linear-CEF.Rmd

# Linear Conditional Expectation Function

## Variance of Error
We first compute the (unconditional) variance of the error vector $\pmb{e}$. The covariance matrix 
$$
\mathbb{V}[\pmb{e}]=\E{\pmb{e}\pmb{e}'}-\E{\pmb{e}}\E{\pmb{e}'}=\E{\pmb{e}\pmb{e}'}\stackrel{\text{def}}{=}\mathbb{D}.
$$
For $i\neq j$, the errors $e_i$,$e_j$ are independent. As a result, $\E{e_ie_j}=\E{e_i}\E{e_j}=0$. So, $\mathbb{D}$ is a diagonal matrix with the $i$-th diagonal element $\sigma_i^2$:
$$
\mathbb{D}=\begin{bmatrix}
\sigma_1^2 & 0 & \ldots & 0 \\
0 & \sigma_2^2 & \ldots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & \sigma_n^2
\end{bmatrix}.
$$

## Variance of OLS Estimators