Unbiasedness of the OLS estimator

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• exogeneity, \(E\left( ε \right|X)=0 \) (or, equivalently, a correctly specified model, \(E\left( y \right|X)=Xβ\) )
• Result: the OLS estimator \(b={\left( X'X \right)}^{-1}X'y\) is unbiased

\[E\left( b \right)=β\]