{'r','e'}); + +::: {.command} +smoother2histval ; smoother2histval(OPTIONS\...); + +The purpose of this command is to construct initial conditions (for a +subsequent simulation) that are the smoothed values of a previous +estimation. + +More precisely, after an estimation run with the `smoother` option, +`smoother2histval` will extract the smoothed values (from +`oo_.SmoothedVariables`, and possibly from `oo_.SmoothedShocks` if there +are lagged exogenous), and will use these values to construct initial +conditions (as if they had been manually entered through `histval`). + +*Options* + +::: {.option} +period = INTEGER + +Period number to use as the starting point for the subsequent +simulation. It should be between 1 and the number of observations that +were used to produce the smoothed values. Default: the last observation. +::: + +::: {.option} +infile = FILENAME + +Load the smoothed values from a `_results.mat` file created by a +previous Dynare run. Default: use the smoothed values currently in the +global workspace. +::: + +::: {.option} +invars = ( VARIABLE\_NAME \[VARIABLE\_NAME \...\] ) + +A list of variables to read from the smoothed values. It can contain +state endogenous variables, and also exogenous variables having a lag. +Default: all the state endogenous variables, and all the exogenous +variables with a lag. +::: + +::: {.option} +outfile = FILENAME + +Write the initial conditions to a file. Default: write the initial +conditions in the current workspace, so that a simulation can be +performed. +::: + +::: {.option} +outvars = ( VARIABLE\_NAME \[VARIABLE\_NAME \...\] ) + +A list of variables which will be given the initial conditions. This +list must have the same length than the list given to `invars`, and +there will be a one-to-one mapping between the two list. Default: same +value as option `invars`. +::: + +*Use cases* + +There are three possible ways of using this command: + +> - Everything in a single file: run an estimation with a smoother, +> then run `smoother2histval` (without the `infile` and `outfile` +> options), then run a stochastic simulation. +> - In two files: in the first file, run the smoother and then run +> `smoother2histval` with the `outfile` option; in the second file, +> run `histval_file` to load the initial conditions, and run a +> (deterministic or stochastic) simulation. +> - In two files: in the first file, run the smoother; in the second +> file, run `smoother2histval` with the `infile` option equal to the +> `_results.mat` file created by the first file, and then run a +> (deterministic or stochastic) simulation. +::: + +Optimal policy +-------------- + +Dynare has tools to compute optimal policies for various types of +objectives. You can either solve for optimal policy under commitment +with `ramsey_model`, for optimal policy under discretion with +`discretionary_policy` or for optimal simple rules with `osr` (also +implying commitment). + +::: {.command} +planner\_objective MODEL\_EXPRESSION ; + +This command declares the policy maker objective, for use with +`ramsey_model` or `discretionary_policy`. + +You need to give the one-period objective, not the discounted lifetime +objective. The discount factor is given by the `planner_discount` option +of `ramsey_model` and `discretionary_policy`. The objective function can +only contain current endogenous variables and no exogenous ones. This +limitation is easily circumvented by defining an appropriate auxiliary +variable in the model. + +With `ramsey_model`, you are not limited to quadratic objectives: you +can give any arbitrary nonlinear expression. + +With `discretionary_policy`, the objective function must be quadratic. +::: + +::: {.command} +evaluate\_planner\_objective; evaluate\_planner\_objective +(OPTIONS\...); + +This command computes, displays, and stores the value of the planner +objective function under Ramsey policy or discretion in +`oo_.planner_objective_value`. It will provide both unconditional +welfare and welfare conditional on the initial (i.e. period 0) values of +the endogenous and exogenous state variables inherited by the planner. +In a deterministic context, the respective initial values are set using +`initval` or `histval` (depending on the exact context). + +In a stochastic context, if no initial state values have been specified +with `histval`, their values are taken to be the steady state values. +Because conditional welfare is computed conditional on optimal policy by +the planner in the first endogenous period (period 1), it is conditional +on the information set in the period 1. This information set includes +both the predetermined states inherited from period 0 (specified via +`histval` for both endogenous and lagged exogenous states) as well as +the period 1 values of the exogenous shocks. The latter are specified +using the perfect foresight syntax of the shocks-block. + +At the current stage, the stochastic context does not support the +`pruning` option. At `order>3`, only the computation of conditional +welfare with steady state Lagrange multipliers is supported. Note that +at [order=2]{.title-ref}, the output is based on the second-order +accurate approximation of the variance stored in [oo\_.var]{.title-ref}. + +*Options* + +::: {.option} +periods = INTEGER + +The value of the option specifies the number of periods to use in the +simulations in the computation of unconditional welfare at higher order. + +Default: `10000`. +::: + +::: {.option} +drop = INTEGER + +The number of burn-in draws out of `periods` discarded before computing +the unconditional welfare at higher order. Default: `1000`. +::: + +*Example (stochastic context)* + +> var a ...; +> varexo u; +> +> model; +> a = rho*a(-1)+u+u(-1); +> ... +> end; +> +> histval; +> u(0)=1; +> a(0)=-1; +> end; +> +> shocks; +> var u; stderr 0.008; +> var u; +> periods 1; +> values 1; +> end; +> +> evaluate_planner_objective; + +::: {.matvar} +[oo]().planner\_objective\_value.unconditional +::: + +Scalar storing the value of unconditional welfare. In a perfect +foresight context, it corresponds to welfare in the long-run, +approximated as welfare in the terminal simulation period. + +::: {.matvar} +[oo]().planner\_objective\_value.conditional +::: + +In a perfect foresight context, this field will be a scalar storing the +value of welfare conditional on the specified initial condition and zero +initial Lagrange multipliers. + +In a stochastic context, it will have two subfields: + +::: {.matvar} +[oo]().planner\_objective\_value.conditional.steady\_initial\_multiplier +::: + +Stores the value of the planner objective when the initial Lagrange +multipliers associated with the planner's problem are set to their +steady state values (see `ramsey_policy`{.interpreted-text +role="comm"}). + +::: {.matvar} +[oo]().planner\_objective\_value.conditional.zero\_initial\_multiplier +::: + +Stores the value of the planner objective when the initial Lagrange +multipliers associated with the planner's problem are set to 0, i.e. it +is assumed that the planner exploits its ability to surprise private +agents in the first period of implementing Ramsey policy. This value +corresponds to the planner implementing optimal policy for the first +time and committing not to re-optimize in the future. +::: + +### Optimal policy under commitment (Ramsey) + +Dynare allows to automatically compute optimal policy choices of a +Ramsey planner who takes the specified private sector equilibrium +conditions into account and commits to future policy choices. Doing so +requires specifying the private sector equilibrium conditions in the +`model`-block and a `planner_objective` as well as potentially some +`instruments` to facilitate computations. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +Be careful when employing forward-looking auxiliary variables in the +context of timeless perspective Ramsey computations. They may alter the +problem the Ramsey planner will solve for the first period, although +they seemingly leave the private sector equilibrium unaffected. The +reason is the planner optimizes with respect to variables dated `t` and +takes the value of time 0 variables as given, because they are +predetermined. This set of initially predetermined variables will change +with forward-looking definitions. Thus, users are strongly advised to +use model-local variables instead. + +*Example* + +> Consider a perfect foresight example where the Euler equation for the +> return to capital is given by +> +> 1/C=beta*1/C(+1)*(R(+1)+(1-delta)) +> +> The job of the Ramsey planner in period `1` is to choose $C_1$ and +> $R_1$, taking as given $C_0$. The above equation may seemingly +> equivalently be written as +> +> 1/C=beta*1/C(+1)*(R_cap); +> R_cap=R(+1)+(1-delta); +> +> due to perfect foresight. However, this changes the problem of the +> Ramsey planner in the first period to choosing $C_1$ and $R_1$, taking +> as given both $C_0$ and $R^{cap}_0$. Thus, the relevant return to +> capital in the Euler equation of the first period is not a choice of +> the planner anymore due to the forward-looking nature of the +> definition in the second line! +> +> A correct specification would be to instead define `R_cap` as a +> model-local variable: +> +> 1/C=beta*1/C(+1)*(R_cap); +> #R_cap=R(+1)+(1-delta); +::: + +::: {.command} +ramsey\_model (OPTIONS\...); + +This command computes the First Order Conditions for maximizing the +policy maker objective function subject to the constraints provided by +the equilibrium path of the private economy. + +The planner objective must be declared with the +`planner_objective`{.interpreted-text role="comm"} command. + +This command only creates the expanded model, it doesn't perform any +computations. It needs to be followed by other instructions to actually +perform desired computations. Examples are calls to `steady` to compute +the steady state of the Ramsey economy, to `stoch_simul` with various +approximation orders to conduct stochastic simulations based on +perturbation solutions, to `estimation` in order to estimate models +under optimal policy with commitment, and to perfect foresight +simulation routines. + +See `aux-variables`{.interpreted-text role="ref"}, for an explanation of +how Lagrange multipliers are automatically created. + +*Options* + +This command accepts the following options: + +::: {.option} +planner\_discount = EXPRESSION + +Declares or reassigns the discount factor of the central planner +`optimal_policy_discount_factor`. Default: `1.0`. +::: + +::: {.option} +planner\_discount\_latex\_name = LATEX\_NAME + +Sets the LaTeX name of the `optimal_policy_discount_factor` parameter. +::: + +::: {.option} +instruments = (VARIABLE\_NAME,\...) + +Declares instrument variables for the computation of the steady state +under optimal policy. Requires a `steady_state_model` block or a +`_steadystate.m` file. See below. +::: + +*Steady state* + +Dynare takes advantage of the fact that the Lagrange multipliers appear +linearly in the equations of the steady state of the model under optimal +policy. Nevertheless, it is in general very difficult to compute the +steady state with simply a numerical guess in `initval` for the +endogenous variables. + +It greatly facilitates the computation, if the user provides an +analytical solution for the steady state (in `steady_state_model` block +or in a `_steadystate.m` file). In this case, it is necessary to provide +a steady state solution CONDITIONAL on the value of the instruments in +the optimal policy problem and declared with the option `instruments`. +The initial value of the instrument for steady state finding in this +case is set with `initval`. Note that computing and displaying steady +state values using the `steady`-command or calls to `resid` must come +after the `ramsey_model` statement and the `initval`-block. + +Note that choosing the instruments is partly a matter of interpretation +and you can choose instruments that are handy from a mathematical point +of view but different from the instruments you would refer to in the +analysis of the paper. A typical example is choosing inflation or +nominal interest rate as an instrument. +::: + +::: {.block} +ramsey\_constraints ; + +This block lets you define constraints on the variables in the Ramsey +problem. The constraints take the form of a variable, an inequality +operator (\> or \<) and a constant. + +*Example* + +> ramsey_constraints; +> i > 0; +> end; +::: + +::: {.command} +ramsey\_policy \[VARIABLE\_NAME\...\]; ramsey\_policy (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +This command is deprecated and formally equivalent to the calling +sequence + +> ramsey_model; +> stoch_simul; +> evaluate_planner_objective; + +It computes an approximation of the policy that maximizes the policy +maker's objective function subject to the constraints provided by the +equilibrium path of the private economy and under commitment to this +optimal policy. The Ramsey policy is computed by approximating the +equilibrium system around the perturbation point where the Lagrange +multipliers are at their steady state, i.e. where the Ramsey planner +acts as if the initial multipliers had been set to 0 in the distant +past, giving them time to converge to their steady state value. +Consequently, the optimal decision rules are computed around this steady +state of the endogenous variables and the Lagrange multipliers. + +Note that the variables in the list after the `ramsey_policy` or +`stoch_simul`-command can also contain multiplier names, but in a +case-sensititve way (e.g. `MULT_1`). In that case, Dynare will for +example display the IRFs of the respective multipliers when `irf>0`. + +The planner objective must be declared with the +`planner_objective`{.interpreted-text role="comm"} command. + +*Options* + +This command accepts all options of `stoch_simul`, plus: + +::: {.option} +planner\_discount = EXPRESSION + +See `planner_discount `{.interpreted-text +role="opt"}. +::: + +::: {.option} +instruments = (VARIABLE\_NAME,\...) + +Declares instrument variables for the computation of the steady state +under optimal policy. Requires a `steady_state_model` block or a +`_steadystate.m` file. See below. +::: + +*Output* + +This command generates all the output variables of `stoch_simul`. For +specifying the initial values for the endogenous state variables (except +for the Lagrange multipliers), see above. + +*Steady state* + +See `Ramsey steady state `{.interpreted-text role="comm"}. +::: + +### Optimal policy under discretion + +::: {.command} +discretionary\_policy \[VARIABLE\_NAME\...\]; discretionary\_policy +(OPTIONS\...) \[VARIABLE\_NAME\...\]; + +This command computes an approximation of the optimal policy under +discretion. The algorithm implemented is essentially an LQ solver, and +is described by *Dennis (2007)*. + +You must ensure that your objective is quadratic. Regarding the model, +it must either be linear or solved at first order with an analytical +steady state provided. In the first case, you should set the `linear` +option of the `model` block. + +It is possible to use the `estimation`{.interpreted-text role="comm"} +command after the `discretionary_policy` command, in order to estimate +the model with optimal policy under discretion and +`evaluate_planner_objective`{.interpreted-text role="comm"} to compute +welfare. + +*Options* + +This command accepts the same options as `ramsey_policy`, plus: + +::: {.option} +discretionary\_tol = NON-NEGATIVE DOUBLE + +Sets the tolerance level used to assess convergence of the solution +algorithm. Default: `1e-7`. +::: + +::: {.option} +maxit = INTEGER + +Maximum number of iterations. Default: `3000`. +::: +::: + +### Optimal Simple Rules (OSR) + +::: {.command} +osr \[VARIABLE\_NAME\...\]; osr (OPTIONS\...) \[VARIABLE\_NAME\...\]; + +This command computes optimal simple policy rules for linear-quadratic +problems of the form: + +> $$\min_\gamma E(y'_tWy_t)$$ + +such that: + +> $$A_1 E_ty_{t+1}+A_2 y_t+ A_3 y_{t-1}+C e_t=0$$ + +where: + +> - $E$ denotes the unconditional expectations operator; +> - $\gamma$ are parameters to be optimized. They must be elements of +> the matrices $A_1$, $A_2$, $A_3$, i.e. be specified as parameters +> in the `params` command and be entered in the `model` block; +> - $y$ are the endogenous variables, specified in the `var` command, +> whose (co)-variance enters the loss function; +> - $e$ are the exogenous stochastic shocks, specified in the +> `varexo`- ommand; +> - $W$ is the weighting matrix; + +The linear quadratic problem consists of choosing a subset of model +parameters to minimize the weighted (co)-variance of a specified subset +of endogenous variables, subject to a linear law of motion implied by +the first order conditions of the model. A few things are worth +mentioning. First, $y$ denotes the selected endogenous variables' +deviations from their steady state, i.e. in case they are not already +mean 0 the variables entering the loss function are automatically +demeaned so that the centered second moments are minimized. Second, +`osr` only solves linear quadratic problems of the type resulting from +combining the specified quadratic loss function with a first order +approximation to the model's equilibrium conditions. The reason is that +the first order state-space representation is used to compute the +unconditional (co)-variances. Hence, `osr` will automatically select +`order=1`. Third, because the objective involves minimizing a weighted +sum of unconditional second moments, those second moments must be +finite. In particular, unit roots in $y$ are not allowed. + +The subset of the model parameters over which the optimal simple rule is +to be optimized, $\gamma$, must be listed with `osr_params`. + +The weighting matrix $W$ used for the quadratic objective function is +specified in the `optim_weights` block. By attaching weights to +endogenous variables, the subset of endogenous variables entering the +objective function, $y$, is implicitly specified. + +The linear quadratic problem is solved using the numerical optimizer +specified with `opt_algo `{.interpreted-text +role="opt"}. + +*Options* + +The `osr` command will subsequently run `stoch_simul` and accepts the +same options, including restricting the endogenous variables by listing +them after the command, as `stoch_simul` (see +`stoch-sol`{.interpreted-text role="ref"}) plus + +::: {.option} +opt\_algo = INTEGER + +Specifies the optimizer for minimizing the objective function. The same +solvers as for `mode_compute` (see +`mode_compute `{.interpreted-text +role="opt"}) are available, except for `5`, `6`, and `10`. +::: + +::: {.option} +optim = (NAME, VALUE, \...) + +A list of NAME\`[ and VALUE pairs. Can be used to set options for the +optimization routines. The set of available options depends on the +selected optimization routine (i.e. on the value of option +:opt:\`opt\_algo \]{.title-ref}). See +`optim `{.interpreted-text role="opt"}. +::: + +::: {.option} +maxit = INTEGER + +Determines the maximum number of iterations used in `opt_algo=4`. This +option is now deprecated and will be removed in a future release of +Dynare. Use `optim` instead to set optimizer-specific values. Default: +`1000`. +::: + +::: {.option} +tolf = DOUBLE + +Convergence criterion for termination based on the function value used +in `opt_algo=4`. Iteration will cease when it proves impossible to +improve the function value by more than tolf. This option is now +deprecated and will be removed in a future release of Dynare. Use +`optim` instead to set optimizer-specific values. Default: `1e-7`. +::: + +::: {.option} +silent\_optimizer + +See `silent_optimizer`{.interpreted-text role="opt"}. +::: + +::: {.option} +huge\_number = DOUBLE + +Value for replacing the infinite bounds on parameters by finite numbers. +Used by some optimizers for numerical reasons (see +`huge_number `{.interpreted-text role="opt"}). +Users need to make sure that the optimal parameters are not larger than +this value. Default: `1e7`. +::: + +The value of the objective is stored in the variable +`oo_.osr.objective_function` and the value of parameters at the optimum +is stored in `oo_.osr.optim_params`. See below for more details. + +After running `osr` the parameters entering the simple rule will be set +to their optimal value so that subsequent runs of `stoch_simul` will be +conducted at these values. +::: + +::: {.command} +osr\_params PARAMETER\_NAME\...; + +This command declares parameters to be optimized by `osr`. +::: + +::: {.block} +optim\_weights ; + +This block specifies quadratic objectives for optimal policy problems. + +More precisely, this block specifies the nonzero elements of the weight +matrix $W$ used in the quadratic form of the objective function in +`osr`. + +An element of the diagonal of the weight matrix is given by a line of +the form: + + VARIABLE_NAME EXPRESSION; + +An off-the-diagonal element of the weight matrix is given by a line of +the form: + + VARIABLE_NAME, VARIABLE_NAME EXPRESSION; +::: + +*Example* + +> var y inflation r; +> varexo y_ inf_; +> +> parameters delta sigma alpha kappa gammarr gammax0 gammac0 gamma_y_ gamma_inf_; +> +> delta = 0.44; +> kappa = 0.18; +> alpha = 0.48; +> sigma = -0.06; +> +> gammarr = 0; +> gammax0 = 0.2; +> gammac0 = 1.5; +> gamma_y_ = 8; +> gamma_inf_ = 3; +> +> model(linear); +> y = delta * y(-1) + (1-delta)*y(+1)+sigma *(r - inflation(+1)) + y_; +> inflation = alpha * inflation(-1) + (1-alpha) * inflation(+1) + kappa*y + inf_; +> r = gammax0*y(-1)+gammac0*inflation(-1)+gamma_y_*y_+gamma_inf_*inf_; +> end; +> +> shocks; +> var y_; stderr 0.63; +> var inf_; stderr 0.4; +> end; +> +> optim_weights; +> inflation 1; +> y 1; +> y, inflation 0.5; +> end; +> +> osr_params gammax0 gammac0 gamma_y_ gamma_inf_; +> osr y; + +::: {.block} +osr\_params\_bounds ; + +This block declares lower and upper bounds for parameters in the optimal +simple rule. If not specified the optimization is unconstrained. + +Each line has the following syntax: + + PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND; + +Note that the use of this block requires the use of a constrained +optimizer, i.e. setting +`opt_algo `{.interpreted-text role="opt"} to `1`, +`2`, `5` or `9`. + +*Example* + +> osr_params_bounds; +> gamma_inf_, 0, 2.5; +> end; +> +> osr(opt_algo=9) y; +::: + +::: {.matvar} +[oo]().osr.objective\_function + +After an execution of the `osr` command, this variable contains the +value of the objective under optimal policy. +::: + +::: {.matvar} +[oo]().osr.optim\_params + +After an execution of the `osr` command, this variable contains the +value of parameters at the optimum, stored in fields of the form +`oo_.osr.optim_params.PARAMETER_NAME`. +::: + +::: {.matvar} +[M]().osr.param\_names + +After an execution of the `osr` command, this cell contains the names of +the parameters. +::: + +::: {.matvar} +[M]().osr.param\_indices + +After an execution of the `osr` command, this vector contains the +indices of the OSR parameters in `M_.params`. +::: + +::: {.matvar} +[M]().osr.param\_bounds + +After an execution of the `osr` command, this two by number of OSR +parameters matrix contains the lower and upper bounds of the parameters +in the first and second column, respectively. +::: + +::: {.matvar} +[M]().osr.variable\_weights + +After an execution of the `osr` command, this sparse matrix contains the +weighting matrix associated with the variables in the objective +function. +::: + +::: {.matvar} +[M]().osr.variable\_indices + +After an execution of the `osr` command, this vector contains the +indices of the variables entering the objective function in +`M_.endo_names`. +::: + + +Displaying and saving results +----------------------------- + +Dynare has comments to plot the results of a simulation and to save the +results. + +::: {.command} +rplot VARIABLE\_NAME\...; + +Plots the simulated path of one or several variables, as stored in +`oo_.endo_simul` by either `perfect_foresight_solver`, `simul` (see +`det-simul`{.interpreted-text role="ref"}) or `stoch_simul` with option +`periods` (see `stoch-sol`{.interpreted-text role="ref"}). The variables +are plotted in levels. +::: + +::: {.command} +dynatype (FILENAME) \[VARIABLE\_NAME\...\]; + +This command prints the listed endogenous or exogenous variables in a +text file named FILENAME. If no VARIABLE\_NAME is listed, all endogenous +variables are printed. +::: + +::: {.command} +dynasave (FILENAME) \[VARIABLE\_NAME\...\]; + +This command saves the listed endogenous or exogenous variables in a +binary file named FILENAME. If no VARIABLE\_NAME is listed, all +endogenous variables are saved. + +In MATLAB or Octave, variables saved with the `dynasave` command can be +retrieved by the command: + + load(FILENAME,'-mat') +::: + +Macro processing language {#macro-proc-lang} +------------------------- + +It is possible to use "macro" commands in the `.mod` file for performing +tasks such as: including modular source files, replicating blocks of +equations through loops, conditionally executing some code, writing +indexed sums or products inside equations\... + +The Dynare macro-language provides a new set of *macro-commands* which +can be used in `.mod` files. It features: + +> - File inclusion +> - Loops (`for` structure) +> - Conditional inclusion (`if/then/else` structures) +> - Expression substitution + +This macro-language is totally independent of the basic Dynare language, +and is processed by a separate component of the Dynare pre-processor. +The macro processor transforms a `.mod` file with macros into a `.mod` +file without macros (doing expansions/inclusions), and then feeds it to +the Dynare parser. The key point to understand is that the macro +processor only does text substitution (like the C preprocessor or the +PHP language). Note that it is possible to see the output of the macro +processor by using the `savemacro` option of the `dynare` command (see +`dyn-invoc`{.interpreted-text role="ref"}). + +The macro processor is invoked by placing *macro directives* in the +`.mod` file. Directives begin with an at-sign followed by a pound sign +(`@#`). They produce no output, but give instructions to the macro +processor. In most cases, directives occupy exactly one line of text. If +needed, two backslashes (`\\`) at the end of the line indicate that the +directive is continued on the next line. Macro directives following `//` +are not interpreted by the macro processor. For historical reasons, +directives in commented blocks, *ie* surrounded by `/*` and `*/`, are +interpreted by the macro processor. The user should not rely on this +behavior. The main directives are: + +> - `@#includepath`, paths to search for files that are to be +> included, +> - `@#include`, for file inclusion, +> - `@#define`, for defining a macro processor variable, +> - `@#if, @#ifdef, @#ifndef, @#elseif, @#else, @#endif` for +> conditional statements, +> - `@#for, @#endfor` for constructing loops. + +The macro processor maintains its own list of variables (distinct from +model variables and MATLAB/Octave variables). These macro-variables are +assigned using the `@#define` directive and can be of the following +basic types: boolean, real, string, tuple, function, and array (of any +of the previous types). + +### Macro expressions {#macro-exp} + +Macro-expressions can be used in two places: + +> - Inside macro directives, directly; +> - In the body of the `.mod` file, between an at-sign and curly +> braces (like `@{expr}`): the macro processor will substitute the +> expression with its value + +It is possible to construct macro-expressions that can be assigned to +macro-variables or used within a macro-directive. The expressions are +constructed using literals of the basic types (boolean, real, string, +tuple, array), comprehensions, macro-variables, macro-functions, and +standard operators. + +::: {.note} +::: {.admonition-title} +Note +::: + +Elsewhere in the manual, MACRO\_EXPRESSION designates an expression +constructed as explained in this section. +::: + +**Boolean** + +The following operators can be used on booleans: + +> - Comparison operators: `==, !=` +> - Logical operators: `&&, ||, !` + +**Real** + +The following operators can be used on reals: + +> - Arithmetic operators: `+, -, *, /, ^` +> - Comparison operators: `<, >, <=, >=, ==, !=` +> - Logical operators: `&&, ||, !` +> - Ranges with an increment of `1`: `REAL1:REAL2` (for example, `1:4` +> is equivalent to real array `[1, 2, 3, 4]`). +> +> ::: {.versionchanged} +> 4.6 Previously, putting brackets around the arguments to the colon +> operator (e.g. `[1:4]`) had no effect. Now, `[1:4]` will create an +> array containing an array (i.e. `[ [1, 2, 3, 4] ]`). +> ::: +> +> - Ranges with user-defined increment: `REAL1:REAL2:REAL3` (for +> example, `6:-2.1:-1` is equivalent to real array +> `[6, 3.9, 1.8, -0.3]`). +> - Functions: +> `max, min, mod, exp, log, log10, sin, cos, tan, asin, acos, atan, sqrt, cbrt, sign, floor, ceil, trunc, erf, erfc, gamma, lgamma, round, normpdf, normcdf`. +> NB `ln` can be used instead of `log` + +**String** + +String literals have to be enclosed by **double** quotes (like +`"name"`). + +The following operators can be used on strings: + +> - Comparison operators: `<, >, <=, >=, ==, !=` +> - Concatenation of two strings: `+` +> - Extraction of substrings: if `s` is a string, then `s[3]` is a +> string containing only the third character of `s`, and `s[4:6]` +> contains the characters from 4th to 6th +> - Function: `length` + +**Tuple** + +Tuples are enclosed by parenthesis and elements separated by commas +(like `(a,b,c)` or `(1,2,3)`). + +The following operators can be used on tuples: + +> - Comparison operators: `==, !=` +> - Functions: `empty, length` + +**Array** + +Arrays are enclosed by brackets, and their elements are separated by +commas (like `[1,[2,3],4]` or `["US", "FR"]`). + +The following operators can be used on arrays: + +> - Comparison operators: `==, !=` +> - Dereferencing: if `v` is an array, then `v[2]` is its 2nd element +> - Concatenation of two arrays: `+` +> - Set union of two arrays: `|` +> - Set intersection of two arrays: `&` +> - Difference `-`: returns the first operand from which the elements +> of the second operand have been removed. +> - Cartesian product of two arrays: `*` +> - Cartesian product of one array N times: `^N` +> - Extraction of sub-arrays: e.g. `v[4:6]` +> - Testing membership of an array: `in` operator (for example: `"b"` +> in `["a", "b", "c"]` returns `1`) +> - Functions: `empty, sum, length` + +**Comprehension** + +Comprehension syntax is a shorthand way to make arrays from other +arrays. There are three different ways the comprehension syntax can be +employed: [filtering]{.title-ref}, [mapping]{.title-ref}, and [filtering +and mapping]{.title-ref}. + +**Filtering** + +Filtering allows one to choose those elements from an array for which a +certain condition hold. + +> *Example* +> +> Create a new array, choosing the even numbers from the array `1:5`: +> +> [ i in 1:5 when mod(i,2) == 0 ] +> +> would result in: +> +> [2, 4] + +**Mapping** + +Mapping allows you to apply a transformation to every element of an +array. + +> *Example* +> +> Create a new array, squaring all elements of the array `1:5`: +> +> [ i^2 for i in 1:5 ] +> +> would result in: +> +> [1, 4, 9, 16, 25] + +**Filtering and Mapping** + +Combining the two preceding ideas would allow one to apply a +transformation to every selected element of an array. + +> *Example* +> +> Create a new array, squaring all even elements of the array `1:5`: +> +> [ i^2 for i in 1:5 when mod(i,2) == 0] +> +> would result in: +> +> [4, 16] +> +> *Further Examples* : +> +> [ (j, i+1) for (i,j) in (1:2)^2 ] +> [ (j, i+1) for (i,j) in (1:2)*(1:2) when i < j ] +> +> would result in: +> +> [(1, 2), (2, 2), (1, 3), (2, 3)] +> [(2, 2)] + +**Function** + +Functions can be defined in the macro processor using the `@#define` +directive (see below). A function is evaluated at the time it is +invoked, not at define time. Functions can be included in expressions +and the operators that can be combined with them depend on their return +type. + +**Checking variable type** + +Given a variable name or literal, you can check the type it evaluates to +using the following functions: `isboolean`, `isreal`, `isstring`, +`istuple`, and `isarray`. + +> *Examples* + + ---------------------------------- + **Code** **Output** + --------------------- ------------ + `isboolean(0)` `false` + + `isboolean(true)` `true` + + `isreal("str")` `false` + ---------------------------------- + +**Casting between types** + +Variables and literals of one type can be cast into another type. Some +type changes are straightforward (e.g. changing a [real]{.title-ref} to +a [string]{.title-ref}) whereas others have certain requirements (e.g. +to cast an [array]{.title-ref} to a [real]{.title-ref} it must be a one +element array containing a type that can be cast to [real]{.title-ref}). + +> *Examples* + + ----------------------------------------------- + **Code** **Output** + ---------------------------------- ------------ + `(bool) -1.1` `true` + + `(bool) 0` `false` + + `(real) "2.2"` `2.2` + + `(tuple) [3.3]` `(3.3)` + + `(array) 4.4` `[4.4]` + + `(real) [5.5]` `5.5` + + `(real) [6.6, 7.7]` `error` + + `(real) "8.8 in a string"` `error` + ----------------------------------------------- + +Casts can be used in expressions: + +> *Examples* + + ---------------------------------------- + **Code** **Output** + --------------------------- ------------ + `(bool) 0 && true` `false` + + `(real) "1" + 2` `3` + + `(string) (3 + 4)` `"7"` + + `(array) 5 + (array) 6` `[5, 6]` + ---------------------------------------- + +### Macro directives + +::: {.macrodir} +@\#includepath \"PATH\" @\#includepath MACRO\_EXPRESSION + +This directive adds the path contained in PATH to the list of those to +search when looking for a `.mod` file specified by `@#include`. If +provided with a MACRO\_EXPRESSION argument, the argument must evaluate +to a string. Note that these paths are added *after* any paths passed +using `-I <-I\<\\>>`{.interpreted-text role="opt"}. + +*Example* + +> @#includepath "/path/to/folder/containing/modfiles" +> @#includepath folders_containing_mod_files +::: + +::: {.macrodir} +@\#include \"FILENAME\" @\#include MACRO\_EXPRESSION + +This directive simply includes the content of another file in its place; +it is exactly equivalent to a copy/paste of the content of the included +file. If provided with a MACRO\_EXPRESSION argument, the argument must +evaluate to a string. Note that it is possible to nest includes (i.e. to +include a file from an included file). The file will be searched for in +the current directory. If it is not found, the file will be searched for +in the folders provided by `-I <-I\<\\>>`{.interpreted-text +role="opt"} and `@#includepath`. + +*Example* + +> @#include "modelcomponent.mod" +> @#include location_of_modfile +::: + +::: {.macrodir} +@\#define MACRO\_VARIABLE @\#define MACRO\_VARIABLE = MACRO\_EXPRESSION +@\#define MACRO\_FUNCTION = MACRO\_EXPRESSION + +Defines a macro-variable or macro function. + +*Example* + +> @#define var // Equals true +> @#define x = 5 // Real +> @#define y = "US" // String +> @#define v = [ 1, 2, 4 ] // Real array +> @#define w = [ "US", "EA" ] // String array +> @#define u = [ 1, ["EA"] ] // Mixed array +> @#define z = 3 + v[2] // Equals 5 +> @#define t = ("US" in w) // Equals true +> @#define f(x) = " " + x + y // Function `f` with argument `x` +> // returns the string ' ' + x + 'US' + +*Example* + +> @#define x = 1 +> @#define y = [ "B", "C" ] +> @#define i = 2 +> @#define f(x) = x + " + " + y[i] +> @#define i = 1 +> +> model; +> A = @{y[i] + f("D")}; +> end; +> +> The latter is strictly equivalent to: +> +> model; +> A = BD + B; +> end; +::: + +::: {.macrodir} +@\#if MACRO\_EXPRESSION @\#ifdef MACRO\_VARIABLE @\#ifndef +MACRO\_VARIABLE @\#elseif MACRO\_EXPRESSION @\#else @\#endif + +Conditional inclusion of some part of the `.mod` file. The lines between +`@#if`, `@#ifdef`, or `@#ifndef` and the next `@#elseif`, `@#else` or +`@#endif` is executed only if the condition evaluates to `true`. +Following the `@#if` body, you can zero or more `@#elseif` branches. An +`@#elseif` condition is only evaluated if the preceding `@#if` or +`@#elseif` condition evaluated to `false`. The `@#else` branch is +optional and is only evaluated if all `@#if` and `@#elseif` statements +evaluate to false. + +Note that when using `@#ifdef`, the condition will evaluate to `true` if +the MACRO\_VARIABLE has been previously defined, regardless of its +value. Conversely, `@#ifndef` will evaluate to true if the +MACRO\_VARIABLE has not yet been defined. + +Note that when using `@#elseif` you can check whether or not a variable +has been defined by using the `defined` operator. Hence, to enter the +body of an `@#elseif` branch if the variable `X` has not been defined, +you would write: `@#elseif !defined(X)`. + +Note that if a real appears as the result of the MACRO\_EXPRESSION, it +will be interpreted as a boolean; a value of `0` is interpreted as +`false`, otherwise it is interpreted as `true`. Further note that +because of the imprecision of reals, extra care must be taken when +testing them in the MACRO\_EXPRESSION. For example, `exp(log(5)) == 5` +will evaluate to `false`. Hence, when comparing real values, you should +generally use a zero tolerance around the value desired, e.g. +`exp(log(5)) > 5-1e-14 && exp(log(5)) < 5+1e-14` + +*Example* + +> Choose between two alternative monetary policy rules using a +> macro-variable: +> +> @#define linear_mon_pol = false // 0 would be treated the same +> ... +> model; +> @#if linear_mon_pol +> i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); +> @#else +> i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; +> @#endif +> ... +> end; +> +> This would result in: +> +> ... +> model; +> i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; +> ... +> end; + +*Example* + +> Choose between two alternative monetary policy rules using a +> macro-variable. The only difference between this example and the +> previous one is the use of `@#ifdef` instead of `@#if`. Even though +> `linear_mon_pol` contains the value `false` because `@#ifdef` only +> checks that the variable has been defined, the linear monetary policy +> is output: +> +> @#define linear_mon_pol = false // 0 would be treated the same +> ... +> model; +> @#ifdef linear_mon_pol +> i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); +> @#else +> i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; +> @#endif +> ... +> end; +> +> This would result in: +> +> ... +> model; +> i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); +> ... +> end; +::: + +::: {.macrodir} +@\#for MACRO\_VARIABLE in MACRO\_EXPRESSION @\#for MACRO\_VARIABLE in +MACRO\_EXPRESSION when MACRO\_EXPRESSION @\#for MACRO\_TUPLE in +MACRO\_EXPRESSION @\#for MACRO\_TUPLE in MACRO\_EXPRESSION when +MACRO\_EXPRESSION @\#endfor + +Loop construction for replicating portions of the `.mod` file. Note that +this construct can enclose variable/parameters declaration, +computational tasks, but not a model declaration. + +*Example* + +> model; +> @#for country in [ "home", "foreign" ] +> GDP_@{country} = A * K_@{country}^a * L_@{country}^(1-a); +> @#endfor +> end; +> +> The latter is equivalent to: +> +> model; +> GDP_home = A * K_home^a * L_home^(1-a); +> GDP_foreign = A * K_foreign^a * L_foreign^(1-a); +> end; + +*Example* + +> model; +> @#for (i, j) in ["GDP"] * ["home", "foreign"] +> @{i}_@{j} = A * K_@{j}^a * L_@{j}^(1-a); +> @#endfor +> end; +> +> The latter is equivalent to: +> +> model; +> GDP_home = A * K_home^a * L_home^(1-a); +> GDP_foreign = A * K_foreign^a * L_foreign^(1-a); +> end; + +*Example* + +> @#define countries = ["US", "FR", "JA"] +> @#define nth_co = "US" +> model; +> @#for co in countries when co != nth_co +> (1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; +> @#endfor +> E_@{nth_co} = 1; +> end; +> +> The latter is equivalent to: +> +> model; +> (1+i_FR) = (1+i_US) * E_FR(+1) / E_FR; +> (1+i_JA) = (1+i_US) * E_JA(+1) / E_JA; +> E_US = 1; +> end; +::: + +::: {.macrodir} +@\#echo MACRO\_EXPRESSION + +Asks the preprocessor to display some message on standard output. The +argument must evaluate to a string. +::: + +::: {.macrodir} +@\#error MACRO\_EXPRESSION + +Asks the preprocessor to display some error message on standard output +and to abort. The argument must evaluate to a string. +::: + +::: {.macrodir} +@\#echomacrovars @\#echomacrovars MACRO\_VARIABLE\_LIST +@\#echomacrovars(save) MACRO\_VARIABLE\_LIST + +Asks the preprocessor to display the value of all macro variables up +until this point. If the `save` option is passed, then values of the +macro variables are saved to `options_.macrovars_line_<>`. +If `NAME_LIST` is passed, only display/save variables and functions with +that name. + +*Example* + +> @#define A = 1 +> @#define B = 2 +> @#define C(x) = x*2 +> @#echomacrovars A C D +> +> The output of the command above is: +> +> Macro Variables: +> A = 1 +> Macro Functions: +> C(x) = (x * 2) +::: + +### Typical usages + +#### Modularization + +The `@#include` directive can be used to split `.mod` files into several +modular components. + +Example setup: + +`modeldesc.mod` + +> Contains variable declarations, model equations, and shocks +> declarations. + +`simul.mod` + +> Includes `modeldesc.mod`, calibrates parameter,s and runs stochastic +> simulations. + +`estim.mod` + +> Includes `modeldesc.mod`, declares priors on parameters, and runs +> Bayesian estimation. + +Dynare can be called on `simul.mod` and `estim.mod` but it makes no +sense to run it on `modeldesc.mod`. + +The main advantage is that you don\'t have to copy/paste the whole model +(at the beginning) or changes to the model (during development). + +#### Indexed sums of products + +The following example shows how to construct a moving average: + + @#define window = 2 + + var x MA_x; + ... + model; + ... + MA_x = @{1/(2*window+1)}*( + @#for i in -window:window + +x(@{i}) + @#endfor + ); + ... + end; + +After macro processing, this is equivalent to: + + var x MA_x; + ... + model; + ... + MA_x = 0.2*( + +x(-2) + +x(-1) + +x(0) + +x(1) + +x(2) + ); + ... + end; + +#### Multi-country models + +Here is a skeleton example for a multi-country model: + + @#define countries = [ "US", "EA", "AS", "JP", "RC" ] + @#define nth_co = "US" + + @#for co in countries + var Y_@{co} K_@{co} L_@{co} i_@{co} E_@{co} ...; + parameters a_@{co} ...; + varexo ...; + @#endfor + + model; + @#for co in countries + Y_@{co} = K_@{co}^a_@{co} * L_@{co}^(1-a_@{co}); + ... + @#if co != nth_co + (1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; // UIP relation + @#else + E_@{co} = 1; + @#endif + @#endfor + end; + +#### Endogeneizing parameters + +When calibrating the model, it may be useful to consider a parameter as +an endogenous variable (and vice-versa). + +For example, suppose production is defined by a CES function: + +> $$y = \left(\alpha^{1/\xi} \ell^{1-1/\xi}+(1-\alpha)^{1/\xi}k^{1-1/\xi}\right)^{\xi/(\xi-1)}$$ + +and the labor share in GDP is defined as: + +> $$\textrm{lab\_rat} = (w \ell)/(p y)$$ + +In the model, $\alpha$ is a (share) parameter and `lab_rat` is an +endogenous variable. + +It is clear that calibrating $\alpha$ is not straightforward; on the +contrary, we have real world data for `lab_rat` and it is clear that +these two variables are economically linked. + +The solution is to use a method called *variable flipping*, which +consists in changing the way of computing the steady state. During this +computation, $\alpha$ will be made an endogenous variable and `lab_rat` +will be made a parameter. An economically relevant value will be +calibrated for `lab_rat`, and the solution algorithm will deduce the +implied value for $\alpha$. + +An implementation could consist of the following files: + +`modeqs.mod` + +> This file contains variable declarations and model equations. The code +> for the declaration of $\alpha$ and `lab_rat` would look like: +> +> @#if steady +> var alpha; +> parameter lab_rat; +> @#else +> parameter alpha; +> var lab_rat; +> @#endif + +`steady.mod` + +> This file computes the steady state. It begins with: +> +> @#define steady = 1 +> @#include "modeqs.mod" +> +> Then it initializes parameters (including `lab_rat`, excluding +> $\alpha$), computes the steady state (using guess values for +> endogenous, including $\alpha$), then saves values of parameters and +> endogenous at steady state in a file, using the +> `save_params_and_steady_state` command. + +`simul.mod` + +> This file computes the simulation. It begins with: +> +> @#define steady = 0 +> @#include "modeqs.mod" +> +> Then it loads values of parameters and endogenous at steady state from +> file, using the `load_params_and_steady_state` command, and computes +> the simulations. + +### MATLAB/Octave loops versus macro processor loops + +Suppose you have a model with a parameter $\rho$ and you want to run +simulations for three values: $\rho = 0.8, 0.9, +1$. There are several ways of doing this: + +*With a MATLAB/Octave loop* + +> rhos = [ 0.8, 0.9, 1]; +> for i = 1:length(rhos) +> rho = rhos(i); +> stoch_simul(order=1); +> end +> +> Here the loop is not unrolled, MATLAB/Octave manages the iterations. +> This is interesting when there are a lot of iterations. + +*With a macro processor loop (case 1)* + +> rhos = [ 0.8, 0.9, 1]; +> @#for i in 1:3 +> rho = rhos(@{i}); +> stoch_simul(order=1); +> @#endfor +> +> This is very similar to the previous example, except that the loop is +> unrolled. The macro processor manages the loop index but not the data +> array (`rhos`). + +*With a macro processor loop (case 2)* + +> @#for rho_val in [ 0.8, 0.9, 1] +> rho = @{rho_val}; +> stoch_simul(order=1); +> @#endfor +> +> The advantage of this method is that it uses a shorter syntax, since +> the list of values is directly given in the loop construct. The +> inconvenience is that you can not reuse the macro array in +> MATLAB/Octave. + +Verbatim inclusion {#verbatim} +------------------ + +Pass everything contained within the verbatim block to the +`.m` file. + +::: {.block} +verbatim ; + +By default, whenever Dynare encounters code that is not understood by +the parser, it is directly passed to the preprocessor output. + +In order to force this behavior you can use the `verbatim` block. This +is useful when the code you want passed to the `.m` file +contains tokens recognized by the Dynare preprocessor. + +*Example* + +> verbatim; +> % Anything contained in this block will be passed +> % directly to the .m file, including comments +> var = 1; +> end; +::: + +Misc commands +------------- + +::: {.command} +set\_dynare\_seed (INTEGER) set\_dynare\_seed (\`default\') +set\_dynare\_seed (\`clock\') set\_dynare\_seed (\`reset\') +set\_dynare\_seed (\`ALGORITHM\', INTEGER) + +Sets the seed used for random number generation. It is possible to set a +given integer value, to use a default value, or to use the clock (by +using the latter, one will therefore get different results across +different Dynare runs). The `reset` option serves to reset the seed to +the value set by the last `set_dynare_seed` command. On MATLAB 7.8 or +above, it is also possible to choose a specific algorithm for random +number generation; accepted values are `mcg16807`, `mlfg6331_64`, +`mrg32k3a`, `mt19937ar` (the default), `shr3cong` and `swb2712`. +::: + +::: {.command} +save\_params\_and\_steady\_state (FILENAME); + +For all parameters, endogenous and exogenous variables, stores their +value in a text file, using a simple name/value associative table. + +> - for parameters, the value is taken from the last parameter +> initialization. +> - for exogenous, the value is taken from the last `initval` block. +> - for endogenous, the value is taken from the last steady state +> computation (or, if no steady state has been computed, from the +> last `initval` block). + +Note that no variable type is stored in the file, so that the values can +be reloaded with `load_params_and_steady_state` in a setup where the +variable types are different. + +The typical usage of this function is to compute the steady-state of a +model by calibrating the steady-state value of some endogenous variables +(which implies that some parameters must be endogeneized during the +steady-state computation). + +You would then write a first `.mod` file which computes the steady state +and saves the result of the computation at the end of the file, using +`save_params_and_steady_state`. + +In a second file designed to perform the actual simulations, you would +use `load_params_and_steady_state` just after your variable +declarations, in order to load the steady state previously computed +(including the parameters which had been endogeneized during the steady +state computation). + +The need for two separate `.mod` files arises from the fact that the +variable declarations differ between the files for steady state +calibration and for simulation (the set of endogenous and parameters +differ between the two); this leads to different `var` and `parameters` +statements. + +Also note that you can take advantage of the `@#include` directive to +share the model equations between the two files (see +`macro-proc-lang`{.interpreted-text role="ref"}). +::: + +::: {.command} +load\_params\_and\_steady\_state (FILENAME); + +For all parameters, endogenous and exogenous variables, loads their +value from a file created with `save_params_and_steady_state`. + +> - for parameters, their value will be initialized as if they had +> been calibrated in the `.mod` file. +> - for endogenous and exogenous variables, their value will be +> initialized as they would have been from an `initval` block . + +This function is used in conjunction with +`save_params_and_steady_state`; see the documentation of that function +for more information. +::: + +::: {.command} +compilation\_setup (OPTIONS); + +When the `use_dll`{.interpreted-text role="opt"} option is present, +Dynare uses the GCC compiler that was distributed with it to compile the +static and dynamic C files produced by the preprocessor. You can use +this option to change the compiler, flags, and libraries used. + +*Options* + +> ::: {.option} +> compiler = FILENAME +> +> The path to the compiler. +> ::: +> +> ::: {.option} +> substitute\_flags = QUOTED\_STRING +> +> The flags to use instead of the default flags. +> ::: +> +> ::: {.option} +> add\_flags = QUOTED\_STRING +> +> The flags to use in addition to the default flags. If +> `substitute_flags` is passed, these flags are added to the flags +> specified there. +> ::: +> +> ::: {.option} +> substitute\_libs = QUOTED\_STRING +> +> The libraries to link against instead of the default libraries. +> ::: +> +> ::: {.option} +> add\_libs = QUOTED\_STRING +> +> The libraries to link against in addition to the default libraries. If +> `substitute_libs` is passed, these libraries are added to the +> libraries specified there. +> ::: +::: + +::: {.matcomm} +dynare\_version ; + +Output the version of Dynare that is currently being used (i.e. the one +that is highest on the MATLAB/Octave path). +::: + +::: {.matcomm} +write\_latex\_definitions ; + +Writes the names, LaTeX names and long names of model variables to +tables in a file named `<>_latex_definitions.tex`. Requires +the following LaTeX packages: `longtable`. +::: + +::: {.matcomm} +write\_latex\_parameter\_table ; + +Writes the LaTeX names, parameter names, and long names of model +parameters to a table in a file named +`<>_latex_parameters.tex.` The command writes the values of +the parameters currently stored. Thus, if parameters are set or changed +in the steady state computation, the command should be called after a +steady-command to make sure the parameters were correctly updated. The +long names can be used to add parameter descriptions. Requires the +following LaTeX packages: `longtable, booktabs`. +::: + +::: {.matcomm} +write\_latex\_prior\_table ; + +Writes descriptive statistics about the prior distribution to a LaTeX +table in a file named `<>_latex_priors_table.tex`. The command +writes the prior definitions currently stored. Thus, this command must +be invoked after the `estimated_params` block. If priors are defined +over the measurement errors, the command must also be preceeded by the +declaration of the observed variables (with `varobs`). The command +displays a warning if no prior densities are defined (ML estimation) or +if the declaration of the observed variables is missing. Requires the +following LaTeX packages: `longtable, booktabs`. +::: + +::: {.matcomm} +collect\_latex\_files ; + +Writes a LaTeX file named `<>_TeX_binder.tex` that collects +all TeX output generated by Dynare into a file. This file can be +compiled using `pdflatex` and automatically tries to load all required +packages. Requires the following LaTeX packages: `breqn`, `psfrag`, +`graphicx`, `epstopdf`, `longtable`, `booktabs`, `caption`, `float,` +`amsmath`, `amsfonts`, and `morefloats`. +::: + +**Footnotes** + +[^1]: A `.mod` file must have lines that end with a line feed character, + which is not commonly visible in text editors. Files created on + Windows and Unix-based systems have always conformed to this + requirement, as have files created on OS X and macOS. Files created + on old, pre-OS X Macs used carriage returns as end of line + characters. If you get a Dynare parsing error of the form + `ERROR: <>: line 1, cols 341-347: syntax error,...` and + there\'s more than one line in your `.mod` file, know that it uses + the carriage return as an end of line character. To get more helpful + error messages, the carriage returns should be changed to line + feeds. + +[^2]: Note that arbitrary MATLAB or Octave expressions can be put in a + `.mod` file, but those expressions have to be on separate lines, + generally at the end of the file for post-processing purposes. They + are not interpreted by Dynare, and are simply passed on unmodified + to MATLAB or Octave. Those constructions are not addresses in this + section. + +[^3]: In particular, for big models, the compilation step can be very + time-consuming, and use of this option may be counter-productive in + those cases. + +[^4]: See options `conf_sig `{.interpreted-text role="ref"} and + `mh_conf_sig `{.interpreted-text role="opt"} + to change the size of the HPD interval. + +[^5]: See options `conf_sig `{.interpreted-text role="ref"} () + and `mh_conf_sig `{.interpreted-text + role="opt"} to qchange the size of the HPD interval. + +[^6]: When the shocks are correlated, it is the decomposition of + orthogonalized shocks via Cholesky decomposition according to the + order of declaration of shocks (see `var-decl`{.interpreted-text + role="ref"}) + +[^7]: See `forecast `{.interpreted-text role="opt"} + for more information. + +[^8]: In case of Excel not being installed, + + may be helpful. + +[^9]: In case of Excel not being installed, + + may be helpful. + +[^10]: See option `conf_sig `{.interpreted-text role="ref"} to + change the size of the HPD interval. + +[^11]: See option `conf_sig `{.interpreted-text role="ref"} to + change the size of the HPD interval. + +[^12]: If you want to align the paper with the description herein, + please note that $A$ is $A^0$ and $F$ is $A^+$. + +[^13]: An example can be found at + . diff --git a/doc/src/orig/time-series.md b/docs/src/orig/time-series.md similarity index 100% rename from doc/src/orig/time-series.md rename to docs/src/orig/time-series.md diff --git a/doc/src/running-dynare.md b/docs/src/running-dynare.md similarity index 100% rename from doc/src/running-dynare.md rename to docs/src/running-dynare.md diff --git a/doc/src/the-model-file.md b/docs/src/the-model-file.md similarity index 100% rename from doc/src/the-model-file.md rename to docs/src/the-model-file.md

