ols regression - classical derivation

Derivation

{% Var(\vec{b}|X) = \sigma^2 (X^T X)^{-1} %}

{% Var(\vec{b}|X) = Var(\vec{b} - \vec{\beta}|X) %}

{% = Var[(X^TX)^{-1} \epsilon|X] %}

{% =(X^TX)^{-1} \, Var(\epsilon|X) \, [(X^TX)^{-1}]^T %}

{% =(X^TX)^{-1} \, \mathbb{E}(\epsilon \epsilon^T|X) \, [(X^TX)^{-1}]^T %}

{% =(X^TX)^{-1} \, (\sigma^2 I |X) \, [(X^TX)^{-1}]^T %}

{% = \sigma^2 (X^TX)^{-1} [(X^TX)^{-1}]^T %}

{% = \sigma^2 (X^TX)^{-1} %}