Variance of the OLS estimator

Establish from the normal equations that $\hat{\beta}_{OLS}=(X'X)^{-1}X'y$. A1-Full Rank needs to hold for this result.

$Var[\hat{\beta}_{OLS}|X]=E[((X'X)^{-1}X'y-E[(X'X)^{1}X'y|X])'((X'X)^{1}X'y-E[(X'X)^{-1}X'y|X])]$.

Using A2-Linearity and >=A3Rsmi (resulting in unbiasedness of $\hat{\beta}_{OLS}\,$), this simplifies to

$=E[(X'X)^{-1}X'\epsilon \epsilon' X(X'X)^{-1}|X] \,$

Case 1: Using A4GM

$=(X'X)^{-1}X'E[\epsilon \epsilon'|X] X(X'X)^{-1} = (X'X)^{-1}X'\sigma^{2}X(X'X)^{-1}=\sigma^{2} (X'X)^{-1}\,$

Case 2: Using A4Ω

$=(X'X)^{-1}X'E[\epsilon \epsilon'|X] X(X'X)^{-1} = (X'X)^{-1}X'\sigma^{2} \Omega X(X'X)^{-1} = \sigma^{2} (X'X)^{-1}X'\Omega X(X'X)^{-1} \,$