Consistency of the OLS estimator

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: \(b\) is a consistent estimator of \(β\)

\[plim b=β\]