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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} \,  $
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