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b/docs/build/development/lre_solution_derivatives.aux new file mode 100644 index 00000000..753dc682 --- /dev/null +++ b/docs/build/development/lre_solution_derivatives.aux @@ -0,0 +1,6 @@ +\relax +\@writefile{toc}{\contentsline {section}{\numberline {1}Derivatives of the steady state}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2}Derivatives of the coefficents of the linear approximation of the model}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {3}Derivatives of the solution of the UQME}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {4}Derivatives of the coefficients of response to the shocks}{2}{}\protected@file@percent } +\gdef \@abspage@last{3} diff --git a/docs/build/orig/the-model-file.md.new b/docs/build/orig/the-model-file.md.new new file mode 100644 index 00000000..a488f595 --- /dev/null +++ b/docs/build/orig/the-model-file.md.new @@ -0,0 +1,10384 @@ +::: {.default-domain} +dynare +::: + +The model file {#model-file} +============== + +Conventions {#conv} +----------- + +A model file contains a list of commands and of blocks. Each command and +each element of a block is terminated by a semicolon (;). Blocks are +terminated by `end;`. + +If Dynare encounters an unknown expression at the beginning of a line or +after a semicolon, it will parse the rest of that line as native MATLAB +code, even if there are more statements separated by semicolons present. +To prevent cryptic error messages, it is strongly recommended to always +only put one statement/command into each line and start a new line after +each semicolon.[^1] + +Lines of codes can be commented out line by line or as a block. +Single-line comments begin with `//` and stop at the end of the line. +Multiline comments are introduced by `/*` and terminated by `*/`. + +*Examples* + +> // This is a single line comment +> +> var x; // This is a comment about x +> +> /* This is another inline comment about alpha */ alpha = 0.3; +> +> /* +> This comment is spanning +> two lines. +> */ + +Note that these comment marks should not be used in native MATLAB code +regions where the [%]{.title-ref} should be preferred instead to +introduce a comment. In a `verbatim` block, see +`verbatim`{.interpreted-text role="ref"}, this would result in a crash +since `//` is not a valid MATLAB statement). + +Most Dynare commands have arguments and several accept options, +indicated in parentheses after the command keyword. Several options are +separated by commas. + +In the description of Dynare commands, the following conventions are +observed: + +- Optional arguments or options are indicated between square brackets: + '\[\]'; +- Repeated arguments are indicated by ellipses: "\..."; +- Mutually exclusive arguments are separated by vertical bars: '\|'; +- INTEGER indicates an integer number; +- INTEGER\_VECTOR indicates a vector of integer numbers separated by + spaces, enclosed by square brackets; +- DOUBLE indicates a double precision number. The following syntaxes + are valid: `1.1e3`, `1.1E3`, `1.1d3`, `1.1D3`. In some places, + infinite Values `Inf` and `-Inf` are also allowed; +- NUMERICAL\_VECTOR indicates a vector of numbers separated by spaces, + enclosed by square brackets; +- EXPRESSION indicates a mathematical expression valid outside the + model description (see `expr`{.interpreted-text role="ref"}); +- MODEL\_EXPRESSION (sometimes MODEL\_EXP) indicates a mathematical + expression valid in the model description (see + `expr`{.interpreted-text role="ref"} and + `model-decl`{.interpreted-text role="ref"}); +- MACRO\_EXPRESSION designates an expression of the macro processor + (see `macro-exp`{.interpreted-text role="ref"}); +- VARIABLE\_NAME (sometimes VAR\_NAME) indicates a variable name + starting with an alphabetical character and can't contain: + '()+-\*/\^=!;:@\#.' or accentuated characters; +- PARAMETER\_NAME (sometimes PARAM\_NAME) indicates a parameter name + starting with an alphabetical character and can't contain: + '()+-\*/\^=!;:@\#.' or accentuated characters; +- LATEX\_NAME (sometimes TEX\_NAME) indicates a valid LaTeX expression + in math mode (not including the dollar signs); +- FUNCTION\_NAME indicates a valid MATLAB function name; +- FILENAME indicates a filename valid in the underlying operating + system; it is necessary to put it between quotes when specifying the + extension or if the filename contains a non-alphanumeric character; +- QUOTED\_STRING indicates an arbitrary string enclosed between + (single) quotes. + +Variable declarations {#var-decl} +--------------------- + +While Dynare allows the user to choose their own variable names, there +are some restrictions to be kept in mind. First, variables and +parameters must not have the same name as Dynare commands or built-in +functions. In this respect, Dynare is not case-sensitive. For example, +do not use `Ln` or `Sigma_e` to name your variable. Not conforming to +this rule might yield hard-to-debug error messages or crashes. Second, +when employing user-defined steady state files it is recommended to +avoid using the name of MATLAB functions as this may cause conflicts. In +particular, when working with user-defined steady state files, do not +use correctly-spelled greek names like [alpha]{.title-ref}, because +there are MATLAB functions of the same name. Rather go for `alppha` or +`alph`. Lastly, please do not name a variable or parameter `i`. This may +interfere with the imaginary number i and the index in many loops. +Rather, name investment `invest`. Using `inv` is also not recommended as +it already denotes the inverse operator. Commands for declaring +variables and parameters are described below. + +::: {.command} +var VAR\_NAME \[\$TEX\_NAME\$\] +\[(long\_name=QUOTED\_STRINGNAME=QUOTED\_STRING)\]\...; +var(deflator=MODEL\_EXPR) VAR\_NAME (\... same options apply) var(log, +deflator=MODEL\_EXPR) VAR\_NAME (\... same options apply) +var(log\_deflator=MODEL\_EXPR) VAR\_NAME (\... same options apply) + +This required command declares the endogenous variables in the model. +See `conv`{.interpreted-text role="ref"} for the syntax of *VAR\_NAME* +and *MODEL\_EXPR*. Optionally it is possible to give a LaTeX name to the +variable or, if it is nonstationary, provide information regarding its +deflator. The variables in the list can be separated by spaces or by +commas. `var` commands can appear several times in the file and Dynare +will concatenate them. Dynare stores the list of declared parameters, in +the order of declaration, in a column cell array `M_.endo_names`. + +If the model is nonstationary and is to be written as such in the +`model` block, Dynare will need the trend deflator for the appropriate +endogenous variables in order to stationarize the model. The trend +deflator must be provided alongside the variables that follow this +trend. + +*Options* + +::: {.option} +log + +In addition to the endogenous variable(s) thus declared, this option +also triggers the creation of auxiliary variable(s) equal to the log of +the corresponding endogenous variable(s). For example, given a +`var(log) y` statement, two endogenous will be created (`y` and +`LOG_y`), and an auxiliary equation linking the two will also be added +(equal to `LOG_y = log(y)`). Moreover, every occurence of `y` in the +model will be replaced by `exp(LOG_y)`. This option is for example +useful when one wants to perform a loglinear approximation of some +variable(s) in the context of a first-order stochastic approximation; or +when one wants to ensure the variable(s) stay(s) in the definition +domain of the function defining the steady state or the dynamic +residuals when the nonlinear solver is used. +::: + +::: {.option} +deflator = MODEL\_EXPR + +The expression used to detrend an endogenous variable. All trend +variables, endogenous variables and parameters referenced in MODEL\_EXPR +must already have been declared by the `trend_var, log_trend_var, var` +and `parameters` commands. The deflator is assumed to be multiplicative; +for an additive deflator, use `log_deflator`. This option can be used +together with the `log` option (the latter must come first). +::: + +::: {.option} +log\_deflator = MODEL\_EXPR + +Same as `deflator`, except that the deflator is assumed to be additive +instead of multiplicative (or, to put it otherwise, the declared +variable is equal to the log of a variable with a multiplicative trend). +This option cannot be used together with the `log` option, because it +would not make much sense from an economic point of view (the +corresponding auxiliary variable would correspond to the log taken two +times on a variable with a multiplicative trend). +::: + +::: {#long-name} +::: {.option} +long\_name = QUOTED\_STRING + +This is the long version of the variable name. Its value is stored in +`M_.endo_names_long` (a column cell array, in the same order as +`M_.endo_names`). In case multiple `long_name` options are provided, the +last one will be used. Default: `VAR_NAME`. +::: +::: + +::: {#partitioning} +::: {.option} +NAME = QUOTED\_STRING + +This is used to create a partitioning of variables. It results in the +direct output in the `.m` file analogous to: +`M_.endo_partitions.NAME = QUOTED_STRING`;. +::: +::: + +*Example (variable partitioning)* + +> var c gnp cva (country=`US', state=`VA') +> cca (country=`US', state=`CA', long_name=`Consumption CA'); +> var(deflator=A) i b; +> var c $C$ (long_name=`Consumption'); +::: + +::: {.command} +varexo\_det VAR\_NAME \[\$TEX\_NAME\$\] +\[(long\_name=QUOTED\_STRING\|NAME=QUOTED\_STRING)\...\]; + +This optional command declares exogenous deterministic variables in a +stochastic model. See `conv`{.interpreted-text role="ref"} for the +syntax of VARIABLE\_NAME. Optionally it is possible to give a LaTeX name +to the variable. The variables in the list can be separated by spaces or +by commas. `varexo_det` commands can appear several times in the file +and Dynare will concatenate them. + +It is possible to mix deterministic and stochastic shocks to build +models where agents know from the start of the simulation about future +exogenous changes. In that case `stoch_simul` will compute the rational +expectation solution adding future information to the state space +(nothing is shown in the output of `stoch_simul`) and forecast will +compute a simulation conditional on initial conditions and future +information. + +Note that exogenous deterministic variables cannot appear with a lead or +a lag in the model. + +*Options* + +::: {.option} +long\_name = QUOTED\_STRING + +Like `long_name `{.interpreted-text role="ref"} but value +stored in `M_.exo_det_names_long`. +::: + +::: {.option} +NAME = QUOTED\_STRING + +Like `partitioning `{.interpreted-text role="ref"} but +QUOTED\_STRING stored in `M_.exo_det_partitions.NAME`. +::: + +*Example* + +> varexo m gov; +> varexo_det tau; +::: + +::: {.command} +var\_remove VAR\_NAME \| PARAM\_NAME\...; + +Removes the listed variables (or parameters) from the model. Removing a +variable that has already been used in a model equation or elsewhere +will lead to an error. +::: + +::: {.command} +predetermined\_variables VAR\_NAME\...; + +In Dynare, the default convention is that the timing of a variable +reflects when this variable is decided. The typical example is for +capital stock: since the capital stock used at current period is +actually decided at the previous period, then the capital stock entering +the production function is `k(-1)`, and the law of motion of capital +must be written: + + k = i + (1-delta)*k(-1) + +Put another way, for stock variables, the default in Dynare is to use a +"stock at the end of the period" concept, instead of a "stock at the +beginning of the period" convention. + +The `predetermined_variables` is used to change that convention. The +endogenous variables declared as predetermined variables are supposed to +be decided one period ahead of all other endogenous variables. For stock +variables, they are supposed to follow a "stock at the beginning of the +period" convention. + +Note that Dynare internally always uses the "stock at the end of the +period" concept, even when the model has been entered using the +`predetermined_variables` command. Thus, when plotting, computing or +simulating variables, Dynare will follow the convention to use variables +that are decided in the current period. For example, when generating +impulse response functions for capital, Dynare will plot `k`, which is +the capital stock decided upon by investment today (and which will be +used in tomorrow's production function). This is the reason that capital +is shown to be moving on impact, because it is `k` and not the +predetermined `k(-1)` that is displayed. It is important to remember +that this also affects simulated time series and output from smoother +routines for predetermined variables. Compared to non-predetermined +variables they might otherwise appear to be falsely shifted to the +future by one period. + +*Example* + +> The following two program snippets are strictly equivalent. +> +> Using default Dynare timing convention: +> +> var y, k, i; +> ... +> model; +> y = k(-1)^alpha; +> k = i + (1-delta)*k(-1); +> ... +> end; +> +> Using the alternative timing convention: +> +> var y, k, i; +> predetermined_variables k; +> ... +> model; +> y = k^alpha; +> k(+1) = i + (1-delta)*k; +> ... +> end; +::: + +::: {.command} +trend\_var (growth\_factor = MODEL\_EXPR) VAR\_NAME +\[\$LATEX\_NAME\$\]\...; + +This optional command declares the trend variables in the model. See +`conv`{.interpreted-text role="ref"} for the syntax of MODEL\_EXPR and +VAR\_NAME. Optionally it is possible to give a LaTeX name to the +variable. + +The variable is assumed to have a multiplicative growth trend. For an +additive growth trend, use `log_trend_var` instead. + +Trend variables are required if the user wants to be able to write a +nonstationary model in the `model` block. The `trend_var` command must +appear before the var command that references the trend variable. + +`trend_var` commands can appear several times in the file and Dynare +will concatenate them. + +If the model is nonstationary and is to be written as such in the +`model` block, Dynare will need the growth factor of every trend +variable in order to stationarize the model. The growth factor must be +provided within the declaration of the trend variable, using the +`growth_factor` keyword. All endogenous variables and parameters +referenced in MODEL\_EXPR must already have been declared by the var and +parameters commands. + +*Example* + +> trend_var (growth_factor=gA) A; +::: + +::: {.command} +model\_local\_variable VARIABLE\_NAME \[LATEX\_NAME\]\... ; + +This optional command declares a model local variable. See +`conv`{.interpreted-text role="ref"} for the syntax of VARIABLE\_NAME. +As you can create model local variables on the fly in the model block +(see `model-decl`{.interpreted-text role="ref"}), the interest of this +command is primarily to assign a LATEX\_NAME to the model local +variable. + +*Example* + +> model_local_variable GDP_US $GDPUS$; +::: + +### On-the-fly Model Variable Declaration {#on-the-fly-declaration} + +Endogenous variables, exogenous variables, and parameters can also be +declared inside the model block. You can do this in two different ways: +either via the equation tag or directly in an equation. + +To declare a variable on-the-fly in an equation tag, simply state the +type of variable to be declared (`endogenous`, `exogenous`, or +`parameter` followed by an equal sign and the variable name in single +quotes. Hence, to declare a variable `c` as endogenous in an equation +tag, you can type `[endogenous='c']`. + +To perform on-the-fly variable declaration in an equation, simply follow +the symbol name with a vertical line (`|`, pipe character) and either an +`e`, an `x`, or a `p`. For example, to declare a parameter named +`alphaa` in the model block, you could write `alphaa|p` directly in an +equation where it appears. Similarly, to declare an endogenous variable +`c` in the model block you could write `c|e`. Note that in-equation +on-the-fly variable declarations must be made on contemporaneous +variables. + +On-the-fly variable declarations do not have to appear in the first +place where this variable is encountered. + +*Example* + +> The following two snippets are equivalent: +> +> > model; +> > [endogenous='k',name='law of motion of capital'] +> > k(+1) = i|e + (1-delta|p)*k; +> > y|e = k^alpha|p; +> > ... +> > end; +> > delta = 0.025; +> > alpha = 0.36; +> > +> > var k, i, y; +> > parameters delta, alpha; +> > delta = 0.025; +> > alpha = 0.36; +> > ... +> > model; +> > [name='law of motion of capital'] +> > k(1) = i|e + (1-delta|p)*k; +> > y|e = k|e^alpha|p; +> > ... +> > end; + +Expressions {#expr} +----------- + +Dynare distinguishes between two types of mathematical expressions: +those that are used to describe the model, and those that are used +outside the model block (e.g. for initializing parameters or variables, +or as command options). In this manual, those two types of expressions +are respectively denoted by MODEL\_EXPRESSION and EXPRESSION. + +Unlike MATLAB or Octave expressions, Dynare expressions are necessarily +scalar ones: they cannot contain matrices or evaluate to matrices.[^2] + +Expressions can be constructed using integers (INTEGER), floating point +numbers (DOUBLE), parameter names (PARAMETER\_NAME), variable names +(VARIABLE\_NAME), operators and functions. + +The following special constants are also accepted in some contexts: + +::: {.constant} +inf + +Represents infinity. +::: + +::: {.constant} +nan + +"Not a number": represents an undefined or unrepresentable value. +::: + +### Parameters and variables + +Parameters and variables can be introduced in expressions by simply +typing their names. The semantics of parameters and variables is quite +different whether they are used inside or outside the model block. + +#### Inside the model + +Parameters used inside the model refer to the value given through +parameter initialization (see `param-init`{.interpreted-text +role="ref"}) or `homotopy_setup` when doing a simulation, or are the +estimated variables when doing an estimation. + +Variables used in a MODEL\_EXPRESSION denote current period values when +neither a lead or a lag is given. A lead or a lag can be given by +enclosing an integer between parenthesis just after the variable name: a +positive integer means a lead, a negative one means a lag. Leads or lags +of more than one period are allowed. For example, if `c` is an +endogenous variable, then `c(+1)` is the variable one period ahead, and +`c(-2)` is the variable two periods before. + +When specifying the leads and lags of endogenous variables, it is +important to respect the following convention: in Dynare, the timing of +a variable reflects when that variable is decided. A control variable +--- which by definition is decided in the current period --- must have +no lead. A predetermined variable --- which by definition has been +decided in a previous period --- must have a lag. A consequence of this +is that all stock variables must use the "stock at the end of the +period" convention. + +Leads and lags are primarily used for endogenous variables, but can be +used for exogenous variables. They have no effect on parameters and are +forbidden for local model variables (see Model declaration). + +#### Outside the model + +When used in an expression outside the model block, a parameter or a +variable simply refers to the last value given to that variable. More +precisely, for a parameter it refers to the value given in the +corresponding parameter initialization (see +`param-init`{.interpreted-text role="ref"}); for an endogenous or +exogenous variable, it refers to the value given in the most recent +`initval` or `endval` block. + +### Operators + +The following operators are allowed in both MODEL\_EXPRESSION and +EXPRESSION: + +- Binary arithmetic operators: `+`, `-`, `*`, `/`, `^` +- Unary arithmetic operators: `+`, `-` +- Binary comparison operators (which evaluate to either 0 or 1): `<`, + `>`, `<=`, `>=`, `==`, `!=` + +Note the binary comparison operators are differentiable everywhere +except on a line of the 2-dimensional real plane. However for +facilitating convergence of Newton-type methods, Dynare assumes that, at +the points of non-differentiability, the partial derivatives of these +operators with respect to both arguments is equal to 0 (since this is +the value of the partial derivatives everywhere else). + +The following special operators are accepted in MODEL\_EXPRESSION (but +not in EXPRESSION): + +::: {.operator} +STEADY\_STATE (MODEL\_EXPRESSION) + +This operator is used to take the value of the enclosed expression at +the steady state. A typical usage is in the Taylor rule, where you may +want to use the value of GDP at steady state to compute the output gap. + +Exogenous and exogenous deterministic variables may not appear in +MODEL\_EXPRESSION. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +The concept of a steady state is ambiguous in a perfect foresight +context with permament and potentially anticipated shocks occuring. +Dynare will use the contents of `oo_.steady_state` as its reference for +calls to the `STEADY_STATE()`-operator. In the presence of `endval`, +this implies that the terminal state provided by the user is used. This +may be a steady state computed by Dynare (if `endval` is followed by +`steady`) or simply the terminal state provided by the user (if `endval` +is not followed by `steady`). Put differently, Dynare will not +automatically compute the steady state conditional on the specificed +value of the exogenous variables in the respective periods. +::: +::: + +::: {.operator} +EXPECTATION (INTEGER) (MODEL\_EXPRESSION) + +This operator is used to take the expectation of some expression using a +different information set than the information available at current +period. For example, `EXPECTATION(-1)(x(+1))` is equal to the expected +value of variable x at next period, using the information set available +at the previous period. See `aux-variables`{.interpreted-text +role="ref"} for an explanation of how this operator is handled +internally and how this affects the output. +::: + +### Functions + +#### Built-in functions + +The following standard functions are supported internally for both +MODEL\_EXPRESSION and EXPRESSION: + +::: {.function} +exp(x) + +Natural exponential. +::: + +::: {.function} +log(x) +::: + +::: {.function} +ln(x) + +Natural logarithm. +::: + +::: {.function} +log10(x) + +Base 10 logarithm. +::: + +::: {.function} +sqrt(x) + +Square root. +::: + +::: {.function} +cbrt(x) + +Cube root. +::: + +::: {.function} +sign(x) + +Signum function, defined as: + +> $$\begin{aligned} +> \textrm{sign}(x) = +> \begin{cases} +> -1 &\quad\text{if }x<0\\ +> 0 &\quad\text{if }x=0\\ +> 1 &\quad\text{if }x>0 +> \end{cases} +> \end{aligned}$$ + +Note that this function is not continuous, hence not differentiable, at +$x=0$. However, for facilitating convergence of Newton-type methods, +Dynare assumes that the derivative at $x=0$ is equal to $0$. This +assumption comes from the observation that both the right- and +left-derivatives at this point exist and are equal to $0$, so we can +remove the singularity by postulating that the derivative at $x=0$ is +$0$. +::: + +::: {.function} +abs(x) + +Absolute value. + +Note that this continuous function is not differentiable at $x=0$. +However, for facilitating convergence of Newton-type methods, Dynare +assumes that the derivative at $x=0$ is equal to $0$ (even if the +derivative does not exist). The rational for this mathematically +unfounded definition, rely on the observation that the derivative of +$\mathrm{abs}(x)$ is equal to $\mathrm{sign}(x)$ for any $x\neq 0$ in +$\mathbb R$ and from the convention for the value of $\mathrm{sign}(x)$ +at $x=0$). +::: + +::: {.function} +sin(x) +::: + +::: {.function} +cos(x) +::: + +::: {.function} +tan(x) +::: + +::: {.function} +asin(x) +::: + +::: {.function} +acos(x) +::: + +::: {.function} +atan(x) + +Trigonometric functions. +::: + +::: {.function} +sinh(x) +::: + +::: {.function} +cosh(x) +::: + +::: {.function} +tanh(x) +::: + +::: {.function} +asinh(x) +::: + +::: {.function} +acosh(x) +::: + +::: {.function} +atanh(x) + +Hyperbolic functions. +::: + +::: {.function} +max(a, b) +::: + +::: {.function} +min(a, b) + +Maximum and minimum of two reals. + +Note that these functions are differentiable everywhere except on a line +of the 2-dimensional real plane defined by $a=b$. However for +facilitating convergence of Newton-type methods, Dynare assumes that, at +the points of non-differentiability, the partial derivative of these +functions with respect to the first (resp. the second) argument is equal +to $1$ (resp. to $0$) (i.e. the derivatives at the kink are equal to the +derivatives observed on the half-plane where the function is equal to +its first argument). +::: + +::: {.function} +normcdf(x) normcdf(x, mu, sigma) + +Gaussian cumulative density function, with mean *mu* and standard +deviation *sigma*. Note that `normcdf(x)` is equivalent to +`normcdf(x,0,1)`. +::: + +::: {.function} +normpdf(x) normpdf(x, mu, sigma) + +Gaussian probability density function, with mean *mu* and standard +deviation *sigma*. Note that `normpdf(x)` is equivalent to +`normpdf(x,0,1)`. +::: + +::: {.function} +erf(x) + +Gauss error function. +::: + +::: {.function} +erfc(x) + +Complementary error function, *i.e.* +$\mathrm{erfc}(x) = 1-\mathrm{erf}(x)$. +::: + +#### External functions + +Any other user-defined (or built-in) MATLAB or Octave function may be +used in both a MODEL\_EXPRESSION and an EXPRESSION, provided that this +function has a scalar argument as a return value. + +To use an external function in a MODEL\_EXPRESSION, one must declare the +function using the `external_function` statement. This is not required +for external functions used in an EXPRESSION outside of a `model` block +or `steady_state_model` block. + +::: {.command} +external\_function (OPTIONS\...); + +This command declares the external functions used in the model block. It +is required for every unique function used in the model block. + +`external_function` commands can appear several times in the file and +must come before the model block. + +*Options* + +::: {.option} +name = NAME + +The name of the function, which must also be the name of the Julia file +implementing it. This option is mandatory. +::: + +::: {.option} +nargs = INTEGER + +The number of arguments of the function. If this option is not provided, +Dynare assumes `nargs = 1`. +::: + +::: {.option} +first\_deriv\_provided \[= NAME\] + +If NAME is provided, this tells Dynare that the Jacobian is provided as +the only output of the Julia file given as the option argument. If NAME +is not provided, this tells Dynare that the Julia file specified by the +argument passed to NAME returns the Jacobian as its second output +argument. When this option is not provided, Dynare will use finite +difference approximations for computing the derivatives of the function, +whenever needed. +::: + +::: {.option} +second\_deriv\_provided \[= NAME\] + +If NAME is provided, this tells Dynare that the Hessian is provided as +the only output of the Julia file given as the option argument. If NAME +is not provided, this tells Dynare that the Julia file specified by the +argument passed to NAME returns the Hessian as its third output +argument. NB: This option can only be used if the `first_deriv_provided` +option is used in the same `external_function` command. When this option +is not provided, Dynare will use finite difference approximations for +computing the Hessian derivatives of the function, whenever needed. +::: + +*Example* + +> external_function(name = funcname); +> external_function(name = otherfuncname, nargs = 2, first_deriv_provided, second_deriv_provided); +> external_function(name = yetotherfuncname, nargs = 3, first_deriv_provided = funcname_deriv); +::: + +### A few words of warning in stochastic context + +The use of the following functions and operators is strongly discouraged +in a stochastic context: `max`, `min`, `abs`, `sign`, `<`, `>`, `<=`, +`>=`, `==`, `!=`. + +The reason is that the local approximation used by `stoch_simul` or +`estimation` will by nature ignore the non-linearities introduced by +these functions if the steady state is away from the kink. And, if the +steady state is exactly at the kink, then the approximation will be +bogus because the derivative of these functions at the kink is bogus (as +explained in the respective documentations of these functions and +operators). + +Note that `extended_path` is not affected by this problem, because it +does not rely on a local approximation of the mode. + +Parameter initialization {#param-init} +------------------------ + +When using Dynare for computing simulations, it is necessary to +calibrate the parameters of the model. This is done through parameter +initialization. + +The syntax is the following: + + PARAMETER_NAME = EXPRESSION; + +Here is an example of calibration: + + parameters alpha, beta; + + beta = 0.99; + alpha = 0.36; + A = 1-alpha*beta; + +Internally, the parameter values are stored in `M_.params`: + +::: {.matvar} +[M]().params + +Contains the values of model parameters. The parameters are in the order +that was used in the `parameters` command, hence ordered as in +`M_.param_names`. +::: + +The parameter names are stored in `M_.param_names`: + +::: {.matvar} +[M]().param\_names + +Cell array containing the names of the model parameters. +::: + +::: {.matcomm} +get\_param\_by\_name (\'PARAMETER\_NAME\'); + +Given the name of a parameter, returns its calibrated value as it is +stored in `M_.params`. +::: + +::: {.matcomm} +set\_param\_value (\'PARAMETER\_NAME\', MATLAB\_EXPRESSION); + +Sets the calibrated value of a parameter to the provided expression. +This does essentially the same as the parameter initialization syntax +described above, except that it accepts arbitrary MATLAB/Octave +expressions, and that it works from MATLAB/Octave scripts. +::: + +Model declaration {#model-decl} +----------------- + +The model is declared inside a `model` block: + +::: {.block} +model ; model (OPTIONS\...); + +The equations of the model are written in a block delimited by `model` +and `end` keywords. + +There must be as many equations as there are endogenous variables in the +model, except when computing the unconstrained optimal policy with +`ramsey_model`, `ramsey_policy` or `discretionary_policy`. + +The syntax of equations must follow the conventions for +MODEL\_EXPRESSION as described in `expr`{.interpreted-text role="ref"}. +Each equation must be terminated by a semicolon (';'). A normal equation +looks like: + +> MODEL\_EXPRESSION = MODEL\_EXPRESSION; + +When the equations are written in homogenous form, it is possible to +omit the '=0' part and write only the left hand side of the equation. A +homogenous equation looks like: + +> MODEL\_EXPRESSION; + +Inside the model block, Dynare allows the creation of *model-local +variables*, which constitute a simple way to share a common expression +between several equations. The syntax consists of a pound sign (\#) +followed by the name of the new model local variable (which must **not** +be declared as in `var-decl`{.interpreted-text role="ref"}, but may have +been declared by `model_local_variable`{.interpreted-text role="comm"}), +an equal sign, and the expression for which this new variable will +stand. Later on, every time this variable appears in the model, Dynare +will substitute it by the expression assigned to the variable. Note that +the scope of this variable is restricted to the model block; it cannot +be used outside. To assign a LaTeX name to the model local variable, use +the declaration syntax outlined by +`model_local_variable`{.interpreted-text role="comm"}. A model local +variable declaration looks like: + +> \#VARIABLE\_NAME = MODEL\_EXPRESSION; + +It is possible to tag equations written in the model block. A tag can +serve different purposes by allowing the user to attach arbitrary +informations to each equation and to recover them at runtime. For +instance, it is possible to name the equations with a `name`-tag, using +a syntax like: + + model; + + [name = 'Budget constraint']; + c + k = k^theta*A; + + end; + +Here, `name` is the keyword indicating that the tag names the equation. +If an equation of the model is tagged with a name, the `resid` command +will display the name of the equations (which may be more informative +than the equation numbers) in addition to the equation number. Several +tags for one equation can be separated using a comma: + + model; + + [name='Taylor rule',mcp = 'r > -1.94478'] + r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e; + + end; + +More information on tags is available at +. + +There can be several `model` blocks, in which case they are simply +concatenated. The set of effective options is also the concatenation of +the options declared in all the blocks, but in that case you may rather +want to use the `model_options`{.interpreted-text role="comm"} command. + +*Options* + +::: {.option} +linear + +Declares the model as being linear. It spares oneself from having to +declare initial values for computing the steady state of a stationary +linear model. This option can't be used with non-linear models, it will +NOT trigger linearization of the model. +::: + +::: {.option} +no\_static + +Don't create the static model file. This can be useful for models which +don't have a steady state. +::: + +::: {.option} +differentiate\_forward\_vars differentiate\_forward\_vars = ( +VARIABLE\_NAME \[VARIABLE\_NAME \...\] ) + +Tells Dynare to create a new auxiliary variable for each endogenous +variable that appears with a lead, such that the new variable is the +time differentiate of the original one. More precisely, if the model +contains `x(+1)`, then a variable `AUX_DIFF_VAR` will be created such +that `AUX_DIFF_VAR=x-x(-1)`, and `x(+1)` will be replaced with +`x+AUX_DIFF_VAR(+1)`. + +The transformation is applied to all endogenous variables with a lead if +the option is given without a list of variables. If there is a list, the +transformation is restricted to endogenous with a lead that also appear +in the list. + +This option can useful for some deterministic simulations where +convergence is hard to obtain. Bad values for terminal conditions in the +case of very persistent dynamics or permanent shocks can hinder correct +solutions or any convergence. The new differentiated variables have +obvious zero terminal conditions (if the terminal condition is a steady +state) and this in many cases helps convergence of simulations. +::: + +::: {.option} +parallel\_local\_files = ( FILENAME \[, FILENAME\]\... ) + +Declares a list of extra files that should be transferred to follower +nodes when doing a parallel computation (see +`paral-conf`{.interpreted-text role="ref"}). +::: + +::: {.option} +balanced\_growth\_test\_tol = DOUBLE + +Tolerance used for determining whether cross-derivatives are zero in the +test for balanced growth path (the latter is documented on +). Default: +`1e-6` +::: + +*Example* (Elementary RBC model) + +> var c k; +> varexo x; +> parameters aa alph bet delt gam; +> +> model; +> c = - k + aa*x*k(-1)^alph + (1-delt)*k(-1); +> c^(-gam) = (aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam)/(1+bet); +> end; + +*Example* (Use of model local variables) + +> The following program: +> +> model; +> # gamma = 1 - 1/sigma; +> u1 = c1^gamma/gamma; +> u2 = c2^gamma/gamma; +> end; +> +> \...is formally equivalent to: +> +> model; +> u1 = c1^(1-1/sigma)/(1-1/sigma); +> u2 = c2^(1-1/sigma)/(1-1/sigma); +> end; + +*Example* (A linear model) + +> model(linear); +> x = a*x(-1)+b*y(+1)+e_x; +> y = d*y(-1)+e_y; +> end; +::: + +::: {.command} +model\_options (OPTIONS\...); + +This command accepts the same options as the `model`{.interpreted-text +role="bck"} block. + +The purpose of this statement is to specify the options that apply to +the whole model, when there are several `model` blocks, so as to restore +the symmetry between those blocks (since otherwise one `model` block +would typically bear the options, while the other ones would typically +have no option). +::: + +::: {.command} +model\_remove (TAGS\...); + +This command removes equations that appeared in a previous +`model`{.interpreted-text role="bck"} block. + +The equations must be specified by a list of tag values, separated by +commas. Each element of the list is either a simple quoted string, in +which case it designates an equation by its `name` tag; or a tag name +(without quotes), followed by an equal sign, then by the tag value +(within quotes). + +Each removed equation must either have an `endogenous` tag, or have a +left hand side containing a single endogenous variable. The +corresponding endogenous variable will be either turned into an +exogenous (if it is still used in somewhere in the model at that point), +otherwise it will be removed from the model. + +*Example* + +> var c k dummy1 dummy2; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1) + dummy1; +> c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> [ name = 'eq:dummy1', endogenous = 'dummy1' ] +> c*k = dummy1; +> [ foo = 'eq:dummy2' ] +> log(dummy2) = k + 2; +> end; +> +> model_remove('eq:dummy1', foo = 'eq:dummy2'); +> +> In the above example, the last two equations will be removed, `dummy1` +> will be turned into an exogenous, and `dummy2` will be removed. +::: + +::: {.block} +model\_replace (TAGS\...); + +This block replaces several equations in the model. It removes the +equations given by the tags list (with the same syntax as in +`model_remove`{.interpreted-text role="comm"}), and it adds equations +given within the block (with the same syntax as +`model`{.interpreted-text role="bck"}). + +No variable is removed or has its type changed in the process. + +*Example* + +> var c k; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); +> [ name = 'dummy' ] +> c*k = 1; +> end; +> +> model_replace('dummy'); +> c^(-gam) = (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> end; +> +> In the above example, the dummy equation is replaced by a proper Euler +> equation. +::: + +Dynare has the ability to output the original list of model equations to +a LaTeX file, using the `write_latex_original_model` command, the list +of transformed model equations using the +`write_latex_dynamic_model command`, and the list of static model +equations using the `write_latex_static_model` command. + +::: {.command} +write\_latex\_original\_model (OPTIONS); + +This command creates two LaTeX files: one containing the model as +defined in the model block and one containing the LaTeX document header +information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/original.tex`, which includes a file called +`FILENAME/latex/original_content.tex` (also created by Dynare) +containing the list of all the original model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Time subscripts (`t`, `t+1`, `t-1`, \...) will be appended to the +variable names, as LaTeX subscripts. + +Compiling the TeX file requires the following LaTeX packages: +`geometry, fullpage, breqn`. + +*Options* + +::: {.option} +write\_equation\_tags + +Write the equation tags in the LaTeX output. The equation tags will be +interpreted with LaTeX markups. +::: +::: + +::: {.command} +write\_latex\_dynamic\_model ; write\_latex\_dynamic\_model (OPTIONS); + +This command creates two LaTeX files: one containing the dynamic model +and one containing the LaTeX document header information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/dynamic.tex`, which includes a file called +`FILENAME/latex/dynamic_content.tex` (also created by Dynare) containing +the list of all the dynamic model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Time subscripts (`t`, `t+1`, `t-1`, \...) will be appended to the +variable names, as LaTeX subscripts. + +Note that the model written in the TeX file will differ from the model +declared by the user in the following dimensions: + +> - The timing convention of predetermined variables (see +> `predetermined_variables`{.interpreted-text role="comm"}) will +> have been changed to the default Dynare timing convention; in +> other words, variables declared as predetermined will be lagged on +> period back, +> - The `EXPECTATION` operators will have been removed, replaced by +> auxiliary variables and new equations (as explained in the +> documentation of +> `EXPECTATION `{.interpreted-text +> role="op"}), +> - Endogenous variables with leads or lags greater or equal than two +> will have been removed, replaced by new auxiliary variables and +> equations, +> - Exogenous variables with leads or lags will also have been +> replaced by new auxiliary variables and equations. + +For the required LaTeX packages, see +`write_latex_original_model`{.interpreted-text role="comm"}. + +*Options* + +::: {.option} +write\_equation\_tags + +See `write_equation_tags`{.interpreted-text role="opt"} +::: +::: + +::: {.command} +write\_latex\_static\_model (OPTIONS); + +This command creates two LaTeX files: one containing the static model +and one containing the LaTeX document header information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/static.tex`, which includes a file called +`FILENAME/latex/static_content.tex` (also created by Dynare) containing +the list of all the steady state model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Note that the model written in the TeX file will differ from the model +declared by the user in the some dimensions (see +`write_latex_dynamic_model`{.interpreted-text role="comm"} for details). + +Also note that this command will not output the contents of the optional +`steady_state_model` block (see `steady_state_model`{.interpreted-text +role="bck"}); it will rather output a static version (i.e. without leads +and lags) of the dynamic `model` declared in the model block. To write +the LaTeX contents of the `steady_state_model` see +`write_latex_steady_state_model`{.interpreted-text role="comm"}. + +For the required LaTeX packages, see +`write_latex_original_model`{.interpreted-text role="comm"}. + +*Options* + +::: {.option} +write\_equation\_tags + +See `write_equation_tags`{.interpreted-text role="opt"}. +::: +::: + +::: {.command} +write\_latex\_steady\_state\_model + +This command creates two LaTeX files: one containing the steady state +model and one containing the LaTeX document header information. + +If your `.mod` file is `FILENAME.mod`, then Dynare will create a file +called `FILENAME/latex/steady_state.tex`, which includes a file called +`FILENAME/latex/steady_state_content.tex` (also created by Dynare) +containing the list of all the steady state model equations. + +If LaTeX names were given for variables and parameters (see +`var-decl`{.interpreted-text role="ref"}), then those will be used; +otherwise, the plain text names will be used. + +Note that the model written in the `.tex` file will differ from the +model declared by the user in some dimensions (see +`write_latex_dynamic_model`{.interpreted-text role="comm"} for details). + +For the required LaTeX packages, see +`write_latex_original_model`{.interpreted-text role="comm"}. +::: + +Auxiliary variables {#aux-variables} +------------------- + +The model which is solved internally by Dynare is not exactly the model +declared by the user. In some cases, Dynare will introduce auxiliary +endogenous variables---along with corresponding auxiliary +equations---which will appear in the final output. + +The main transformation concerns leads and lags. Dynare will perform a +transformation of the model so that there is only one lead and one lag +on endogenous variables and no leads/lags on exogenous variables. + +This transformation is achieved by the creation of auxiliary variables +and corresponding equations. For example, if `x(+2)` exists in the +model, Dynare will create one auxiliary variable +`AUX_ENDO_LEAD = x(+1)`, and replace `x(+2)` by `AUX_ENDO_LEAD(+1)`. + +A similar transformation is done for lags greater than 2 on endogenous +(auxiliary variables will have a name beginning with `AUX_ENDO_LAG`), +and for exogenous with leads and lags (auxiliary variables will have a +name beginning with `AUX_EXO_LEAD` or `AUX_EXO_LAG` respectively). + +Another transformation is done for the `EXPECTATION` operator. For each +occurrence of this operator, Dynare creates an auxiliary variable +defined by a new equation, and replaces the expectation operator by a +reference to the new auxiliary variable. For example, the expression +`EXPECTATION(-1)(x(+1))` is replaced by `AUX_EXPECT_LAG_1(-1)`, and the +new auxiliary variable is declared as `AUX_EXPECT_LAG_1 = x(+2)`. + +Auxiliary variables are also introduced by the preprocessor for the +`ramsey_model` and `ramsey_policy` commands. In this case, they are used +to represent the Lagrange multipliers when first order conditions of the +Ramsey problem are computed. The new variables take the form `MULT_i`, +where *i* represents the constraint with which the multiplier is +associated (counted from the order of declaration in the model block). + +Auxiliary variables are also introduced by the +`differentiate_forward_vars` option of the model block. The new +variables take the form `AUX_DIFF_FWRD_i`, and are equal to `x-x(-1)` +for some endogenous variable `x`. + +Finally, auxiliary variables will arise in the context of employing the +`diff`-operator. + +Once created, all auxiliary variables are included in the set of +endogenous variables. The output of decision rules (see below) is such +that auxiliary variable names are replaced by the original variables +they refer to. + +The number of endogenous variables