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