When are the OLS estimators unbiased and consistent?

Summary

Setup

• The LRM with random sampling

\[y_i=β_1+β_2x_i+ε_i i=1,…,n\]

Unbiased and consistent

• An estimator \(b\) of a parameter \(β\) is called unbiased if

\[E\left( b \right)=β\]

• An estimator \(b\) of a parameter \(β\) is called consistent if \(b\) becomes equal to \(β\) as \(n→∞\)
• The OLS estimators \(b_1,b_2\) are unbiased estimators of \(β_1,β_2\) if \(x\) is exogenous
• The OLS estimators \(b_1,b_2\) are consistent estimators of \(β_1,β_2\) if \(x\) is exogenous and other mild conditions apply