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