The Gauss Markov theorem

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• the Gauss-Markov assumptions, \(E\left( ε \right|X)=0\) and \(Var\left( ε \right|X)=σ^2I\)
• The OLS estimator is the Best Linear Unbiased Estimator (BLUE): the OLS estimator has the smallest variance among all linear unbiased estimators.
• Formally, if   \(\tilde{b}=Ay\) is a linear estimator of \(β\) which is unbiased, \(E\left( \tilde{b} \right)=β\) then

\[Var\left( \tilde{b}|X \right)-Var\left( b \right|X)\]

• is a positive semidefinite matrix .