before the creation of auxiliary +variables is stored in `M_.orig_endo_nbr`, and the number of endogenous +variables after the creation of auxiliary variables is stored in +`M_.endo_nbr`. + +See +for more technical details on auxiliary variables. + +Initial and terminal conditions {#init-term-cond} +------------------------------- + +For most simulation exercises, it is necessary to provide initial (and +possibly terminal) conditions. It is also necessary to provide initial +guess values for non-linear solvers. This section describes the +statements used for those purposes. + +In many contexts (deterministic or stochastic), it is necessary to +compute the steady state of a non-linear model: `initval` then specifies +numerical initial values for the non-linear solver. The command `resid` +can be used to compute the equation residuals for the given initial +values. + +Used in perfect foresight mode, the types of forward-looking models for +which Dynare was designed require both initial and terminal conditions. +Most often these initial and terminal conditions are static equilibria, +but not necessarily. + +One typical application is to consider an economy at the equilibrium at +time 0, trigger a shock in first period, and study the trajectory of +return to the initial equilibrium. To do that, one needs `initval` and +`shocks` (see `shocks-exo`{.interpreted-text role="ref"}). + +Another one is to study how an economy, starting from arbitrary initial +conditions at time 0 converges towards equilibrium. In this case models, +the command `histval` permits to specify different historical initial +values for variables with lags for the periods before the beginning of +the simulation. Due to the design of Dynare, in this case `initval` is +used to specify the terminal conditions. + +::: {.block} +initval ; initval(OPTIONS\...); + +The `initval` block has two main purposes: providing guess values for +non-linear solvers in the context of perfect foresight simulations and +providing guess values for steady state computations in both perfect +foresight and stochastic simulations. Depending on the presence of +`histval` and `endval` blocks it is also used for declaring the initial +and terminal conditions in a perfect foresight simulation exercise. +Because of this interaction of the meaning of an `initval` block with +the presence of `histval` and `endval` blocks in perfect foresight +simulations, it is strongly recommended to check that the constructed +`oo_.endo_simul` and `oo_.exo_simul` variables contain the desired +values after running `perfect_foresight_setup` and before running +`perfect_foresight_solver`. In the presence of leads and lags, these +subfields of the results structure will store the historical values for +the lags in the first column/row and the terminal values for the leads +in the last column/row. + +The `initval` block is terminated by `end;` and contains lines of the +form: + +> VARIABLE\_NAME = EXPRESSION; + +*In a deterministic (i.e. perfect foresight) model* + +First, both the `oo_.endo_simul` and `oo_.exo_simul` variables storing +the endogenous and exogenous variables will be filled with the values +provided by this block. If there are no other blocks present, it will +therefore provide the initial and terminal conditions for all the +endogenous and exogenous variables, because it will also fill the last +column/row of these matrices. For the intermediate simulation periods it +thereby provides the starting values for the solver. In the presence of +a `histval` block (and therefore absence of an `endval` block), this +`histval` block will provide/overwrite the historical values for the +state variables (lags) by setting the first column/row of +`oo_.endo_simul` and `oo_.exo_simul`. This implies that the `initval` +block in the presence of `histval` only sets the terminal values for the +variables with leads and provides initial values for the perfect +foresight solver. + +Because of these various functions of `initval` it is often necessary to +provide values for all the endogenous variables in an `initval` block. +Initial and terminal conditions are strictly necessary for lagged/leaded +variables, while feasible starting values are required for the solver. +It is important to be aware that if some variables, endogenous or +exogenous, are not mentioned in the `initval` block, a zero value is +assumed. It is particularly important to keep this in mind when +specifying exogenous variables using `varexo` that are not allowed to +take on the value of zero, like e.g. TFP. + +Note that if the `initval` block is immediately followed by a `steady` +command, its semantics are slightly changed. The `steady` command will +compute the steady state of the model for all the endogenous variables, +assuming that exogenous variables are kept constant at the value +declared in the `initval` block. These steady state values conditional +on the declared exogenous variables are then written into +`oo_.endo_simul` and take up the potential roles as historical and +terminal conditions as well as starting values for the solver. An +`initval` block followed by `steady` is therefore formally equivalent to +an `initval` block with the specified values for the exogenous +variables, and the endogenous variables set to the associated steady +state values conditional on the exogenous variables. + +*In a stochastic model* + +The main purpose of `initval` is to provide initial guess values for the +non-linear solver in the steady state computation. Note that if the +`initval` block is not followed by `steady`, the steady state +computation will still be triggered by subsequent commands +(`stoch_simul`, `estimation`\...). + +As such, `initval` allows specifying the initial instrument value for +steady state finding when providing an analytical conditional steady +state file for `ramsey_model`-computations. + +It is not necessary to declare 0 as initial value for exogenous +stochastic variables, since it is the only possible value. + +The subsequently computed steady state (not the initial values, use +histval for this) will be used as the initial condition at all the +periods preceeding the first simulation period for the three possible +types of simulations in stochastic mode: + +> - `stoch_simul`{.interpreted-text role="comm"}, if the `periods` +> option is specified. +> - `forecast`{.interpreted-text role="comm"} as the initial point at +> which the forecasts are computed. +> - `conditional_forecast`{.interpreted-text role="comm"} as the +> initial point at which the conditional forecasts are computed. + +To start simulations at a particular set of starting values that are not +a computed steady state, use `histval`{.interpreted-text role="bck"}. + +*Options* + +::: {.option} +all\_values\_required + +Issues an error and stops processing the .mod file if there is at least +one endogenous or exogenous variable that has not been set in the +initval block. +::: + +*Example* + +: initval; + c = 1.2; + k = 12; + x = 1; + end; + + steady; +::: + +::: {.block} +endval ; endval (OPTIONS\...); + +This block is terminated by `end;` and contains lines of the form: + +> VARIABLE\_NAME = EXPRESSION; + +The `endval` block makes only sense in a deterministic model and cannot +be used together with `histval`. Similar to the `initval` command, it +will fill both the `oo_.endo_simul` and `oo_.exo_simul` variables +storing the endogenous and exogenous variables with the values provided +by this block. If no `initval` block is present, it will fill the whole +matrices, therefore providing the initial and terminal conditions for +all the endogenous and exogenous variables, because it will also fill +the first and last column/row of these matrices. Due to also filling the +intermediate simulation periods it will provide the starting values for +the solver as well. + +If an `initval` block is present, `initval` will provide the historical +values for the variables (if there are states/lags), while `endval` will +fill the remainder of the matrices, thereby still providing *i*) the +terminal conditions for variables entering the model with a lead and +*ii*) the initial guess values for all endogenous variables at all the +simulation dates for the perfect foresight solver. + +Note that if some variables, endogenous or exogenous, are NOT mentioned +in the `endval` block, the value assumed is that of the last `initval` +block or `steady` command (if present). Therefore, in contrast to +`initval`, omitted variables are not automatically assumed to be 0 in +this case. Again, it is strongly recommended to check the constructed +`oo_.endo_simul` and `oo_.exo_simul` variables after running +`perfect_foresight_setup` and before running `perfect_foresight_solver` +to see whether the desired outcome has been achieved. + +Like `initval`, if the `endval` block is immediately followed by a +`steady` command, its semantics are slightly changed. The `steady` +command will compute the steady state of the model for all the +endogenous variables, assuming that exogenous variables are kept +constant to the value declared in the `endval` block. These steady state +values conditional on the declared exogenous variables are then written +into `oo_.endo_simul` and therefore take up the potential roles as +historical and terminal conditions as well as starting values for the +solver. An `endval` block followed by `steady` is therefore formally +equivalent to an `endval` block with the specified values for the +exogenous variables, and the endogenous variables set to the associated +steady state values. + +*Options* + +::: {.option} +all\_values\_required + +See `all_values_required`{.interpreted-text role="opt"}. +::: + +*Example* + +> var c k; +> varexo x; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); +> c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> end; +> +> initval; +> c = 1.2; +> k = 12; +> x = 1; +> end; +> +> steady; +> +> endval; +> c = 2; +> k = 20; +> x = 2; +> end; +> +> steady; +> +> perfect_foresight_setup(periods=200); +> perfect_foresight_solver; +> +> In this example, the problem is finding the optimal path for +> consumption and capital for the periods $t=1$ to $T=200$, given the +> path of the exogenous technology level `x`. `c` is a forward-looking +> variable and the exogenous variable `x` appears with a lead in the +> expected return of physical capital, while `k` is a purely +> backward-looking (state) variable. +> +> The initial equilibrium is computed by `steady` conditional on `x=1`, +> and the terminal one conditional on `x=2`. The `initval` block sets +> the initial condition for `k` (since it is the only backward-looking +> variable), while the `endval` block sets the terminal condition for +> `c` (since it is the only forward-looking endogenous variable). The +> starting values for the perfect foresight solver are given by the +> `endval` block. See below for more details. + +*Example* + +> var c k; +> varexo x; +> +> model; +> c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); +> c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); +> end; +> +> initval; +> k = 12; +> end; +> +> endval; +> c = 2; +> x = 1.1; +> end; +> +> perfect_foresight_setup(periods=200); +> perfect_foresight_solver; +> +> In this example, there is no [steady]{.title-ref} command, hence the +> conditions are exactly those specified in the [initval]{.title-ref} +> and [endval]{.title-ref} blocks. We need terminal conditions for `c` +> and `x`, since both appear with a lead, and an initial condition for +> `k`, since it appears with a lag. +> +> Setting `x=1.1` in the `endval` block without a `shocks` block implies +> that technology is at $1.1$ in $t=1$ and stays there forever, because +> `endval` is filling all entries of `oo_.endo_simul` and +> `oo_.exo_simul` except for the very first one, which stores the +> initial conditions and was set to $0$ by the `initval` block when not +> explicitly specifying a value for it. +> +> Because the law of motion for capital is backward-looking, we need an +> initial condition for `k` at time $0$. Due to the presence of +> `endval`, this cannot be done via a `histval` block, but rather must +> be specified in the `initval` block. Similarly, because the Euler +> equation is forward-looking, we need a terminal condition for `c` at +> $t=201$, which is specified in the `endval` block. +> +> As can be seen, it is not necessary to specify `c` and `x` in the +> `initval` block and `k` in the `endval` block, because they have no +> impact on the results. Due to the optimization problem in the first +> period being to choose `c,k` at $t=1$ given the predetermined capital +> stock `k` inherited from $t=0$ as well as the current and future +> values for technology `x`, the values for `c` and `x` at time $t=0$ +> play no role. The same applies to the choice of `c,k` at time $t=200$, +> which does not depend on `k` at $t=201$. As the Euler equation shows, +> that choice only depends on current capital as well as future +> consumption `c` and technology `x`, but not on future capital `k`. The +> intuitive reason is that those variables are the consequence of +> optimization problems taking place in at periods $t=0$ and $t=201$, +> respectively, which are not modeled here. + +*Example* + +> initval; +> c = 1.2; +> k = 12; +> x = 1; +> end; +> +> endval; +> c = 2; +> k = 20; +> x = 1.1; +> end; +> +> In this example, initial conditions for the forward-looking variables +> `x` and `c` are provided, together with a terminal condition for the +> backward-looking variable `k`. As shown in the previous example, these +> values will not affect the simulation results. Dynare simply takes +> them as given and basically assumes that there were realizations of +> exogenous variables and states that make those choices equilibrium +> values (basically initial/terminal conditions at the unspecified time +> periods $t<0$ and $t>201$). +> +> The above example suggests another way of looking at the use of +> `steady` after `initval` and `endval`. Instead of saying that the +> implicit unspecified conditions before and after the simulation range +> have to fit the initial/terminal conditions of the endogenous +> variables in those blocks, steady specifies that those conditions at +> $t<0$ and $t>201$ are equal to being at the steady state given the +> exogenous variables in the `initval` and `endval` blocks. The +> endogenous variables at $t=0$ and $t=201$ are then set to the +> corresponding steady state equilibrium values. +> +> The fact that `c` at $t=0$ and `k` at $t=201$ specified in `initval` +> and `endval` are taken as given has an important implication for +> plotting the simulated vector for the endogenous variables, i.e. the +> rows of `oo_.endo_simul`: this vector will also contain the initial +> and terminal conditions and thus is 202 periods long in the example. +> When you specify arbitrary values for the initial and terminal +> conditions for forward- and backward-looking variables, respectively, +> these values can be very far away from the endogenously determined +> values at $t=1$ and $t=200$. While the values at $t=0$ and $t=201$ are +> unrelated to the dynamics for $0 strange-looking large jumps. In the example above, consumption will +> display a large jump from $t=0$ to $t=1$ and capital will jump from +> $t=200$ to $t=201$ when using `rplot`{.interpreted-text role="comm"} +> or manually plotting `oo_.endo_val`. +::: + +::: {.block} +histval ; histval (OPTIONS\...); + +*In a deterministic perfect foresight context* + +In models with lags on more than one period, the `histval` block permits +to specify different historical initial values for different periods of +the state variables. In this case, the `initval` block takes over the +role of specifying terminal conditions and starting values for the +solver. Note that the `histval` block does not take non-state variables. + +This block is terminated by `end;` and contains lines of the form: + +> VARIABLE\_NAME(INTEGER) = EXPRESSION; + +EXPRESSION is any valid expression returning a numerical value and can +contain already initialized variable names. + +By convention in Dynare, period 1 is the first period of the simulation. +Going backward in time, the first period before the start of the +simulation is period 0, then period -1, and so on. + +State variables not initialized in the `histval` block are assumed to +have a value of zero at period 0 and before. Note that `histval` cannot +be followed by `steady`. + +*Example* + +> model; +> x=1.5*x(-1)-0.6*x(-2)+epsilon; +> log(c)=0.5*x+0.5*log(c(+1)); +> end; +> +> histval; +> x(0)=-1; +> x(-1)=0.2; +> end; +> +> initval; +> c=1; +> x=1; +> end; +> +> In this example, `histval` is used to set the historical conditions +> for the two lags of the endogenous variable `x`, stored in the first +> column of `oo_.endo_simul`. The `initval` block is used to set the +> terminal condition for the forward looking variable `c`, stored in the +> last column of `oo_.endo_simul`. Moreover, the `initval` block defines +> the starting values for the perfect foresight solver for both +> endogenous variables `c` and `x`. + +*In a stochastic simulation context* + +In the context of stochastic simulations, `histval` allows setting the +starting point of those simulations in the state space. As for the case +of perfect foresight simulations, all not explicitly specified variables +are set to 0. Moreover, as only states enter the recursive policy +functions, all values specified for control variables will be ignored. +This can be used + +> - In `stoch_simul`{.interpreted-text role="comm"}, if the `periods` +> option is specified. Note that this only affects the starting +> point for the simulation, but not for the impulse response +> functions. When using the `loglinear `{.interpreted-text +> role="ref"} option, the `histval` block nevertheless takes the +> unlogged starting values. +> - In `forecast`{.interpreted-text role="comm"} as the initial point +> at which the forecasts are computed. When using the `loglinear +> `{.interpreted-text role="ref"} option, the `histval` block +> nevertheless takes the unlogged starting values. +> - In `conditional_forecast`{.interpreted-text role="comm"} for a +> calibrated model as the initial point at which the conditional +> forecasts are computed. When using the +> `loglinear `{.interpreted-text role="ref"} option, the +> histval-block nevertheless takes the unlogged starting values. +> - In `Ramsey policy `{.interpreted-text role="comm"}, +> where it also specifies the values of the endogenous states +> (including lagged exogenous) at which the objective function of +> the planner is computed. Note that the initial values of the +> Lagrange multipliers associated with the planner's problem cannot +> be set (see `evaluate_planner_objective`{.interpreted-text +> role="comm"}). + +*Options* + +::: {.option} +all\_values\_required + +See `all_values_required`{.interpreted-text role="opt"}. +::: + +*Example* + +> var x y; +> varexo e; +> +> model; +> x = y(-1)^alpha*y(-2)^(1-alpha)+e; +> +> end; +> +> initval; +> x = 1; +> y = 1; +> e = 0.5; +> end; +> +> steady; +> +> histval; +> y(0) = 1.1; +> y(-1) = 0.9; +> end; +> +> stoch_simul(periods=100); +::: + +::: {.command} +resid ; + +This command will display the residuals of the static equations of the +model, using the values given for the endogenous in the last `initval` +or `endval` block (or the steady state file if you provided one, see +`st-st`{.interpreted-text role="ref"}). + +*Options* + +::: {.option} +non\_zero + +Only display non-zero residuals. +::: +::: + +::: {.command} +initval\_file (OPTIONS\...); + +In a deterministic setup, this command is used to specify a path for all +endogenous and exogenous variables. The length of these paths must be +equal to the number of simulation periods, plus the number of leads and +the number of lags of the model (for example, with 50 simulation +periods, in a model with 2 lags and 1 lead, the paths must have a length +of 53). Note that these paths cover two different things: + +> - The constraints of the problem, which are given by the path for +> exogenous and the initial and terminal values for endogenous +> - The initial guess for the non-linear solver, which is given by the +> path for endogenous variables for the simulation periods +> (excluding initial and terminal conditions) + +In perfect foresight and stochastic contexts, `steady` uses the first +observation loaded by `initval_file` as guess value to solve for the +steady state of the model. This first observation is determined by the +`first_obs` option when it is used. + +Don't mix `initval_file` with `initval` statements. However, after +`initval_file`, you can modify the historical initial values with +`histval` or `histval_file` statement. + +There can be several `initval_file` statements in a model file. Each +statement resets `oo_.initval_series`. + +*Options* + +::: {.option} +datafile = FILENAME filename = FILENAME (deprecated) + +The name of the file containing the data. It must be included in quotes +if the filename contains a path or an extension. The command accepts the +following file formats: + +- M-file (extension `.m`): for each endogenous and exogenous variable, + the file must contain a row or column vector of the same name. +- MAT-file (extension `.mat`): same as for M-files. +- Excel file (extension `.xls` or `.xlsx`): for each endogenous and + exogenous variable, the file must contain a column of the same name. + NB: Octave only supports the `.xlsx` file extension and must have + the [io](https://octave.sourceforge.io/io/) package installed + (easily done via octave by typing '`pkg install -forge io`'). The + first column may contain the date +::: + +> of each observation. +> +> : - CSV files (extension `.csv`): for each endogenous and +> exogenous variable, the file must contain a column of the same +> name. The first column may contain the date of each +> observation. +> +::: {.option} +first\_obs = {INTEGER \| DATE} + +The observation number or the date (see +::: + +`dates-members`{.interpreted-text role="ref"}) of the first observation +to be used in the file + +::: {.option} +first\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates `{.interpreted-text role="ref"}) at which the +simulation (or the forecast) is starting. This option avoids to have to +compute the maximum number of lags in the model. The observation +corresponding to the first period of simulation doesn't need to exist in +the file as the only dates necessary for initialization are before that +date. + +::: {.option} +last\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates `{.interpreted-text role="ref"}) at which the +simulation (or the forecast) is ending. This option avoids to have to +compute the maximum number of leads in the model. + +::: {.option} +last\_obs = {INTEGER \| DATE} + +The observaton number or the date (see +::: + +`dates-members`{.interpreted-text role="ref"}) of the last observation +to be used in the file. + +::: {.option} +nobs = INTEGER +::: + +The number of observations to be used in the file (starting with first +of `first_obs` observation). + +::: {.option} +series = DSERIES NAME + +The name of a DSERIES containing the data (see +`dseries-members`{.interpreted-text role="ref"}) +::: + +*Example 1* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv` +(nothing guarantees that these vales are the steady state of the model). +The guess value for the trajectories are also taken from the file. The +file must contain at least 203 observations of variables `c`, `x` and +`e`. If there are more than 203 observations available in the file, the +first 203 are used by `perfect_foresight_setup(periods=200)`. Note that +the values for the auxiliary variable corresponding to `x(-2)` are +automatically computed by `initval_file`. + +*Example 2* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs=10); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv` +starting with the 10th observation in the file. There must be at least +212 observations in the file. + +*Example 3* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> ds = dseries(mydata.csv); lds = log(ds); +> +> initval\_file(series=lds, +> +> : first\_obs=2010Q1); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from dseries `lds`. All +observations are loaded starting with the 1st quarter of 2010 until the +end of the file. There must be data available at least until 2050Q3. + +*Example 4* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_simulation\_period=2010Q1); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv`. The +observations in the file must have dates. All observations are loaded +from the 3rd quarter of 2009 until the end of the file. There must be +data available in the file at least until 2050Q1. + +*Example 5* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : last\_obs = 212); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv`. The +first 212 observations are loaded and the first 203 observations will be +used by `perfect_foresight_setup(periods=200)`. + +*Example 6* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs = 10, +> +> > nobs = 203); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +The initial and terminal values are taken from file `mydata.csv`. +Observations 10 to 212 are loaded. + +*Example 7* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs = 10); +> +> steady; +> +> The values of the 10th observation of `mydata.csv` are used + +as guess value to compute the steady state. The exogenous variables are +set to values found in the file or zero if these variables aren\'t +present. +::: + +::: {.command} +histval\_file (OPTIONS\...); + +This command is equivalent to `histval`, except that it reads its input +from a file, and is typically used in conjunction with +`smoother2histval`. + +> *Options* + +::: {.option} +datafile = FILENAME filename = FILENAME (deprecated) + +The name of the file containing the data. The command accepts +::: + +the following file formats: + +> - M-file (extension `.m`): for each endogenous and exogenous +> variable, the file must contain a row or column vector of the same +> name. +> - MAT-file (extension `.mat`): same as for M-files. +> - Excel file (extension `.xls` or `.xlsx`): for each endogenous and +> exogenous variable, the file must contain a column of the same +> name. NB: Octave only supports the `.xlsx` file extension and must +> have the [io](https://octave.sourceforge.io/io/) package installed +> (easily done via octave by typing '`pkg install -forge io`'). The +> first column may contain the date of each observation. +> - CSV files (extension `.csv`): for each endogenous and exogenous +> variable, the file must contain a column of the same name. The +> first column may contain the date of each observation. + +::: {.option} +first\_obs = {INTEGER \| DATE} + +The observation number or the date (see +`dates-members`{.interpreted-text role="ref"}) of +::: + +the first observation to be used in the file + +::: {.option} +first\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates-members`{.interpreted-text role="ref"}) at which the simulation +(or the forecast) is starting. This option avoids to have to compute the +maximum number of lags in the model. The observation corresponding to +the first period of simulation doesn't need to exist in the file as the +only dates necessary for initialization are before that date. + +::: {.option} +last\_simulation\_period = {INTEGER \| DATE} +::: + +The observation number in the file or the date (see +`dates `{.interpreted-text role="ref"}) at which the +simulation (or the forecast) is ending. This option avoids to have to +compute the maximum number of leads in the model. + +::: {.option} +last\_obs = {INTEGER \| DATE} + +The observation number or the date (see +`dates-members`{.interpreted-text role="ref"}) of the +::: + +last observation to be used in the file. + +::: {.option} +nobs = INTEGER +::: + +The number of observations to be used in the file (starting with first +of `first_obs` observation). + +::: {.option} +series = DSERIES NAME + +The name of a DSERIES containing the data (see +`dseries-members`{.interpreted-text role="ref"}) +::: + +*Example 1* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> steady\_state\_model; x = 0; c = exp(c\*x/(1 - d)); end; +> +> histval\_file(datafile=mydata.csv); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from the two +first rows of file `mydata.csv`. + +*Example 2* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : first\_obs=10); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from rows 10 +and 11 of file `mydata.csv`. + +*Example 3* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : first\_obs=2010Q1); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from +observations 2010Q1 and 2010Q2 of file `mydata.csv`. + +*Example 4* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : first\_simulation\_period=2010Q1) +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from +observations 2009Q3 and 2009Q4 of file `mydata.csv`. + +*Example 5* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> histval\_file(datafile=mydata.csv, +> +> : last\_obs = 4); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from the two +first rows of file `mydata.csv`. + +*Example 6* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs = 10, +> +> > nobs = 4); +> +> stoch\_simul(order=1,periods=100); + +The initial values for the stochastic simulation are taken from rows 10 +and 11 of file `mydata.csv`. + +*Example 7* + +> var c x; +> varexo e; +> +> parameters a b c d; +> +> a = 1.5; b = -0,6; c = 0.5; d = 0.5; +> +> > model; x = a\*x(-1) + b\*x(-2) + e; log(c) = c\*x + d\*log(c(+1)); +> > end; +> +> initval\_file(datafile=mydata.csv, +> +> : first\_obs=10); +> +> > histval\_file(datafile=myotherdata.csv); +> +> perfect\_foresight\_setup(periods=200); perfect\_foresight\_solver; + +Historical initial values for the simulation are taken from the two +first rows of file `myotherdata.csv`. + +> Terminal values and guess values for the simulation are taken + +from file `mydata.csv` starting with the 12th observation in the file. +There must be at least 212 observations in the file. +::: + +Shocks on exogenous variables {#shocks-exo} +----------------------------- + +In a deterministic context, when one wants to study the transition of +one equilibrium position to another, it is equivalent to analyze the +consequences of a permanent shock and this in done in Dynare through the +proper use of `initval` and `endval`. + +Another typical experiment is to study the effects of a temporary shock +after which the system goes back to the original equilibrium (if the +model is stable\...). A temporary shock is a temporary change of value +of one or several exogenous variables in the model. Temporary shocks are +specified with the command `shocks`. + +In a stochastic framework, the exogenous variables take random values in +each period. In Dynare, these random values follow a normal distribution +with zero mean, but it belongs to the user to specify the variability of +these shocks. The non-zero elements of the matrix of variance-covariance +of the shocks can be entered with the `shocks` command. Or, the entire +matrix can be directly entered with `Sigma_e` (this use is however +deprecated). + +If the variance of an exogenous variable is set to zero, this variable +will appear in the report on policy and transition functions, but isn't +used in the computation of moments and of Impulse Response Functions. +Setting a variance to zero is an easy way of removing an exogenous +shock. + +Note that, by default, if there are several `shocks` or `mshocks` blocks +in the same `.mod` file, then they are cumulative: all the shocks +declared in all the blocks are considered; however, if a `shocks` or +`mshocks` block is declared with the `overwrite` option, then it +replaces all the previous `shocks` and `mshocks` blocks. + +::: {.block} +shocks ; shocks(overwrite); + +See above for the meaning of the `overwrite` option. + +*In deterministic context* + +For deterministic simulations, the `shocks` block specifies temporary +changes in the value of exogenous variables. For permanent shocks, use +an `endval` block. + +The block should contain one or more occurrences of the following group +of three lines: + + var VARIABLE_NAME; + periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; + values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...; + +It is possible to specify shocks which last several periods and which +can vary over time. The `periods` keyword accepts a list of several +dates or date ranges, which must be matched by as many shock values in +the `values` keyword. Note that a range in the `periods` keyword can be +matched by only one value in the `values` keyword. If `values` +represents a scalar, the same value applies to the whole range. If +`values` represents a vector, it must have as many elements as there are +periods in the range. + +Note that shock values are not restricted to numerical constants: +arbitrary expressions are also allowed, but you have to enclose them +inside parentheses. + +The feasible range of `periods` is from 0 to the number of `periods` +specified in `perfect_foresight_setup`. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +Note that the first endogenous simulation period is period 1. Thus, a +shock value specified for the initial period 0 may conflict with (i.e. +may overwrite or be overwritten by) values for the initial period +specified with `initval` or `endval` (depending on the exact context). +Users should always verify the correct setting of `oo_.exo_simul` after +`perfect_foresight_setup`. +::: + +*Example* (with scalar values) + + shocks; + + var e; + periods 1; + values 0.5; + var u; + periods 4:5; + values 0; + var v; + periods 4:5 6 7:9; + values 1 1.1 0.9; + var w; + periods 1 2; + values (1+p) (exp(z)); + + end; + +*Example* (with vector values) + + xx = [1.2; 1.3; 1]; + + shocks; + var e; + periods 1:3; + values (xx); + end; + +*In stochastic context* + +For stochastic simulations, the `shocks` block specifies the non zero +elements of the covariance matrix of the shocks of exogenous variables. + +You can use the following types of entries in the block: + +- Specification of the standard error of an exogenous variable. + + var VARIABLE_NAME; stderr EXPRESSION; + +- Specification of the variance of an exogenous variable. + + var VARIABLE_NAME = EXPRESSION; + +- Specification the covariance of two exogenous variables. + + var VARIABLE_NAME, VARIABLE_NAME = EXPRESSION; + +- Specification of the correlation of two exogenous variables. + + corr VARIABLE_NAME, VARIABLE_NAME = EXPRESSION; + +In an estimation context, it is also possible to specify variances and +covariances on endogenous variables: in that case, these values are +interpreted as the calibration of the measurement errors on these +variables. This requires the `varobs` command to be specified before the +`shocks` block. + +*Example* + + shocks; + var e = 0.000081; + var u; stderr 0.009; + corr e, u = 0.8; + var v, w = 2; + end; + +*In stochastic optimal policy context* + +When computing conditional welfare in a `ramsey_model` or +`discretionary_policy` context, welfare is conditional on the state +values inherited by planner when making choices in the first period. The +information set of the first period includes the respective exogenous +shock realizations. Thus, their known value can be specified using the +perfect foresight syntax. Note that i) all other values specified for +periods than period 1 will be ignored and ii) the value of lagged shocks +(e.g. in the case of news shocks) is specified with `histval`. + +*Example* + + shocks; + var u; stderr 0.008; + var u; + periods 1; + values 1; + end; + +*Mixing deterministic and stochastic shocks* + +It is possible to mix deterministic and stochastic shocks to build +models where agents know from the start of the simulation about future +exogenous changes. In that case `stoch_simul` will compute the rational +expectation solution adding future information to the state space +(nothing is shown in the output of `stoch_simul`) and `forecast` will +compute a simulation conditional on initial conditions and future +information. + +*Example* + + varexo_det tau; + varexo e; + ... + shocks; + var e; stderr 0.01; + var tau; + periods 1:9; + values -0.15; + end; + + stoch_simul(irf=0); + + forecast; +::: + + + +::: {.matcomm} +get\_shock\_stderr\_by\_name (\'EXOGENOUS\_NAME\'); + +Given the name of an exogenous variable, returns its standard deviation, +as set by a previous `shocks` block. +::: + +::: {.matcomm} +set\_shock\_stderr\_value (\'EXOGENOUS\_NAME\', MATLAB\_EXPRESSION); + +Sets the standard deviation of an exgonous variable. This does +essentially the same as setting the standard error via a `shocks` block, +except that it accepts arbitrary Julia expressions, and that it +works from Julia scripts. +::: + +Steady state {#st-st} +------------ + +There are two ways of computing the steady state (i.e. the static +equilibrium) of a model. The first way is to let Dynare compute the +steady state using a nonlinear Newton-type solver; this should work for +most models, and is relatively simple to use. The second way is to give +more guidance to Dynare, using your knowledge of the model, by providing +it with a method to compute the steady state, either using a +[steady\_state\_model]{.title-ref} block or writing matlab routine. + +### Finding the steady state with Dynare nonlinear solver + +::: {.command} +steady ; steady (OPTIONS\...); + +This command computes the steady state of a model using a nonlinear +Newton-type solver and displays it. When a steady state file is used +`steady` displays the steady state and checks that it is a solution of +the static model. + +More precisely, it computes the equilibrium value of the endogenous +variables for the value of the exogenous variables specified in the +previous `initval` or `endval` block. + +`steady` uses an iterative procedure and takes as initial guess the +value of the endogenous variables set in the previous `initval` or +`endval` block. + +For complicated models, finding good numerical initial values for the +endogenous variables is the trickiest part of finding the equilibrium of +that model. Often, it is better to start with a smaller model and add +new variables one by one. + +*Options* + +::: {#steady_maxit} +::: {.option} +maxit = INTEGER + +Determines the maximum number of iterations used in the non-linear +solver. The default value of `maxit` is 50. +::: +::: + +::: {#steady_tolf} +::: {.option} +tolf = DOUBLE + +Convergence criterion for termination based on the function value. +Iteration will cease when the residuals are smaller than `tolf`. +Default: `eps^(1/3)` +::: +::: + +::: {.option} +tolx = DOUBLE + +Convergence criterion for termination based on the step tolerance along. +Iteration will cease when the attempted step size is smaller than +`tolx`. Default: `eps^(2/3)` +::: + + +::: {.option} +homotopy\_mode = INTEGER + +Use a homotopy (or divide-and-conquer) technique to solve for the steady +state. If you use this option, you must specify a `homotopy_setup` +block. This option can take three possible values: + +> `1` +> +> > In this mode, all the parameters are changed simultaneously, and the +> > distance between the boundaries for each parameter is divided in as +> > many intervals as there are steps (as defined by the +> > `homotopy_steps` option); the problem is solved as many times as +> > there are steps. +> +> `2` +> +> > Same as mode `1`, except that only one parameter is changed at a +> > time; the problem is solved as many times as steps times number of +> > parameters. +> +> `3` +> +> > Dynare tries first the most extreme values. If it fails to compute +> > the steady state, the interval between initial and desired values is +> > divided by two for all parameters. Every time that it is impossible +> > to find a steady state, the previous interval is divided by two. +> > When it succeeds to find a steady state, the previous interval is +> > multiplied by two. In that last case `homotopy_steps` contains the +> > maximum number of computations attempted before giving up. +::: + +::: {.option} +homotopy\_steps = INTEGER + +Defines the number of steps when performing a homotopy. See +`homotopy_mode` option for more details. +::: + +::: {.option} +homotopy\_force\_continue = INTEGER + +This option controls what happens when homotopy fails. + +> `0` +> +> > `steady` fails with an error message +> +> `1` +> +> > `steady` keeps the values of the last homotopy step that was +> > successful and continues. **BE CAREFUL**: parameters and/or +> > exogenous variables are NOT at the value expected by the user + +Default is `0`. +::: + +::: {.option} +nocheck + +Don't check the steady state values when they are provided explicitly +either by a steady state file or a `steady_state_model` block. This is +useful for models with unit roots as, in this case, the steady state is +not unique or doesn't exist. +::: + + +*Example* + +See `init-term-cond`{.interpreted-text role="ref"}. +::: + +After computation, the steady state is available in the following +variable: + +::: {.matvar} +[oo]().steady\_state + +Contains the computed steady state. Endogenous variables are ordered in +the order of declaration used in the `var` command (which is also the +order used in `M_.endo_names`). +::: + +::: {.matcomm} +get\_mean (\'ENDOGENOUS\_NAME\' \[, \'ENDOGENOUS\_NAME\'\]\... ); + +Returns the steady of state of the given endogenous variable(s), as it +is stored in `oo_.steady_state`. Note that, if the steady state has not +yet been computed with `steady`, it will first try to compute it. +::: + +::: {.block} +homotopy\_setup ; + +This block is used to declare initial and final values when using a +homotopy method. It is used in conjunction with the option +`homotopy_mode` of the steady command. + +The idea of homotopy (also called divide-and-conquer by some authors) is +to subdivide the problem of finding the steady state into smaller +problems. It assumes that you know how to compute the steady state for a +given set of parameters, and it helps you finding the steady state for +another set of parameters, by incrementally moving from one to another +set of parameters. + +The purpose of the `homotopy_setup` block is to declare the final (and +possibly also the initial) values for the parameters or exogenous that +will be changed during the homotopy. It should contain lines of the +form: + + VARIABLE_NAME, EXPRESSION, EXPRESSION; + +This syntax specifies the initial and final values of a given +parameter/exogenous. + +There is an alternative syntax: + + VARIABLE_NAME, EXPRESSION; + +Here only the final value is specified for a given parameter/exogenous; +the initial value is taken from the preceeding `initval` block. + +A necessary condition for a successful homotopy is that Dynare must be +able to solve the steady state for the initial parameters/exogenous +without additional help (using the guess values given in the `initval` +block). + +If the homotopy fails, a possible solution is to increase the number of +steps (given in `homotopy_steps` option of `steady`). + +*Example* + +In the following example, Dynare will first compute the steady state for +the initial values (`gam=0.5` and `x=1`), and then subdivide the problem +into 50 smaller problems to find the steady state for the final values +(`gam=2` and `x=2`): + + var c k; + varexo x; + + parameters alph gam delt bet aa; + alph=0.5; + delt=0.02; + aa=0.5; + bet=0.05; + + model; + c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); + c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); + end; + + initval; + x = 1; + k = ((delt+bet)/(aa*x*alph))^(1/(alph-1)); + c = aa*x*k^alph-delt*k; + end; + + homotopy_setup; + gam, 0.5, 2; + x, 2; + end; + + steady(homotopy_mode = 1, homotopy_steps = 50); +::: + +### Providing the steady state to Dynare + +If you know how to compute the steady state for your model, you can +provide a MATLAB/Octave function doing the computation instead of using +`steady`. Again, there are two options for doing that: + +> - The easiest way is to write a `steady_state_model` block, which is +> described below in more details. See also `fs2000.mod` in the +> `examples` directory for an example. The steady state file +> generated by Dynare will be called `+FILENAME/steadystate.m.` +> - You can write the corresponding MATLAB function by hand. If your +> MOD-file is called `FILENAME.mod`, the steady state file must be +> called `FILENAME_steadystate.m`. See `NK_baseline_steadystate.m` +> in the examples directory for an example. This option gives a bit +> more flexibility (loops and conditional structures can be used), +> at the expense of a heavier programming burden and a lesser +> efficiency. + +Note that both files allow to update parameters in each call of the +function. This allows for example to calibrate a model to a labor supply +of 0.2 in steady state by setting the labor disutility parameter to a +corresponding value (see `NK_baseline_steadystate.m` in the `examples` +directory). They can also be used in estimation where some parameter may +be a function of an estimated parameter and needs to be updated for +every parameter draw. For example, one might want to set the capital +utilization cost parameter as a function of the discount rate to ensure +that capacity utilization is 1 in steady state. Treating both parameters +as independent or not updating one as a function of the other would lead +to wrong results. But this also means that care is required. Do not +accidentally overwrite your parameters with new values as it will lead +to wrong results. + +::: {.block} +steady\_state\_model ; + +When the analytical solution of the model is known, this command can be +used to help Dynare find the steady state in a more efficient and +reliable way, especially during estimation where the steady state has to +be recomputed for every point in the parameter space. + +Each line of this block consists of a variable (either an endogenous, a +temporary variable or a parameter) which is assigned an expression +(which can contain parameters, exogenous at the steady state, or any +endogenous or temporary variable already declared above). Each line +therefore looks like: + + VARIABLE_NAME = EXPRESSION; + +Note that it is also possible to assign several variables at the same +time, if the main function in the right hand side is a MATLAB/Octave +function returning several arguments: + + [ VARIABLE_NAME, VARIABLE_NAME... ] = EXPRESSION; + +Dynare will automatically generate a steady state file (of the form +`+FILENAME/steadystate.m`) using the information provided in this block. + +*Steady state file for deterministic models* + +The `steady_state_model` block also works with deterministic models. An +`initval` block and, when necessary, an `endval` block, is used to set +the value of the exogenous variables. Each `initval` or `endval` block +must be followed by `steady` to execute the function created by +`steady_state_model` and set the initial, respectively terminal, steady +state. + +*Example* + +> var m P c e W R k d n l gy_obs gp_obs y dA; +> varexo e_a e_m; +> +> parameters alp bet gam mst rho psi del; +> +> ... +> // parameter calibration, (dynamic) model declaration, shock calibration... +> ... +> +> steady_state_model; +> dA = exp(gam); +> gst = 1/dA; // A temporary variable +> m = mst; +> +> // Three other temporary variables +> khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1)); +> xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1); +> nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp ); +> +> n = xist/(nust+xist); +> P = xist + nust; +> k = khst*n; +> +> l = psi*mst*n/( (1-psi)*(1-n) ); +> c = mst/P; +> d = l - mst + 1; +> y = k^alp*n^(1-alp)*gst^alp; +> R = mst/bet; +> +> // You can use MATLAB functions which return several arguments +> [W, e] = my_function(l, n); +> +> gp_obs = m/dA; +> gy_obs = dA; +> end; +> +> steady; +::: + +### Replace some equations during steady state computations {#eq-tag-ss} + +When there is no steady state file, Dynare computes the steady state by +solving the static model, i.e. the model from the `.mod` file from which +leads and lags have been removed. + +In some specific cases, one may want to have more control over the way +this static model is created. Dynare therefore offers the possibility to +explicitly give the form of equations that should be in the static +model. + +More precisely, if an equation is prepended by a `[static]` tag, then it +will appear in the static model used for steady state computation, but +that equation will not be used for other computations. For every +equation tagged in this way, you must tag another equation with +`[dynamic]`: that equation will not be used for steady state +computation, but will be used for other computations. + +This functionality can be useful on models with a unit root, where there +is an infinity of steady states. An equation (tagged `[dynamic]`) would +give the law of motion of the nonstationary variable (like a random +walk). To pin down one specific steady state, an equation tagged +`[static]` would affect a constant value to the nonstationary variable. +Another situation where the `[static]` tag can be useful is when one has +only a partial closed form solution for the steady state. + +*Example* + +This is a trivial example with two endogenous variables. The second +equation takes a different form in the static model: + + var c k; + varexo x; + ... + model; + c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); + [dynamic] c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); + [static] k = ((delt+bet)/(x*aa*alph))^(1/(alph-1)); + end; + +Getting information about the model +----------------------------------- + +::: {.command} +check ; check (OPTIONS\...); + +Computes the eigenvalues of the model linearized around the values +specified by the last `initval`, `endval` or `steady` statement. +Generally, the eigenvalues are only meaningful if the linearization is +done around a steady state of the model. It is a device for local +analysis in the neighborhood of this steady state. + +A necessary condition for the uniqueness of a stable equilibrium in the +neighborhood of the steady state is that there are as many eigenvalues +larger than one in modulus as there are forward looking variables in the +system. An additional rank condition requires that the square submatrix +of the right Schur vectors corresponding to the forward looking +variables (jumpers) and to the explosive eigenvalues must have full +rank. + +Note that the outcome may be different from what would be suggested by +`sum(abs(oo_.dr.eigval))` when eigenvalues are very close to +`qz_criterium `{.interpreted-text role="opt"}. + +*Options* + +::: {.option} +solve\_algo = INTEGER + +See `solve_algo `{.interpreted-text role="ref"}, for the +possible values and their meaning. +::: + +::: {.option} +qz\_zero\_threshold = DOUBLE + +Value used to test if a generalized eigenvalue is $0/0$ in the +generalized Schur decomposition (in which case the model does not admit +a unique solution). Default: `1e-6`. +::: + +*Output* + +`check` returns the eigenvalues in the global variable `oo_.dr.eigval`. +::: + +::: {.matvar} +[oo]().dr.eigval + +Contains the eigenvalues of the model, as computed by the `check` +command. +::: + +::: {.command} +model\_diagnostics ; + +This command performs various sanity checks on the model, and prints a +message if a problem is detected (missing variables at current period, +invalid steady state, singular Jacobian of static model). +::: + +::: {.command} +model\_info ; model\_info (OPTIONS\...); + +This command provides information about the model. + +When used outside the context of the `block` option of the `model` +block, it will provide a list of predetermined state variables, +forward-looking variables, and purely static variables. + + +Deterministic simulation {#det-simul} +------------------------ + +### Perfect foresight + +When the framework is deterministic, Dynare can be used for models with +the assumption of perfect foresight. Typically, the system is supposed +to be in a state of equilibrium before a period `1` when the news of a +contemporaneous or of a future shock is learned by the agents in the +model. The purpose of the simulation is to describe the reaction in +anticipation of, then in reaction to the shock, until the system returns +to the old or to a new state of equilibrium. In most models, this return +to equilibrium is only an asymptotic phenomenon, which one must +approximate by an horizon of simulation far enough in the future. +Another exercise for which Dynare is well suited is to study the +transition path to a new equilibrium following a permanent shock. For +deterministic simulations, the numerical problem consists of solving a +nonlinear system of simultaneous equations in `n` endogenous variables +in `T` periods. Dynare offers several algorithms for solving this +problem, which can be chosen via the `stack_solve_algo` option. By +default (`stack_solve_algo=0`), Dynare uses a Newton-type method to +solve the simultaneous equation system. Because the resulting Jacobian +is in the order of `n` by `T` and hence will be very large for long +simulations with many variables, Dynare makes use of the sparse matrix +capacities of MATLAB/Octave. A slower but potentially less memory +consuming alternative (`stack_solve_algo=1`) is based on a Newton-type +algorithm first proposed by *Laffargue (1990)* and *Boucekkine (1995)*, +which avoids ever storing the full Jacobian. The details of the +algorithm can be found in *Juillard (1996)*. The third type of +algorithms makes use of block decomposition techniques +(divide-and-conquer methods) that exploit the structure of the model. +The principle is to identify recursive and simultaneous blocks in the +model structure and use this information to aid the solution process. +These solution algorithms can provide a significant speed-up on large +models. