Estimating the variance of the OLS estimator:

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample of size \(n\)
• the Gauss-Markov assumptions, \(E\left( ε \right|X)=0\) and \(Var\left( ε \right|X)=σ^2I\)
• \(b={\left( X'X \right)}^{-1}X'y\) is the OLS estimator of \(β\) (both are \(k×1\) )
• \(e=y-Xb\) are the OLS residuals.
• \(s^2=e'e/(n-k)\)
• \(Var\left( b \right|X)=σ^2{\left( X'X \right)}^{-1}\)
• We define the estimated (conditional) variance of \(b\) as

\[\widehat{Var}\left( b \right|X)=s^2{\left( X'X \right)}^{-1}\]

• Result: \(\widehat{Var}\left( b \right|X)\) is an unbiased estimator of \(Var\left( b \right|X)\) .