Estimating the variance of the error terms

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.
• Definition of \(s^2\) :

\[s^2= \frac{1}{n-k}\sum_{i=1}^{n}{ e_i^2 }=e'e/(n-k) \]

• Result: \(s^2\) is an unbiased estimator of \(σ^2\) ,

\[E\left( s^2 \right)=σ^2\]