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +Be careful when employing auxiliary variables in the context of perfect +foresight computations. The same model may work for stochastic +simulations, but fail for perfect foresight simulations. The issue +arises when an equation suddenly only contains variables dated `t+1` (or +`t-1` for that matter). In this case, the derivative in the last (first) +period with respect to all variables will be 0, rendering the stacked +Jacobian singular. + +*Example* + +> Consider the following specification of an Euler equation with log +> utility: +> +> Lambda = beta*C(-1)/C; +> Lambda(+1)*R(+1)= 1; +> +> Clearly, the derivative of the second equation with respect to all +> endogenous variables at time `t` is zero, causing +> `perfect_foresight_solver` to generally fail. This is due to the use +> of the Lagrange multiplier `Lambda` as an auxiliary variable. Instead, +> employing the identical +> +> beta*C/C(+1)*R(+1)= 1; +> +> will work. +::: + +::: {.command} +perfect\_foresight\_setup ; perfect\_foresight\_setup (OPTIONS\...); + +Prepares a perfect foresight simulation, by extracting the information +in the `initval`, `endval` and `shocks` blocks and converting them into +simulation paths for exogenous and endogenous variables. + +This command must always be called before running the simulation with +`perfect_foresight_solver`. + +*Options* + +::: {.option} +periods = INTEGER + +Number of periods of the simulation. +::: + +::: {.option} +datafile = FILENAME + +Used to specify path for all endogenous and exogenous variables. +Strictly equivalent to `initval_file`{.interpreted-text role="comm"}. +::: + +*Output* + +The paths for the exogenous variables are stored into `oo_.exo_simul`. + +The initial and terminal conditions for the endogenous variables and the +initial guess for the path of endogenous variables are stored into +`oo_.endo_simul`. +::: + +::: {.command} +perfect\_foresight\_solver ; perfect\_foresight\_solver (OPTIONS\...); + +Computes the perfect foresight (or deterministic) simulation of the +model. + +Note that `perfect_foresight_setup` must be called before this command, +in order to setup the environment for the simulation. + +*Options* + +::: {.option} +maxit = INTEGER + +Determines the maximum number of iterations used in the non-linear +solver. The default value of `maxit` is `50`. +::: + +::: {.option} +tolf = DOUBLE + +Convergence criterion for termination based on the function value. +Iteration will cease when it proves impossible to improve the function +value by more than `tolf`. Default: `1e-5` +::: + +::: {.option} +tolx = DOUBLE + +Convergence criterion for termination based on the change in the +function argument. Iteration will cease when the solver attempts to take +a step that is smaller than `tolx`. Default: `1e-5` +::: + +::: {.option} +noprint + +Don't print anything. Useful for loops. +::: + +::: {.option} +print + +Print results (opposite of `noprint`). +::: + + +::: {.option} +lmmcp + +Solves the perfect foresight model with a Levenberg-Marquardt mixed +complementarity problem (LMMCP) solver (*Kanzow and Petra (2004)*), +which allows to consider inequality constraints on the endogenous +variables (such as a ZLB on the nominal interest rate or a model with +irreversible investment). This option is equivalent to +`stack_solve_algo=7` **and** `solve_algo=10`. Using the LMMCP solver +requires a particular model setup as the goal is to get rid of any +min/max operators and complementary slackness conditions that might +introduce a singularity into the Jacobian. This is done by attaching an +equation tag (see `model-decl`{.interpreted-text role="ref"}) with the +`mcp` keyword to affected equations. This tag states that the equation +to which the tag is attached has to hold unless the expression within +the tag is binding. For instance, a ZLB on the nominal interest rate +would be specified as follows in the model block: + + model; + ... + [mcp = 'r > -1.94478'] + r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e; + ... + end; + +where `1.94478` is the steady state level of the nominal interest rate +and `r` is the nominal interest rate in deviation from the steady state. +This construct implies that the Taylor rule is operative, unless the +implied interest rate `r<=-1.94478`, in which case the `r` is fixed at +`-1.94478` (thereby being equivalent to a complementary slackness +condition). By restricting the value of `r` coming out of this equation, +the `mcp` tag also avoids using `max(r,-1.94478)` for other occurrences +of `r` in the rest of the model. It is important to keep in mind that, +because the `mcp` tag effectively replaces a complementary slackness +condition, it cannot be simply attached to any equation. Rather, it must +be attached to the correct affected equation as otherwise the solver +will solve a different problem than originally intended. Also, since the +problem to be solved is nonlinear, the sign of the residuals of the +dynamic equation matters. In the previous example, for the nominal +interest rate rule, if the LHS and RHS are reversed the sign of the +residuals (the difference between the LHS and the RHS) will change and +it may happen that solver fails to identify the solution path. More +generally, convergence of the nonlinear solver is not guaranteed when +using mathematically equivalent representations of the same equation. + +Note that in the current implementation, the content of the `mcp` +equation tag is not parsed by the preprocessor. The inequalities must +therefore be as simple as possible: an endogenous variable, followed by +a relational operator, followed by a number (not a variable, parameter +or expression). +::: + +::: {.option} +endogenous\_terminal\_period + +The number of periods is not constant across Newton iterations when +solving the perfect foresight model. The size of the nonlinear system of +equations is reduced by removing the portion of the paths (and +associated equations) for which the solution has already been identified +(up to the tolerance parameter). This strategy can be interpreted as a +mix of the shooting and relaxation approaches. Note that round off +errors are more important with this mixed strategy (user should check +the reported value of the maximum absolute error). Only available with +option `stack_solve_algo==0`. +::: + +::: {.option} +linear\_approximation + +Solves the linearized version of the perfect foresight model. The model +must be stationary. Only available with option `stack_solve_algo==0` or +`stack_solve_algo==7`. +::: + +*Output* + +The simulated endogenous variables are available in global matrix +`oo_.endo_simul`. +::: + + +This variable stores the path of exogenous variables during a simulation +(computed by `perfect_foresight_solver`, `simul`, `stoch_simul` or +`extended_path`). The variables are arranged in columns, in order of +declaration (as in `M_.exo_names`). Periods are in rows. Note that this +convention regarding columns and rows is the opposite of the convention +for `oo_.endo_simul`! +::: + + +Stochastic solution and simulation {#stoch-sol} +---------------------------------- + +In a stochastic context, Dynare computes one or several simulations +corresponding to a random draw of the shocks. + +The main algorithm for solving stochastic models relies on a Taylor +approximation, up to third order, of the expectation functions (see +*Judd (1996)*, *Collard and Juillard (2001a, 2001b)*, and *Schmitt-Grohé +and Uríbe (2004)*). The details of the Dynare implementation of the +first order solution are given in *Villemot (2011)*. Such a solution is +computed using the `stoch_simul` command. + +As an alternative, it is possible to compute a simulation to a +stochastic model using the *extended path* method presented by *Fair and +Taylor (1983)*. This method is especially useful when there are strong +nonlinearities or binding constraints. Such a solution is computed using +the `extended_path` command. + +### Computing the stochastic solution {#stoch-sol-simul} + +::: {.command} +stoch\_simul \[VARIABLE\_NAME\...\]; stoch\_simul (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +Solves a stochastic (i.e. rational expectations) model, using +perturbation techniques. + +More precisely, `stoch_simul` computes a Taylor approximation of the +model around the deterministic steady state and solves of the the +decision and transition functions for the approximated model. Using +this, it computes impulse response functions and various descriptive +statistics (moments, variance decomposition, correlation and +autocorrelation coefficients). For correlated shocks, the variance +decomposition is computed as in the VAR literature through a Cholesky +decomposition of the covariance matrix of the exogenous variables. When +the shocks are correlated, the variance decomposition depends upon the +order of the variables in the `varexo` command. + +The Taylor approximation is computed around the steady state (see +`st-st`{.interpreted-text role="ref"}). + +The IRFs are computed as the difference between the trajectory of a +variable following a shock at the beginning of period `1` and its steady +state value. More details on the computation of IRFs can be found at +. + +Variance decomposition, correlation, autocorrelation are only displayed +for variables with strictly positive variance. Impulse response +functions are only plotted for variables with response larger than +$10^{-10}$. + +Variance decomposition is computed relative to the sum of the +contribution of each shock. Normally, this is of course equal to +aggregate variance, but if a model generates very large variances, it +may happen that, due to numerical error, the two differ by a significant +amount. Dynare issues a warning if the maximum relative difference +between the sum of the contribution of each shock and aggregate variance +is larger than `0.01%`. + +The covariance matrix of the shocks is specified with the `shocks` +command (see `shocks-exo`{.interpreted-text role="ref"}). + +When a list of `VARIABLE_NAME` is specified, results are displayed only +for these variables. + +The `stoch_simul` command with a first order approximation can benefit +from the block decomposition of the model (see `block`{.interpreted-text +role="opt"}). + +*Options* + +::: {.option} +ar = INTEGER + +Order of autocorrelation coefficients to compute and to print. Default: +`5`. +::: + +::: {.option} +drop = INTEGER + +Number of points (burnin) dropped at the beginning of simulation before +computing the summary statistics. Note that this option does not affect +the simulated series stored in `oo_.endo_simul` and the workspace. Here, +no periods are dropped. Default: `100`. +::: + +::: {.option} +hp\_filter = DOUBLE + +Uses HP filter with $\lambda =$ `DOUBLE` before computing moments. If +theoretical moments are requested, the spectrum of the model solution is +filtered following the approach outlined in Uhlig (2001). Default: no +filter. +::: + +::: {.option} +one\_sided\_hp\_filter = DOUBLE + +Uses the one-sided HP filter with $\lambda =$ `DOUBLE` described in +*Stock and Watson (1999)* before computing moments. This option is only +available with simulated moments. Default: no filter. +::: + +::: {.option} +bandpass\_filter + +Uses a bandpass filter with the default passband before computing +moments. If theoretical moments are requested, the spectrum of the model +solution is filtered using an ideal bandpass filter. If empirical +moments are requested, the *Baxter and King (1999)* filter is used. +Default: no filter. +::: + +::: {.option} +bandpass\_filter = \[HIGHEST\_PERIODICITY LOWEST\_PERIODICITY\] + +Uses a bandpass filter before computing moments. The passband is set to +a periodicity of to LOWEST\_PERIODICITY, e.g. $6$ to $32$ quarters if +the model frequency is quarterly. Default: `[6,32]`. +::: + +::: {.option} +filtered\_theoretical\_moments\_grid = INTEGER + +When computing filtered theoretical moments (with either option +`hp_filter` or option `bandpass_filter`), this option governs the number +of points in the grid for the discrete Inverse Fast Fourier Transform. +It may be necessary to increase it for highly autocorrelated processes. +Default: `512`. +::: + +::: {.option} +irf = INTEGER + +Number of periods on which to compute the IRFs. Setting `irf=0` +suppresses the plotting of IRFs. Default: `40`. +::: + +::: {.option} +irf\_shocks = ( VARIABLE\_NAME \[\[,\] VARIABLE\_NAME \...\] ) + +The exogenous variables for which to compute IRFs. Default: all. +::: + +::: {.option} +relative\_irf + +Requests the computation of normalized IRFs. At first order, the normal +shock vector of size one standard deviation is divided by the standard +deviation of the current shock and multiplied by 100. The impulse +responses are hence the responses to a unit shock of size 1 (as opposed +to the regular shock size of one standard deviation), multiplied by 100. +Thus, for a loglinearized model where the variables are measured in +percent, the IRFs have the interpretation of the percent responses to a +100 percent shock. For example, a response of 400 of output to a TFP +shock shows that output increases by 400 percent after a 100 percent TFP +shock (you will see that TFP increases by 100 on impact). Given +linearity at `order=1`, it is straightforward to rescale the IRFs stored +in `oo_.irfs` to any desired size. At higher order, the interpretation +is different. The `relative_irf` option then triggers the generation of +IRFs as the response to a 0.01 unit shock (corresponding to 1 percent +for shocks measured in percent) and no multiplication with 100 is +performed. That is, the normal shock vector of size one standard +deviation is divided by the standard deviation of the current shock and +divided by 100. For example, a response of 0.04 of log output (thus +measured in percent of the steady state output level) to a TFP shock +also measured in percent then shows that output increases by 4 percent +after a 1 percent TFP shock (you will see that TFP increases by 0.01 on +impact). +::: + +::: {.option} +irf\_plot\_threshold = DOUBLE + +Threshold size for plotting IRFs. All IRFs for a particular variable +with a maximum absolute deviation from the steady state smaller than +this value are not displayed. Default: `1e-10`. +::: + +::: {.option} +nocorr + +Don't print the correlation matrix (printing them is the default). +::: + +::: {.option} +nodecomposition + +Don't compute (and don't print) unconditional variance decomposition. +::: + +::: {.option} +nofunctions + +Don't print the coefficients of the approximated solution (printing them +is the default). +::: + +::: {.option} +nomoments + +Don't print moments of the endogenous variables (printing them is the +default). +::: + +::: {.option} +nograph + +Do not create graphs (which implies that they are not saved to the disk +nor displayed). If this option is not used, graphs will be saved to disk +(to the format specified by `graph_format` option, except if +`graph_format=none`) and displayed to screen (unless `nodisplay` option +is used). +::: + +::: {.option} +graph + +Re-enables the generation of graphs previously shut off with `nograph`. +::: + +::: {.option} +nodisplay + +Do not display the graphs, but still save them to disk (unless `nograph` +is used). +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +Specify the file format(s) for graphs saved to disk. Possible values are +`eps` (the default), `pdf`, `fig` and `none` (under Octave, `fig` is +unavailable). If the file format is set equal to `none`, the graphs are +displayed but not saved to the disk. +::: + +::: {.option} +noprint + +See `noprint`{.interpreted-text role="opt"}. +::: + +::: {.option} +print + +See `print`{.interpreted-text role="opt"}. +::: + +::: {.option} +order = INTEGER + +Order of Taylor approximation. Note that for third order and above, the +`k_order_solver` option is implied and only empirical moments are +available (you must provide a value for `periods` option). Default: `2` +(except after an `estimation` command, in which case the default is the +value used for the estimation). +::: + +::: {.option} +k\_order\_solver + +Use a k-order solver (implemented in C++) instead of the default Dynare +solver. This option is not yet compatible with the `bytecode` option +(see `model-decl`{.interpreted-text role="ref"}). Default: disabled for +order 1 and 2, enabled for order 3 and above. +::: + +::: {.option} +periods = INTEGER + +If different from zero, empirical moments will be computed instead of +theoretical moments. The value of the option specifies the number of +periods to use in the simulations. Values of the initval block, possibly +recomputed by `steady`, will be used as starting point for the +simulation. The simulated endogenous variables are made available to the +user in a vector for each variable and in the global matrix +`oo_.endo_simul` (see `oo_.endo_simul`{.interpreted-text role="mvar"}). +The simulated exogenous variables are made available in `oo_.exo_simul` +(see `oo_.exo_simul`{.interpreted-text role="mvar"}). Default: `0`. +::: + +::: {.option} +qz\_criterium = DOUBLE + +Value used to split stable from unstable eigenvalues in reordering the +Generalized Schur decomposition used for solving first order problems. +Default: `1.000001` (except when estimating with `lik_init` option equal +to `1`: the default is `0.999999` in that case; see +`estim`{.interpreted-text role="ref"}). +::: + +::: {.option} +qz\_zero\_threshold = DOUBLE + +See `qz_zero_threshold `{.interpreted-text +role="opt"}. +::: + +::: {.option} +replic = INTEGER + +Number of simulated series used to compute the IRFs. Default: `1` if +`order=1`, and `50` otherwise. +::: + +::: {.option} +simul\_replic = INTEGER + +Number of series to simulate when empirical moments are requested (i.e. +`periods` $>$ 0). Note that if this option is greater than 1, the +additional series will not be used for computing the empirical moments +but will simply be saved in binary form to the file `FILENAME_simul` in +the `FILENAME/Output`-folder. Default: `1`. +::: + +::: {.option} +solve\_algo = INTEGER + +See `solve_algo `{.interpreted-text role="ref"}, for the +possible values and their meaning. +::: + + +::: {.option} +conditional\_variance\_decomposition = INTEGER +conditional\_variance\_decomposition = \[INTEGER1:INTEGER2\] +conditional\_variance\_decomposition = \[INTEGER1 INTEGER2 \...\] + +Computes a conditional variance decomposition for the specified +period(s). The periods must be strictly positive. Conditional variances +are given by $var(y_{t+k}\vert t)$. For period 1, the conditional +variance decomposition provides the decomposition of the effects of +shocks upon impact. + +The results are stored in `oo_.conditional_variance_decomposition` (see +`oo_.conditional_variance_decomposition`{.interpreted-text +role="mvar"}). In the presence of measurement error, the +`oo_.conditional_variance_decomposition` field will contain the variance +contribution after measurement error has been taken out, i.e. the +decomposition will be conducted of the actual as opposed to the measured +variables. The variance decomposition of the measured variables will be +stored in `oo_.conditional_variance_decomposition_ME` (see +`oo_.conditional_variance_decomposition_ME`{.interpreted-text +role="mvar"}). The variance decomposition is only conducted, if +theoretical moments are requested, *i.e.* using the `periods=0`-option. +Only available at `order<3` and without +`pruning''. In case of`order=2`, Dynare provides a second-order accurate approximation to the true second moments based on the linear terms of the second-order solution (see *Kim, Kim, Schaumburg and Sims (2008)*). Note that the unconditional variance decomposition *i.e.* at horizon infinity) is automatically conducted if theoretical moments are requested and if`nodecomposition\`[ +is not set (see :mvar:\`oo\_.variance\_decomposition]{.title-ref}). +::: + +::: {.option} +pruning + +Discard higher order terms when iteratively computing simulations of the +solution. At second order, Dynare uses the algorithm of *Kim, Kim, +Schaumburg and Sims (2008)*, while at third order its generalization by +*Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018)* is used. Not +available above third order. When specified, theoretical moments are +based on the pruned state space, i.e. the computation of second moments +uses all terms as in *Andreasen, Fernández-Villaverde and Rubio-Ramírez +(2018), page 10* as opposed to simply providing a second-order accurate +result based on the linear solution as in *Kim, Kim, Schaumburg and Sims +(2008)*. +::: + + +::: {.option} +sylvester = OPTION + +Determines the algorithm used to solve the Sylvester equation for block +decomposed model. Possible values for OPTION are: + +> `default` +> +> > Uses the default solver for Sylvester equations (`gensylv`) based on +> > Ondra Kamenik's algorithm (see +> > [here](https://www.dynare.org/assets/dynare++/sylvester.pdf) for +> > more information). +> +> `fixed_point` +> +> > Uses a fixed point algorithm to solve the Sylvester equation +> > (`gensylv_fp`). This method is faster than the default one for large +> > scale models. + +Default value is `default`. +::: + +::: {.option} +sylvester\_fixed\_point\_tol = DOUBLE + +The convergence criterion used in the fixed point Sylvester solver. Its +default value is `1e-12`. +::: + +::: {.option} +dr = OPTION + +Determines the method used to compute the decision rule. Possible values +for OPTION are: + +> `default` +> +> > Uses the default method to compute the decision rule based on the +> > generalized Schur decomposition (see *Villemot (2011)* for more +> > information). +> +> `cycle_reduction` +> +> > Uses the cycle reduction algorithm to solve the polynomial equation +> > for retrieving the coefficients associated to the endogenous +> > variables in the decision rule. This method is faster than the +> > default one for large scale models. +> + +Default value is `default`. +::: + +::: {.option} +dr\_cycle\_reduction\_tol = DOUBLE + +The convergence criterion used in the cycle reduction algorithm. Its +default value is `1e-7`. +::: + +::: {.option} +tex + +Requests the printing of results and graphs in TeX tables and graphics +that can be later directly included in LaTeX files. +::: + +::: {.option} +dr\_display\_tol = DOUBLE + +Tolerance for the suppression of small terms in the display of decision +rules. Rows where all terms are smaller than `dr_display_tol` are not +displayed. Default value: `1e-6`. +::: + +::: {.option} +contemporaneous\_correlation + +Saves the contemporaneous correlation between the endogenous variables +in `oo_.contemporaneous_correlation`. Requires the `nocorr` option not +to be set. +::: + +::: {.option} +spectral\_density + +Triggers the computation and display of the theoretical spectral density +of the (filtered) model variables. Results are stored in +`oo_.SpectralDensity`, defined below. Default: do not request spectral +density estimates. +::: + +::: {.option} +hp\_ngrid = INTEGER + +Deprecated option. It has the same effect as +`filtered_theoretical_moments_grid `{.interpreted-text +role="opt"}. +::: + +*Output* + +This command sets `oo_.dr`, `oo_.mean`, `oo_.var`, `oo_.var_list`, and +`oo_.autocorr`, which are described below. + +If the `periods` option is present, sets `oo_.skewness`, `oo_.kurtosis`, +and `oo_.endo_simul` (see `oo_.endo_simul`{.interpreted-text +role="mvar"}), and also saves the simulated variables in MATLAB/Octave +vectors of the global workspace with the same name as the endogenous +variables. + +If option `irf` is different from zero, sets `oo_.irfs` (see below) and +also saves the IRFs in MATLAB/Octave vectors of the global workspace +(this latter way of accessing the IRFs is deprecated and will disappear +in a future version). + +If the option `contemporaneous_correlation` is different from `0`, sets +`oo_.contemporaneous_correlation`, which is described below. + +*Example* + +> shocks; +> var e; +> stderr 0.0348; +> end; +> +> stoch_simul; +> +> Performs the simulation of the 2nd-order approximation of a model with +> a single stochastic shock `e`, with a standard error of `0.0348`. + +*Example* + +> stoch_simul(irf=60) y k; +> +> Performs the simulation of a model and displays impulse response +> functions on 60 periods for variables `y` and `k`. +::: + +::: {.matvar} +[oo]().mean + +After a run of `stoch_simul`, contains the mean of the endogenous +variables. Contains theoretical mean if the `periods` option is not +present, and simulated mean otherwise. The variables are arranged in +declaration order. +::: + +::: {.matvar} +[oo]().var + +After a run of `stoch_simul`, contains the variance-covariance of the +endogenous variables. Contains theoretical variance if the `periods` +option is not present and simulated variance otherwise. Only available +for `order<4`. At `order=2` it will be be a second-order accurate +approximation (i.e. ignoring terms of order 3 and 4 that would arise +when using the full second-order policy function). At `order=3`, +theoretical moments are only available with `pruning`. The variables are +arranged in declaration order. +::: + +::: {.matvar} +[oo]().var\_list + +The list of variables for which results are displayed. +::: + +::: {.matvar} +[oo]().skewness + +After a run of `stoch_simul` contains the skewness (standardized third +moment) of the simulated variables if the `periods` option is present. +The variables are arranged in declaration order. +::: + +::: {.matvar} +[oo]().kurtosis + +After a run of `stoch_simul` contains the excess kurtosis (standardized +fourth moment) of the simulated variables if the `periods` option is +present. The variables are arranged in declaration order. +::: + +::: {.matvar} +[oo]().autocorr + +After a run of `stoch_simul`, contains a cell array of the +autocorrelation matrices of the endogenous variables. The element number +of the matrix in the cell array corresponds to the order of +autocorrelation. The option ar specifies the number of autocorrelation +matrices available. Contains theoretical autocorrelations if the +`periods` option is not present and simulated autocorrelations +otherwise. Only available for `order<4`. At `order=2` it will be be a +second-order accurate approximation. At `order=3`, theoretical moments +are only available with `pruning`. The field is only created if +stationary variables are present. + +The element `oo_.autocorr{i}(k,l)` is equal to the correlation between +$y^k_t$ and $y^l_{t-i}$, where $y^k$ (resp. $y^l$) is the $k$-th (resp. +$l$-th) endogenous variable in the declaration order. + +Note that if theoretical moments have been requested, `oo_.autocorr{i}` +is the same than `oo_.gamma_y{i+1}`. +::: + +::: {.matvar} +[oo]().gamma\_y + +After a run of `stoch_simul`, if theoretical moments have been requested +(i.e. if the `periods` option is not present), this variable contains a +cell array with the following values (where `ar` is the value of the +option of the same name): + +> `oo_.gamma{1}` +> +> > Variance/covariance matrix. +> +> `oo_.gamma{i+1}` (for i=1:ar) +> +> > Autocorrelation function. See `oo_.autocorr`{.interpreted-text +> > role="mvar"} for more details. **Beware**, this is the +> > autocorrelation function, not the autocovariance function. +> +> `oo_.gamma{ar+2}` +> +> > Unconditional variance decomposition, see +> > `oo_.variance_decomposition`{.interpreted-text role="mvar"}. +> +> `oo_.gamma{ar+3}` +> +> > If a second order approximation has been requested, contains the +> > vector of the mean correction terms. +> > +> > Only available at `order<4`. In case `order=2`, the theoretical +> > second moments are a second order accurate approximation of the true +> > second moments. See conditional\_variance\_decomposition. At +> > `order=3`, theoretical moments are only available with `pruning`. +::: + +::: {.matvar} +[oo]().variance\_decomposition + +After a run of `stoch_simul` when requesting theoretical moments +(`periods=0`), contains a matrix with the result of the unconditional +variance decomposition (i.e. at horizon infinity). The first dimension +corresponds to the endogenous variables (in the order of declaration +after the command or in `M_.endo_names`) and the second dimension +corresponds to exogenous variables (in the order of declaration). +Numbers are in percent and sum up to 100 across columns. In the presence +of measurement error, the field will contain the variance contribution +after measurement error has been taken out, *i.e.* the decomposition +will be conducted of the actual as opposed to the measured variables. +::: + +::: {.matvar} +[oo]().variance\_decomposition\_ME + +Field set after a run of `stoch_simul` when requesting theoretical +moments (`periods=0`) if measurement error is present. It is similar to +`oo_.variance_decomposition`{.interpreted-text role="mvar"}, but the +decomposition will be conducted of the measured variables. The field +contains a matrix with the result of the unconditional variance +decomposition (*i.e.* at horizon infinity). The first dimension +corresponds to the observed endoogenous variables (in the order of +declaration after the command) and the second dimension corresponds to +exogenous variables (in the order of declaration), with the last column +corresponding to the contribution of measurement error. Numbers are in +percent and sum up to 100 across columns. +::: + +::: {.matvar} +[oo]().conditional\_variance\_decomposition + +After a run of `stoch_simul` with the +`conditional_variance_decomposition` option, contains a +three-dimensional array with the result of the decomposition. The first +dimension corresponds to the endogenous variables (in the order of +declaration after the command or in `M_.endo_names` if not specified), +the second dimension corresponds to the forecast horizons (as declared +with the option), and the third dimension corresponds to the exogenous +variables (in the order of declaration). In the presence of measurement +error, the field will contain the variance contribution after +measurement error has been taken out, *i.e.* the decomposition will be +conductedof the actual as opposed to the measured variables. +::: + +::: {.matvar} +[oo]().conditional\_variance\_decomposition\_ME + +Field set after a run of `stoch_simul` with the +`conditional_variance_decomposition` option if measurement error is +present. It is similar to +`oo_.conditional_variance_decomposition`{.interpreted-text role="mvar"}, +but the decomposition will be conducted of the measured variables. It +contains a three-dimensional array with the result of the decomposition. +The first dimension corresponds to the endogenous variables (in the +order of declaration after the command or in `M_.endo_names` if not +specified), the second dimension corresponds to the forecast horizons +(as declared with the option), and the third dimension corresponds to +the exogenous variables (in the order of declaration), with the last +column corresponding to the contribution of the measurement error. +::: + +::: {.matvar} +[oo]().contemporaneous\_correlation + +After a run of `stoch_simul` with the +`contemporaneous_correlation option`, contains theoretical +contemporaneous correlations if the `periods` option is not present, and +simulated contemporaneous correlations otherwise. Only available for +`order<4`. At `order=2` it will be be a second-order accurate +approximation. At `order=3`, theoretical moments are only available with +`pruning`. The variables are arranged in declaration order. +::: + +::: {.matvar} +[oo]().SpectralDensity + +After a run of `stoch_simul` with option `spectral_density`, contains +the spectral density of the model variables. There will be a `nvars` by +`nfrequencies` subfield `freqs` storing the respective frequency grid +points ranging from $0$ to $2\pi$ and a same sized subfield `density` +storing the corresponding density. +::: + +::: {.matvar} +[oo]().irfs + +After a run of `stoch_simul` with option `irf` different from zero, +contains the impulse responses, with the following naming convention: +[VARIABLE\_NAME\_SHOCK\_NAME]{.title-ref}. + +> For example, `oo_.irfs.gnp_ea` contains the effect on `gnp` of a +> one-standard deviation shock on `ea`. +::: + +::: {.matcomm} +get\_irf (\'EXOGENOUS\_NAME\' \[, \'ENDOGENOUS\_NAME\'\]\... ); + +Given the name of an exogenous variables, returns the IRFs for the +requested endogenous variable(s), as they are stored in `oo_.irfs`. +::: + +The approximated solution of a model takes the form of a set of decision +rules or transition equations expressing the current value of the +endogenous variables of the model as function of the previous state of +the model and shocks observed at the beginning of the period. The +decision rules are stored in the structure `oo_.dr` which is described +below. + +::: {.matvar} +[oo]().dr + +Structure storing the decision rules. The subfields for different orders +of approximation are explained below. +::: + +::: {.command} +extended\_path ; extended\_path (OPTIONS\...); + +Simulates a stochastic (i.e. rational expectations) model, using the +extended path method presented by *Fair and Taylor (1983)*. Time series +for the endogenous variables are generated by assuming that the agents +believe that there will no more shocks in the following periods. + +This function first computes a random path for the exogenous variables +(stored in `oo_.exo_simul`, see `oo_.exo_simul`{.interpreted-text +role="mvar"}) and then computes the corresponding path for endogenous +variables, taking the steady state as starting point. The result of the +simulation is stored in `oo_.endo_simul` (see +`oo_.endo_simul`{.interpreted-text role="mvar"}). Note that this +simulation approach does not solve for the policy and transition +equations but for paths for the endogenous variables. + +*Options* + +::: {.option} +periods = INTEGER + +The number of periods for which the simulation is to be computed. No +default value, mandatory option. +::: + +::: {.option} +solver\_periods = INTEGER + +The number of periods used to compute the solution of the perfect +foresight at every iteration of the algorithm. Default: `200`. +::: + +::: {.option} +order = INTEGER + +If order is greater than `0` Dynare uses a gaussian quadrature to take +into account the effects of future uncertainty. If `order` $=S$ then the +time series for the endogenous variables are generated by assuming that +the agents believe that there will no more shocks after period $t+S$. +This is an experimental feature and can be quite slow. A non-zero value +is not compatible with either the `bytecode` or the `block` option of +the `model` block. Default: `0`. +::: + +::: {.option} +hybrid + +Use the constant of the second order perturbation reduced form to +correct the paths generated by the (stochastic) extended path algorithm. +::: + +::: {.option} +lmmcp + +Solves the perfect foresight model with a Levenberg-Marquardt mixed +complementarity problem (LMMCP) solver (*Kanzow and Petra (2004)*), +which allows to consider inequality constraints on the endogenous +variables (such as a ZLB on the nominal interest rate or a model with +irreversible investment). For specifying the necessary `mcp`-tag, see +`lmmcp`{.interpreted-text role="opt"}. +::: +::: + +### Typology and ordering of variables + +Dynare distinguishes four types of endogenous variables: + +*Purely backward (or purely predetermined) variables* + +> Those that appear only at current and past period in the model, but +> not at future period (i.e. at $t$ and $t-1$ but not $t+1$). The number +> of such variables is equal to `M_.npred`. + +*Purely forward variables* + +> Those that appear only at current and future period in the model, but +> not at past period (i.e. at $t$ and $t+1$ but not $t-1$). The number +> of such variables is stored in `M_.nfwrd`. + +*Mixed variables* + +> Those that appear at current, past and future period in the model +> (i.e. at $t$, $t+1$ and $t-1$). The number of such variables is stored +> in `M_.nboth`. + +*Static variables* + +> Those that appear only at current, not past and future period in the +> model (i.e. only at $t$, not at $t+1$ or $t-1$). The number of such +> variables is stored in `M_.nstatic`. + +Note that all endogenous variables fall into one of these four +categories, since after the creation of auxiliary variables (see +`aux-variables`{.interpreted-text role="ref"}), all endogenous have at +most one lead and one lag. We therefore have the following identity: + +> ``` {.sourceCode .matlab} +> M_.npred + M_.both + M_.nfwrd + M_.nstatic = M_.endo_nbr +> ``` + +::: {.matvar} +[M]().state\_var + +Vector of numerical indices identifying the state variables in the +vector of declared variables. `M_.endo_names(M_.state_var)` therefore +yields the name of all variables that are states in the model +declaration, i.e. that show up with a lag. +::: + +Internally, Dynare uses two orderings of the endogenous variables: the +order of declaration (which is reflected in `M_.endo_names`), and an +order based on the four types described above, which we will call the +DR-order ("DR" stands for decision rules). Most of the time, the +declaration order is used, but for elements of the decision rules, the +DR-order is used. + +The DR-order is the following: static variables appear first, then +purely backward variables, then mixed variables, and finally purely +forward variables. Inside each category, variables are arranged +according to the declaration order. + +::: {.matvar} +[oo]().dr.order\_var + +This variables maps DR-order to declaration order. +::: + +::: {.matvar} +[oo]().dr.inv\_order\_var + +This variable contains the inverse map. +::: + +In other words, the k-th variable in the DR-order corresponds to the +endogenous variable numbered `oo_.dr.order_var(k)` in declaration order. +Conversely, k-th declared variable is numbered `oo_.dr.inv_order_var(k)` +in DR-order. + +Finally, the state variables of the model are the purely backward +variables and the mixed variables. They are ordered in DR-order when +they appear in decision rules elements. There are +`M_.nspred = M_.npred + M_.nboth` such variables. Similarly, one has +`M_.nsfwrd = M_.nfwrd + M_.nboth`, and +`M_.ndynamic = M_.nfwrd + M_.nboth + M_.npred`. + +### First-order approximation + +The approximation has the stylized form: + +$$y_t = y^s + A y^h_{t-1} + B u_t$$ + +where $y^s$ is the steady state value of $y$ and $y^h_t=y_t-y^s$. + +::: {.matvar} +oo.dr.state\_var + +Vector of numerical indices identifying the state variables in the +vector of declared variables, *given the current parameter values* for +which the decision rules have been computed. It may differ from +`M_.state_var` in case a state variable drops from the model given the +current parameterization, because it only gets 0 coefficients in the +decision rules. See `M_.state_var`{.interpreted-text role="mvar"}. +::: + +The coefficients of the decision rules are stored as follows: + +- $y^s$ is stored in `oo_.dr.ys`. The vector rows correspond to all + endogenous in the declaration order. +- $A$ is stored in `oo_.dr.ghx`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to state + variables in DR-order, as given by `oo_.dr.state_var`. (N.B.: if the + `block` option to the `model` block has been specified, then rows + are in declaration order, and columns are ordered according to + `oo_.dr.state_var` which may differ from DR-order.) +- $B$ is stored `oo_.dr.ghu`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to exogenous + variables in declaration order. (N.B.: if the `block` option to the + `model` block has been specified, then rows are in declaration + order.) + +Of course, the shown form of the approximation is only stylized, because +it neglects the required different ordering in $y^s$ and $y^h_t$. The +precise form of the approximation that shows the way Dynare deals with +differences between declaration and DR-order, is + +$$y_t(\mathrm{oo\_.dr.order\_var}) = +y^s(\mathrm{oo\_.dr.order\_var}) + A \cdot +y_{t-1}(\mathrm{oo\_.dr.order\_var(k2)}) - +y^s(\mathrm{oo\_.dr.order\_var(k2)}) + B\cdot u_t$$ + +where $\mathrm{k2}$ selects the state variables, $y_t$ and $y^s$ are in +declaration order and the coefficient matrices are in DR-order. +Effectively, all variables on the right hand side are brought into DR +order for computations and then assigned to $y_t$ in declaration order. + +### Second-order approximation + +The approximation has the form: + +$$y_t = y^s + 0.5 \Delta^2 + A y^h_{t-1} + B u_t + 0.5 C (y^h_{t-1}\otimes y^h_{t-1}) + 0.5 D (u_t \otimes u_t) + E (y^h_{t-1} \otimes u_t)$$ + +where $y^s$ is the steady state value of $y$, $y^h_t=y_t-y^s$, and +$\Delta^2$ is the shift effect of the variance of future shocks. For the +reordering required due to differences in declaration and DR order, see +the first order approximation. + +The coefficients of the decision rules are stored in the variables +described for first order approximation, plus the following variables: + +- $\Delta^2$ is stored in `oo_.dr.ghs2`. The vector rows correspond to + all endogenous in DR-order. +- $C$ is stored in `oo_.dr.ghxx`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of the vector of state variables in DR-order. +- $D$ is stored in `oo_.dr.ghuu`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of exogenous variables in declaration order. +- $E$ is stored in `oo_.dr.ghxu`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of the vector of state variables (in DR-order) by + the vector of exogenous variables (in declaration order). + +### Third-order approximation + +The approximation has the form: + +$$y_t = y^s + G_0 + G_1 z_t + G_2 (z_t \otimes z_t) + G_3 (z_t \otimes z_t \otimes z_t)$$ + +where $y^s$ is the steady state value of $y$, and $z_t$ is a vector +consisting of the deviation from the steady state of the state variables +(in DR-order) at date $t-1$ followed by the exogenous variables at date +$t$ (in declaration order). The vector $z_t$ is therefore of size $n_z$ += `M_.nspred` + `M_.exo_nbr`. + +The coefficients of the decision rules are stored as follows: + +- $y^s$ is stored in `oo_.dr.ys`. The vector rows correspond to all + endogenous in the declaration order. +- $G_0$ is stored in `oo_.dr.g_0`. The vector rows correspond to all + endogenous in DR-order. +- $G_1$ is stored in `oo_.dr.g_1`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to state + variables in DR-order, followed by exogenous in declaration order. +- $G_2$ is stored in `oo_.dr.g_2`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the + Kronecker product of state variables (in DR-order), followed by + exogenous (in declaration order). Note that the Kronecker product is + stored in a folded way, i.e. symmetric elements are stored only + once, which implies that the matrix has $n_z(n_z+1)/2$ columns. More + precisely, each column of this matrix corresponds to a pair + $(i_1, i_2)$ where each index represents an element of $z_t$ and is + therefore between $1$ and $n_z$. Only non-decreasing pairs are + stored, i.e. those for which $i_1 + \leq i_2$. The columns are arranged in the lexicographical order of + non-decreasing pairs. Also note that for those pairs where + $i_1 \neq i_2$, since the element is stored only once but appears + two times in the unfolded $G_2$ matrix, it must be multiplied by 2 + when computing the decision rules. +- $G_3$ is stored in `oo_.dr.g_3`. The matrix rows correspond to all + endogenous in DR-order. The matrix columns correspond to the third + Kronecker power of state variables (in DR-order), followed by + exogenous (in declaration order). Note that the third Kronecker + power is stored in a folded way, i.e. symmetric elements are stored + only once, which implies that the matrix has $n_z(n_z+1)(n_z+2)/6$ + columns. More precisely, each column of this matrix corresponds to a + tuple $(i_1, i_2, i_3)$ where each index represents an element of + $z_t$ and is therefore between $1$ and $n_z$. Only non-decreasing + tuples are stored, i.e. those for which $i_1 \leq i_2 \leq i_3$. The + columns are arranged in the lexicographical order of non-decreasing + tuples. Also note that for tuples that have three distinct indices + (i.e. $i_1 \neq i_2$ and $i_1 \neq i_3$ and $i_2 + \neq i_3$), since these elements are stored only once but appears + six times in the unfolded $G_3$ matrix, they must be multiplied by 6 + when computing the decision rules. Similarly, for those tuples that + have two equal indices (i.e. of the form $(a,a,b)$ or $(a,b,a)$ or + $(b,a,a)$), since these elements are stored only once but appears + three times in the unfolded $G_3$ matrix, they must be multiplied by + 3 when computing the decision rules. + +### Higher-order approximation + +Higher-order approximations are simply a generalization of what is done +at order 3. + +The steady state is stored in `oo_.dr.ys` and the constant correction is +stored in `oo_.dr.g_0`. The coefficient for orders 1, 2, 3, 4... are +respectively stored in `oo_.dr.g_0`, `oo_.dr.g_1`, `oo_.dr.g_2`, +`oo_.dr.g_3`, `oo_.dr.g_4`... The columns of those matrices correspond +to multidimensional indices of state variables, in such a way that +symmetric elements are never repeated (for more details, see the +description of `oo_.dr.g_3` in the third-order case). + + +Estimation based on likelihood {#estim} +------------------------------ + +Provided that you have observations on some endogenous variables, it is +possible to use Dynare to estimate some or all parameters. Both maximum +likelihood (as in *Ireland (2004)*) and Bayesian techniques (as in +*Fernández-Villaverde and Rubio-Ramírez (2004)*, *Rabanal and +Rubio-Ramirez (2003)*, *Schorfheide (2000)* or *Smets and Wouters +(2003)*) are available. Using Bayesian methods, it is possible to +estimate DSGE models, VAR models, or a combination of the two techniques +called DSGE-VAR. + +Note that in order to avoid stochastic singularity, you must have at +least as many shocks or measurement errors in your model as you have +observed variables. + +The estimation using a first order approximation can benefit from the +block decomposition of the model (see `block`{.interpreted-text +role="opt"}). + +::: {#varobs} +::: {.command} +varobs VARIABLE\_NAME\...; + +This command lists the name of observed endogenous variables for the +estimation procedure. These variables must be available in the data file +(see `estimation_cmd `{.interpreted-text role="ref"}). + +Alternatively, this command is also used in conjunction with the +`partial_information` option of `stoch_simul`, for declaring the set of +observed variables when solving the model under partial information. + +Only one instance of `varobs` is allowed in a model file. If one needs +to declare observed variables in a loop, the macro processor can be used +as shown in the second example below. + +*Example* + +> varobs C y rr; +> +> Declares endogenous variables `C`, `y` and `rr` as observed variables. + +*Example* (with a macro processor loop) + +> varobs +> @#for co in countries +> GDP_@{co} +> @#endfor +> ; +::: +::: + +::: {.block} +observation\_trends ; + +This block specifies linear trends for observed variables as functions +of model parameters. In case the `loglinear` option is used, this +corresponds to a linear trend in the logged observables, i.e. an +exponential trend in the level of the observables. + +Each line inside of the block should be of the form: + + VARIABLE_NAME(EXPRESSION); + +In most cases, variables shouldn't