7.1 Variance of Error

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