Greene (econometric analysis 7th ed. p 53) states that OLS is the "optimal linear predictor":
Then on the next page, he states that OLS is also the BLUE estimator (Gauss-Markov Theorem):
I understand the proof of the Gauss Markov theorem. However, I am not entirely clear why Greene separates "optimal linear predictor" and "linear unbiased estimator" as two separate concepts. So long as the optimal linear predictor is unbiased, then these two concepts are the same, are they not?