Limit of √n(b-β)

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• the Gauss-Markov assumptions, \(E\left( ε \right|X)=0\) and \(Var\left( ε \right|X)=σ^2I\)
• \(b={\left( X'X \right)}^{-1}X'y\) is the OLS estimator of \(β\)
• \(Σ_{xx'}=E\left( x_ix'_i \right)\) is invertible
• Result:

\[E\left( \sqrt{n}\left( b-β \right) \right)=0\]

• Result:

\[Var\left( \sqrt{n}\left( b-β \right) \right)=σ^2Σ_{xx'}^{-1}\]

• Result

\[\sqrt{n}\left( b-β \right)→N\left( 0,σ^2Σ_{xx'}^{-1} \right)\]

• \(σ^2Σ_{xx'}^{-1}\) is called the asymptotic variance matrix or the asymptotic covariance matrix of \(b\) .
• Approximately , for \(n\) large

\[b \sim N\left( β,σ^2{\left( X'X \right)}^{-1} \right)\]

• Approximately , for \(n\) large

\[b \sim N\left( β,s^2{\left( X'X \right)}^{-1} \right)\]

• All inference based on the normal distribution of \(b\) is therefore approximately correct.