be centered when `observation_trends` +is used. + +*Example* + +> observation_trends; +> Y (eta); +> P (mu/eta); +> end; +::: + +::: {.block} +estimated\_params ; estimated\_params (overwrite) ; + +This block lists all parameters to be estimated and specifies bounds and +priors as necessary. + +Each line corresponds to an estimated parameter. + +In a maximum likelihood or a method of moments estimation, each line +follows this syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME + , INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ]; + +In a Bayesian MCMC or a penalized method of moments estimation, each +line follows this syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT + [, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE, + PRIOR_MEAN, PRIOR_STANDARD_ERROR [, PRIOR_3RD_PARAMETER [, + PRIOR_4TH_PARAMETER [, SCALE_PARAMETER ] ] ]; + +The first part of the line consists of one of the four following +alternatives: + +- `stderr VARIABLE_NAME` + + Indicates that the standard error of the exogenous variable + VARIABLE\_NAME, or of the observation error/measurement errors + associated with endogenous observed variable VARIABLE\_NAME, is to + be estimated. + +- `corr VARIABLE_NAME1, VARIABLE_NAME2` + + Indicates that the correlation between the exogenous variables + VARIABLE\_NAME1 and VARIABLE\_NAME2, or the correlation of the + observation errors/measurement errors associated with endogenous + observed variables VARIABLE\_NAME1 and VARIABLE\_NAME2, is to be + estimated. Note that correlations set by previous shocks-blocks or + estimation-commands are kept at their value set prior to estimation + if they are not estimated again subsequently. Thus, the treatment is + the same as in the case of deep parameters set during model + calibration and not estimated. + +- `PARAMETER_NAME` + + The name of a model parameter to be estimated + +- `DSGE_PRIOR_WEIGHT` + + Special name for the weigh of the DSGE model in DSGE-VAR model. + +The rest of the line consists of the following fields, some of them +being optional: + +::: {.option} +INITIAL\_VALUE + +Specifies a starting value for the posterior mode optimizer or the +maximum likelihood estimation. If unset, defaults to the prior mean. +::: + +::: {.option} +LOWER\_BOUND + +Specifies a lower bound for the parameter value in maximum likelihood +estimation. In a Bayesian estimation context, sets a lower bound only +effective while maximizing the posterior kernel. This lower bound does +not modify the shape of the prior density, and is only aimed at helping +the optimizer in identifying the posterior mode (no consequences for the +MCMC). For some prior densities (namely inverse gamma, gamma, uniform, +beta or Weibull) it is possible to shift the support of the prior +distributions to the left or the right using +`prior_3rd_parameter `{.interpreted-text +role="opt"}. In this case the prior density is effectively modified +(note that the truncated Gaussian density is not implemented in Dynare). +If unset, defaults to minus infinity (ML) or the natural lower bound of +the prior (Bayesian estimation). +::: + +::: {.option} +UPPER\_BOUND + +Same as `lower_bound`, but specifying an upper bound instead. +::: + +::: {.option} +PRIOR\_SHAPE + +A keyword specifying the shape of the prior density. The possible values +are: `beta_pdf`, `gamma_pdf`, `normal_pdf`, `uniform_pdf`, +`inv_gamma_pdf`, `inv_gamma1_pdf`, `inv_gamma2_pdf` and `weibull_pdf`. +Note that `inv_gamma_pdf` is equivalent to `inv_gamma1_pdf`. +::: + +::: {.option} +PRIOR\_MEAN + +The mean of the prior distribution. +::: + +::: {.option} +PRIOR\_STANDARD\_ERROR + +The standard error of the prior distribution. +::: + +::: {.option} +PRIOR\_3RD\_PARAMETER + +A third parameter of the prior used for generalized beta distribution, +generalized gamma, generalized Weibull and for the uniform distribution. +Default: `0`. +::: + +::: {.option} +PRIOR\_4TH\_PARAMETER + +A fourth parameter of the prior used for generalized beta distribution +and for the uniform distribution. Default: `1`. +::: + +::: {.option} +SCALE\_PARAMETER + +A parameter specific scale parameter for the jumping distribution's +covariance matrix of the Metropolis-Hasting algorithm. +::: + +Note that INITIAL\_VALUE, LOWER\_BOUND, UPPER\_BOUND, PRIOR\_MEAN, +PRIOR\_STANDARD\_ERROR, PRIOR\_3RD\_PARAMETER, PRIOR\_4TH\_PARAMETER and +SCALE\_PARAMETER can be any valid EXPRESSION. Some of them can be empty, +in which Dynare will select a default value depending on the context and +the prior shape. + +In case of the uniform distribution, it can be specified either by +providing an upper and a lower bound using +`PRIOR_3RD_PARAMETER`{.interpreted-text role="opt"} and +`PRIOR_4TH_PARAMETER`{.interpreted-text role="opt"} or via mean and +standard deviation using `PRIOR_MEAN`{.interpreted-text role="opt"}, +`PRIOR_STANDARD_ERROR`{.interpreted-text role="opt"}. The other two will +automatically be filled out. Note that providing both sets of +hyperparameters will yield an error message. + +As one uses options more towards the end of the list, all previous +options must be filled: for example, if you want to specify +SCALE\_PARAMETER, you must specify `PRIOR_3RD_PARAMETER` and +`PRIOR_4TH_PARAMETER`. Use empty values, if these parameters don't +apply. + +*Example* + +> corr eps_1, eps_2, 0.5, , , beta_pdf, 0, 0.3, -1, 1; +> +> Sets a generalized beta prior for the correlation between `eps_1` and +> `eps_2` with mean `0` and variance `0.3`. By setting +> `PRIOR_3RD_PARAMETER` to `-1` and `PRIOR_4TH_PARAMETER` to `1` the +> standard beta distribution with support `[0,1]` is changed to a +> generalized beta with support `[-1,1]`. Note that LOWER\_BOUND and +> UPPER\_BOUND are left empty and thus default to `-1` and `1`, +> respectively. The initial value is set to `0.5`. + +*Example* + +> corr eps_1, eps_2, 0.5, -0.5, 1, beta_pdf, 0, 0.3, -1, 1; +> +> Sets the same generalized beta distribution as before, but now +> truncates this distribution to `[-0.5,1]` through the use of +> LOWER\_BOUND and UPPER\_BOUND. + +*Parameter transformation* + +Sometimes, it is desirable to estimate a transformation of a parameter +appearing in the model, rather than the parameter itself. It is of +course possible to replace the original parameter by a function of the +estimated parameter everywhere is the model, but it is often +unpractical. + +In such a case, it is possible to declare the parameter to be estimated +in the parameters statement and to define the transformation, using a +pound sign (\#) expression (see `model-decl`{.interpreted-text +role="ref"}). + +*Example* + +> parameters bet; +> +> model; +> # sig = 1/bet; +> c = sig*c(+1)*mpk; +> end; +> +> estimated_params; +> bet, normal_pdf, 1, 0.05; +> end; + +It is possible to have several `estimated_params` blocks. By default, +subsequent blocks are concatenated with the previous ones; this can be +useful when building models in a modular fashion (see also +`estimated_params_remove`{.interpreted-text role="bck"} for that use +case). However, if an `estimated_params` block has the `overwrite` +option, its contents becomes the new list of estimated parameters, +cancelling previous blocks; this can be useful when doing several +estimations in a single `.mod` file. +::: + +::: {.block} +estimated\_params\_init ; estimated\_params\_init (OPTIONS\...); + +This block declares numerical initial values for the optimizer when +these ones are different from the prior mean. It should be specified +after the `estimated_params` block as otherwise the specified starting +values are overwritten by the latter. + +Each line has the following syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, INITIAL_VALUE; + +*Options* + +::: {.option} +use\_calibration + +For not specifically initialized parameters, use the deep parameters and +the elements of the covariance matrix specified in the `shocks` block +from calibration as starting values for estimation. For components of +the `shocks` block that were not explicitly specified during calibration +or which violate the prior, the prior mean is used. +::: + +See `estimated_params`{.interpreted-text role="bck"}, for the meaning +and syntax of the various components. +::: + +::: {.block} +estimated\_params\_bounds ; + +This block declares lower and upper bounds for parameters in maximum +likelihood estimation. + +Each line has the following syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND; + +See `estimated_params`{.interpreted-text role="bck"}, for the meaning +and syntax of the various components. +::: + +::: {.block} +estimated\_params\_remove ; + +This block partially undoes the effect of a previous +`estimated_params`{.interpreted-text role="bck"} block, by removing some +parameters from the estimation. + +Each line has the following syntax: + + stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME; +::: + +::: {#estim-comm} +::: {.command} +estimation \[VARIABLE\_NAME\...\]; estimation (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +This command runs Bayesian or maximum likelihood estimation. + +The following information will be displayed by the command: + +- Results from posterior optimization (also for maximum likelihood) +- Marginal log data density +- Posterior mean and highest posterior density interval (shortest + credible set) from posterior simulation +- Convergence diagnostic table when only one MCM chain is used or + Metropolis-Hastings convergence graphs documented in + *Pfeifer (2014)* in case of multiple MCM chains +- Table with numerical inefficiency factors of the MCMC +- Graphs with prior, posterior, and mode +- Graphs of smoothed shocks, smoothed observation errors, smoothed and + historical variables + +Note that the posterior moments, smoothed variables, k-step ahead +filtered variables and forecasts (when requested) will only be computed +on the variables listed after the `estimation` command. Alternatively, +one can choose to compute these quantities on all endogenous or on all +observed variables (see `consider_all_endogenous`, +`consider_all_endogenous_and_auxiliary`, and `consider_only_observed` +options below). If no variable is listed after the estimation command, +then Dynare will interactively ask which variable set to use. + +Also, during the MCMC (Bayesian estimation with `mh_replic` $>0$) a +(graphical or text) waiting bar is displayed showing the progress of the +Monte-Carlo and the current value of the acceptance ratio. Note that if +the `load_mh_file` option is used (see below) the reported acceptance +ratio does not take into account the draws from the previous MCMC. In +the literature there is a general agreement for saying that the +acceptance ratio should be close to one third or one quarter. If this +not the case, you can stop the MCMC (`Ctrl-C`) and change the value of +option `mh_jscale` (see below). + +Note that by default Dynare generates random numbers using the algorithm +`mt199937ar` (i.e. Mersenne Twister method) with a seed set equal to +`0`. Consequently the MCMCs in Dynare are deterministic: one will get +exactly the same results across different Dynare runs (*ceteris +paribus*). For instance, the posterior moments or posterior densities +will be exactly the same. This behaviour allows to easily identify the +consequences of a change on the model, the priors or the estimation +options. But one may also want to check that across multiple runs, with +different sequences of proposals, the returned results are almost +identical. This should be true if the number of iterations (i.e. the +value of `mh_replic`) is important enough to ensure the convergence of +the MCMC to its ergodic distribution. In this case the default behaviour +of the random number generators in not wanted, and the user should set +the seed according to the system clock before the estimation command +using the following command: + + set_dynare_seed('clock'); + +so that the sequence of proposals will be different across different +runs. + +Finally, Dynare does not always properly distinguish between maximum +likelihood and Bayesian estimation in its field names. While there is an +important conceptual distinction between frequentist confidence +intervals and Bayesian highest posterior density intervals (HPDI) as +well as between posterior density and likelilhood, Dynare sometimes uses +the Bayesian terms as a stand-in in its display of maximum likelihood +results. An example is the storage of the output of the +`forecast`-option of `estimation` with ML, which will use +`HPDinf/HPDsup` to denote the confidence interval. + +*Algorithms* + +The Monte Carlo Markov Chain (MCMC) diagnostics are generated by the +estimation command if +`mh_replic `{.interpreted-text role="opt"} is +larger than 2000 and if option `nodiagnostic`{.interpreted-text +role="opt"} is not used. If +`mh_nblocks `{.interpreted-text role="opt"} is +equal to one, the convergence diagnostics of *Geweke (1992,1999)* is +computed. It uses a chi-square test to compare the means of the first +and last draws specified by `geweke_interval +`{.interpreted-text role="opt"} after +discarding the burn-in of `mh_drop `{.interpreted-text +role="opt"}. The test is computed using variance estimates under the +assumption of no serial correlation as well as using tapering windows +specified in `taper_steps +`{.interpreted-text role="opt"}. +If `mh_nblocks +`{.interpreted-text role="opt"} is larger than 1, +the convergence diagnostics of *Brooks and Gelman (1998)* are used +instead. As described in section 3 of *Brooks and Gelman (1998)* the +univariate convergence diagnostics are based on comparing pooled and +within MCMC moments (Dynare displays the second and third order moments, +and the length of the Highest Probability Density interval covering 80% +of the posterior distribution). Due to computational reasons, the +multivariate convergence diagnostic does not follow *Brooks and Gelman +(1998)* strictly, but rather applies their idea for univariate +convergence diagnostics to the range of the posterior likelihood +function instead of the individual parameters. The posterior kernel is +used to aggregate the parameters into a scalar statistic whose +convergence is then checked using the *Brooks and Gelman (1998)* +univariate convergence diagnostic. + +The inefficiency factors are computed as in *Giordano et al.(2011)* +based on Parzen windows as in e.g. *Andrews (1991)*. + +*Options* + +::: {#dataf} +::: {.option} +datafile = FILENAME + +The datafile: a `.m` file, a `.mat` file, a `.csv` file, or a +`.xls/.xlsx` file (under Octave, the +[io](https://octave.sourceforge.io/io/) package from Octave-Forge is +required for the `.csv` and `.xlsx` formats and the `.xls` file +extension is not supported). Note that the base name (i.e. without +extension) of the datafile has to be different from the base name of the +model file. If there are several files named FILENAME, but with +different file endings, the file name must be included in quoted strings +and provide the file ending like: + + estimation(datafile='../fsdat_simul.mat',...); +::: +::: + +::: {.option} +dirname = FILENAME + +Directory in which to store `estimation` output. To pass a subdirectory +of a directory, you must quote the argument. Default: ``. +::: + +::: {.option} +xls\_sheet = QUOTED\_STRING + +The name of the sheet with the data in an Excel file. +::: + +::: {.option} +xls\_range = RANGE + +The range with the data in an Excel file. For example, +`xls_range=B2:D200`. +::: + +::: {.option} +nobs = INTEGER + +The number of observations following `first_obs `{.interpreted-text role="opt"} to be used. +Default: all observations in the file after `first_obs`. +::: + +::: {.option} +nobs = \[INTEGER1:INTEGER2\] + +Runs a recursive estimation and forecast for samples of size ranging of +`INTEGER1` to `INTEGER2`. Option `forecast` must also be specified. The +forecasts are stored in the `RecursiveForecast` field of the results +structure (see +`RecursiveForecast `{.interpreted-text +role="mvar"}). The respective results structures `oo_` are saved in +`oo_recursive_` (see `oo_recursive_`{.interpreted-text role="mvar"}) and +are indexed with the respective sample length. +::: + +::: {.option} +first\_obs = INTEGER + +The number of the first observation to be used. In case of estimating a +DSGE-VAR, `first_obs` needs to be larger than the number of lags. +Default: `1`. +::: + +::: {.option} +first\_obs = \[INTEGER1:INTEGER2\] + +Runs a rolling window estimation and forecast for samples of fixed size +`nobs` starting with the first observation ranging from `INTEGER1` to +`INTEGER2`. Option `forecast` must also be specified. This option is +incompatible with requesting recursive forecasts using an expanding +window (see `nobs +`{.interpreted-text role="opt"}). The +respective results structures `oo_` are saved in `oo_recursive_` (see +`oo_recursive_`{.interpreted-text role="mvar"}) and are indexed with the +respective first observation of the rolling window. +::: + +::: {.option} +prefilter = INTEGER + +A value of 1 means that the estimation procedure will demean each data +series by its empirical mean. If the `loglinear +`{.interpreted-text role="ref"} option without the +`logdata`{.interpreted-text role="opt"} option is requested, the data +will first be logged and then demeaned. Default: `0`, i.e. no +prefiltering. +::: + +::: {.option} +presample = INTEGER + +The number of observations after `first_obs `{.interpreted-text role="opt"} to be skipped before +evaluating the likelihood. These presample observations do not enter the +likelihood, but are used as a training sample for starting the Kalman +filter iterations. This option is incompatible with estimating a +DSGE-VAR. Default: `0`. +::: + +::: {#logl} +::: {.option} +loglinear + +Computes a log-linear approximation of the model instead of a linear +approximation. As always in the context of estimation, the data must +correspond to the definition of the variables used in the model (see +*Pfeifer (2013)* for more details on how to correctly specify +observation equations linking model variables and the data). If you +specify the loglinear option, Dynare will take the logarithm of both +your model variables and of your data as it assumes the data to +correspond to the original non-logged model variables. The displayed +posterior results like impulse responses, smoothed variables, and +moments will be for the logged variables, not the original un-logged +ones. Default: computes a linear approximation. +::: +::: + +::: {.option} +logdata + +Dynare applies the $log$ transformation to the provided data if a +log-linearization of the model is requested +(`loglinear`{.interpreted-text role="opt"}) unless `logdata` option is +used. This option is necessary if the user provides data already in +logs, otherwise the $log$ transformation will be applied twice (this may +result in complex data). +::: + +::: {.option} +plot\_priors = INTEGER + +Control the plotting of priors. + +> `0` +> +> > No prior plot. +> +> `1` +> +> > Prior density for each estimated parameter is plotted. It is +> > important to check that the actual shape of prior densities matches +> > what you have in mind. Ill-chosen values for the prior standard +> > density can result in absurd prior densities. + +Default value is `1`. +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +posterior\_nograph + +Suppresses the generation of graphs associated with Bayesian IRFs +(`bayesian_irf`{.interpreted-text role="opt"}), posterior smoothed +objects (`smoother`{.interpreted-text role="opt"}), and posterior +forecasts (`forecast`{.interpreted-text role="opt"}). +::: + +::: {.option} +posterior\_graph + +Re-enables the generation of graphs previously shut off with +`posterior_nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See +`graph_format `{.interpreted-text +role="opt"}. +::: + +::: {.option} +no\_init\_estimation\_check\_first\_obs + +Do not check for stochastic singularity in first period. If used, +[ESTIMATION CHECKS]{.title-ref} does not return an error if the check +fails only in first observation. This should only be used when observing +stock variables (e.g. capital) in first period, on top of their +associated flow (e.g. investment). Using this option may lead to a crash +or provide undesired/wrong results for badly specified problems (e.g. +the additional variable observed in first period is not predetermined). + +For advanced use only. +::: + +::: {.option} +lik\_init = INTEGER + +Type of initialization of Kalman filter: + +> `1` +> +> > For stationary models, the initial matrix of variance of the error +> > of forecast is set equal to the unconditional variance of the state +> > variables. +> +> `2` +> +> > For nonstationary models: a wide prior is used with an initial +> > matrix of variance of the error of forecast diagonal with 10 on the +> > diagonal (follows the suggestion of *Harvey and Phillips(1979)*). +> +> `3` +> +> > For nonstationary models: use a diffuse filter (use rather the +> > `diffuse_filter` option). +> +> `4` +> +> > The filter is initialized with the fixed point of the Riccati +> > equation. +> +> `5` +> +> > Use i) option 2 for the non-stationary elements by setting their +> > initial variance in the forecast error matrix to 10 on the diagonal +> > and all covariances to 0 and ii) option 1 for the stationary +> > elements. + +Default value is 1. For advanced use only. +::: + +::: {.option} +lik\_algo = INTEGER + +For internal use and testing only. +::: + +::: {.option} +conf\_sig = DOUBLE + +Level of significance of the confidence interval used for classical +forecasting after estimation. Default: 0.9. +::: + +::: {.option} +mh\_conf\_sig = DOUBLE + +Confidence/HPD interval used for the computation of prior and posterior +statistics like: parameter distributions, prior/posterior moments, +conditional variance decomposition, impulse response functions, Bayesian +forecasting. Default: `0.9`. +::: + +::: {.option} +mh\_replic = INTEGER + +Number of replications for each chain of the Metropolis-Hastings +algorithm. The number of draws should be sufficient to achieve +convergence of the MCMC and to meaningfully compute posterior objects. +Default: `20000`. +::: + +::: {.option} +sub\_draws = INTEGER + +Number of draws from the MCMC that are used to compute posterior +distribution of various objects (smoothed variable, smoothed shocks, +forecast, moments, IRF). The draws used to compute these posterior +moments are sampled uniformly in the estimated empirical posterior +distribution (i.e. draws of the MCMC). `sub_draws` should be smaller +than the total number of MCMC draws available. Default: +`min(posterior_max_subsample_draws, (Total number of draws)*(number of chains) )`. +::: + +::: {.option} +posterior\_max\_subsample\_draws = INTEGER + +Maximum number of draws from the MCMC used to compute posterior +distribution of various objects (smoothed variable, smoothed shocks, +forecast, moments, IRF), if not overriden by option `sub_draws`. +Default: `1200`. +::: + +::: {.option} +mh\_nblocks = INTEGER + +Number of parallel chains for Metropolis-Hastings algorithm. Default: +`2`. +::: + +::: {.option} +mh\_drop = DOUBLE + +The fraction of initially generated parameter vectors to be dropped as a +burn-in before using posterior simulations. Default: `0.5`. +::: + +::: {.option} +mh\_jscale = DOUBLE + +The scale parameter of the jumping distribution's covariance matrix +(Metropolis-Hastings or TaRB-algorithm). The default value is rarely +satisfactory. This option must be tuned to obtain, ideally, an +acceptance ratio of 25%-33%. Basically, the idea is to increase the +variance of the jumping distribution if the acceptance ratio is too +high, and decrease the same variance if the acceptance ratio is too low. +In some situations it may help to consider parameter-specific values for +this scale parameter. This can be done in the +`estimated_params`{.interpreted-text role="bck"} block. + +Note that `mode_compute=6` will tune the scale parameter to achieve an +acceptance rate of `AcceptanceRateTarget`{.interpreted-text +role="ref"}. The resulting scale parameter will be saved into a file +named `MODEL_FILENAME_mh_scale.mat` in the `FILENAME/Output`-folder. +This file can be loaded in subsequent runs via the +`posterior_sampler_options` option +`scale_file `{.interpreted-text role="ref"}. Both +`mode_compute=6` and `scale_file` will overwrite any value specified in +`estimated_params` with the tuned value. Default: `0.2`. + +Note also that for the Random Walk Metropolis Hastings algorithm, it is +possible to use option `mh_tune_jscale +`{.interpreted-text role="opt"}, to +automatically tune the value of `mh_jscale`. In this case, the +`mh_jscale` option must not be used. +::: + +::: {.option} +mh\_init\_scale = DOUBLE + +The scale to be used for drawing the initial value of the +Metropolis-Hastings chain. Generally, the starting points should be +overdispersed for the *Brooks and Gelman (1998)* convergence diagnostics +to be meaningful. Default: `2*mh_jscale.` + +It is important to keep in mind that `mh_init_scale` is set at the +beginning of Dynare execution, i.e. the default will not take into +account potential changes in `mh_jscale` introduced by either +`mode_compute=6` or the `posterior_sampler_options` option `scale_file +`{.interpreted-text role="ref"}. If `mh_init_scale` is too +wide during initalization of the posterior sampler so that 100 tested +draws are inadmissible (e.g. Blanchard-Kahn conditions are always +violated), Dynare will request user input of a new `mh_init_scale` value +with which the next 100 draws will be drawn and tested. If the +`nointeractive`{.interpreted-text role="opt"} option has been invoked, +the program will instead automatically decrease `mh_init_scale` by 10 +percent after 100 futile draws and try another 100 draws. This iterative +procedure will take place at most 10 times, at which point Dynare will +abort with an error message. +::: + +::: {.option} +mh\_tune\_jscale \[= DOUBLE\] + +Automatically tunes the scale parameter of the jumping distribution\'s +covariance matrix (Metropolis-Hastings), so that the overall acceptance +ratio is close to the desired level. Default value is `0.33`. It is not +possible to match exactly the desired acceptance ratio because of the +stochastic nature of the algorithm (the proposals and the initial +conditions of the markov chains if `mh_nblocks>1`). This option is only +available for the Random Walk Metropolis Hastings algorithm. Must not be +used in conjunction with `mh_jscale = DOUBLE`{.interpreted-text +role="opt"}. +::: + +::: {.option} +mh\_tune\_guess = DOUBLE + +Specifies the initial value for the `mh_tune_jscale +`{.interpreted-text role="opt"} option. +Default: `0.2`. Must not be set if +`mh_tune_jscale `{.interpreted-text +role="opt"} is not used. +::: + +::: {.option} +mh\_recover + +Attempts to recover a Metropolis-Hastings simulation that crashed +prematurely, starting with the last available saved `mh`-file. Shouldn't +be used together with `load_mh_file` or a different `mh_replic` than in +the crashed run. Since Dynare 4.5 the proposal density from the previous +run will automatically be loaded. In older versions, to assure a neat +continuation of the chain with the same proposal density, you should +provide the `mode_file` used in the previous run or the same +user-defined `mcmc_jumping_covariance` when using this option. Note that +under Octave, a neat continuation of the crashed chain with the +respective last random number generator state is currently not +supported. +::: + +::: {.option} +mh\_posterior\_mode\_estimation + +Skip optimizer-based mode-finding and instead compute the mode based on +a run of a MCMC. The MCMC will start at the prior mode and use the prior +variances to compute the inverse Hessian. +::: + +::: {.option} +mode\_file = FILENAME + +Name of the file containing previous value for the mode. When computing +the mode, Dynare stores the mode (`xparam1`) and the hessian (`hh`, only +if `cova_compute=1`) in a file called `MODEL_FILENAME_mode.mat` in the +`FILENAME/Output`-folder. After a successful run of the estimation +command, the `mode_file` will be disabled to prevent other function +calls from implicitly using an updated mode-file. Thus, if the `.mod` +file contains subsequent `estimation` commands, the `mode_file` option, +if desired, needs to be specified again. +::: + + +::: {.option} +silent\_optimizer + +Instructs Dynare to run mode computing/optimization silently without +displaying results or saving files in between. Useful when running +loops. +::: + +::: {.option} +mcmc\_jumping\_covariance = OPTION + +Tells Dynare which covariance to use for the proposal density of the +MCMC sampler. OPTION can be one of the following: + +> `hessian` +> +> > Uses the Hessian matrix computed at the mode. +> +> `prior_variance` +> +> > Uses the prior variances. No infinite prior variances are allowed in +> > this case. +> +> `identity_matrix` +> +> > Uses an identity matrix. +> +> `FILENAME` +> +> > Loads an arbitrary user-specified covariance matrix from +> > `FILENAME.mat`. The covariance matrix must be saved in a variable +> > named `jumping_covariance`, must be square, positive definite, and +> > have the same dimension as the number of estimated parameters. + +Note that the covariance matrices are still scaled with +`mh_jscale `{.interpreted-text role="opt"}. Default +value is `hessian`. +::: + +::: {.option} +mode\_check + +Tells Dynare to plot the posterior density for values around the +computed mode for each estimated parameter in turn. This is helpful to +diagnose problems with the optimizer. Note that for `order>1` the +likelihood function resulting from the particle filter is not +differentiable anymore due to the resampling step. For this reason, the +`mode_check` plot may look wiggly. +::: + +::: {.option} +mode\_check\_neighbourhood\_size = DOUBLE + +Used in conjunction with option `mode_check`, gives the width of the +window around the posterior mode to be displayed on the diagnostic +plots. This width is expressed in percentage deviation. The `Inf` value +is allowed, and will trigger a plot over the entire domain (see also +`mode_check_symmetric_plots`). Default:`0.5`. +::: + +::: {.option} +mode\_check\_symmetric\_plots = INTEGER + +Used in conjunction with option `mode_check`, if set to `1`, tells +Dynare to ensure that the check plots are symmetric around the posterior +mode. A value of `0` allows to have asymmetric plots, which can be +useful if the posterior mode is close to a domain boundary, or in +conjunction with `mode_check_neighbourhood_size = Inf` when the domain +in not the entire real line. Default: `1`. +::: + +::: {.option} +mode\_check\_number\_of\_points = INTEGER + +Number of points around the posterior mode where the posterior kernel is +evaluated (for each parameter). Default is `20`. +::: + +::: {.option} +prior\_trunc = DOUBLE + +Probability of extreme values of the prior density that is ignored when +computing bounds for the parameters. Default: `1e-32`. +::: + +::: {.option} +huge\_number = DOUBLE + +Value for replacing infinite values in the definition of (prior) bounds +when finite values are required for computational reasons. Default: +`1e7`. +::: + +::: {.option} +load\_mh\_file + +Tells Dynare to add to previous Metropolis-Hastings simulations instead +of starting from scratch. Since Dynare 4.5 the proposal density from the +previous run will automatically be loaded. In older versions, to assure +a neat continuation of the chain with the same proposal density, you +should provide the `mode_file` used in the previous run or the same +user-defined `mcmc_jumping_covariance` when using this option. Shouldn't +be used together with `mh_recover`. Note that under Octave, a neat +continuation of the chain with the last random number generator state of +the already present draws is currently not supported. +::: + +::: {.option} +load\_results\_after\_load\_mh + +This option is available when loading a previous MCMC run without adding +additional draws, i.e. when `load_mh_file` is specified with +`mh_replic=0`. It tells Dynare to load the previously computed +convergence diagnostics, marginal data density, and posterior statistics +from an existing `_results` file instead of recomputing them. +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc + +This option allows to pick initial values for new MCMC from a previous +one, where the model specification, the number of estimated parameters, +(some) prior might have changed (so a situation where `load_mh_file` +would not work). If an additional parameter is estimated, it is +automatically initialized from prior\_draw. Note that, if this option is +used to skip the optimization step, you should use a sampling method +which does not require a proposal density, like slice. Otherwise, +optimization should always be done beforehand or a mode file with an +appropriate posterior covariance matrix should be used. +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc\_directory = FILENAME + +If `mh_initialize_from_previous_mcmc` is set, users must provide here +the path to the standard FNAME folder from where to load prior +definitions and last MCMC values to be used to initialize the new MCMC. + +Example: if previous project directory is `/my_previous_dir` and FNAME +is `mymodel`, users should set the option as + +`mh_initialize_from_previous_mcmc_directory = '/my_previous_dir/mymodel'` + +Dynare will then look for the last record file into + +`/my_previous_dir/mymodel/metropolis/mymodel_mh_history_.mat` + +and for the prior definition file into + +`/my_previous_dir/mymodel/prior/definition.mat` +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc\_record = FILENAME + +If `mh_initialize_from_previous_mcmc` is set, and whenever the standard +file or directory tree is not applicable to load initial values, users +may directly provide here the path to the record file from which to load +values to be used to initialize the new MCMC. +::: + +::: {.option} +mh\_initialize\_from\_previous\_mcmc\_prior = FILENAME + +If `mh_initialize_from_previous_mcmc` is set, and whenever the standard +file or directory tree is not applicable to load initial values, users +may directly provide here the path to the prior definition file, to get +info in the priors used in previous MCMC. +::: + +::: {.option} +optim = (NAME, VALUE, \...) + +A list of NAME and VALUE pairs. Can be used to set options for the +optimization routines. The set of available options depends on the +selected optimization routine (i.e. on the value of option +`mode_compute `{.interpreted-text role="opt"}): + +> `1, 3, 7, 12, 13` +> +> > Available options are given in the documentation of the MATLAB +> > Optimization Toolbox or in Octave's documentation. +> +> `2` +> +> > Available options are: +> > +> > > `'initial_step_length'` +> > > +> > > > Initial step length. Default: `1`. +> > > +> > > `'initial_temperature'` +> > > +> > > > Initial temperature. Default: `15`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of function evaluations. Default: `100000`. +> > > +> > > `'neps'` +> > > +> > > > Number of final function values used to decide upon termination. +> > > > Default: `10`. +> > > +> > > `'ns'` +> > > +> > > > Number of cycles. Default: `10`. +> > > +> > > `'nt'` +> > > +> > > > Number of iterations before temperature reduction. Default: +> > > > `10`. +> > > +> > > `'step_length_c'` +> > > +> > > > Step length adjustment. Default: `0.1`. +> > > +> > > `'TolFun'` +> > > +> > > > Stopping criteria. Default: `1e-8`. +> > > +> > > `'rt'` +> > > +> > > > Temperature reduction factor. Default: `0.1`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization, ranging from +> > > > `0` (silent) to `3` (each function evaluation). Default: `1` +> +> `4` +> +> > Available options are: +> > +> > > `'InitialInverseHessian'` +> > > +> > > > Initial approximation for the inverse of the Hessian matrix of +> > > > the posterior kernel (or likelihood). Obviously this +> > > > approximation has to be a square, positive definite and +> > > > symmetric matrix. Default: `'1e-4*eye(nx)'`, where nx is the +> > > > number of parameters to be estimated. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `1000`. +> > > +> > > `'NumgradAlgorithm'` +> > > +> > > > Possible values are `2`, `3` and `5`, respectively, +> > > > corresponding to the two, three and five points formula used to +> > > > compute the gradient of the objective function (see *Abramowitz +> > > > and Stegun (1964)*). Values `13` and `15` are more experimental. +> > > > If perturbations on the right and the left increase the value of +> > > > the objective function (we minimize this function) then we force +> > > > the corresponding element of the gradient to be zero. The idea +> > > > is to temporarily reduce the size of the optimization problem. +> > > > Default: `2`. +> > > +> > > `'NumgradEpsilon'` +> > > +> > > > Size of the perturbation used to compute numerically the +> > > > gradient of the objective function. Default: `1e-6`. +> > > +> > > `'TolFun'` +> > > +> > > > Stopping criteria. Default: `1e-7`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> > > +> > > `'SaveFiles'` +> > > +> > > > Controls saving of intermediate results during optimization. Set +> > > > to `0` to shut off saving. Default: `1`. +> +> `5` +> +> > Available options are: +> > +> > `'Hessian'` +> > +> > > Triggers three types of Hessian computations. `0`: outer product +> > > gradient; `1`: default Dynare Hessian routine; `2`: 'mixed' outer +> > > product gradient, where diagonal elements are obtained using +> > > second order derivation formula and outer product is used for +> > > correlation structure. Both {0} and {2} options require univariate +> > > filters, to ensure using maximum number of individual densities +> > > and a positive definite Hessian. Both {0} and {2} are quicker than +> > > default Dynare numeric Hessian, but provide decent starting values +> > > for Metropolis for large models (option {2} being more accurate +> > > than {0}). Default: `1`. +> > +> > `'MaxIter'` +> > +> > > Maximum number of iterations. Default: `1000`. +> > +> > `'TolFun'` +> > +> > > Stopping criteria. Default: `1e-5` for numerical derivatives, +> > > `1e-7` for analytic derivatives. +> > +> > `'verbosity'` +> > +> > > Controls verbosity of display during optimization. Set to `0` to +> > > set to silent. Default: `1`. +> > +> > `'SaveFiles'` +> > +> > > Controls saving of intermediate results during optimization. Set +> > > to `0` to shut off saving. Default: `1`. +> +> `6` +> +> > Available options are: +> > +> > > ::: {#art} +> > > `'AcceptanceRateTarget'` +> > > ::: +> > > +> > > > A real number between zero and one. The scale parameter of the +> > > > jumping distribution is adjusted so that the effective +> > > > acceptance rate matches the value of option +> > > > `'AcceptanceRateTarget'`. Default: `1.0/3.0`. +> > > +> > > `'InitialCovarianceMatrix'` +> > > +> > > > Initial covariance matrix of the jumping distribution. Default +> > > > is `'previous'` if option `mode_file` is used, `'prior'` +> > > > otherwise. +> > > +> > > `'nclimb-mh'` +> > > +> > > > Number of iterations in the last MCMC (climbing mode). Default: +> > > > `200000`. +> > > +> > > `'ncov-mh'` +> > > +> > > > Number of iterations used for updating the covariance matrix of +> > > > the jumping distribution. Default: `20000`. +> > > +> > > `'nscale-mh'` +> > > +> > > > Maximum number of iterations used for adjusting the scale +> > > > parameter of the jumping distribution. Default: `200000`. +> > > +> > > `'NumberOfMh'` +> > > +> > > > Number of MCMC run sequentially. Default: `3`. +> +> `8` +> +> > Available options are: +> > +> > > `'InitialSimplexSize'` +> > > +> > > > Initial size of the simplex, expressed as percentage deviation +> > > > from the provided initial guess in each direction. Default: +> > > > `.05`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `5000`. +> > > +> > > `'MaxFunEvals'` +> > > +> > > > Maximum number of objective function evaluations. No default. +> > > +> > > `'MaxFunvEvalFactor'` +> > > +> > > > Set `MaxFunvEvals` equal to `MaxFunvEvalFactor` times the number +> > > > of estimated parameters. Default: `500`. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-4`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-4`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> +> `9` +> +> > Available options are: +> > +> > > `'CMAESResume'` +> > > +> > > > Resume previous run. Requires the `variablescmaes.mat` from the +> > > > last run. Set to `1` to enable. Default: `0`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. +> > > +> > > `'MaxFunEvals'` +> > > +> > > > Maximum number of objective function evaluations. Default: +> > > > `Inf`. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-7`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-7`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> > > +> > > `'SaveFiles'` +> > > +> > > > Controls saving of intermediate results during optimization. Set +> > > > to `0` to shut off saving. Default: `1`. +> +> `10` +> +> > Available options are: +> > +> > > `'EndTemperature'` +> > > +> > > > Terminal condition w.r.t the temperature. When the temperature +> > > > reaches `EndTemperature`, the temperature is set to zero and the +> > > > algorithm falls back into a standard simplex algorithm. Default: +> > > > `0.1`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `5000`. +> > > +> > > `'MaxFunvEvals'` +> > > +> > > > Maximum number of objective function evaluations. No default. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-4`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-4`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> +> `101` +> +> > Available options are: +> > +> > > `'LBGradientStep'` +> > > +> > > > Lower bound for the stepsize used for the difference +> > > > approximation of gradients. Default: `1e-11`. +> > > +> > > `'MaxIter'` +> > > +> > > > Maximum number of iterations. Default: `15000` +> > > +> > > `'SpaceDilation'` +> > > +> > > > Coefficient of space dilation. Default: `2.5`. +> > > +> > > `'TolFun'` +> > > +> > > > Tolerance parameter (w.r.t the objective function). Default: +> > > > `1e-6`. +> > > +> > > `'TolX'` +> > > +> > > > Tolerance parameter (w.r.t the instruments). Default: `1e-6`. +> > > +> > > `'verbosity'` +> > > +> > > > Controls verbosity of display during optimization. Set to `0` to +> > > > set to silent. Default: `1`. +> +> `102` +> +> > Available options are given in the documentation of the MATLAB +> > Global Optimization Toolbox. + +*Example* + +> To change the defaults of `csminwel` (`mode_compute=4`): +> +> estimation(..., mode_compute=4,optim=('NumgradAlgorithm',3,'TolFun',1e-5),...); +::: + +::: {.option} +nodiagnostic + +Does not compute the convergence diagnostics for Metropolis-Hastings. +Default: diagnostics are computed and displayed. +::: + +::: {.option} +bayesian\_irf + +Triggers the computation of the posterior distribution of IRFs. The +length of the IRFs are controlled by the `irf` option. Results are +stored in `oo_.PosteriorIRF.dsge` (see below for a description of this +variable). +::: + +::: {.option} +relative\_irf + +See `relative_irf`{.interpreted-text role="opt"}. +::: + + +::: {.option} +posterior\_sampling\_method = NAME + +Selects the sampler used to sample from the posterior distribution +during Bayesian estimation. Default:`’random_walk_metropolis_hastings’`. + +> `'random_walk_metropolis_hastings'` +> +> > Instructs Dynare to use the Random-Walk Metropolis-Hastings. In this +> > algorithm, the proposal density is recentered to the previous draw +> > in every step. +> +> `'tailored_random_block_metropolis_hastings'` +> +> > Instructs Dynare to use the Tailored randomized block (TaRB) +> > Metropolis-Hastings algorithm proposed by *Chib and Ramamurthy +> > (2010)* instead of the standard Random-Walk Metropolis-Hastings. In +> > this algorithm, at each iteration the estimated parameters are +> > randomly assigned to different blocks. For each of these blocks a +> > mode-finding step is conducted. The inverse Hessian at this mode is +> > then used as the covariance of the proposal density for a +> > Random-Walk Metropolis-Hastings step. If the numerical Hessian is +> > not positive definite, the generalized Cholesky decomposition of +> > *Schnabel and Eskow (1990)* is used, but without pivoting. The +> > TaRB-MH algorithm massively reduces the autocorrelation in the MH +> > draws and thus reduces the number of draws required to +> > representatively sample from the posterior. However, this comes at a +> > computational cost as the algorithm takes more time to run. +> +> `'independent_metropolis_hastings'` +> +> > Use the Independent Metropolis-Hastings algorithm where the proposal +> > distribution - in contrast to the Random Walk Metropolis-Hastings +> > algorithm - does not depend on the state of the chain. +> +> `'slice'` +> +> > Instructs Dynare to use the Slice sampler of *Planas, Ratto, and +> > Rossi (2015)*. Note that `'slice'` is incompatible with +> > `prior_trunc=0`. +::: + +::: {.option} +posterior\_sampler\_options = (NAME, VALUE, \...) + +A list of NAME and VALUE pairs. Can be used to set options for the +posterior sampling methods. The set of available options depends on the +selected posterior sampling routine (i.e. on the value of option +`posterior_sampling_method +`{.interpreted-text role="opt"}): + +> `'random_walk_metropolis_hastings'` +> +> > Available options are: +> +> ::: {#prop_distrib} +> `'proposal_distribution'` +> ::: +> +> > Specifies the statistical distribution used for the proposal +> > density. +> +> `'rand_multivariate_normal'` +> +> > Use a multivariate normal distribution. This is the default. +> +> `'rand_multivariate_student'` +> +> > Use a multivariate student distribution. +> +> `'student_degrees_of_freedom'` +> +> > Specifies the degrees of freedom to be used with the multivariate +> > student distribution. Default: `3`. +> +> ::: {#usemhcov} +> `'use_mh_covariance_matrix'` +> ::: +> +> > Indicates to use the covariance matrix of the draws from a previous +> > MCMC run to define the covariance of the proposal distribution. +> > Requires the `load_mh_file`{.interpreted-text role="opt"} option to +> > be specified. Default: `0`. +> +> ::: {#scale-file} +> `'scale_file'` +> ::: +> +> > Provides the name of a `_mh_scale.mat` file storing the tuned scale +> > factor from a previous run of `mode_compute=6`. +> +> ::: {#savetmp} +> `'save_tmp_file'` +> ::: +> +> > Save the MCMC draws into a `_mh_tmp_blck` file at the refresh rate +> > of the status bar instead of just saving the draws when the current +> > `_mh*_blck` file is full. Default: `0` +> +> `'independent_metropolis_hastings'` +> +> > Takes the same options as in the case of +> > `random_walk_metropolis_hastings`. +> +> `'slice'` +> +> `'rotated'` +> +> > Triggers rotated slice iterations using a covariance matrix from +> > initial burn-in iterations. Requires either +> > `use_mh_covariance_matrix` or `slice_initialize_with_mode`. Default: +> > `0`. +> +> `'mode_files'` +> +> > For multimodal posteriors, provide the name of a file containing a +> > `nparam` by `nmodes` variable called `xparams` storing the different +> > modes. This array must have one column vector per mode and the +> > estimated parameters along the row dimension. With this info, the +> > code will automatically trigger the `rotated` and `mode` options. +> > Default: `[]`. +> +> `'slice_initialize_with_mode'` +> +> > The default for slice is to set `mode_compute=0` and start the +> > chain(s) from a random location in the prior space. This option +> > first runs the mode-finder and then starts the chain from the mode. +> > Together with `rotated`, it will use the inverse Hessian from the +> > mode to perform rotated slice iterations. Default: `0`. +> +> `'initial_step_size'` +> +> > Sets the initial size of the interval in the stepping-out procedure +> > as fraction of the prior support, i.e. the size will be +> > `initial_step_size * (UB-LB)`. `initial_step_size` must be a real +> > number in the interval `[0,1]`. Default: `0.8`. +> +> `'use_mh_covariance_matrix'` +> +> > See `use_mh_covariance_matrix `{.interpreted-text +> > role="ref"}. Must be used with `'rotated'`. Default: `0`. +> +> `'save_tmp_file'` +> +> > See `save_tmp_file `{.interpreted-text role="ref"}. +> > Default: `1`. +> +> `'tailored_random_block_metropolis_hastings'` +> +> `'proposal_distribution'` +> +> > Specifies the statistical distribution used for the proposal +> > density. See +> > `proposal_distribution `{.interpreted-text +> > role="ref"}. +> +> `new_block_probability = DOUBLE` +> +> > Specifies the probability of the next parameter belonging to a new +> > block when the random blocking in the TaRB Metropolis-Hastings +> > algorithm is conducted. The higher this number, the smaller is the +> > average block size and the more random blocks are formed during each +> > parameter sweep. Default: `0.25`. +> +> `mode_compute = INTEGER` +> +> > Specifies the mode-finder run in every iteration for every block of +> > the TaRB Metropolis-Hastings algorithm. See +> > `mode_compute > INTEGER | FUNCTION_NAME>`{.interpreted-text role="opt"}. Default: +> > `4`. +> +> `optim = (NAME, VALUE,...)` +> +> > Specifies the options for the mode-finder used in the TaRB +> > Metropolis-Hastings algorithm. See `optim +> > `{.interpreted-text role="opt"}. +> +> `'scale_file'` +> +> > See `scale_file `{.interpreted-text role="ref"}.. +> +> `'save_tmp_file'` +> +> > See `save_tmp_file `{.interpreted-text role="ref"}. +> > Default: `1`. +::: + +::: {.option} +moments\_varendo + +Triggers the computation of the posterior distribution of the +theoretical moments of the endogenous variables. Results are stored in +`oo_.PosteriorTheoreticalMoments` (see +`oo_.PosteriorTheoreticalMoments`{.interpreted-text role="mvar"}). The +number of lags in the autocorrelation function is controlled by the `ar` +option. +::: + +::: {.option} +contemporaneous\_correlation + +See `contemporaneous_correlation`{.interpreted-text role="opt"}. Results +are stored in `oo_.PosteriorTheoreticalMoments`. Note that the `nocorr` +option has no effect. +::: + +::: {.option} +no\_posterior\_kernel\_density + +Shuts off the computation of the kernel density estimator for the +posterior objects (see `density `{.interpreted-text role="ref"} +field). +::: + +::: {.option} +conditional\_variance\_decomposition = INTEGER +conditional\_variance\_decomposition = \[INTEGER1:INTEGER2\] +conditional\_variance\_decomposition = \[INTEGER1 INTEGER2 \...