Unbiasedness OLS Estimator

$$\hat{\beta}_{OLS}=(X'X)^{-1}X'y$$ By A1-Full Rank $$E[\hat{\beta}_{OLS}|X]=E[(X'X)^{-1}X'y|X]$$ By A2-Linearity $$=E[(X'X)^{1}X'(X\beta+\epsilon)|X] =E[\beta+(X'X)^{1}X'\epsilon |X] \,$$ $$=\beta + E[(X'X)^{1}X'\epsilon|X] \,$$

Case 1: >A3Rsru
$$E[\hat{\beta}_{OLS}|X]=\beta+0=\beta$$ Under A3Rmi or stronger, the OLS estimator is unbiased.

Case 2: <=A3Rsru
$$E[\hat{\beta}_{OLS}|X]=\beta+E[(X'X)^{1}X'\epsilon|X]$$

Under A3Rsru or weaker, the OLS estimator is biased.