\] + +Computes the posterior distribution of the conditional variance +decomposition for the specified period(s). The periods must be strictly +positive. Conditional variances are given by $var(y_{t+k}\vert t)$. For +period 1, the conditional variance decomposition provides the +decomposition of the effects of shocks upon impact. The results are +stored in +`oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecomposition`.. +Note that this option requires the option `moments_varendo` to be +specified. In the presence of measurement error, the field will contain +the variance contribution after measurement error has been taken out, +*i.e.* the decomposition will be conducted of the actual as opposed to +the measured variables. The variance decomposition of the measured +variables will be stored in +`oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecompositionME`. +::: + +::: {.option} +filtered\_vars + +Triggers the computation of the posterior distribution of filtered +endogenous variables/one-step ahead forecasts, i.e. $E_{t}{y_{t+1}}$. +Results are stored in `oo_.FilteredVariables` (see below for a +description of this variable) +::: + +::: {.option} +smoother + +Triggers the computation of the posterior distribution of smoothed +endogenous variables and shocks, i.e. the expected value of variables +and shocks given the information available in all observations up to the +final date ($E_{T}{y_t}$). Results are stored in +`oo_.SmoothedVariables`, `oo_.SmoothedShocks` and +`oo_.SmoothedMeasurementErrors`. Also triggers the computation of +`oo_.UpdatedVariables`, which contains the estimation of the expected +value of variables given the information available at the current date +($E_{t}{y_t}$). See below for a description of all these variables. +::: + +::: {.option} +smoother\_redux + +Triggers a faster computation of the smoothed endogenous variables and +shocks for large models. It runs the smoother only for the state +variables (i.e. with the same representation used for likelihood +computations) and computes the remaining variables ex-post. Static +unobserved objects (filtered, smoothed, updated, k-step ahead) are +recovered, but there are exceptions to a full recovery, depending on how +static unobserved variables depend on the restricted state space +adopted. For example, lagged shocks which are ONLY used to recover +NON-observed static variables will not be recovered). For such +exceptions, only the following output is provided: + +> `FilteredVariablesKStepAhead`: will be fully recovered +> +> `SmoothedVariables`, `FilteredVariables`, `UpdatedVariables`: recovered for all periods beyond period `d+1`, +> +> : where `d` denotes the number of diffuse filtering steps. +> +> `FilteredVariablesKStepAheadVariances`, `Variance`, and +> `State_uncertainty` cannot be recovered, and ZERO is provided as +> output. + +If you need variances for those variables, either do not set the option, +or declare the variable as observed, using NaNs as data points. +::: + +::: {.option} +forecast = INTEGER + +Computes the posterior distribution of a forecast on INTEGER periods +after the end of the sample used in estimation. If no +Metropolis-Hastings is computed, the result is stored in variable +`oo_.forecast` and corresponds to the forecast at the posterior mode. If +a Metropolis-Hastings is computed, the distribution of forecasts is +stored in variables `oo_.PointForecast` and `oo_.MeanForecast`. See +`fore`{.interpreted-text role="ref"}, for a description of these +variables. +::: + +::: {.option} +tex + +See `tex`{.interpreted-text role="opt"}. +::: + +::: {.option} +kalman\_algo = INTEGER + +`0` + +> Automatically use the Multivariate Kalman Filter for stationary models +> and the Multivariate Diffuse Kalman Filter for non-stationary models. + +`1` + +> Use the Multivariate Kalman Filter. + +`2` + +> Use the Univariate Kalman Filter. + +`3` + +> Use the Multivariate Diffuse Kalman Filter. + +`4` + +> Use the Univariate Diffuse Kalman Filter. +::: + +> Default value is `0`. In case of missing observations of single or all +> series, Dynare treats those missing values as unobserved states and +> uses the Kalman filter to infer their value (see e.g. *Durbin and +> Koopman (2012)*, Ch. 4.10) This procedure has the advantage of being +> capable of dealing with observations where the forecast error variance +> matrix becomes singular for some variable(s). If this happens, the +> respective observation enters with a weight of zero in the +> log-likelihood, i.e. this observation for the respective variable(s) +> is dropped from the likelihood computations (for details see *Durbin +> and Koopman (2012)*, Ch. 6.4 and 7.2.5 and *Koopman and Durbin +> (2000)*). If the use of a multivariate Kalman filter is specified and +> a singularity is encountered, Dynare by default automatically switches +> to the univariate Kalman filter for this parameter draw. This behavior +> can be changed via the +> `use_univariate_filters_if_singularity_is_detected +> `{.interpreted-text +> role="opt"} option. + +::: {.option} +fast\_kalman\_filter + +Select the fast Kalman filter using Chandrasekhar recursions as +described by `Herbst (2015)`. This setting is only used with +`kalman_algo=1` or `kalman_algo=3`. In case of using the diffuse Kalman +filter (`kalman_algo=3/lik_init=3`), the observables must be stationary. +This option is not yet compatible with +`analytic_derivation`{.interpreted-text role="opt"}. +::: + +::: {.option} +kalman\_tol = DOUBLE + +Numerical tolerance for determining the singularity of the covariance +matrix of the prediction errors during the Kalman filter (minimum +allowed reciprocal of the matrix condition number). Default value is +`1e-10`. +::: + +::: {.option} +diffuse\_kalman\_tol = DOUBLE + +Numerical tolerance for determining the singularity of the covariance +matrix of the prediction errors ($F_{\infty}$) and the rank of the +covariance matrix of the non-stationary state variables ($P_{\infty}$) +during the Diffuse Kalman filter. Default value is `1e-6`. +::: + +::: {.option} +filter\_covariance + +Saves the series of one step ahead error of forecast covariance +matrices. With Metropolis, they are saved in +`oo_.FilterCovariance`{.interpreted-text role="mvar"}, otherwise in +`oo_.Smoother.Variance`{.interpreted-text role="mvar"}. Saves also +k-step ahead error of forecast covariance matrices if +`filter_step_ahead` is set. +::: + +::: {.option} +filter\_step\_ahead = \[INTEGER1:INTEGER2\] filter\_step\_ahead = +\[INTEGER1 INTEGER2 \...\] + +Triggers the computation k-step ahead filtered values, i.e. +$E_{t}{y_{t+k}}$. Stores results in `oo_.FilteredVariablesKStepAhead`. +Also stores 1-step ahead values in `oo_.FilteredVariables`. +`oo_.FilteredVariablesKStepAheadVariances` is stored if +`filter_covariance`. +::: + +::: {.option} +filter\_decomposition + +Triggers the computation of the shock decomposition of the above k-step +ahead filtered values. Stores results in +`oo_.FilteredVariablesShockDecomposition`. +::: + +::: {.option} +smoothed\_state\_uncertainty + +Triggers the computation of the variance of smoothed estimates, i.e. +$var_T(y_t)$. Stores results in `oo_.Smoother.State_uncertainty`. +::: + +::: {.option} +diffuse\_filter + +Uses the diffuse Kalman filter (as described in *Durbin and Koopman +(2012)* and *Koopman and Durbin (2003)* for the multivariate and +*Koopman and Durbin (2000)* for the univariate filter) to estimate +models with non-stationary observed variables. This option will also +reset the `qz_criterium` to count unit root variables towards the stable +variables. Trying to estimate a model with unit roots will otherwise +result in a Blanchard-Kahn error. + +When `diffuse_filter` is used the `lik_init` option of `estimation` has +no effect. + +When there are nonstationary exogenous variables in a model, there is no +unique deterministic steady state. For instance, if productivity is a +pure random walk: + +> $$a_t = a_{t-1} + e_t$$ + +any value of $\bar a$ of $a$ is a deterministic steady state for +productivity. Consequently, the model admits an infinity of steady +states. In this situation, the user must help Dynare in selecting one +steady state, except if zero is a trivial model's steady state, which +happens when the `linear` option is used in the model declaration. The +user can either provide the steady state to Dynare using a +`steady_state_model` block (or writing a steady state file) if a closed +form solution is available, see `steady_state_model`{.interpreted-text +role="bck"}, or specify some constraints on the steady state, see +`equation_tag_for_conditional_steady_state `{.interpreted-text +role="ref"}, so that Dynare computes the steady state conditionally on +some predefined levels for the non stationary variables. In both cases, +the idea is to use dummy values for the steady state level of the +exogenous non stationary variables. + +Note that the nonstationary variables in the model must be integrated +processes (their first difference or k-difference must be stationary). +::: + + +::: {.option} +selected\_variables\_only + +Only run the classical smoother on the variables listed just after the +`estimation` command. This option is incompatible with requesting +classical frequentist forecasts and will be overridden in this case. +When using Bayesian estimation, the smoother is by default only run on +the declared endogenous variables. Default: run the smoother on all the +declared endogenous variables. +::: + +::: {.option} +cova\_compute = INTEGER + +When `0`, the covariance matrix of estimated parameters is not computed +after the computation of posterior mode (or maximum likelihood). This +increases speed of computation in large models during development, when +this information is not always necessary. Of course, it will break all +successive computations that would require this covariance matrix. +Otherwise, if this option is equal to `1`, the covariance matrix is +computed and stored in variable `hh` of `MODEL_FILENAME_mode.mat`. +Default is `1`. +::: + +::: {.option} +solve\_algo = INTEGER + +See `solve_algo `{.interpreted-text role="ref"}. +::: + +::: {.option} +order = INTEGER + +Order of approximation around the deterministic steady state. When +greater than 1, the likelihood is evaluated with a particle or nonlinear +filter (see *Fernández-Villaverde and Rubio-Ramírez (2005)*). Default is +`1`, i.e. the likelihood of the linearized model is evaluated using a +standard Kalman filter. +::: + +::: {.option} +irf = INTEGER + +See `irf `{.interpreted-text role="opt"}. Only used if +`bayesian_irf`{.interpreted-text role="opt"} is passed. +::: + +::: {.option} +irf\_shocks = ( VARIABLE\_NAME \[\[,\] VARIABLE\_NAME \...\] ) + +See `irf_shocks `{.interpreted-text role="opt"}. Only used if +`bayesian_irf`{.interpreted-text role="opt"} is passed. +::: + +::: {.option} +irf\_plot\_threshold = DOUBLE + +See `irf_plot_threshold `{.interpreted-text role="opt"}. Only used if +`bayesian_irf`{.interpreted-text role="opt"} is passed. +::: + +::: {.option} +aim\_solver + +See `aim_solver`{.interpreted-text role="opt"}. +::: + +::: {.option} +sylvester = OPTION + +See `sylvester `{.interpreted-text role="opt"}. +::: + +::: {.option} +sylvester\_fixed\_point\_tol = DOUBLE + +See `sylvester_fixed_point_tol `{.interpreted-text role="opt"} . +::: + +::: {.option} +lyapunov = OPTION + +Determines the algorithm used to solve the Lyapunov equation to +initialized the variance-covariance matrix of the Kalman filter using +the steady-state value of state variables. Possible values for OPTION +are: + +> `default` +> +> > Uses the default solver for Lyapunov equations based on +> > Bartels-Stewart algorithm. +> +> `fixed_point` +> +> > Uses a fixed point algorithm to solve the Lyapunov equation. This +> > method is faster than the `default` one for large scale models, but +> > it could require a large amount of iterations. +> +> `doubling` +> +> > Uses a doubling algorithm to solve the Lyapunov equation +> > (`disclyap_fast`). This method is faster than the two previous one +> > for large scale models. +> +> `square_root_solver` +> +> > Uses a square-root solver for Lyapunov equations (`dlyapchol`). This +> > method is fast for large scale models (available under MATLAB if the +> > Control System Toolbox is installed; available under Octave if the +> > [control](https://octave.sourceforge.io/control/) package from +> > Octave-Forge is installed) + +Default value is `default`. +::: + +::: {.option} +lyapunov\_fixed\_point\_tol = DOUBLE + +This is the convergence criterion used in the fixed point Lyapunov +solver. Its default value is `1e-10`. +::: + +::: {.option} +lyapunov\_doubling\_tol = DOUBLE + +This is the convergence criterion used in the doubling algorithm to +solve the Lyapunov equation. Its default value is `1e-16`. +::: + +::: {.option} +use\_penalized\_objective\_for\_hessian + +Use the penalized objective instead of the objective function to compute +numerically the hessian matrix at the mode. The penalties decrease the +value of the posterior density (or likelihood) when, for some +perturbations, Dynare is not able to solve the model (issues with steady +state existence, Blanchard and Kahn conditions, \...). In pratice, the +penalized and original objectives will only differ if the posterior mode +is found to be near a region where the model is ill-behaved. By default +the original objective function is used. +::: + +::: {.option} +analytic\_derivation + +Triggers estimation with analytic gradient at `order=1`. The final +hessian at the mode is also computed analytically. Only works for +stationary models without missing observations, i.e. for +`kalman_algo<3`. Optimizers that rely on analytic gradients are +`mode_compute=1,3,4,5,101`. +::: + +::: {.option} +ar = INTEGER + +See `ar `{.interpreted-text role="opt"}. Only useful in +conjunction with option `moments_varendo`. +::: + +::: {.option} +endogenous\_prior + +Use endogenous priors as in *Christiano, Trabandt and Walentin (2011)*. +The procedure is motivated by sequential Bayesian learning. Starting +from independent initial priors on the parameters, specified in the +`estimated_params` block, the standard deviations observed in a +\"pre-sample\", taken to be the actual sample, are used to update the +initial priors. Thus, the product of the initial priors and the +pre-sample likelihood of the standard deviations of the observables is +used as the new prior (for more information, see the technical appendix +of *Christiano, Trabandt and Walentin (2011)*). This procedure helps in +cases where the regular posterior estimates, which minimize in-sample +forecast errors, result in a large overprediction of model variable +variances (a statistic that is not explicitly targeted, but often of +particular interest to researchers). +::: + +::: {.option} +use\_univariate\_filters\_if\_singularity\_is\_detected = INTEGER + +Decide whether Dynare should automatically switch to univariate filter +if a singularity is encountered in the likelihood computation (this is +the behaviour if the option is equal to `1`). Alternatively, if the +option is equal to `0`, Dynare will not automatically change the filter, +but rather use a penalty value for the likelihood when such a +singularity is encountered. Default: `1`. +::: + +::: {.option} +keep\_kalman\_algo\_if\_singularity\_is\_detected + +With the default `use_univariate_filters_if_singularity_is_detected=1 +`{.interpreted-text +role="opt"}, Dynare will switch to the univariate Kalman filter when it +encounters a singular forecast error variance matrix during Kalman +filtering. Upon encountering such a singularity for the first time, all +subsequent parameter draws and computations will automatically rely on +univariate filter, i.e. Dynare will never try the multivariate filter +again. Use the `keep_kalman_algo_if_singularity_is_detected` option to +have the `use_univariate_filters_if_singularity_is_detected` only affect +the behavior for the current draw/computation. +::: + +::: {.option} +rescale\_prediction\_error\_covariance + +Rescales the prediction error covariance in the Kalman filter to avoid +badly scaled matrix and reduce the probability of a switch to univariate +Kalman filters (which are slower). By default no rescaling is done. +::: + +::: {.option} +qz\_zero\_threshold = DOUBLE + +See `qz_zero_threshold `{.interpreted-text +role="opt"}. +::: + +::: {.option} +taper\_steps = \[INTEGER1 INTEGER2 \...\] + +Percent tapering used for the spectral window in the *Geweke +(1992,1999)* convergence diagnostics (requires +`mh_nblocks=1 `{.interpreted-text role="opt"}). +The tapering is used to take the serial correlation of the posterior +draws into account. Default: `[4 8 15]`. +::: + +::: {.option} +geweke\_interval = \[DOUBLE DOUBLE\] + +Percentage of MCMC draws at the beginning and end of the MCMC chain +taken to compute the *Geweke (1992,1999)* convergence diagnostics +(requires `mh_nblocks=1 `{.interpreted-text role="opt"}) after discarding the first +`mh_drop = DOUBLE +`{.interpreted-text role="opt"} percent of draws as a burnin. +Default: \[0.2 0.5\]. +::: + +::: {.option} +raftery\_lewis\_diagnostics + +Triggers the computation of the *Raftery and Lewis (1992)* convergence +diagnostics. The goal is deliver the number of draws required to +estimate a particular quantile of the CDF `q` with precision `r` with a +probability `s`. Typically, one wants to estimate the `q=0.025` +percentile (corresponding to a 95 percent HPDI) with a precision of 0.5 +percent (`r=0.005`) with 95 percent certainty (`s=0.95`). The defaults +can be changed via `raftery_lewis_qrs +`{.interpreted-text +role="opt"}. Based on the theory of first order Markov Chains, the +diagnostics will provide a required burn-in (`M`), the number of draws +after the burnin (`N`) as well as a thinning factor that would deliver a +first order chain (`k`). The last line of the table will also deliver +the maximum over all parameters for the respective values. +::: + +::: {.option} +raftery\_lewis\_qrs = \[DOUBLE DOUBLE DOUBLE\] + +Sets the quantile of the CDF `q` that is estimated with precision `r` +with a probability `s` in the *Raftery and Lewis (1992)* convergence +diagnostics. Default: `[0.025 0.005 0.95]`. +::: + +::: {.option} +consider\_all\_endogenous + +Compute the posterior moments, smoothed variables, k-step ahead filtered +variables and forecasts (when requested) on all the endogenous +variables. This is equivalent to manually listing all the endogenous +variables after the `estimation` command. +::: + +::: {.option} +consider\_all\_endogenous\_and\_auxiliary + +Compute the posterior moments, smoothed variables, k-step ahead filtered +variables and forecasts (when requested) on all the endogenous variables +and the auxiliary variables introduced by the preprocessor. This option +is useful when e.g. running `smoother2histval` on the results of the +Kalman smoother. +::: + +::: {.option} +consider\_only\_observed + +Compute the posterior moments, smoothed variables, k-step ahead filtered +variables and forecasts (when requested) on all the observed variables. +This is equivalent to manually listing all the observed variables after +the `estimation` command. +::: + + +*"Endogenous" prior restrictions* + +It is also possible to impose implicit "endogenous" priors about IRFs +and moments on the model during estimation. For example, one can specify +that all valid parameter draws for the model must generate fiscal +multipliers that are bigger than 1 by specifying how the IRF to a +government spending shock must look like. The prior restrictions can be +imposed via `irf_calibration` and `moment_calibration` blocks (see +`irf-momcal`{.interpreted-text role="ref"}). The way it works internally +is that any parameter draw that is inconsistent with the "calibration" +provided in these blocks is discarded, i.e. assigned a prior density of +0. When specifying these blocks, it is important to keep in mind that +one won't be able to easily do `model_comparison` in this case, because +the prior density will not integrate to 1. + +*Output* + +After running estimation, the parameters `M_.params` and the variance +matrix `M_.Sigma_e` of the shocks are set to the mode for maximum +likelihood estimation or posterior mode computation without Metropolis +iterations. After estimation with Metropolis iterations (option +`mh_replic > 0` or option `load_mh_file` set) the parameters `M_.params` +and the variance matrix `M_.Sigma_e` of the shocks are set to the +posterior mean. + +Depending on the options, `estimation` stores results in various fields +of the `oo_` structure, described below. In the following variables, we +will adopt the following shortcuts for specific field names: + +> `MOMENT_NAME` +> +> > This field can take the following values: +> > +> > `HPDinf` +> > +> > > Lower bound of a 90% HPD interval.[^4] +> > +> > `HPDsup` +> > +> > > Upper bound of a 90% HPD interval. +> > +> > `HPDinf_ME` +> > +> > > Lower bound of a 90% HPD interval[^5] for observables when taking +> > > measurement error into account (see e.g. *Christoffel et al. +> > > (2010*), p.17). +> > +> > `HPDsup_ME` +> > +> > > Upper bound of a 90% HPD interval for observables when taking +> > > measurement error into account. +> > +> > `Mean` +> > +> > > Mean of the posterior distribution. +> > +> > `Median` +> > +> > > Median of the posterior distribution. +> > +> > `Std` +> > +> > > Standard deviation of the posterior distribution. +> > +> > `Variance` +> > +> > > Variance of the posterior distribution. +> > +> > `deciles` +> > +> > > Deciles of the distribution. +> > +> > ::: {#dens} +> > `density` +> > ::: +> > +> > > Non parametric estimate of the posterior density following the +> > > approach outlined in *Skoeld and Roberts (2003)*. First and second +> > > columns are respectively abscissa and ordinate coordinates. +> +> `ESTIMATED_OBJECT` +> +> > This field can take the following values: +> > +> > `measurement_errors_corr` +> > +> > > Correlation between two measurement errors. +> > +> > `measurement_errors_std` +> > +> > > Standard deviation of measurement errors. +> > +> > `parameters` +> > +> > > Parameters. +> > +> > `shocks_corr` +> > +> > > Correlation between two structural shocks. +> > +> > `shocks_std` +> > +> > > Standard deviation of structural shocks. + +::: {.matvar} +[oo]().MarginalDensity.LaplaceApproximation + +Variable set by the `estimation` command. Stores the marginal data +density based on the Laplace Approximation. +::: + +::: {.matvar} +[oo]().MarginalDensity.ModifiedHarmonicMean + +Variable set by the `estimation command`, if it is used with +`mh_replic > 0` or `load_mh_file` option. Stores the marginal data +density based on *Geweke (1999)* Modified Harmonic Mean estimator. +::: + +::: {.matvar} +[oo]().posterior.optimization + +Variable set by the `estimation` command if mode-finding is used. Stores +the results at the mode. Fields are of the form: + + oo_.posterior.optimization.OBJECT + +where OBJECT is one of the following: + +> `mode` +> +> > Parameter vector at the mode. +> +> `Variance` +> +> > Inverse Hessian matrix at the mode or MCMC jumping covariance matrix +> > when used with the `MCMC_jumping_covariance > = OPTION>`{.interpreted-text role="opt"} option. +> +> `log_density` +> +> > Log likelihood (ML)/log posterior density (Bayesian) at the mode +> > when used with `mode_compute>0`. +::: + +::: {.matvar} +[oo]().posterior.metropolis + +Variable set by the `estimation` command if `mh_replic>0` is used. +Fields are of the form: + + oo_.posterior.metropolis.OBJECT + +where OBJECT is one of the following: + +> `mean` +> +> > Mean parameter vector from the MCMC. +> +> `Variance` +> +> > Covariance matrix of the parameter draws in the MCMC. +::: + +::: {.matvar} +[oo]().FilteredVariables + +Variable set by the `estimation` command, if it is used with the +`filtered_vars` option. + +After an estimation without Metropolis, fields are of the form: + + oo_.FilteredVariables.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.FilteredVariables.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().FilteredVariablesKStepAhead + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead` option. The k-steps are stored along the rows while +the columns indicate the respective variables. The third dimension of +the array provides the observation for which the forecast has been made. +For example, if `filter_step_ahead=[1 2 4]` and `nobs=200`, the element +(3,5,204) stores the four period ahead filtered value of variable 5 +computed at time t=200 for time t=204. The periods at the beginning and +end of the sample for which no forecasts can be made, e.g. entries +(1,5,1) and (1,5,204) in the example, are set to zero. Note that in case +of Bayesian estimation the variables will be ordered in the order of +declaration after the estimation command (or in general declaration +order if no variables are specified here). In case of running the +classical smoother, the variables will always be ordered in general +declaration order. If the `selected_variables_only`{.interpreted-text +role="opt"} option is specified with the classical smoother, +non-requested variables will be simply left out in this order. +::: + +::: {.matvar} +[oo]().FilteredVariablesKStepAheadVariances + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead option`. It is a 4 dimensional array where the +k-steps are stored along the first dimension, while the fourth dimension +of the array provides the observation for which the forecast has been +made. The second and third dimension provide the respective variables. +For example, if `filter_step_ahead=[1 2 4]` and `nobs=200`, the element +(3,4,5,204) stores the four period ahead forecast error covariance +between variable 4 and variable 5, computed at time t=200 for time +t=204. Padding with zeros and variable ordering is analogous to +`oo_.FilteredVariablesKStepAhead`. +::: + +::: {.matvar} +[oo]().Filtered\_Variables\_X\_step\_ahead + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead option` in the context of Bayesian estimation. Fields +are of the form: + + oo_.Filtered_Variables_X_step_ahead.VARIABLE_NAME + +The n-th entry stores the k-step ahead filtered variable computed at +time n for time n+k. +::: + +::: {.matvar} +[oo]().FilteredVariablesShockDecomposition + +Variable set by the `estimation` command, if it is used with the +`filter_step_ahead` option. The k-steps are stored along the rows while +the columns indicate the respective variables. The third dimension +corresponds to the shocks in declaration order. The fourth dimension of +the array provides the observation for which the forecast has been made. +For example, if `filter_step_ahead=[1 2 4]` and `nobs=200`, the element +(3,5,2,204) stores the contribution of the second shock to the four +period ahead filtered value of variable 5 (in deviations from the mean) +computed at time t=200 for time t=204. The periods at the beginning and +end of the sample for which no forecasts can be made, e.g. entries +(1,5,1) and (1,5,204) in the example, are set to zero. Padding with +zeros and variable ordering is analogous to +`oo_.FilteredVariablesKStepAhead`. +::: + +::: {.matvar} +[oo]().PosteriorIRF.dsge + +Variable set by the `estimation` command, if it is used with the +`bayesian_irf` option. Fields are of the form: + + oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME +::: + +::: {.matvar} +[oo]().SmoothedMeasurementErrors + +Variable set by the `estimation` command, if it is used with the +`smoother` option. Fields are of the form: + + oo_.SmoothedMeasurementErrors.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().SmoothedShocks + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. + +After an estimation without Metropolis, or if computed by +`calib_smoother`, fields are of the form: + + oo_.SmoothedShocks.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.SmoothedShocks.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().SmoothedVariables + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. + +After an estimation without Metropolis, or if computed by +`calib_smoother`, fields are of the form: + + oo_.SmoothedVariables.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.SmoothedVariables.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matcomm} +get\_smooth (\'VARIABLE\_NAME\' \[, \'VARIABLE\_NAME\'\]\...); + +Returns the smoothed values of the given endogenous or exogenous +variable(s), as they are stored in the `oo_.SmoothedVariables` and +`oo_.SmoothedShocks` variables. +::: + +::: {.matvar} +[oo]().UpdatedVariables + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. Contains the estimation of +the expected value of variables given the information available at the +current date. + +After an estimation without Metropolis, or if computed by +`calib_smoother`, fields are of the form: + + oo_.UpdatedVariables.VARIABLE_NAME + +After an estimation with Metropolis, fields are of the form: + + oo_.UpdatedVariables.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matcomm} +get\_update (\'VARIABLE\_NAME\' \[, \'VARIABLE\_NAME\'\]\...); + +Returns the updated values of the given variable(s), as they are stored +in the `oo_.UpdatedVariables` variable. +::: + +::: {.matvar} +[oo]().FilterCovariance + +Three-dimensional array set by the `estimation` command if used with the +`smoother` and Metropolis, if the `filter_covariance` option has been +requested. Contains the series of one-step ahead forecast error +covariance matrices from the Kalman smoother. The `M_.endo_nbr` times +`M_.endo_nbr` times `T+1` array contains the variables in declaration +order along the first two dimensions. The third dimension of the array +provides the observation for which the forecast has been made. Fields +are of the form: + + oo_.FilterCovariance.MOMENT_NAME + +Note that density estimation is not supported. +::: + +::: {.matvar} +[oo]().Smoother.Variance + +Three-dimensional array set by the `estimation` command (if used with +the `smoother`) without Metropolis, or by the `calib_smoother` command, +if the `filter_covariance` option has been requested. Contains the +series of one-step ahead forecast error covariance matrices from the +Kalman smoother. The `M_.endo_nbr` times `M_.endo_nbr` times `T+1` array +contains the variables in declaration order along the first two +dimensions. The third dimension of the array provides the observation +for which the forecast has been made. +::: + +::: {.matvar} +[oo]().Smoother.State\_uncertainty + +Three-dimensional array set by the `estimation` command (if used with +the `smoother` option) without Metropolis, or by the `calib_smoother` +command, if the `smoothed_state_uncertainty` option has been requested. +Contains the series of covariance matrices for the state estimate given +the full data from the Kalman smoother. The `M_.endo_nbr` times +`M_.endo_nbr` times `T` array contains the variables in declaration +order along the first two dimensions. The third dimension of the array +provides the observation for which the smoothed estimate has been made. +::: + +::: {.matvar} +[oo]().Smoother.SteadyState + +Variable set by the `estimation` command (if used with the `smoother`) +without Metropolis, or by the `calib_smoother` command. Contains the +steady state component of the endogenous variables used in the smoother +in order of variable declaration. +::: + +::: {.matvar} +[oo]().Smoother.TrendCoeffs + +Variable set by the `estimation` command (if used with the `smoother`) +without Metropolis, or by the `calib_smoother` command. Contains the +trend coefficients of the observed variables used in the smoother in +order of declaration of the observed variables. +::: + +::: {.matvar} +[oo]().Smoother.Trend + +Variable set by the `estimation command` (if used with the `smoother` +option), or by the `calib_smoother` command. Contains the trend +component of the variables used in the smoother. + +Fields are of the form: + + oo_.Smoother.Trend.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().Smoother.Constant + +Variable set by the `estimation` command (if used with the `smoother` +option), or by the `calib_smoother` command. Contains the constant part +of the endogenous variables used in the smoother, accounting e.g. for +the data mean when using the prefilter option. + +Fields are of the form: + + oo_.Smoother.Constant.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().Smoother.loglinear + +Indicator keeping track of whether the smoother was run with the +`loglinear `{.interpreted-text role="ref"} option and thus whether +stored smoothed objects are in logs. +::: + +::: {.matvar} +[oo]().PosteriorTheoreticalMoments + +Variable set by the `estimation` command, if it is used with the +`moments_varendo` option. Fields are of the form: + + oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_NAME.VARIABLE_NAME + +where *THEORETICAL\_MOMENT* is one of the following: + +> `covariance` +> +> > Variance-covariance of endogenous variables. +> +> `contemporaneous_correlation` +> +> > Contemporaneous correlation of endogenous variables when the +> > `contemporaneous_correlation`{.interpreted-text role="opt"} option +> > is specified. +> +> `correlation` +> +> > Auto- and cross-correlation of endogenous variables. Fields are +> > vectors with correlations from 1 up to order `options_.ar`. +> +> ::: {#VarianceDecomposition} +> `VarianceDecomposition` +> ::: +> +> > Decomposition of variance (unconditional variance, i.e. at horizon +> > infinity).[^6] +> +> `VarianceDecompositionME` +> +> > Same as [VarianceDecomposition](), but contains the decomposition of +> > the measured as opposed to the actual variable. The joint +> > contribution of the measurement error will be saved in a field named +> > `ME`. +> +> ::: {#ConditionalVarianceDecomposition} +> `ConditionalVarianceDecomposition` +> ::: +> +> > Only if the `conditional_variance_decomposition` option has been +> > specified. In the presence of measurement error, the field will +> > contain the variance contribution after measurement error has been +> > taken out, i.e. the decomposition will be conducted of the actual as +> > opposed to the measured variables. +> +> `ConditionalVarianceDecompositionME` +> +> > Only if the `conditional_variance_decomposition` option has been +> > specified. Same as [ConditionalVarianceDecomposition](), but +> > contains the decomposition of the measured as opposed to the actual +> > variable. The joint contribution of the measurement error will be +> > saved in a field names `ME`. +::: + +::: {.matvar} +[oo]().posterior\_density + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_density.PARAMETER_NAME +::: + +::: {.matvar} +[oo]().posterior\_hpdinf + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_hpdsup + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_mean + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_mode + +Variable set by the `estimation` command during mode-finding. Fields are +of the form: + + oo_.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_std\_at\_mode + +Variable set by the `estimation` command during mode-finding. It is +based on the inverse Hessian at `oo_.posterior_mode`. Fields are of the +form: + + oo_.posterior_std_at_mode.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_std + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_var + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_var.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().posterior\_median + +Variable set by the `estimation` command, if it is used with +`mh_replic > 0` or `load_mh_file` option. Fields are of the form: + + oo_.posterior_median.ESTIMATED_OBJECT.VARIABLE_NAME +::: + +*Example* + +Here are some examples of generated variables: + + oo_.posterior_mode.parameters.alp + oo_.posterior_mean.shocks_std.ex + oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso + +::: {.matvar} +[oo]().dsge\_var.posterior\_mode + +Structure set by the `dsge_var` option of the `estimation` command after +mode\_compute. + +The following fields are saved: + +> `PHI_tilde` +> +> > Stacked posterior DSGE-BVAR autoregressive matrices at the mode +> > (equation (28) of *Del Negro and Schorfheide (2004)*). +> +> `SIGMA_u_tilde` +> +> > Posterior covariance matrix of the DSGE-BVAR at the mode (equation +> > (29) of *Del Negro and Schorfheide (2004)*). +> +> `iXX` +> +> > Posterior population moments in the DSGE-BVAR at the mode ( +> > $inv(\lambda T \Gamma_{XX}^*+ X'X)$). +> +> `prior` +> +> > Structure storing the DSGE-BVAR prior. +> +> `PHI_star` +> +> > Stacked prior DSGE-BVAR autoregressive matrices at the mode +> > (equation (22) of *Del Negro and Schorfheide (2004)*). +> +> `SIGMA_star` +> +> > Prior covariance matrix of the DSGE-BVAR at the mode (equation (23) +> > of *Del Negro and Schorfheide (2004)*). +> +> `ArtificialSampleSize` +> +> > Size of the artifical prior sample ( $inv(\lambda T)$). +> +> `DF` +> +> > Prior degrees of freedom ( $inv(\lambda T-k-n)$). +> +> `iGXX_star` +> +> > Inverse of the theoretical prior "covariance" between X and X +> > ($\Gamma_{xx}^*$ in *Del Negro and Schorfheide (2004)*). +::: + +::: {.matvar} +[oo]().RecursiveForecast + +Variable set by the `forecast` option of the `estimation` command when +used with the nobs = \[INTEGER1:INTEGER2\] option (see +`nobs `{.interpreted-text role="opt"}). + +Fields are of the form: + + oo_.RecursiveForecast.FORECAST_OBJECT.VARIABLE_NAME + +where `FORECAST_OBJECT` is one of the following[^7] : + +`Mean` + +> Mean of the posterior forecast distribution. + +`HPDinf/HPDsup` + +> Upper/lower bound of the 90% HPD interval taking into account only +> parameter uncertainty (corresponding to +> `oo_.MeanForecast`{.interpreted-text role="mvar"}). + +`HPDTotalinf/HPDTotalsup`. + +> Upper/lower bound of the 90% HPD interval taking into account both +> parameter and future shock uncertainty (corresponding to +> `oo_.PointForecast`{.interpreted-text role="mvar"}) + +`VARIABLE_NAME` contains a matrix of the following size: number of time +periods for which forecasts are requested using the +`nobs = [INTEGER1:INTEGER2]` option times the number of forecast +horizons requested by the forecast option. i.e., the row indicates the +period at which the forecast is performed and the column the respective +k-step ahead forecast. The starting periods are sorted in ascending +order, not in declaration order. +::: + +::: {.matvar} +[oo]().convergence.geweke + +Variable set by the convergence diagnostics of the `estimation` command +when used with `mh_nblocks=1` option (see +`mh_nblocks `{.interpreted-text role="opt"}). + +Fields are of the form: + + oo_.convergence.geweke.VARIABLE_NAME.DIAGNOSTIC_OBJECT + +where *DIAGNOSTIC\_OBJECT* is one of the following: + +`posteriormean` + +> Mean of the posterior parameter distribution. + +`posteriorstd` + +> Standard deviation of the posterior parameter distribution. + +`nse_iid` + +> Numerical standard error (NSE) under the assumption of iid draws. + +`rne_iid` + +> Relative numerical efficiency (RNE) under the assumption of iid draws. + +`nse_x` + +> Numerical standard error (NSE) when using an x% taper. + +`rne_x` + +> Relative numerical efficiency (RNE) when using an x% taper. + +`pooled_mean` + +> Mean of the parameter when pooling the beginning and end parts of the +> chain specified in `geweke_interval +> `{.interpreted-text role="opt"} and +> weighting them with their relative precision. It is a vector +> containing the results under the iid assumption followed by the ones +> using the `taper_steps` option (see `taper_steps = [INTEGER1 INTEGER2 ...]>`{.interpreted-text role="opt"}). + +`pooled_nse` + +> NSE of the parameter when pooling the beginning and end parts of the +> chain and weighting them with their relative precision. See +> `pooled_mean`. + +`prob_chi2_test` + +> p-value of a chi-squared test for equality of means in the beginning +> and the end of the MCMC chain. See `pooled_mean`. A value above 0.05 +> indicates that the null hypothesis of equal means and thus convergence +> cannot be rejected at the 5 percent level. Differing values along the +> `taper_steps` signal the presence of significant autocorrelation in +> draws. In this case, the estimates using a higher tapering are usually +> more reliable. +::: +::: +::: + +::: {.command} +unit\_root\_vars VARIABLE\_NAME\...; + +This command is deprecated. Use `estimation` option `diffuse_filter` +instead for estimating a model with non-stationary observed variables or +`steady` option `nocheck` to prevent `steady` to check the steady state +returned by your steady state file. +::: + +Dynare also has the ability to estimate Bayesian VARs: + +::: {.command} +bvar\_density ; + +Computes the marginal density of an estimated BVAR model, using +Minnesota priors. + +See `bvar-a-la-sims.pdf`, which comes with Dynare distribution, for more +information on this command. +::: + + +Shock Decomposition +------------------- + +::: {.command} +shock\_decomposition \[VARIABLE\_NAME\]\...; shock\_decomposition +(OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command computes the historical shock decomposition for a given +sample based on the Kalman smoother, i.e. it decomposes the historical +deviations of the endogenous variables from their respective steady +state values into the contribution coming from the various shocks. The +`variable_names` provided govern for which variables the decomposition +is plotted. + +Note that this command must come after either `estimation` (in case of +an estimated model) or `stoch_simul` (in case of a calibrated model). + +*Options* + +::: {.option} +parameter\_set = OPTION + +Specify the parameter set to use for running the smoother. Possible +values for OPTION are: + +> - `calibration` +> - `prior_mode` +> - `prior_mean` +> - `posterior_mode` +> - `posterior_mean` +> - `posterior_median` +> - `mle_mode` + +Note that the parameter set used in subsequent commands like +`stoch_simul` will be set to the specified `parameter_set`. Default +value: `posterior_mean` if Metropolis has been run, `mle_mode` if MLE +has been run. +::: + +::: {.option} +datafile = FILENAME + +See `datafile `{.interpreted-text role="ref"}. Useful when +computing the shock decomposition on a calibrated model. +::: + +::: {.option} +first\_obs = INTEGER + +See `first_obs `{.interpreted-text role="opt"}. +::: + +::: {.option} +nobs = INTEGER + +See `nobs `{.interpreted-text role="opt"}. +::: + +::: {.option} +prefilter = INTEGER + +See `prefilter `{.interpreted-text role="opt"}. +::: + +::: {.option} +loglinear + +See `loglinear `{.interpreted-text role="opt"}. +::: + +::: {.option} +diffuse\_kalman\_tol = DOUBLE + +See `diffuse_kalman_tol `{.interpreted-text +role="opt"}. +::: + +::: {.option} +diffuse\_filter + +See `diffuse_filter `{.interpreted-text role="opt"}. +::: + +::: {.option} +xls\_sheet = QUOTED\_STRING + +See `xls_sheet `{.interpreted-text +role="opt"}. +::: + +::: {.option} +xls\_range = RANGE + +See `xls_range `{.interpreted-text role="opt"}. +::: + +::: {.option} +use\_shock\_groups \[= NAME\] + +Uses shock grouping defined by the string instead of individual shocks +in the decomposition. The groups of shocks are defined in the +`shock_groups`{.interpreted-text role="bck"} block. If no group name is +given, `default` is assumed. +::: + +::: {.option} +colormap = VARIABLE\_NAME + +Controls the `colormap` used for the shocks decomposition graphs. +VARIABLE\_NAME must be the name of a MATLAB/Octave variable that has +been declared beforehand and whose value will be passed to the +MATLAB/Octave `colormap` function (see the MATLAB/Octave manual for the +list of acceptable values). +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. Suppresses the display and +creation only within the `shock_decomposition` command, but does not +affect other commands. See `plot_shock_decomposition`{.interpreted-text +role="comm"} for plotting graphs. +::: + +::: {.option} +init\_state = BOOLEAN + +If equal to 0, the shock decomposition is computed conditional on the +smoothed state variables in period `0`, i.e. the smoothed shocks +starting in period 1 are used. If equal to `1`, the shock decomposition +is computed conditional on the smoothed state variables in period 1. +Default: `0`. +::: + +::: {.option} +with\_epilogue + +If set, then also compute the decomposition for variables declared in +the `epilogue` block (see `epilogue`{.interpreted-text role="ref"}). +::: + +*Output* + +::: {.matvar} +[oo]().shock\_decomposition + +The results are stored in the field `oo_.shock_decomposition`, which is +a three dimensional array. The first dimension contains the +`M_.endo_nbr` endogenous variables. The second dimension stores in the +first `M_.exo_nbr` columns the contribution of the respective shocks. +Column `M_.exo_nbr+1` stores the contribution of the initial conditions, +while column `M_.exo_nbr+2` stores the smoothed value of the respective +endogenous variable in deviations from their steady state, i.e. the mean +and trends are subtracted. The third dimension stores the time periods. +Both the variables and shocks are stored in the order of declaration, +i.e. `M_.endo_names` and `M_.exo_names`, respectively. +::: +::: + +::: {.block} +shock\_groups ; shock\_groups(OPTIONS\...); + +Shocks can be regrouped for the purpose of shock decomposition. The +composition of the shock groups is written in a block delimited by +`shock_groups` and `end`. + +Each line defines a group of shocks as a list of exogenous variables: + + SHOCK_GROUP_NAME = VARIABLE_1 [[,] VARIABLE_2 [,]...]; + 'SHOCK GROUP NAME' = VARIABLE_1 [[,] VARIABLE_2 [,]...]; + +*Options* + +::: {.option} +name = NAME + +Specifies a name for the following definition of shock groups. It is +possible to use several `shock_groups` blocks in a model file, each +grouping being identified by a different name. This name must in turn be +used in the `shock_decomposition` command. If no name is given, +`default` is used. +::: + +*Example* + +> varexo e_a, e_b, e_c, e_d; +> ... +> +> shock_groups(name=group1); +> supply = e_a, e_b; +> 'aggregate demand' = e_c, e_d; +> end; +> +> shock_decomposition(use_shock_groups=group1); +> +> This example defines a shock grouping with the name `group1`, +> containing a set of supply and demand shocks and conducts the shock +> decomposition for these two groups. +::: + +::: {.command} +realtime\_shock\_decomposition \[VARIABLE\_NAME\]\...; +realtime\_shock\_decomposition (OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command computes the realtime historical shock decomposition for a +given sample based on the Kalman smoother. For each period +$T=[\texttt{presample},\ldots,\texttt{nobs}]$, it recursively computes +three objects: + +> - Real-time historical shock decomposition $Y(t\vert T)$ for +> $t=[1,\ldots,T]$, i.e. without observing data in +> $[T+1,\ldots,\texttt{nobs}]$. This results in a standard shock +> decomposition being computed for each additional datapoint +> becoming available after `presample`. +> - Forecast shock decomposition $Y(T+k\vert T)$ for +> $k=[1,\ldots,forecast]$, i.e. the $k$-step ahead forecast made for +> every $T$ is decomposed in its shock contributions. +> - Real-time conditional shock decomposition of the difference +> between the real-time historical shock decomposition and the +> forecast shock decomposition. If `vintage INTEGER>`{.interpreted-text role="opt"} is equal to `0`, it +> computes the effect of shocks realizing in period $T$, i.e. +> decomposes $Y(T\vert T)-Y(T\vert T-1)$. Put differently, it +> conducts a $1$-period ahead shock decomposition from $T-1$ to $T$, +> by decomposing the update step of the Kalman filter. If +> `vintage>0` and smaller than `nobs`, the decomposition is +> conducted of the forecast revision +> $Y(T+k\vert T+k)-Y(T+k\vert T)$. + +Like `shock_decomposition`{.interpreted-text role="comm"} it decomposes +the historical deviations of the endogenous variables from their +respective steady state values into the contribution coming from the +various shocks. The `variable_names` provided govern for which variables +the decomposition is plotted. + +Note that this command must come after either `estimation` (in case of +an estimated model) or `stoch_simul` (in case of a calibrated model). + +*Options* + +::: {.option} +parameter\_set = OPTION + +See `parameter_set `{.interpreted-text +role="opt"} for possible values. +::: + +::: {.option} +datafile = FILENAME + +See `datafile `{.interpreted-text role="ref"}. +::: + +::: {.option} +first\_obs = INTEGER + +See `first_obs `{.interpreted-text role="opt"}. +::: + +::: {.option} +nobs = INTEGER + +See `nobs `{.interpreted-text role="opt"}. +::: + +::: {.option} +use\_shock\_groups \[= NAME\] + +See `use_shock_groups `{.interpreted-text +role="opt"}. +::: + +::: {.option} +colormap = VARIABLE\_NAME + +See `colormap `{.interpreted-text role="opt"}. +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. Only shock decompositions +are computed and stored in `oo_.realtime_shock_decomposition`, +`oo_.conditional_shock_decomposition` and +`oo_.realtime_forecast_shock_decomposition` but no plot is made (See +`plot_shock_decomposition`{.interpreted-text role="comm"}). +::: + +::: {.option} +presample = INTEGER + +Data point above which recursive realtime shock decompositions are +computed, *i.e.* for $T=[\texttt{presample+1} \ldots \texttt{nobs}]$. +::: + +::: {.option} +forecast = INTEGER + +Compute shock decompositions up to $T+k$ periods, i.e. get shock +contributions to k-step ahead forecasts. +::: + +::: {.option} +save\_realtime = INTEGER\_VECTOR + +Choose for which vintages to save the full realtime shock decomposition. +Default: `0`. +::: + +::: {.option} +fast\_realtime = INTEGER fast\_realtime = \[INTEGER1:INTEGER2\] +fast\_realtime = \[INTEGER1 INTEGER2 \...\] + +Runs the smoother only for the data vintages provided by the specified +integer (vector). +::: + +::: {.option} +with\_epilogue + +See `with_epilogue`{.interpreted-text role="opt"}. +::: + +*Output* + +::: {.matvar} +[oo]().realtime\_shock\_decomposition + +Structure storing the results of realtime historical decompositions. +Fields are three-dimensional arrays with the first two dimension equal +to the ones of `oo_.shock_decomposition`{.interpreted-text role="mvar"}. +The third dimension stores the time periods and is therefore of size +`T+forecast`. Fields are of the form: + + oo_.realtime_shock_decomposition.OBJECT + +where OBJECT is one of the following: + +> `pool` +> +> > Stores the pooled decomposition, i.e. for every real-time shock +> > decomposition terminal period +> > $T=[\texttt{presample},\ldots,\texttt{nobs}]$ it collects the last +> > period's decomposition $Y(T\vert T)$ (see also +> > `plot_shock_decomposition`{.interpreted-text role="comm"}). The +> > third dimension of the array will have size `nobs+forecast`. +> +> `time_*` +> +> > Stores the vintages of realtime historical shock decompositions if +> > `save_realtime` is used. For example, if `save_realtime=[5]` and +> > `forecast=8`, the third dimension will be of size `13`. +::: + +::: {.matvar} +[oo]().realtime\_conditional\_shock\_decomposition + +Structure storing the results of real-time conditional decompositions. +Fields are of the form: + + oo_.realtime_conditional_shock_decomposition.OBJECT + +where OBJECT is one of the following: + +> `pool` +> +> > Stores the pooled real-time conditional shock decomposition, i.e. +> > collects the decompositions of $Y(T\vert T)-Y(T\vert T-1)$ for the +> > terminal periods $T=[\texttt{presample},\ldots,\texttt{nobs}]$. The +> > third dimension is of size `nobs`. +> +> `time_*` +> +> > Store the vintages of $k$-step conditional forecast shock +> > decompositions $Y(t\vert T+k)$, for $t=[T \ldots T+k]$. See `vintage +> > `{.interpreted-text role="opt"}. The third +> > dimension is of size `1+forecast`. +::: + +::: {.matvar} +[oo]().realtime\_forecast\_shock\_decomposition + +Structure storing the results of realtime forecast decompositions. +Fields are of the form: + + oo_.realtime_forecast_shock_decomposition.OBJECT + +where `OBJECT` is one of the following: + +> `pool` +> +> > Stores the pooled real-time forecast decomposition of the $1$-step +> > ahead effect of shocks on the $1$-step ahead prediction, i.e. +> > $Y(T\vert +> > T-1)$. +> +> `time_*` +> +> > Stores the vintages of $k$-step out-of-sample forecast shock +> > decompositions, i.e. $Y(t\vert +> > T)$, for $t=[T \ldots T+k]$. See `vintage +> > `{.interpreted-text role="opt"}. +::: +::: + +::: {.command} +plot\_shock\_decomposition \[VARIABLE\_NAME\]\...; +plot\_shock\_decomposition (OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command plots the historical shock decomposition already computed +by `shock_decomposition` or `realtime_shock_decomposition`. For that +reason, it must come after one of these commands. The `variable_names` +provided govern which variables the decomposition is plotted for. + +Further note that, unlike the majority of Dynare commands, the options +specified below are overwritten with their defaults before every call to +`plot_shock_decomposition`. Hence, if you want to reuse an option in a +subsequent call to `plot_shock_decomposition`, you must pass it to the +command again. + +*Options* + +::: {.option} +use\_shock\_groups \[= NAME\] + +See `use_shock_groups `{.interpreted-text +role="opt"}. +::: + +::: {.option} +colormap = VARIABLE\_NAME + +See `colormap `{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See `graph_format `{.interpreted-text +role="opt"}. +::: + +::: {.option} +detail\_plot + +Plots shock contributions using subplots, one per shock (or group of +shocks). Default: not activated +::: + +::: {.option} +interactive + +Under MATLAB, add uimenus for detailed group plots. Default: not +activated +::: + +::: {.option} +screen\_shocks + +For large models (i.e. for models with more than 16 shocks), plots only +the shocks that have the largest historical contribution for chosen +selected `variable_names`. Historical contribution is ranked by the mean +absolute value of all historical contributions. +::: + +::: {.option} +steadystate + +If passed, the the $y$-axis value of the zero line in the shock +decomposition plot is translated to the steady state level. Default: not +activated +::: + +::: {.option} +type = qoq \| yoy \| aoa + +For quarterly data, valid arguments are: `qoq` for quarter-on-quarter +plots, `yoy` for year-on-year plots of growth rates, `aoa` for +annualized variables, i.e. the value in the last quarter for each year +is plotted. Default value: empty, i.e. standard period-on-period plots +(`qoq` for quarterly data). +::: + +::: {.option} +fig\_name = STRING + +Specifies a user-defined keyword to be appended to the default figure +name set by `plot_shock_decomposition`. This can avoid to overwrite +plots in case of sequential calls to `plot_shock_decomposition`. +::: + +::: {.option} +write\_xls + +Saves shock decompositions to Excel-file in the main directory, named +`FILENAME_shock_decomposition_TYPE_FIG_NAME.xls`. This option requires +your system to be configured to be able to write Excel files.[^8] +::: + +::: {.option} +realtime = INTEGER + +Which kind of shock decomposition to plot. INTEGER can take the +following values: + +> - `0`: standard historical shock decomposition. See +> `shock_decomposition`{.interpreted-text role="comm"}. +> - `1`: realtime historical shock decomposition. See +> `realtime_shock_decomposition`{.interpreted-text role="comm"}. +> - `2`: conditional realtime shock decomposition. See +> `realtime_shock_decomposition`{.interpreted-text role="comm"}. +> - `3`: realtime forecast shock decomposition. See +> `realtime_shock_decomposition`{.interpreted-text role="comm"}. + +If no vintage is requested, i.e. `vintage=0` then the pooled objects +from `realtime_shock_decomposition`{.interpreted-text role="comm"} will +be plotted and the respective vintage otherwise. Default: `0`. +::: + +::: {.option} +vintage = INTEGER + +Selects a particular data vintage in $[presample,\ldots,nobs]$ for which +to plot the results from +`realtime_shock_decomposition`{.interpreted-text role="comm"} selected +via the `realtime `{.interpreted-text role="opt"} +option. If the standard historical shock decomposition is selected +(`realtime=0`), `vintage` will have no effect. If `vintage=0` the pooled +objects from `realtime_shock_decomposition`{.interpreted-text +role="comm"} will be plotted. If `vintage>0`, it plots the shock +decompositions for vintage $T=\texttt{vintage}$ under the following +scenarios: + +> - `realtime=1`: the full vintage shock decomposition $Y(t\vert T)$ +> for $t=[1,\ldots,T]$ +> - `realtime=2`: the conditional forecast shock decomposition from +> $T$, i.e. plots $Y(T+j\vert T+j)$ and the shock contributions +> needed to get to the data $Y(T+j)$ conditional on $T=$ vintage, +> with $j=[0,\ldots,\texttt{forecast}]$. +> - `realtime=3`: plots unconditional forecast shock decomposition +> from $T$, i.e. $Y(T+j\vert +> T)$, where $T=\texttt{vintage}$ and +> $j=[0,\ldots,\texttt{forecast}]$. + +Default: `0`. +::: + +::: {.option} +plot\_init\_date = DATE + +If passed, plots decomposition using `plot_init_date` as initial period. +Default: first observation in estimation +::: + +::: {.option} +plot\_end\_date = DATE + +If passed, plots decomposition using `plot_end_date` as last period. +Default: last observation in estimation +::: + +::: {.option} +diff + +If passed, plot the decomposition of the first difference of the list of +variables. If used in combination with `flip`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: + +::: {.option} +flip + +If passed, plot the decomposition of the opposite of the list of +variables. If used in combination with `diff`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: + +::: {.option} +max\_nrows + +Maximum number of rows in the subplot layout of detailed shock +decomposition graphs. Note that columns are always 3. Default: 6 +::: + +::: {.option} +with\_epilogue + +See `with_epilogue`{.interpreted-text role="opt"}. +::: + +::: {.option} +init2shocks init2shocks = NAME + +Use the information contained in an `init2shocks`{.interpreted-text +role="bck"} block, in order to attribute initial conditions to shocks. +The name of the block can be explicitly given, otherwise it defaults to +the `default` block. +::: +::: + +::: {.block} +init2shocks ; init2shocks (OPTIONS\...); + +This blocks gives the possibility of attributing the initial condition +of endogenous variables to the contribution of exogenous variables in +the shock decomposition. + +For example, in an AR(1) process, the contribution of the initial +condition on the process variable can naturally be assigned to the +innovation of the process. + +Each line of the block should have the syntax: + + VARIABLE_1 [,] VARIABLE_2; + +Where VARIABLE\_1 is an endogenous variable whose initial condition will +be attributed to the exogenous VARIABLE\_2. + +The information contained in this block is used by the +`plot_shock_decomposition`{.interpreted-text role="comm"} command when +given the `init2shocks` option. + +*Options* + +::: {.option} +name = NAME + +Specifies a name for the block, that can be referenced from +`plot_shock_decomposition`, so that several such blocks can coexist in a +single model file. If the name is unspecified, it defaults to `default`. +::: + +*Example* + +> var y y_s R pie dq pie_s de A y_obs pie_obs R_obs; +> varexo e_R e_q e_ys e_pies e_A; +> ... +> +> model; +> dq = rho_q*dq(-1)+e_q; +> A = rho_A*A(-1)+e_A; +> ... +> end; +> +> ... +> +> init2shocks; +> dq e_q; +> A e_A; +> end; +> +> shock_decomposition(nograph); +> +> plot_shock_decomposition(init2shocks) y_obs R_obs pie_obs dq de; +> +> In this example, the initial conditions of `dq` and `A` will be +> respectively attributed to `e_q` and `e_A`. +::: + +::: {.command} +initial\_condition\_decomposition \[VARIABLE\_NAME\]\...; +initial\_condition\_decomposition (OPTIONS\...) \[VARIABLE\_NAME\]\...; + +This command computes and plots the decomposition of the effect of +smoothed initial conditions of state variables. The `variable_names` +provided govern which variables the decomposition is plotted for. + +Further note that, unlike the majority of Dynare commands, the options +specified below are overwritten with their defaults before every call to +`initial_condition_decomposition`. Hence, if you want to reuse an option +in a subsequent call to `initial_condition_decomposition`, you must pass +it to the command again. + +*Options* + +::: {.option} +colormap = VARIABLE\_NAME + +See `colormap `{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See `graph_format `{.interpreted-text +role="opt"}. +::: + +::: {.option} +detail\_plot + +Plots shock contributions using subplots, one per shock (or group of +shocks). Default: not activated +::: + +::: {.option} +steadystate + +If passed, the the $y$-axis value of the zero line in the shock +decomposition plot is translated to the steady state level. Default: not +activated +::: + +::: {.option} +type = qoq \| yoy \| aoa + +For quarterly data, valid arguments are: `qoq` for quarter-on-quarter +plots, `yoy` for year-on-year plots of growth rates, `aoa` for +annualized variables, i.e. the value in the last quarter for each year +is plotted. Default value: empty, i.e. standard period-on-period plots +(`qoq` for quarterly data). +::: + +::: {.option} +fig\_name = STRING + +Specifies a user-defined keyword to be appended to the default figure +name set by `plot_shock_decomposition`. This can avoid to overwrite +plots in case of sequential calls to `plot_shock_decomposition`. +::: + +::: {.option} +write\_xls + +Saves shock decompositions to Excel-file in the main directory, named +`FILENAME_shock_decomposition_TYPE_FIG_NAME_initval.xls`. This option +requires your system to be configured to be able to write Excel +files.[^9] +::: + +::: {.option} +plot\_init\_date = DATE + +If passed, plots decomposition using `plot_init_date` as initial period. +Default: first observation in estimation +::: + +::: {.option} +plot\_end\_date = DATE + +If passed, plots decomposition using `plot_end_date` as last period. +Default: last observation in estimation +::: + +::: {.option} +diff + +If passed, plot the decomposition of the first difference of the list of +variables. If used in combination with `flip`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: + +::: {.option} +flip + +If passed, plot the decomposition of the opposite of the list of +variables. If used in combination with `diff`{.interpreted-text +role="opt"}, the `diff` operator is first applied. Default: not +activated +::: +::: + +::: {.command} +squeeze\_shock\_decomposition \[VARIABLE\_NAME\]\...; + +For large models, the size of the information stored by shock +decompositions (especially various settings of realtime decompositions) +may become huge. This command allows to squeeze this information in two +possible ways: + +> - Automatic (default): only the variables for which plotting has +> been explicitly required with `plot_shock_decomposition` will have +> their decomposition left in `oo_` after this command is run; +> - If a list of variables is passed to the command, then only those +> variables will have their decomposition left in `oo_` after this +> command is run. +::: + +Calibrated Smoother +------------------- + +Dynare can also run the smoother on a calibrated model: + +::: {.command} +calib\_smoother \[VARIABLE\_NAME\]\...; calib\_smoother (OPTIONS\...) +\[VARIABLE\_NAME\]\...; + +This command computes the smoothed variables (and possible the filtered +variables) on a calibrated model. + +A datafile must be provided, and the observable variables declared with +`varobs`. The smoother is based on a first-order approximation of the +model. + +By default, the command computes the smoothed variables and shocks and +stores the results in `oo_.SmoothedVariables` and `oo_.SmoothedShocks`. +It also fills `oo_.UpdatedVariables`. + +*Options* + +::: {.option} +datafile = FILENAME + +See `datafile `{.interpreted-text role="ref"}. +::: + +::: {.option} +filtered\_vars + +Triggers the computation of filtered variables. See +`filtered_vars`{.interpreted-text role="opt"}, for more details. +::: + +::: {.option} +filter\_step\_ahead = \[INTEGER1:INTEGER2\] + +See +`filter_step_ahead `{.interpreted-text +role="opt"}. +::: + +::: {.option} +prefilter = INTEGER + +See `prefilter `{.interpreted-text role="opt"}. +::: + +::: {.option} +parameter\_set = OPTION + +See `parameter_set `{.interpreted-text +role="opt"} for possible values. Default: `calibration`. +::: + +::: {.option} +loglinear + +See `loglinear `{.interpreted-text role="ref"}. +::: + +::: {.option} +first\_obs = INTEGER + +See `first_obs `{.interpreted-text role="opt"}. +::: + +::: {.option} +filter\_decomposition + +See `filter_decomposition`{.interpreted-text role="opt"}. +::: + +::: {.option} +filter\_covariance + +See `filter_covariance`{.interpreted-text role="opt"}. +::: + +::: {.option} +smoother\_redux + +See `smoother_redux`{.interpreted-text role="opt"}. +::: + +::: {.option} +kalman\_algo = INTEGER + +See `kalman_algo `{.interpreted-text role="opt"}. +::: + +::: {.option} +diffuse\_filter = INTEGER + +See `diffuse_filter`{.interpreted-text role="opt"}. +::: + +::: {.option} +diffuse\_kalman\_tol = DOUBLE + +See `diffuse_kalman_tol `{.interpreted-text +role="opt"}. +::: + +::: {.option} +xls\_sheet = QUOTED\_STRING + +See `xls_sheet `{.interpreted-text +role="opt"}. +::: + +::: {.option} +xls\_range = RANGE + +See `xls_range `{.interpreted-text role="opt"}. +::: +::: + +Forecasting {#fore} +----------- + +On a calibrated model, forecasting is done using the `forecast` command. +On an estimated model, use the `forecast` option of `estimation` +command. + +It is also possible to compute forecasts on a calibrated or estimated +model for a given constrained path of the future endogenous variables. +This is done, from the reduced form representation of the DSGE model, by +finding the structural shocks that are needed to match the restricted +paths. Use `conditional_forecast`{.interpreted-text role="comm"}, +`conditional_forecast_paths`{.interpreted-text role="bck"} and +`plot_conditional_forecast`{.interpreted-text role="comm"} for that +purpose. + +Finally, it is possible to do forecasting with a Bayesian VAR using the +`bvar_forecast`{.interpreted-text role="comm"} command. + +::: {.command} +forecast \[VARIABLE\_NAME\...\]; forecast (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +This command computes a simulation of a stochastic model from an +arbitrary initial point. + +When the model also contains deterministic exogenous shocks, the +simulation is computed conditionally to the agents knowing the future +values of the deterministic exogenous variables. + +`forecast` must be called after `stoch_simul`. + +`forecast` plots the trajectory of endogenous variables. When a list of +variable names follows the command, only those variables are plotted. A +90% confidence interval is plotted around the mean trajectory. Use +option `conf_sig` to change the level of the confidence interval. + +*Options* + +::: {.option} +periods = INTEGER + +Number of periods of the forecast. Default: `5`. +::: + +::: {#confsig} +::: {.option} +conf\_sig = DOUBLE + +Level of significance for confidence interval. Default: `0.90`. +::: +::: + +::: {.option} +nograph + +See `nograph`{.interpreted-text role="opt"}. +::: + +::: {.option} +nodisplay + +See `nodisplay`{.interpreted-text role="opt"}. +::: + +::: {.option} +graph\_format = FORMAT graph\_format = ( FORMAT, FORMAT\... ) + +See `graph_format = FORMAT`{.interpreted-text role="opt"}. +::: + +*Initial Values* + +`forecast` computes the forecast taking as initial values the values +specified in `histval` (see `histval`{.interpreted-text role="bck"}). +When no `histval` block is present, the initial values are the one +stated in `initval`. When `initval` is followed by command `steady`, the +initial values are the steady state (see `steady`{.interpreted-text +role="comm"}). + +*Output* + +The results are stored in `oo_.forecast`, which is described below. + +*Example* + +> varexo_det tau; +> +> varexo e; +> ... +> shocks; +> var e; stderr 0.01; +> var tau; +> periods 1:9; +> values -0.15; +> end; +> +> stoch_simul(irf=0); +> +> forecast; + +::: {.matvar} +[oo]().forecast + +Variable set by the `forecast` command, or by the `estimation` command +if used with the `forecast` option and ML or if no Metropolis-Hastings +has been computed (in that case, the forecast is computed for the +posterior mode). Fields are of the form: + + oo_.forecast.FORECAST_MOMENT.VARIABLE_NAME + +where `FORECAST_MOMENT` is one of the following: + +> `HPDinf` +> +> > Lower bound of a 90% HPD interval[^10] of forecast due to parameter +> > uncertainty, but ignoring the effect of measurement error on +> > observed variables. In case of ML, it stores the lower bound of the +> > confidence interval. +> +> `HPDsup` +> +> > Upper bound of a 90% HPD forecast interval due to parameter +> > uncertainty, but ignoring the effect of measurement error on +> > observed variables. In case of ML, it stores the upper bound of the +> > confidence interval. +> +> `HPDinf_ME` +> +> > Lower bound of a 90% HPD interval[^11] of forecast for observed +> > variables due to parameter uncertainty and measurement error. In +> > case of ML, it stores the lower bound of the confidence interval. +> +> `HPDsup_ME` +> +> > Upper bound of a 90% HPD interval of forecast for observed variables +> > due to parameter uncertainty and measurement error. In case of ML, +> > it stores the upper bound of the confidence interval. +> +> `Mean` +> +> > Mean of the posterior distribution of forecasts. +::: + +::: {.matvar} +[oo]().PointForecast + +Set by the `estimation` command, if it is used with the `forecast` +option and if either `mh_replic > 0` or the `load_mh_file` option are +used. + +Contains the distribution of forecasts taking into account the +uncertainty about both parameters and shocks. + +Fields are of the form: + + oo_.PointForecast.MOMENT_NAME.VARIABLE_NAME +::: + +::: {.matvar} +[oo]().MeanForecast + +Set by the `estimation` command, if it is used with the `forecast` +option and if either `mh_replic > 0` or `load_mh_file` option are used. + +Contains the distribution of forecasts where the uncertainty about +shocks is averaged out. The distribution of forecasts therefore only +represents the uncertainty about parameters. + +Fields are of the form: + + oo_.MeanForecast.MOMENT_NAME.VARIABLE_NAME +::: +::: + +::: {.command} +conditional\_forecast (OPTIONS\...); + +This command computes forecasts on an estimated or calibrated model for +a given constrained path of some future endogenous variables. This is +done using the reduced form first order state-space representation of +the DSGE model by finding the structural shocks that are needed to match +the restricted paths. Consider the augmented state space representation +that stacks both predetermined and non-predetermined variables into a +vector $y_{t}$: + +> $$y_t=Ty_{t-1}+R\varepsilon_t$$ + +Both $y_t$ and $\varepsilon_t$ are split up into controlled and +uncontrolled ones, and we assume without loss of generality that the +constrained endogenous variables and the controlled shocks come first : + +> $$\begin{aligned} +> \begin{pmatrix} +> y_{c,t}\\ +> y_{u,t} +> \end{pmatrix} +> = +> \begin{pmatrix} +> T_{c,c} & T_{c,u}\\ +> T_{u,c} & T_{u,u} +> \end{pmatrix} +> \begin{pmatrix} +> y_{c,t-1}\\ +> y_{u,t-1} +> \end{pmatrix} +> + +> \begin{pmatrix} +> R_{c,c} & R_{c,u}\\ +> R_{u,c} & R_{u,u} +> \end{pmatrix} +> \begin{pmatrix} +> \varepsilon_{c,t}\\ +> \varepsilon_{u,t} +> \end{pmatrix} +> \end{aligned}$$ + +where matrices $T$ and $R$ are partitioned consistently with the vectors +of endogenous variables and innovations. Provided that matrix $R_{c,c}$ +is square and full rank (a necessary condition is that the number of +free endogenous variables matches the number of free innovations), given +$y_{c,t}$, $\varepsilon_{u,t}$ and $y_{t-1}$ the first block of +equations can be solved for $\varepsilon_{c,t}$: + +> $$\varepsilon_{c,t} = R_{c,c}^{-1}\bigl( y_{c,t} - T_{c,c}y_{c,t} - T_{c,u}y_{u,t} - R_{c,u}\varepsilon_{u,t}\bigr)$$ + +and $y_{u,t}$ can be updated by evaluating the second block of +equations: + +> $$y_{u,t} = T_{u,c}y_{c,t-1} + T_{u,u}y_{u,t-1} + R_{u,c}\varepsilon_{c,t} + R_{u,u}\varepsilon_{u,t}$$ + +By iterating over these two blocks of equations, we can build a forecast +for all the endogenous variables in the system conditional on paths for +a subset of the endogenous variables. If the distribution of the free +innovations $\varepsilon_{u,t}$ is provided (*i.e.* some of them have +positive variances) this exercise is replicated (the number of +replication is controlled by the option `replic`{.interpreted-text +role="opt"} described below) by drawing different sequences of free +innovations. The result is a predictive distribution for the +uncontrolled endogenous variables, $y_{u,t}$, that Dynare will use to +report confidence bands around the point conditional forecast. + +A few things need to be noted. First, the controlled exogenous variables +are set to zero for the uncontrolled periods. This implies that there is +no forecast uncertainty arising from these exogenous variables in +uncontrolled periods. Second, by making use of the first order state +space solution, even if a higher-order approximation was performed, the +conditional forecasts will be based on a first order approximation. +Since the controlled exogenous variables are identified on the basis of +the reduced form model (*i.e.* after solving for the expectations), they +are unforeseen shocks from the perspective of the agents in the model. +That is, agents expect the endogenous variables to return to their +respective steady state levels but are surprised in each period by the +realisation of shocks keeping the endogenous variables along a +predefined (unexpected) path. Fourth, if the structural innovations are +correlated, because the calibrated or estimated covariance matrix has +non zero off diagonal elements, the results of the conditional forecasts +will depend on the ordering of the innovations (as declared after +`varexo`). As in VAR models, a Cholesky decomposition is used to +factorise the covariance matrix and identify orthogonal impulses. It is +preferable to declare the correlations in the model block (explicitly +imposing the identification restrictions), unless you are satisfied with +the implicit identification restrictions implied by the Cholesky +decomposition. + +This command has to be called after `estimation` or `stoch_simul`. + +Use `conditional_forecast_paths`{.interpreted-text role="bck"} block to +give the list of constrained endogenous, and their constrained future +path. Option `controlled_varexo` is used to specify the structural +shocks which will be matched to generate the constrained path. + +Use `plot_conditional_forecast`{.interpreted-text role="comm"} to graph +the results. + +*Options* + +::: {.option} +parameter\_set = OPTION + +See `parameter_set `{.interpreted-text +role="opt"} for possible values. No default value, mandatory option. +::: + +::: {.option} +controlled\_varexo = (VARIABLE\_NAME\...) + +Specify the exogenous variables to use as control variables. No default +value, mandatory option. +::: + +::: {.option} +periods = INTEGER + +Number of periods of the forecast. Default: `40`. `periods` cannot be +smaller than the number of constrained periods. +::: + +::: {.option} +replic = INTEGER + +Number of simulations used to compute the conditional forecast +uncertainty. Default: `5000`. +::: + +::: {.option} +conf\_sig = DOUBLE + +Level of significance for confidence interval. Default: `0.80`. +::: + +*Output* + +The results are stored in `oo_.conditional_forecast`, which is described +below. + +*Example* + +> var y a; +> varexo e u; +> ... +> estimation(...); +> +> conditional_forecast_paths; +> var y; +> periods 1:3, 4:5; +> values 2, 5; +> var a; +> periods 1:5; +> values 3; +> end; +> +> conditional_forecast(parameter_set = calibration, controlled_varexo = (e, u), replic = 3000); +> +> plot_conditional_forecast(periods = 10) a y; + +::: {.matvar} +[oo]().conditional\_forecast.cond + +Variable set by the `conditional_forecast` command. It stores the +conditional forecasts. Fields are `periods+1` by `1` vectors storing the +steady state (time 0) and the subsequent `periods` forecasts periods. +Fields are of the form: + + oo_.conditional_forecast.cond.FORECAST_MOMENT.VARIABLE_NAME + +where FORECAST\_MOMENT is one of the following: + +> `Mean` +> +> > Mean of the conditional forecast distribution. +> +> `ci` +> +> > Confidence interval of the conditional forecast distribution. The +> > size corresponds to `conf_sig`. +::: + +::: {.matvar} +[oo]().conditional\_forecast.uncond + +Variable set by the `conditional_forecast` command. It stores the +unconditional forecasts. Fields are of the form: + + oo_.conditional_forecast.uncond.FORECAST_MOMENT.VARIABLE_NAME +::: + +::: {.matvar} +forecasts.instruments + +Variable set by the `conditional_forecast command`. Stores the names of +the exogenous instruments. +::: + +::: {.matvar} +[oo]().conditional\_forecast.controlled\_variables + +Variable set by the `conditional_forecast` command. Stores the position +of the constrained endogenous variables in declaration order. +::: + +::: {.matvar} +[oo]().conditional\_forecast.controlled\_exo\_variables + +Variable set by the `conditional_forecast` command. Stores the values of +the controlled exogenous variables underlying the conditional forecasts +to achieve the constrained endogenous variables. Fields are +`[number of constrained periods]` by `1` vectors and are of the form: + + oo_.conditional_forecast.controlled_exo_variables.FORECAST_MOMENT.SHOCK_NAME +::: + +::: {.matvar} +[oo]().conditional\_forecast.graphs + +Variable set by the `conditional_forecast` command. Stores the +information for generating the conditional forecast plots. +::: +::: + +::: {.block} +conditional\_forecast\_paths ; + +Describes the path of constrained endogenous, before calling +`conditional_forecast`. The syntax is similar to deterministic shocks in +`shocks`, see `conditional_forecast` for an example. + +The syntax of the block is the same as for the deterministic shocks in +the `shocks` blocks (see `shocks-exo`{.interpreted-text role="ref"}). +Note that you need to specify the full path for all constrained +endogenous variables between the first and last specified period. If an +intermediate period is not specified, a value of 0 is assumed. That is, +if you specify only values for periods 1 and 3, the values for period 2 +will be 0. Currently, it is not possible to have uncontrolled +intermediate periods. + +It is however possible to have different number of controlled periods +for different variables. In that case, the order of declaration of +endogenous controlled variables and of `controlled_varexo` matters: if +the second endogenous variable is controlled for less periods than the +first one, the second `controlled_varexo` isn\'t set for the last +periods. + +In case of the presence of `observation_trends`, the specified +controlled path for these variables needs to include the trend +component. When using the `loglinear `{.interpreted-text +role="ref"} option, it is necessary to specify the logarithm of the +controlled variables. +::: + +::: {.block} +filter\_initial\_state ; + +This block specifies the initial values of the endogenous states at the +beginning of the Kalman filter recursions. That is, if the Kalman filter +recursion starts with time t=1 being the first observation, this block +provides the state estimate at time 0 given information at time 0, +$E_0(x_0)$. If nothing is specified, the initial condition is assumed to +be at the steady state (which is the unconditional mean for a stationary +model). + +This block is terminated by `end;`. + +Each line inside of the block should be of the form: + + VARIABLE_NAME(INTEGER)=EXPRESSION; + +`EXPRESSION` is any valid expression returning a numerical value and can +contain parameter values. This allows specifying relationships that will +be honored during estimation. `INTEGER` refers to the lag with which a +variable appears. By convention in Dynare, period 1 is the first period. +Going backwards in time, the first period before the start of the +simulation is period 0, then period -1, and so on. Note that the +`filter_initial_state` block does not take non-state variables. + +*Example* + +> filter_initial_state; +> k(0)= ((1/bet-(1-del))/alp)^(1/(alp-1))*l_ss; +> P(0)=2.5258; +> m(0)= mst; +> end; +::: + +::: {.command} +plot\_conditional\_forecast \[VARIABLE\_NAME\...\]; +plot\_conditional\_forecast (periods = INTEGER) \[VARIABLE\_NAME\...\]; + +Plots the conditional (plain lines) and unconditional (dashed lines) +forecasts. + +To be used after `conditional_forecast`. + +*Options* + +::: {.option} +periods = INTEGER + +Number of periods to be plotted. Default: equal to periods in +`conditional_forecast`. The number of periods declared in +`plot_conditional_forecast` cannot be greater than the one declared in +`conditional_forecast`. +::: +::: + +::: {.command} +bvar\_forecast ; + +This command computes (out-of-sample) forecasts for an estimated BVAR +model, using Minnesota priors. + +See `bvar-a-la-sims.pdf`, which comes with Dynare distribution, for more +information on this command. +::: + +If the model contains strong non-linearities or if some perfectly +expected shocks are considered, the forecasts and the conditional +forecasts can be computed using an extended path method. The forecast +scenario describing the shocks and/or the constrained paths on some +endogenous variables should be build. The first step is the forecast +scenario initialization using the function `init_plan`: + +::: {.matcomm} +HANDLE = init\_plan (DATES); + +Creates a new forecast scenario for a forecast period (indicated as a +dates class, see `dates class members +`{.interpreted-text role="ref"}). This function return a +handle on the new forecast scenario. +::: + +The forecast scenario can contain some simple shocks on the exogenous +variables. This shocks are described using the function `basic_plan`: + +::: {.matcomm} +HANDLE = basic\_plan (HANDLE, \`VAR\_NAME\', \`SHOCK\_TYPE\', DATES, +MATLAB VECTOR OF DOUBLE \| \[DOUBLE \| EXPR \[DOUBLE \| EXPR\] \] ); + +Adds to the forecast scenario a shock on the exogenous variable +indicated between quotes in the second argument. The shock type has to +be specified in the third argument between quotes: 'surprise' in case of +an unexpected shock or 'perfect\_foresight' for a perfectly anticipated +shock. The fourth argument indicates the period of the shock using a +dates class (see `dates class +members `{.interpreted-text role="ref"}). The last +argument is the shock path indicated as a MATLAB vector of double. This +function return the handle of the updated forecast scenario. +::: + +The forecast scenario can also contain a constrained path on an +endogenous variable. The values of the related exogenous variable +compatible with the constrained path are in this case computed. In other +words, a conditional forecast is performed. This kind of shock is +described with the function `flip_plan`: + +::: {.matcomm} +HANDLE = flip\_plan (HANDLE, \`VAR\_NAME\', \`VAR\_NAME\', +\`SHOCK\_TYPE\', DATES, MATLAB VECTOR OF DOUBLE \| \[DOUBLE \| EXPR +\[DOUBLE \| EXPR\] \] ); + +Adds to the forecast scenario a constrained path on the endogenous +variable specified between quotes in the second argument. The associated +exogenous variable provided in the third argument between quotes, is +considered as an endogenous variable and its values compatible with the +constrained path on the endogenous variable will be computed. The nature +of the expectation on the constrained path has to be specified in the +fourth argument between quotes: 'surprise' in case of an unexpected path +or 'perfect\_foresight' for a perfectly anticipated path. The fifth +argument indicates the period where the path of the endogenous variable +is constrained using a dates class (see `dates class +members `{.interpreted-text role="ref"}). The last +argument contains the constrained path as a MATLAB vector of double. +This function return the handle of the updated forecast scenario. +::: + +Once the forecast scenario if fully described, the forecast is computed +with the command `det_cond_forecast`: + +::: {.matcomm} +DSERIES = det\_cond\_forecast (HANDLE\[, DSERIES \[, DATES\]\]); + +Computes the forecast or the conditional forecast using an extended path +method for the given forecast scenario (first argument). The past values +of the endogenous and exogenous variables provided with a dseries class +(see `dseries class +members `{.interpreted-text role="ref"}) can be +indicated in the second argument. By default, the past values of the +variables are equal to their steady-state values. The initial date of +the forecast can be provided in the third argument. By default, the +forecast will start at the first date indicated in the +`init_plan command`. This function returns a dset containing the +historical and forecast values for the endogenous and exogenous +variables. +::: + +*Example* + +> % conditional forecast using extended path method +> % with perfect foresight on r path +> +> var y r; +> varexo e u; +> ... +> smoothed = dseries('smoothed_variables.csv'); +> +> fplan = init_plan(2013Q4:2029Q4); +> fplan = flip_plan(fplan, 'y', 'u', 'surprise', 2013Q4:2014Q4, [1 1.1 1.2 1.1 ]); +> fplan = flip_plan(fplan, 'r', 'e', 'perfect_foresight', 2013Q4:2014Q4, [2 1.9 1.9 1.9 ]); +> +> dset_forecast = det_cond_forecast(fplan, smoothed); +> +> plot(dset_forecast.{'y','u'}); +> plot(dset_forecast.{'r','e'}); + +::: {.command} +smoother2histval ; smoother2histval(OPTIONS\...); + +The purpose of this command is to construct initial conditions (for a +subsequent simulation) that are the smoothed values of a previous +estimation. + +More precisely, after an estimation run with the `smoother` option, +`smoother2histval` will extract the smoothed values (from +`oo_.SmoothedVariables`, and possibly from `oo_.SmoothedShocks` if there +are lagged exogenous), and will use these values to construct initial +conditions (as if they had been manually entered through `histval`). + +*Options* + +::: {.option} +period = INTEGER + +Period number to use as the starting point for the subsequent +simulation. It should be between 1 and the number of observations that +were used to produce the smoothed values. Default: the last observation. +::: + +::: {.option} +infile = FILENAME + +Load the smoothed values from a `_results.mat` file created by a +previous Dynare run. Default: use the smoothed values currently in the +global workspace. +::: + +::: {.option} +invars = ( VARIABLE\_NAME \[VARIABLE\_NAME \...\] ) + +A list of variables to read from the smoothed values. It can contain +state endogenous variables, and also exogenous variables having a lag. +Default: all the state endogenous variables, and all the exogenous +variables with a lag. +::: + +::: {.option} +outfile = FILENAME + +Write the initial conditions to a file. Default: write the initial +conditions in the current workspace, so that a simulation can be +performed. +::: + +::: {.option} +outvars = ( VARIABLE\_NAME \[VARIABLE\_NAME \...\] ) + +A list of variables which will be given the initial conditions. This +list must have the same length than the list given to `invars`, and +there will be a one-to-one mapping between the two list. Default: same +value as option `invars`. +::: + +*Use cases* + +There are three possible ways of using this command: + +> - Everything in a single file: run an estimation with a smoother, +> then run `smoother2histval` (without the `infile` and `outfile` +> options), then run a stochastic simulation. +> - In two files: in the first file, run the smoother and then run +> `smoother2histval` with the `outfile` option; in the second file, +> run `histval_file` to load the initial conditions, and run a +> (deterministic or stochastic) simulation. +> - In two files: in the first file, run the smoother; in the second +> file, run `smoother2histval` with the `infile` option equal to the +> `_results.mat` file created by the first file, and then run a +> (deterministic or stochastic) simulation. +::: + +Optimal policy +-------------- + +Dynare has tools to compute optimal policies for various types of +objectives. You can either solve for optimal policy under commitment +with `ramsey_model`, for optimal policy under discretion with +`discretionary_policy` or for optimal simple rules with `osr` (also +implying commitment). + +::: {.command} +planner\_objective MODEL\_EXPRESSION ; + +This command declares the policy maker objective, for use with +`ramsey_model` or `discretionary_policy`. + +You need to give the one-period objective, not the discounted lifetime +objective. The discount factor is given by the `planner_discount` option +of `ramsey_model` and `discretionary_policy`. The objective function can +only contain current endogenous variables and no exogenous ones. This +limitation is easily circumvented by defining an appropriate auxiliary +variable in the model. + +With `ramsey_model`, you are not limited to quadratic objectives: you +can give any arbitrary nonlinear expression. + +With `discretionary_policy`, the objective function must be quadratic. +::: + +::: {.command} +evaluate\_planner\_objective; evaluate\_planner\_objective +(OPTIONS\...); + +This command computes, displays, and stores the value of the planner +objective function under Ramsey policy or discretion in +`oo_.planner_objective_value`. It will provide both unconditional +welfare and welfare conditional on the initial (i.e. period 0) values of +the endogenous and exogenous state variables inherited by the planner. +In a deterministic context, the respective initial values are set using +`initval` or `histval` (depending on the exact context). + +In a stochastic context, if no initial state values have been specified +with `histval`, their values are taken to be the steady state values. +Because conditional welfare is computed conditional on optimal policy by +the planner in the first endogenous period (period 1), it is conditional +on the information set in the period 1. This information set includes +both the predetermined states inherited from period 0 (specified via +`histval` for both endogenous and lagged exogenous states) as well as +the period 1 values of the exogenous shocks. The latter are specified +using the perfect foresight syntax of the shocks-block. + +At the current stage, the stochastic context does not support the +`pruning` option. At `order>3`, only the computation of conditional +welfare with steady state Lagrange multipliers is supported. Note that +at [order=2]{.title-ref}, the output is based on the second-order +accurate approximation of the variance stored in [oo\_.var]{.title-ref}. + +*Options* + +::: {.option} +periods = INTEGER + +The value of the option specifies the number of periods to use in the +simulations in the computation of unconditional welfare at higher order. + +Default: `10000`. +::: + +::: {.option} +drop = INTEGER + +The number of burn-in draws out of `periods` discarded before computing +the unconditional welfare at higher order. Default: `1000`. +::: + +*Example (stochastic context)* + +> var a ...; +> varexo u; +> +> model; +> a = rho*a(-1)+u+u(-1); +> ... +> end; +> +> histval; +> u(0)=1; +> a(0)=-1; +> end; +> +> shocks; +> var u; stderr 0.008; +> var u; +> periods 1; +> values 1; +> end; +> +> evaluate_planner_objective; + +::: {.matvar} +[oo]().planner\_objective\_value.unconditional +::: + +Scalar storing the value of unconditional welfare. In a perfect +foresight context, it corresponds to welfare in the long-run, +approximated as welfare in the terminal simulation period. + +::: {.matvar} +[oo]().planner\_objective\_value.conditional +::: + +In a perfect foresight context, this field will be a scalar storing the +value of welfare conditional on the specified initial condition and zero +initial Lagrange multipliers. + +In a stochastic context, it will have two subfields: + +::: {.matvar} +[oo]().planner\_objective\_value.conditional.steady\_initial\_multiplier +::: + +Stores the value of the planner objective when the initial Lagrange +multipliers associated with the planner's problem are set to their +steady state values (see `ramsey_policy`{.interpreted-text +role="comm"}). + +::: {.matvar} +[oo]().planner\_objective\_value.conditional.zero\_initial\_multiplier +::: + +Stores the value of the planner objective when the initial Lagrange +multipliers associated with the planner's problem are set to 0, i.e. it +is assumed that the planner exploits its ability to surprise private +agents in the first period of implementing Ramsey policy. This value +corresponds to the planner implementing optimal policy for the first +time and committing not to re-optimize in the future. +::: + +### Optimal policy under commitment (Ramsey) + +Dynare allows to automatically compute optimal policy choices of a +Ramsey planner who takes the specified private sector equilibrium +conditions into account and commits to future policy choices. Doing so +requires specifying the private sector equilibrium conditions in the +`model`-block and a `planner_objective` as well as potentially some +`instruments` to facilitate computations. + +::: {.warning} +::: {.admonition-title} +Warning +::: + +Be careful when employing forward-looking auxiliary variables in the +context of timeless perspective Ramsey computations. They may alter the +problem the Ramsey planner will solve for the first period, although +they seemingly leave the private sector equilibrium unaffected. The +reason is the planner optimizes with respect to variables dated `t` and +takes the value of time 0 variables as given, because they are +predetermined. This set of initially predetermined variables will change +with forward-looking definitions. Thus, users are strongly advised to +use model-local variables instead. + +*Example* + +> Consider a perfect foresight example where the Euler equation for the +> return to capital is given by +> +> 1/C=beta*1/C(+1)*(R(+1)+(1-delta)) +> +> The job of the Ramsey planner in period `1` is to choose $C_1$ and +> $R_1$, taking as given $C_0$. The above equation may seemingly +> equivalently be written as +> +> 1/C=beta*1/C(+1)*(R_cap); +> R_cap=R(+1)+(1-delta); +> +> due to perfect foresight. However, this changes the problem of the +> Ramsey planner in the first period to choosing $C_1$ and $R_1$, taking +> as given both $C_0$ and $R^{cap}_0$. Thus, the relevant return to +> capital in the Euler equation of the first period is not a choice of +> the planner anymore due to the forward-looking nature of the +> definition in the second line! +> +> A correct specification would be to instead define `R_cap` as a +> model-local variable: +> +> 1/C=beta*1/C(+1)*(R_cap); +> #R_cap=R(+1)+(1-delta); +::: + +::: {.command} +ramsey\_model (OPTIONS\...); + +This command computes the First Order Conditions for maximizing the +policy maker objective function subject to the constraints provided by +the equilibrium path of the private economy. + +The planner objective must be declared with the +`planner_objective`{.interpreted-text role="comm"} command. + +This command only creates the expanded model, it doesn't perform any +computations. It needs to be followed by other instructions to actually +perform desired computations. Examples are calls to `steady` to compute +the steady state of the Ramsey economy, to `stoch_simul` with various +approximation orders to conduct stochastic simulations based on +perturbation solutions, to `estimation` in order to estimate models +under optimal policy with commitment, and to perfect foresight +simulation routines. + +See `aux-variables`{.interpreted-text role="ref"}, for an explanation of +how Lagrange multipliers are automatically created. + +*Options* + +This command accepts the following options: + +::: {.option} +planner\_discount = EXPRESSION + +Declares or reassigns the discount factor of the central planner +`optimal_policy_discount_factor`. Default: `1.0`. +::: + +::: {.option} +planner\_discount\_latex\_name = LATEX\_NAME + +Sets the LaTeX name of the `optimal_policy_discount_factor` parameter. +::: + +::: {.option} +instruments = (VARIABLE\_NAME,\...) + +Declares instrument variables for the computation of the steady state +under optimal policy. Requires a `steady_state_model` block or a +`_steadystate.m` file. See below. +::: + +*Steady state* + +Dynare takes advantage of the fact that the Lagrange multipliers appear +linearly in the equations of the steady state of the model under optimal +policy. Nevertheless, it is in general very difficult to compute the +steady state with simply a numerical guess in `initval` for the +endogenous variables. + +It greatly facilitates the computation, if the user provides an +analytical solution for the steady state (in `steady_state_model` block +or in a `_steadystate.m` file). In this case, it is necessary to provide +a steady state solution CONDITIONAL on the value of the instruments in +the optimal policy problem and declared with the option `instruments`. +The initial value of the instrument for steady state finding in this +case is set with `initval`. Note that computing and displaying steady +state values using the `steady`-command or calls to `resid` must come +after the `ramsey_model` statement and the `initval`-block. + +Note that choosing the instruments is partly a matter of interpretation +and you can choose instruments that are handy from a mathematical point +of view but different from the instruments you would refer to in the +analysis of the paper. A typical example is choosing inflation or +nominal interest rate as an instrument. +::: + +::: {.block} +ramsey\_constraints ; + +This block lets you define constraints on the variables in the Ramsey +problem. The constraints take the form of a variable, an inequality +operator (\> or \<) and a constant. + +*Example* + +> ramsey_constraints; +> i > 0; +> end; +::: + +::: {.command} +ramsey\_policy \[VARIABLE\_NAME\...\]; ramsey\_policy (OPTIONS\...) +\[VARIABLE\_NAME\...\]; + +This command is deprecated and formally equivalent to the calling +sequence + +> ramsey_model; +> stoch_simul; +> evaluate_planner_objective; + +It computes an approximation of the policy that maximizes the policy +maker's objective function subject to the constraints provided by the +equilibrium path of the private economy and under commitment to this +optimal policy. The Ramsey policy is computed by approximating the +equilibrium system around the perturbation point where the Lagrange +multipliers are at their steady state, i.e. where the Ramsey planner +acts as if the initial multipliers had been set to 0 in the distant +past, giving them time to converge to their steady state value. +Consequently, the optimal decision rules are computed around this steady +state of the endogenous variables and the Lagrange multipliers. + +Note that the variables in the list after the `ramsey_policy` or +`stoch_simul`-command can also contain multiplier names, but in a +case-sensititve way (e.g. `MULT_1`). In that case, Dynare will for +example display the IRFs of the respective multipliers when `irf>0`. + +The planner objective must be declared with the +`planner_objective`{.interpreted-text role="comm"} command. + +*Options* + +This command accepts all options of `stoch_simul`, plus: + +::: {.option} +planner\_discount = EXPRESSION + +See `planner_discount `{.interpreted-text +role="opt"}. +::: + +::: {.option} +instruments = (VARIABLE\_NAME,\...) + +Declares instrument variables for the computation of the steady state +under optimal policy. Requires a `steady_state_model` block or a +`_steadystate.m` file. See below. +::: + +*Output* + +This command generates all the output variables of `stoch_simul`. For +specifying the initial values for the endogenous state variables (except +for the Lagrange multipliers), see above. + +*Steady state* + +See `Ramsey steady state `{.interpreted-text role="comm"}. +::: + +### Optimal policy under discretion + +::: {.command} +discretionary\_policy \[VARIABLE\_NAME\...\]; discretionary\_policy +(OPTIONS\...) \[VARIABLE\_NAME\...\]; + +This command computes an approximation of the optimal policy under +discretion. The algorithm implemented is essentially an LQ solver, and +is described by *Dennis (2007)*. + +You must ensure that your objective is quadratic. Regarding the model, +it must either be linear or solved at first order with an analytical +steady state provided. In the first case, you should set the `linear` +option of the `model` block. + +It is possible to use the `estimation`{.interpreted-text role="comm"} +command after the `discretionary_policy` command, in order to estimate +the model with optimal policy under discretion and +`evaluate_planner_objective`{.interpreted-text role="comm"} to compute +welfare. + +*Options* + +This command accepts the same options as `ramsey_policy`, plus: + +::: {.option} +discretionary\_tol = NON-NEGATIVE DOUBLE + +Sets the tolerance level used to assess convergence of the solution +algorithm. Default: `1e-7`. +::: + +::: {.option} +maxit = INTEGER + +Maximum number of iterations. Default: `3000`. +::: +::: + +### Optimal Simple Rules (OSR) + +::: {.command} +osr \[VARIABLE\_NAME\...\]; osr (OPTIONS\...) \[VARIABLE\_NAME\...\]; + +This command computes optimal simple policy rules for linear-quadratic +problems of the form: + +> $$\min_\gamma E(y'_tWy_t)$$ + +such that: + +> $$A_1 E_ty_{t+1}+A_2 y_t+ A_3 y_{t-1}+C e_t=0$$ + +where: + +> - $E$ denotes the unconditional expectations operator; +> - $\gamma$ are parameters to be optimized. They must be elements of +> the matrices $A_1$, $A_2$, $A_3$, i.e. be specified as parameters +> in the `params` command and be entered in the `model` block; +> - $y$ are the endogenous variables, specified in the `var` command, +> whose (co)-variance enters the loss function; +> - $e$ are the exogenous stochastic shocks, specified in the +> `varexo`- ommand; +> - $W$ is the weighting matrix; + +The linear quadratic problem consists of choosing a subset of model +parameters to minimize the weighted (co)-variance of a specified subset +of endogenous variables, subject to a linear law of motion implied by +the first order conditions of the model. A few things are worth +mentioning. First, $y$ denotes the selected endogenous variables' +deviations from their steady state, i.e. in case they are not already +mean 0 the variables entering the loss function are automatically +demeaned so that the centered second moments are minimized. Second, +`osr` only solves linear quadratic problems of the type resulting from +combining the specified quadratic loss function with a first order +approximation to the model's equilibrium conditions. The reason is that +the first order state-space representation is used to compute the +unconditional (co)-variances. Hence, `osr` will automatically select +`order=1`. Third, because the objective involves minimizing a weighted +sum of unconditional second moments, those second moments must be +finite. In particular, unit roots in $y$ are not allowed. + +The subset of the model parameters over which the optimal simple rule is +to be optimized, $\gamma$, must be listed with `osr_params`. + +The weighting matrix $W$ used for the quadratic objective function is +specified in the `optim_weights` block. By attaching weights to +endogenous variables, the subset of endogenous variables entering the +objective function, $y$, is implicitly specified. + +The linear quadratic problem is solved using the numerical optimizer +specified with `opt_algo `{.interpreted-text +role="opt"}. + +*Options* + +The `osr` command will subsequently run `stoch_simul` and accepts the +same options, including restricting the endogenous variables by listing +them after the command, as `stoch_simul` (see +`stoch-sol`{.interpreted-text role="ref"}) plus + +::: {.option} +opt\_algo = INTEGER + +Specifies the optimizer for minimizing the objective function. The same +solvers as for `mode_compute` (see +`mode_compute `{.interpreted-text +role="opt"}) are available, except for `5`, `6`, and `10`. +::: + +::: {.option} +optim = (NAME, VALUE, \...) + +A list of NAME\`[ and VALUE pairs. Can be used to set options for the +optimization routines. The set of available options depends on the +selected optimization routine (i.e. on the value of option +:opt:\`opt\_algo \]{.title-ref}). See +`optim `{.interpreted-text role="opt"}. +::: + +::: {.option} +maxit = INTEGER + +Determines the maximum number of iterations used in `opt_algo=4`. This +option is now deprecated and will be removed in a future release of +Dynare. Use `optim` instead to set optimizer-specific values. Default: +`1000`. +::: + +::: {.option} +tolf = DOUBLE + +Convergence criterion for termination based on the function value used +in `opt_algo=4`. Iteration will cease when it proves impossible to +improve the function value by more than tolf. This option is now +deprecated and will be removed in a future release of Dynare. Use +`optim` instead to set optimizer-specific values. Default: `1e-7`. +::: + +::: {.option} +silent\_optimizer + +See `silent_optimizer`{.interpreted-text role="opt"}. +::: + +::: {.option} +huge\_number = DOUBLE + +Value for replacing the infinite bounds on parameters by finite numbers. +Used by some optimizers for numerical reasons (see +`huge_number `{.interpreted-text role="opt"}). +Users need to make sure that the optimal parameters are not larger than +this value. Default: `1e7`. +::: + +The value of the objective is stored in the variable +`oo_.osr.objective_function` and the value of parameters at the optimum +is stored in `oo_.osr.optim_params`. See below for more details. + +After running `osr` the parameters entering the simple rule will be set +to their optimal value so that subsequent runs of `stoch_simul` will be +conducted at these values. +::: + +::: {.command} +osr\_params PARAMETER\_NAME\...; + +This command declares parameters to be optimized by `osr`. +::: + +::: {.block} +optim\_weights ; + +This block specifies quadratic objectives for optimal policy problems. + +More precisely, this block specifies the nonzero elements of the weight +matrix $W$ used in the quadratic form of the objective function in +`osr`. + +An element of the diagonal of the weight matrix is given by a line of +the form: + + VARIABLE_NAME EXPRESSION; + +An off-the-diagonal element of the weight matrix is given by a line of +the form: + + VARIABLE_NAME, VARIABLE_NAME EXPRESSION; +::: + +*Example* + +> var y inflation r; +> varexo y_ inf_; +> +> parameters delta sigma alpha kappa gammarr gammax0 gammac0 gamma_y_ gamma_inf_; +> +> delta = 0.44; +> kappa = 0.18; +> alpha = 0.48; +> sigma = -0.06; +> +> gammarr = 0; +> gammax0 = 0.2; +> gammac0 = 1.5; +> gamma_y_ = 8; +> gamma_inf_ = 3; +> +> model(linear); +> y = delta * y(-1) + (1-delta)*y(+1)+sigma *(r - inflation(+1)) + y_; +> inflation = alpha * inflation(-1) + (1-alpha) * inflation(+1) + kappa*y + inf_; +> r = gammax0*y(-1)+gammac0*inflation(-1)+gamma_y_*y_+gamma_inf_*inf_; +> end; +> +> shocks; +> var y_; stderr 0.63; +> var inf_; stderr 0.4; +> end; +> +> optim_weights; +> inflation 1; +> y 1; +> y, inflation 0.5; +> end; +> +> osr_params gammax0 gammac0 gamma_y_ gamma_inf_; +> osr y; + +::: {.block} +osr\_params\_bounds ; + +This block declares lower and upper bounds for parameters in the optimal +simple rule. If not specified the optimization is unconstrained. + +Each line has the following syntax: + + PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND; + +Note that the use of this block requires the use of a constrained +optimizer, i.e. setting +`opt_algo `{.interpreted-text role="opt"} to `1`, +`2`, `5` or `9`. + +*Example* + +> osr_params_bounds; +> gamma_inf_, 0, 2.5; +> end; +> +> osr(opt_algo=9) y; +::: + +::: {.matvar} +[oo]().osr.objective\_function + +After an execution of the `osr` command, this variable contains the +value of the objective under optimal policy. +::: + +::: {.matvar} +[oo]().osr.optim\_params + +After an execution of the `osr` command, this variable contains the +value of parameters at the optimum, stored in fields of the form +`oo_.osr.optim_params.PARAMETER_NAME`. +::: + +::: {.matvar} +[M]().osr.param\_names + +After an execution of the `osr` command, this cell contains the names of +the parameters. +::: + +::: {.matvar} +[M]().osr.param\_indices + +After an execution of the `osr` command, this vector contains the +indices of the OSR parameters in `M_.params`. +::: + +::: {.matvar} +[M]().osr.param\_bounds + +After an execution of the `osr` command, this two by number of OSR +parameters matrix contains the lower and upper bounds of the parameters +in the first and second column, respectively. +::: + +::: {.matvar} +[M]().osr.variable\_weights + +After an execution of the `osr` command, this sparse matrix contains the +weighting matrix associated with the variables in the objective +function. +::: + +::: {.matvar} +[M]().osr.variable\_indices + +After an execution of the `osr` command, this vector contains the +indices of the variables entering the objective function in +`M_.endo_names`. +::: + + +Displaying and saving results +----------------------------- + +Dynare has comments to plot the results of a simulation and to save the +results. + +::: {.command} +rplot VARIABLE\_NAME\...; + +Plots the simulated path of one or several variables, as stored in +`oo_.endo_simul` by either `perfect_foresight_solver`, `simul` (see +`det-simul`{.interpreted-text role="ref"}) or `stoch_simul` with option +`periods` (see `stoch-sol`{.interpreted-text role="ref"}). The variables +are plotted in levels. +::: + +::: {.command} +dynatype (FILENAME) \[VARIABLE\_NAME\...\]; + +This command prints the listed endogenous or exogenous variables in a +text file named FILENAME. If no VARIABLE\_NAME is listed, all endogenous +variables are printed. +::: + +::: {.command} +dynasave (FILENAME) \[VARIABLE\_NAME\...\]; + +This command saves the listed endogenous or exogenous variables in a +binary file named FILENAME. If no VARIABLE\_NAME is listed, all +endogenous variables are saved. + +In MATLAB or Octave, variables saved with the `dynasave` command can be +retrieved by the command: + + load(FILENAME,'-mat') +::: + +Macro processing language {#macro-proc-lang} +------------------------- + +It is possible to use "macro" commands in the `.mod` file for performing +tasks such as: including modular source files, replicating blocks of +equations through loops, conditionally executing some code, writing +indexed sums or products inside equations\... + +The Dynare macro-language provides a new set of *macro-commands* which +can be used in `.mod` files. It features: + +> - File inclusion +> - Loops (`for` structure) +> - Conditional inclusion (`if/then/else` structures) +> - Expression substitution + +This macro-language is totally independent of the basic Dynare language, +and is processed by a separate component of the Dynare pre-processor. +The macro processor transforms a `.mod` file with macros into a `.mod` +file without macros (doing expansions/inclusions), and then feeds it to +the Dynare parser. The key point to understand is that the macro +processor only does text substitution (like the C preprocessor or the +PHP language). Note that it is possible to see the output of the macro +processor by using the `savemacro` option of the `dynare` command (see +`dyn-invoc`{.interpreted-text role="ref"}). + +The macro processor is invoked by placing *macro directives* in the +`.mod` file. Directives begin with an at-sign followed by a pound sign +(`@#`). They produce no output, but give instructions to the macro +processor. In most cases, directives occupy exactly one line of text. If +needed, two backslashes (`\\`) at the end of the line indicate that the +directive is continued on the next line. Macro directives following `//` +are not interpreted by the macro processor. For historical reasons, +directives in commented blocks, *ie* surrounded by `/*` and `*/`, are +interpreted by the macro processor. The user should not rely on this +behavior. The main directives are: + +> - `@#includepath`, paths to search for files that are to be +> included, +> - `@#include`, for file inclusion, +> - `@#define`, for defining a macro processor variable, +> - `@#if, @#ifdef, @#ifndef, @#elseif, @#else, @#endif` for +> conditional statements, +> - `@#for, @#endfor` for constructing loops. + +The macro processor maintains its own list of variables (distinct from +model variables and MATLAB/Octave variables). These macro-variables are +assigned using the `@#define` directive and can be of the following +basic types: boolean, real, string, tuple, function, and array (of any +of the previous types). + +### Macro expressions {#macro-exp} + +Macro-expressions can be used in two places: + +> - Inside macro directives, directly; +> - In the body of the `.mod` file, between an at-sign and curly +> braces (like `@{expr}`): the macro processor will substitute the +> expression with its value + +It is possible to construct macro-expressions that can be assigned to +macro-variables or used within a macro-directive. The expressions are +constructed using literals of the basic types (boolean, real, string, +tuple, array), comprehensions, macro-variables, macro-functions, and +standard operators. + +::: {.note} +::: {.admonition-title} +Note +::: + +Elsewhere in the manual, MACRO\_EXPRESSION designates an expression +constructed as explained in this section. +::: + +**Boolean** + +The following operators can be used on booleans: + +> - Comparison operators: `==, !=` +> - Logical operators: `&&, ||, !` + +**Real** + +The following operators can be used on reals: + +> - Arithmetic operators: `+, -, *, /, ^` +> - Comparison operators: `<, >, <=, >=, ==, !=` +> - Logical operators: `&&, ||, !` +> - Ranges with an increment of `1`: `REAL1:REAL2` (for example, `1:4` +> is equivalent to real array `[1, 2, 3, 4]`). +> +> ::: {.versionchanged} +> 4.6 Previously, putting brackets around the arguments to the colon +> operator (e.g. `[1:4]`) had no effect. Now, `[1:4]` will create an +> array containing an array (i.e. `[ [1, 2, 3, 4] ]`). +> ::: +> +> - Ranges with user-defined increment: `REAL1:REAL2:REAL3` (for +> example, `6:-2.1:-1` is equivalent to real array +> `[6, 3.9, 1.8, -0.3]`). +> - Functions: +> `max, min, mod, exp, log, log10, sin, cos, tan, asin, acos, atan, sqrt, cbrt, sign, floor, ceil, trunc, erf, erfc, gamma, lgamma, round, normpdf, normcdf`. +> NB `ln` can be used instead of `log` + +**String** + +String literals have to be enclosed by **double** quotes (like +`"name"`). + +The following operators can be used on strings: + +> - Comparison operators: `<, >, <=, >=, ==, !=` +> - Concatenation of two strings: `+` +> - Extraction of substrings: if `s` is a string, then `s[3]` is a +> string containing only the third character of `s`, and `s[4:6]` +> contains the characters from 4th to 6th +> - Function: `length` + +**Tuple** + +Tuples are enclosed by parenthesis and elements separated by commas +(like `(a,b,c)` or `(1,2,3)`). + +The following operators can be used on tuples: + +> - Comparison operators: `==, !=` +> - Functions: `empty, length` + +**Array** + +Arrays are enclosed by brackets, and their elements are separated by +commas (like `[1,[2,3],4]` or `["US", "FR"]`). + +The following operators can be used on arrays: + +> - Comparison operators: `==, !=` +> - Dereferencing: if `v` is an array, then `v[2]` is its 2nd element +> - Concatenation of two arrays: `+` +> - Set union of two arrays: `|` +> - Set intersection of two arrays: `&` +> - Difference `-`: returns the first operand from which the elements +> of the second operand have been removed. +> - Cartesian product of two arrays: `*` +> - Cartesian product of one array N times: `^N` +> - Extraction of sub-arrays: e.g. `v[4:6]` +> - Testing membership of an array: `in` operator (for example: `"b"` +> in `["a", "b", "c"]` returns `1`) +> - Functions: `empty, sum, length` + +**Comprehension** + +Comprehension syntax is a shorthand way to make arrays from other +arrays. There are three different ways the comprehension syntax can be +employed: [filtering]{.title-ref}, [mapping]{.title-ref}, and [filtering +and mapping]{.title-ref}. + +**Filtering** + +Filtering allows one to choose those elements from an array for which a +certain condition hold. + +> *Example* +> +> Create a new array, choosing the even numbers from the array `1:5`: +> +> [ i in 1:5 when mod(i,2) == 0 ] +> +> would result in: +> +> [2, 4] + +**Mapping** + +Mapping allows you to apply a transformation to every element of an +array. + +> *Example* +> +> Create a new array, squaring all elements of the array `1:5`: +> +> [ i^2 for i in 1:5 ] +> +> would result in: +> +> [1, 4, 9, 16, 25] + +**Filtering and Mapping** + +Combining the two preceding ideas would allow one to apply a +transformation to every selected element of an array. + +> *Example* +> +> Create a new array, squaring all even elements of the array `1:5`: +> +> [ i^2 for i in 1:5 when mod(i,2) == 0] +> +> would result in: +> +> [4, 16] +> +> *Further Examples* : +> +> [ (j, i+1) for (i,j) in (1:2)^2 ] +> [ (j, i+1) for (i,j) in (1:2)*(1:2) when i < j ] +> +> would result in: +> +> [(1, 2), (2, 2), (1, 3), (2, 3)] +> [(2, 2)] + +**Function** + +Functions can be defined in the macro processor using the `@#define` +directive (see below). A function is evaluated at the time it is +invoked, not at define time. Functions can be included in expressions +and the operators that can be combined with them depend on their return +type. + +**Checking variable type** + +Given a variable name or literal, you can check the type it evaluates to +using the following functions: `isboolean`, `isreal`, `isstring`, +`istuple`, and `isarray`. + +> *Examples* + + ---------------------------------- + **Code** **Output** + --------------------- ------------ + `isboolean(0)` `false` + + `isboolean(true)` `true` + + `isreal("str")` `false` + ---------------------------------- + +**Casting between types** + +Variables and literals of one type can be cast into another type. Some +type changes are straightforward (e.g. changing a [real]{.title-ref} to +a [string]{.title-ref}) whereas others have certain requirements (e.g. +to cast an [array]{.title-ref} to a [real]{.title-ref} it must be a one +element array containing a type that can be cast to [real]{.title-ref}). + +> *Examples* + + ----------------------------------------------- + **Code** **Output** + ---------------------------------- ------------ + `(bool) -1.1` `true` + + `(bool) 0` `false` + + `(real) "2.2"` `2.2` + + `(tuple) [3.3]` `(3.3)` + + `(array) 4.4` `[4.4]` + + `(real) [5.5]` `5.5` + + `(real) [6.6, 7.7]` `error` + + `(real) "8.8 in a string"` `error` + ----------------------------------------------- + +Casts can be used in expressions: + +> *Examples* + + ---------------------------------------- + **Code** **Output** + --------------------------- ------------ + `(bool) 0 && true` `false` + + `(real) "1" + 2` `3` + + `(string) (3 + 4)` `"7"` + + `(array) 5 + (array) 6` `[5, 6]` + ---------------------------------------- + +### Macro directives + +::: {.macrodir} +@\#includepath \"PATH\" @\#includepath MACRO\_EXPRESSION + +This directive adds the path contained in PATH to the list of those to +search when looking for a `.mod` file specified by `@#include`. If +provided with a MACRO\_EXPRESSION argument, the argument must evaluate +to a string. Note that these paths are added *after* any paths passed +using `-I <-I\<\\>>`{.interpreted-text role="opt"}. + +*Example* + +> @#includepath "/path/to/folder/containing/modfiles" +> @#includepath folders_containing_mod_files +::: + +::: {.macrodir} +@\#include \"FILENAME\" @\#include MACRO\_EXPRESSION + +This directive simply includes the content of another file in its place; +it is exactly equivalent to a copy/paste of the content of the included +file. If provided with a MACRO\_EXPRESSION argument, the argument must +evaluate to a string. Note that it is possible to nest includes (i.e. to +include a file from an included file). The file will be searched for in +the current directory. If it is not found, the file will be searched for +in the folders provided by `-I <-I\<\\>>`{.interpreted-text +role="opt"} and `@#includepath`. + +*Example* + +> @#include "modelcomponent.mod" +> @#include location_of_modfile +::: + +::: {.macrodir} +@\#define MACRO\_VARIABLE @\#define MACRO\_VARIABLE = MACRO\_EXPRESSION +@\#define MACRO\_FUNCTION = MACRO\_EXPRESSION + +Defines a macro-variable or macro function. + +*Example* + +> @#define var // Equals true +> @#define x = 5 // Real +> @#define y = "US" // String +> @#define v = [ 1, 2, 4 ] // Real array +> @#define w = [ "US", "EA" ] // String array +> @#define u = [ 1, ["EA"] ] // Mixed array +> @#define z = 3 + v[2] // Equals 5 +> @#define t = ("US" in w) // Equals true +> @#define f(x) = " " + x + y // Function `f` with argument `x` +> // returns the string ' ' + x + 'US' + +*Example* + +> @#define x = 1 +> @#define y = [ "B", "C" ] +> @#define i = 2 +> @#define f(x) = x + " + " + y[i] +> @#define i = 1 +> +> model; +> A = @{y[i] + f("D")}; +> end; +> +> The latter is strictly equivalent to: +> +> model; +> A = BD + B; +> end; +::: + +::: {.macrodir} +@\#if MACRO\_EXPRESSION @\#ifdef MACRO\_VARIABLE @\#ifndef +MACRO\_VARIABLE @\#elseif MACRO\_EXPRESSION @\#else @\#endif + +Conditional inclusion of some part of the `.mod` file. The lines between +`@#if`, `@#ifdef`, or `@#ifndef` and the next `@#elseif`, `@#else` or +`@#endif` is executed only if the condition evaluates to `true`. +Following the `@#if` body, you can zero or more `@#elseif` branches. An +`@#elseif` condition is only evaluated if the preceding `@#if` or +`@#elseif` condition evaluated to `false`. The `@#else` branch is +optional and is only evaluated if all `@#if` and `@#elseif` statements +evaluate to false. + +Note that when using `@#ifdef`, the condition will evaluate to `true` if +the MACRO\_VARIABLE has been previously defined, regardless of its +value. Conversely, `@#ifndef` will evaluate to true if the +MACRO\_VARIABLE has not yet been defined. + +Note that when using `@#elseif` you can check whether or not a variable +has been defined by using the `defined` operator. Hence, to enter the +body of an `@#elseif` branch if the variable `X` has not been defined, +you would write: `@#elseif !defined(X)`. + +Note that if a real appears as the result of the MACRO\_EXPRESSION, it +will be interpreted as a boolean; a value of `0` is interpreted as +`false`, otherwise it is interpreted as `true`. Further note that +because of the imprecision of reals, extra care must be taken when +testing them in the MACRO\_EXPRESSION. For example, `exp(log(5)) == 5` +will evaluate to `false`. Hence, when comparing real values, you should +generally use a zero tolerance around the value desired, e.g. +`exp(log(5)) > 5-1e-14 && exp(log(5)) < 5+1e-14` + +*Example* + +> Choose between two alternative monetary policy rules using a +> macro-variable: +> +> @#define linear_mon_pol = false // 0 would be treated the same +> ... +> model; +> @#if linear_mon_pol +> i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); +> @#else +> i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; +> @#endif +> ... +> end; +> +> This would result in: +> +> ... +> model; +> i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; +> ... +> end; + +*Example* + +> Choose between two alternative monetary policy rules using a +> macro-variable. The only difference between this example and the +> previous one is the use of `@#ifdef` instead of `@#if`. Even though +> `linear_mon_pol` contains the value `false` because `@#ifdef` only +> checks that the variable has been defined, the linear monetary policy +> is output: +> +> @#define linear_mon_pol = false // 0 would be treated the same +> ... +> model; +> @#ifdef linear_mon_pol +> i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); +> @#else +> i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; +> @#endif +> ... +> end; +> +> This would result in: +> +> ... +> model; +> i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); +> ... +> end; +::: + +::: {.macrodir} +@\#for MACRO\_VARIABLE in MACRO\_EXPRESSION @\#for MACRO\_VARIABLE in +MACRO\_EXPRESSION when MACRO\_EXPRESSION @\#for MACRO\_TUPLE in +MACRO\_EXPRESSION @\#for MACRO\_TUPLE in MACRO\_EXPRESSION when +MACRO\_EXPRESSION @\#endfor + +Loop construction for replicating portions of the `.mod` file. Note that +this construct can enclose variable/parameters declaration, +computational tasks, but not a model declaration. + +*Example* + +> model; +> @#for country in [ "home", "foreign" ] +> GDP_@{country} = A * K_@{country}^a * L_@{country}^(1-a); +> @#endfor +> end; +> +> The latter is equivalent to: +> +> model; +> GDP_home = A * K_home^a * L_home^(1-a); +> GDP_foreign = A * K_foreign^a * L_foreign^(1-a); +> end; + +*Example* + +> model; +> @#for (i, j) in ["GDP"] * ["home", "foreign"] +> @{i}_@{j} = A * K_@{j}^a * L_@{j}^(1-a); +> @#endfor +> end; +> +> The latter is equivalent to: +> +> model; +> GDP_home = A * K_home^a * L_home^(1-a); +> GDP_foreign = A * K_foreign^a * L_foreign^(1-a); +> end; + +*Example* + +> @#define countries = ["US", "FR", "JA"] +> @#define nth_co = "US" +> model; +> @#for co in countries when co != nth_co +> (1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; +> @#endfor +> E_@{nth_co} = 1; +> end; +> +> The latter is equivalent to: +> +> model; +> (1+i_FR) = (1+i_US) * E_FR(+1) / E_FR; +> (1+i_JA) = (1+i_US) * E_JA(+1) / E_JA; +> E_US = 1; +> end; +::: + +::: {.macrodir} +@\#echo MACRO\_EXPRESSION + +Asks the preprocessor to display some message on standard output. The +argument must evaluate to a string. +::: + +::: {.macrodir} +@\#error MACRO\_EXPRESSION + +Asks the preprocessor to display some error message on standard output +and to abort. The argument must evaluate to a string. +::: + +::: {.macrodir} +@\#echomacrovars @\#echomacrovars MACRO\_VARIABLE\_LIST +@\#echomacrovars(save) MACRO\_VARIABLE\_LIST + +Asks the preprocessor to display the value of all macro variables up +until this point. If the `save` option is passed, then values of the +macro variables are saved to `options_.macrovars_line_<>`. +If `NAME_LIST` is passed, only display/save variables and functions with +that name. + +*Example* + +> @#define A = 1 +> @#define B = 2 +> @#define C(x) = x*2 +> @#echomacrovars A C D +> +> The output of the command above is: +> +> Macro Variables: +> A = 1 +> Macro Functions: +> C(x) = (x * 2) +::: + +### Typical usages + +#### Modularization + +The `@#include` directive can be used to split `.mod` files into several +modular components. + +Example setup: + +`modeldesc.mod` + +> Contains variable declarations, model equations, and shocks +> declarations. + +`simul.mod` + +> Includes `modeldesc.mod`, calibrates parameter,s and runs stochastic +> simulations. + +`estim.mod` + +> Includes `modeldesc.mod`, declares priors on parameters, and runs +> Bayesian estimation. + +Dynare can be called on `simul.mod` and `estim.mod` but it makes no +sense to run it on `modeldesc.mod`. + +The main advantage is that you don\'t have to copy/paste the whole model +(at the beginning) or changes to the model (during development). + +#### Indexed sums of products + +The following example shows how to construct a moving average: + + @#define window = 2 + + var x MA_x; + ... + model; + ... + MA_x = @{1/(2*window+1)}*( + @#for i in -window:window + +x(@{i}) + @#endfor + ); + ... + end; + +After macro processing, this is equivalent to: + + var x MA_x; + ... + model; + ... + MA_x = 0.2*( + +x(-2) + +x(-1) + +x(0) + +x(1) + +x(2) + ); + ... + end; + +#### Multi-country models + +Here is a skeleton example for a multi-country model: + + @#define countries = [ "US", "EA", "AS", "JP", "RC" ] + @#define nth_co = "US" + + @#for co in countries + var Y_@{co} K_@{co} L_@{co} i_@{co} E_@{co} ...; + parameters a_@{co} ...; + varexo ...; + @#endfor + + model; + @#for co in countries + Y_@{co} = K_@{co}^a_@{co} * L_@{co}^(1-a_@{co}); + ... + @#if co != nth_co + (1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; // UIP relation + @#else + E_@{co} = 1; + @#endif + @#endfor + end; + +#### Endogeneizing parameters + +When calibrating the model, it may be useful to consider a parameter as +an endogenous variable (and vice-versa). + +For example, suppose production is defined by a CES function: + +> $$y = \left(\alpha^{1/\xi} \ell^{1-1/\xi}+(1-\alpha)^{1/\xi}k^{1-1/\xi}\right)^{\xi/(\xi-1)}$$ + +and the labor share in GDP is defined as: + +> $$\textrm{lab\_rat} = (w \ell)/(p y)$$ + +In the model, $\alpha$ is a (share) parameter and `lab_rat` is an +endogenous variable. + +It is clear that calibrating $\alpha$ is not straightforward; on the +contrary, we have real world data for `lab_rat` and it is clear that +these two variables are economically linked. + +The solution is to use a method called *variable flipping*, which +consists in changing the way of computing the steady state. During this +computation, $\alpha$ will be made an endogenous variable and `lab_rat` +will be made a parameter. An economically relevant value will be +calibrated for `lab_rat`, and the solution algorithm will deduce the +implied value for $\alpha$. + +An implementation could consist of the following files: + +`modeqs.mod` + +> This file contains variable declarations and model equations. The code +> for the declaration of $\alpha$ and `lab_rat` would look like: +> +> @#if steady +> var alpha; +> parameter lab_rat; +> @#else +> parameter alpha; +> var lab_rat; +> @#endif + +`steady.mod` + +> This file computes the steady state. It begins with: +> +> @#define steady = 1 +> @#include "modeqs.mod" +> +> Then it initializes parameters (including `lab_rat`, excluding +> $\alpha$), computes the steady state (using guess values for +> endogenous, including $\alpha$), then saves values of parameters and +> endogenous at steady state in a file, using the +> `save_params_and_steady_state` command. + +`simul.mod` + +> This file computes the simulation. It begins with: +> +> @#define steady = 0 +> @#include "modeqs.mod" +> +> Then it loads values of parameters and endogenous at steady state from +> file, using the `load_params_and_steady_state` command, and computes +> the simulations. + +### MATLAB/Octave loops versus macro processor loops + +Suppose you have a model with a parameter $\rho$ and you want to run +simulations for three values: $\rho = 0.8, 0.9, +1$. There are several ways of doing this: + +*With a MATLAB/Octave loop* + +> rhos = [ 0.8, 0.9, 1]; +> for i = 1:length(rhos) +> rho = rhos(i); +> stoch_simul(order=1); +> end +> +> Here the loop is not unrolled, MATLAB/Octave manages the iterations. +> This is interesting when there are a lot of iterations. + +*With a macro processor loop (case 1)* + +> rhos = [ 0.8, 0.9, 1]; +> @#for i in 1:3 +> rho = rhos(@{i}); +> stoch_simul(order=1); +> @#endfor +> +> This is very similar to the previous example, except that the loop is +> unrolled. The macro processor manages the loop index but not the data +> array (`rhos`). + +*With a macro processor loop (case 2)* + +> @#for rho_val in [ 0.8, 0.9, 1] +> rho = @{rho_val}; +> stoch_simul(order=1); +> @#endfor +> +> The advantage of this method is that it uses a shorter syntax, since +> the list of values is directly given in the loop construct. The +> inconvenience is that you can not reuse the macro array in +> MATLAB/Octave. + +Verbatim inclusion {#verbatim} +------------------ + +Pass everything contained within the verbatim block to the +`.m` file. + +::: {.block} +verbatim ; + +By default, whenever Dynare encounters code that is not understood by +the parser, it is directly passed to the preprocessor output. + +In order to force this behavior you can use the `verbatim` block. This +is useful when the code you want passed to the `.m` file +contains tokens recognized by the Dynare preprocessor. + +*Example* + +> verbatim; +> % Anything contained in this block will be passed +> % directly to the .m file, including comments +> var = 1; +> end; +::: + +Misc commands +------------- + +::: {.command} +set\_dynare\_seed (INTEGER) set\_dynare\_seed (\`default\') +set\_dynare\_seed (\`clock\') set\_dynare\_seed (\`reset\') +set\_dynare\_seed (\`ALGORITHM\', INTEGER) + +Sets the seed used for random number generation. It is possible to set a +given integer value, to use a default value, or to use the clock (by +using the latter, one will therefore get different results across +different Dynare runs). The `reset` option serves to reset the seed to +the value set by the last `set_dynare_seed` command. On MATLAB 7.8 or +above, it is also possible to choose a specific algorithm for random +number generation; accepted values are `mcg16807`, `mlfg6331_64`, +`mrg32k3a`, `mt19937ar` (the default), `shr3cong` and `swb2712`. +::: + +::: {.command} +save\_params\_and\_steady\_state (FILENAME); + +For all parameters, endogenous and exogenous variables, stores their +value in a text file, using a simple name/value associative table. + +> - for parameters, the value is taken from the last parameter +> initialization. +> - for exogenous, the value is taken from the last `initval` block. +> - for endogenous, the value is taken from the last steady state +> computation (or, if no steady state has been computed, from the +> last `initval` block). + +Note that no variable type is stored in the file, so that the values can +be reloaded with `load_params_and_steady_state` in a setup where the +variable types are different. + +The typical usage of this function is to compute the steady-state of a +model by calibrating the steady-state value of some endogenous variables +(which implies that some parameters must be endogeneized during the +steady-state computation). + +You would then write a first `.mod` file which computes the steady state +and saves the result of the computation at the end of the file, using +`save_params_and_steady_state`. + +In a second file designed to perform the actual simulations, you would +use `load_params_and_steady_state` just after your variable +declarations, in order to load the steady state previously computed +(including the parameters which had been endogeneized during the steady +state computation). + +The need for two separate `.mod` files arises from the fact that the +variable declarations differ between the files for steady state +calibration and for simulation (the set of endogenous and parameters +differ between the two); this leads to different `var` and `parameters` +statements. + +Also note that you can take advantage of the `@#include` directive to +share the model equations between the two files (see +`macro-proc-lang`{.interpreted-text role="ref"}). +::: + +::: {.command} +load\_params\_and\_steady\_state (FILENAME); + +For all parameters, endogenous and exogenous variables, loads their +value from a file created with `save_params_and_steady_state`. + +> - for parameters, their value will be initialized as if they had +> been calibrated in the `.mod` file. +> - for endogenous and exogenous variables, their value will be +> initialized as they would have been from an `initval` block . + +This function is used in conjunction with +`save_params_and_steady_state`; see the documentation of that function +for more information. +::: + +::: {.command} +compilation\_setup (OPTIONS); + +When the `use_dll`{.interpreted-text role="opt"} option is present, +Dynare uses the GCC compiler that was distributed with it to compile the +static and dynamic C files produced by the preprocessor. You can use +this option to change the compiler, flags, and libraries used. + +*Options* + +> ::: {.option} +> compiler = FILENAME +> +> The path to the compiler. +> ::: +> +> ::: {.option} +> substitute\_flags = QUOTED\_STRING +> +> The flags to use instead of the default flags. +> ::: +> +> ::: {.option} +> add\_flags = QUOTED\_STRING +> +> The flags to use in addition to the default flags. If +> `substitute_flags` is passed, these flags are added to the flags +> specified there. +> ::: +> +> ::: {.option} +> substitute\_libs = QUOTED\_STRING +> +> The libraries to link against instead of the default libraries. +> ::: +> +> ::: {.option} +> add\_libs = QUOTED\_STRING +> +> The libraries to link against in addition to the default libraries. If +> `substitute_libs` is passed, these libraries are added to the +> libraries specified there. +> ::: +::: + +::: {.matcomm} +dynare\_version ; + +Output the version of Dynare that is currently being used (i.e. the one +that is highest on the MATLAB/Octave path). +::: + +::: {.matcomm} +write\_latex\_definitions ; + +Writes the names, LaTeX names and long names of model variables to +tables in a file named `<>_latex_definitions.tex`. Requires +the following LaTeX packages: `longtable`. +::: + +::: {.matcomm} +write\_latex\_parameter\_table ; + +Writes the LaTeX names, parameter names, and long names of model +parameters to a table in a file named +`<>_latex_parameters.tex.` The command writes the values of +the parameters currently stored. Thus, if parameters are set or changed +in the steady state computation, the command should be called after a +steady-command to make sure the parameters were correctly updated. The +long names can be used to add parameter descriptions. Requires the +following LaTeX packages: `longtable, booktabs`. +::: + +::: {.matcomm} +write\_latex\_prior\_table ; + +Writes descriptive statistics about the prior distribution to a LaTeX +table in a file named `<>_latex_priors_table.tex`. The command +writes the prior definitions currently stored. Thus, this command must +be invoked after the `estimated_params` block. If priors are defined +over the measurement errors, the command must also be preceeded by the +declaration of the observed variables (with `varobs`). The command +displays a warning if no prior densities are defined (ML estimation) or +if the declaration of the observed variables is missing. Requires the +following LaTeX packages: `longtable, booktabs`. +::: + +::: {.matcomm} +collect\_latex\_files ; + +Writes a LaTeX file named `<>_TeX_binder.tex` that collects +all TeX output generated by Dynare into a file. This file can be +compiled using `pdflatex` and automatically tries to load all required +packages. Requires the following LaTeX packages: `breqn`, `psfrag`, +`graphicx`, `epstopdf`, `longtable`, `booktabs`, `caption`, `float,` +`amsmath`, `amsfonts`, and `morefloats`. +::: + +**Footnotes** + +[^1]: A `.mod` file must have lines that end with a line feed character, + which is not commonly visible in text editors. Files created on + Windows and Unix-based systems have always conformed to this + requirement, as have files created on OS X and macOS. Files created + on old, pre-OS X Macs used carriage returns as end of line + characters. If you get a Dynare parsing error of the form + `ERROR: <>: line 1, cols 341-347: syntax error,...` and + there\'s more than one line in your `.mod` file, know that it uses + the carriage return as an end of line character. To get more helpful + error messages, the carriage returns should be changed to line + feeds. + +[^2]: Note that arbitrary MATLAB or Octave expressions can be put in a + `.mod` file, but those expressions have to be on separate lines, + generally at the end of the file for post-processing purposes. They + are not interpreted by Dynare, and are simply passed on unmodified + to MATLAB or Octave. Those constructions are not addresses in this + section. + +[^3]: In particular, for big models, the compilation step can be very + time-consuming, and use of this option may be counter-productive in + those cases. + +[^4]: See options `conf_sig `{.interpreted-text role="ref"} and + `mh_conf_sig `{.interpreted-text role="opt"} + to change the size of the HPD interval. + +[^5]: See options `conf_sig `{.interpreted-text role="ref"} () + and `mh_conf_sig `{.interpreted-text + role="opt"} to qchange the size of the HPD interval. + +[^6]: When the shocks are correlated, it is the decomposition of + orthogonalized shocks via Cholesky decomposition according to the + order of declaration of shocks (see `var-decl`{.interpreted-text + role="ref"}) + +[^7]: See `forecast `{.interpreted-text role="opt"} + for more information. + +[^8]: In case of Excel not being installed, + + may be helpful. + +[^9]: In case of Excel not being installed, + + may be helpful. + +[^10]: See option `conf_sig `{.interpreted-text role="ref"} to + change the size of the HPD interval. + +[^11]: See option `conf_sig `{.interpreted-text role="ref"} to + change the size of the HPD interval. + +[^12]: If you want to align the paper with the description herein, + please note that $A$ is $A^0$ and $F$ is $A^+$. + +[^13]: An example can be found at + . diff --git a/doc/make.jl b/docs/make.jl similarity index 100% rename from doc/make.jl rename to docs/make.jl diff --git a/docs/src/development/Outputs.md b/docs/src/development/Outputs.md new file mode 100644 index 00000000..65e95587 --- /dev/null +++ b/docs/src/development/Outputs.md @@ -0,0 +1,17 @@ +# Results generated by Dynare + +Dynare stores results in the `context` structure. The main entry +points are +- `trends`: `context.results.model_results[1].trends` +- `LREresults`: `context.results.model_results[1].linearrationalresults` +| Object | Storage | Generated by | Accessor | Remark | +|--------------------|--------------------------------|------------------|----------|--------| +| steady state | trends.endogenous_steady_state | Steady_State.jl | | | +| | trends.exogenous_steady_state | | | | +| 1st order solution | LREresults.g1 | perturbations.jl | | | +| | LREresults.g1_1 | | | | +| | LREresults.g1_2 | | | | +| | LREresults.gns1 | | | | +| | LREresults.gs1 | | | | +| | LREresults.hns1 | | | | +| | LREresults.hs1 | | | | diff --git a/docs/src/development/RecursiveBlocks.blocks.md b/docs/src/development/RecursiveBlocks.blocks.md new file mode 100644 index 00000000..3ceafc53 --- /dev/null +++ b/docs/src/development/RecursiveBlocks.blocks.md @@ -0,0 +1,286 @@ +# Computing recursive blocks + +This notes present the identification of recursive blocks and the +construction of function to compute residuals and Jacobian by blocks + +## Identification of recrusive blocks + +Assuming that we have the sparse representation of the Jacobian matrix +of the model where the variables are in columns and the equations in rows + +### Getting the static and dynamic Jacobian matrices +``` +using Dynare + +context = @dynare "models/example1/example1.mod"; + +model = context.models[1] +results = context.results.model_results[1] +wsd = Dynare.DynamicWs(context) +wss = Dynare.StaticWs(context) +params = context.work.params +steadystate = results.trends.endogenous_steady_state +endogenous = repeat(steadystate, 3) +exogenous = results.trends.exogenous_steady_state +Jdynamic = Dynare.get_dynamic_jacobian!(wsd, params, endogenous, exogenous, steadystate, model, 200) +Jstatic = Dynare.get_static_jacobian!(wss, params, steadystate, exogenous, model) +``` + +### The incidence matrix + +The incidence matrix of the model is a boolean matrix indicating which +variable appears in which equation. + +#### Incidence matrix in the static model +``` +using SparseArrays + +ONES = Vector{Bool}(undef, length(Jstatic.rowval)) # all ONES elements must be true +fill!(ONES, true) +IncidenceStatic = SparseMatrixCSC(size(Jstatic, 1), size(Jstatic,2), Jstatic.colptr, Jstatic.rowval, ONES) +``` +The incidence matrix of the static model is a square matrix and its +order corresponds to the number of endogenous variables. + +#### Incidence matrix in the dynamic model +``` +using SparseArrays + +ONES = Vector{Bool}(undef, length(Jdynamic.rowval)) # all ONES elements must be true +fill!(ONES, true) +IncidenceDynamic = SparseMatrixCSC(size(Jdynamic, 1), size(Jdynamic,2), Jdynamic.colptr, Jdynamic.rowval, ONES) +``` +The incidence matrix of the dynamic model is a rectangular matrix +$n\times 3n$ as variables in period $t-1$, $t$, and $t+1$ are treated +as different variables. Depending on how we are going to use the +blocks, we can consider different kind of unknowns. + +When we treat variables identically at all leads and lags, we compute +the incidence matrix in the static model. This is appropriate when we +want to compute dynamic simulations block by block. + +If we want to compute solution functions block by block, we consider +that variables in previous periods are predetermined and we consider +only variables appearing in an equation at the current or future periods. + + +### Normalization of the model + +We assign each variable to one and only one equation that "determines" +this variable. Several assignements are possible. For our purpose any +one is fine. If no normalization is available, that means that the +model is ill specified, the Jacobian is singular, and two or more +equations are linearly dependent. + +Normalization is obtained by computing the maximum cardinality +matching of the graph of the model + +``` +using BipartiteMatching +using Graphs + +n = model.endogenous_nbr + +U1 = BitMatrix(IncidenceStatic) #bipartite graph +matching1, matched1 = findmaxcardinalitybipartitematching(U1) #maximum cardinality matching of the graph +!all(matched1) && error("Model can't be normalized") + +U2 = BitMatrix(IncidenceDynamic[:,n+1:2*n] .| IncidenceDynamic[:,2n+1:3*n]) #bipartite graph +matching2, matched2 = findmaxcardinalitybipartitematching(U2) #maximum cardinality matching of the graph +!all(matched2) && error("Model can't be normalized") +``` + + + +### Recursive blocks + +Once we have the normalization of the model, we can assign each +equation to a different variable and obtain a simple graph describing +which variable enters directly in the determination of which variable. +Equations must be in the same order as variables. As the purpose of +this procedure is to partition equations, we keep equations in the +original order and reorder the variables (columns) of the incidence matrix. + +``` +#Reorder columns of incidence matrix in the static model +iorder1 = [p[2] for p in sort(collect(pairs(matching1)), by=x -> x[1])] + +#Reorder columns of incidence matrix in the dynamic model +iorder2 = [p[2] for p in sort(collect(pairs(matching2)), by=x -> x[1])] +``` + +The strongly connected components of this simple graph provide the +subsets of variables that need to be evaluated or solved for +simultaneously. These are the recursive blocks. + +``` +# Strongly connected components in the static model +g1 = SimpleDiGraph(U1[:, iorder1]) +sccStatic = strongly_connected_components(g1) + +# Strongly connected components in the dynamic model +g2 = SimpleDiGraph(U2[:, iorder2]) +sccDynamic = strongly_connected_components(g2) +``` + + +## Writing block functions + +The general idea is to take advantage of the already parsed version of +the functions for the model as a whole + +### Functions as expressions + +``` +e = :(function f(x); x=1; return x; end) #:(function f(x) +``` +Expression `e` is made of two fields: `head` and `args`: +``` +e.head #:function +e.args #2-element Vector{Any}: + +``` +- `e.head` indicates that expression `e` represents a function +- `e.args[1]` is the calling sequence +- `e.args[2]` contains the body of the function +``` +e.args[2].head #:block +e.args[2].args #5-element Vector{Any}: +``` +- `e.args[2].args[[1, 2,4]]` indicates where the lines are coming from +- `e.args[2].args[3]` is `:(x[1] = 1]` +- `e.args[2].args[5]` is `:(return x)` + +and + +``` +e.args[2].args[3].head #:(=) +e.args[2].args[3].args #2-element Vector{Any}: +e.args[2].args[3].args[1].head #ERROR: type Symbol has no field head +e.args[2].args[3].args[1].args #ERROR: type Symbol has no field args +``` +Observe that the element of `x` that is modified by the function is +indicated in `e.args[2].args[3].args[1].args[2]` + +### The SparseDynamicResid! function as an expression + +``` +julia> Dynare.DFunctions.SparseDynamicResid! +RuntimeGeneratedFunction(#=in Dynare.DFunctions=#, #=using Dynare.DFunctions=#, :((T, residual, y, x, params, steady_state)->begin + #= none:1 =# + #= none:2 =# + #= none:2 =# @assert length(T) >= 5 + #= none:3 =# + #= none:3 =# @assert length(residual) == 6 + #= none:4 =# + #= none:4 =# @assert length(y) == 18 + #= none:5 =# + #= none:5 =# @assert length(x) == 2 + #= none:6 =# + #= none:6 =# @assert length(params) == 7 + #= none:7 =# + #= none:7 =# @inbounds begin + #= none:8 =# + residual[1] = y[8] * params[5] * T[1] - (1 - params[3]) * y[7] + #= none:9 =# + residual[2] = y[9] - params[1] * T[2] * (params[3] * exp(y[18]) * y[13] + y[9] * (1 - params[4])) + #= none:10 =# + residual[3] = y[7] - T[5] + #= none:11 =# + residual[4] = y[9] - (exp(y[12]) * (y[7] - y[8]) + (1 - params[4]) * y[3]) + #= none:12 =# + residual[5] = y[10] - (params[2] * y[4] + params[7] * y[6] + x[1]) + #= none:13 =# + residual[6] = y[12] - (y[4] * params[7] + params[2] * y[6] + x[2]) + end + #= none:15 =# + return nothing + end)) +``` +Using the same logic as above, we get the residual of the first +equation of the model as +``` +julia> Dynare.DFunctions.SparseDynamicResid!.body.args[13].args[3].args[2] +:(residual[1] = y[8] * params[5] * T[1] - (1 - params[3]) * y[7]) +``` +and the number of the first equation is revealed by +``` +julia> Dynare.DFunctions.SparseDynamicResid!.body.args[13].args[3].args[2].args[1].args[2] +1 +``` +It is possible to exploit this technique to dispatch the computation +of the residuals belonging to each block in different functions + +``` +using RuntimeGeneratedFunctions + +sdr = Dynare.DFunctions.SparseDynamicResid! +equs = sdr.body.args[13].args[3].args + +sdr_block_1 = :((T, residual, y, x, params, steady_state) -> @inbounds begin; end) +@show sdr_block_1.args + +function make_block_function(equs, block_component) + # make function template + func = :((T, residual, y, x, params, steady_state) -> @inbounds begin; end) + # add equations belonging to this block + for (i, ie) in enumerate(block_component) + # in sdr each equations is the second of two elements + eq = copy(equs[2*ie]) + # change the index of the residual + eq.args[1].args[2] = i + push!(func.args[2].args[2].args[3].args, eq) + end + return @RuntimeGeneratedFunction(func) +end + +function make_blocks(func0, block_components) + block_functions = Function[] + equs = func0.body.args[13].args[3].args + for block in block_components + push!(block_functions, make_block_function(equs, block)) + end + return block_functions +end + +@show make_blocks(sdr, sccDynamic) + +#Get Sparse Dynamic Resid, G1 and G2 +sdr = Dynare.DFunctions.SparseDynamicResid! +sdg1 = Dynare.DFunctions.SparseDynamicG1! +sdg2 = Dynare.DFunctions.SparseDynamicG2! + +#Blocks by function (tuples) +function blocks_by_functions(funcs0, block_components) + model_functions = Dict()#Vector{Function}[] + for (name, func) in pairs(funcs0) + try + model_functions[name] = make_blocks(func, block_components) + catch + model_functions[name] = "Has no field body" + end + end + return model_functions +end + +funcs0 = (sdr=sdr, sdg1=sdg1, sdg2=sdg2) +model_functions = blocks_by_functions(funcs0, sccDynamic) + +#Blocks by function (Array) +function blocks_by_functions2(funcs0, block_components) + model_functions = Any[] + for func in funcs0 + try + push!(model_functions, make_blocks(func, block_components)) + catch + push!(model_functions, "Has no field body") + end + end + return model_functions +end + +funcs0 = [sdr, sdg1, sdg2] +model_functions = blocks_by_functions2(funcs0, sccDynamic) + +``` + diff --git a/docs/src/RecursiveBlocks.md b/docs/src/development/RecursiveBlocks.md similarity index 100% rename from docs/src/RecursiveBlocks.md rename to docs/src/development/RecursiveBlocks.md diff --git a/docs/src/development/algorighms_derivatives.aux b/docs/src/development/algorighms_derivatives.aux new file mode 100644 index 00000000..64146fde --- /dev/null +++ b/docs/src/development/algorighms_derivatives.aux @@ -0,0 +1,6 @@ +\relax +\@writefile{toc}{\contentsline {section}{\numberline {1}Derivatives of the steady state}{1}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2}Derivatives of the coefficents of the linear approximation of the model}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {3}Derivatives of the solution of the UQME}{2}{}\protected@file@percent } 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