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?
It can be shown (prove it) that $ MSE(\hat{\theta}) = Var(\hat{\theta}) + Bias^2(\hat{\theta})$. If we restrict ourselves to unbiased estimators, the second term on the right disappears and we search for the estimator with minimum variance. This is what 4.2.3 seeks.
However, if we do not insist that the estimator be unbiased, we can at times find an estimator whose overall $MSE$ is smaller than that of any unbiased estimator. In the sense that it minimizes $MSE$ it is optimal.
Greene states his explanation of the "optimal linear predictor" in the quoted paragraph. He says that, "[f]or this criterion [of optimality], we will use the mean squared error rule, so we seek the minimum squared error linear predictor of $y$." Since the mean-squared error is equal to the variance of the estimator plus the square of its bias, you are correct that an unbiased predictor that is "optimal" in this sense, will also be a best linear unbiased estimator (BLUE).
Presumably, Greene separates these concepts because there may be broader estimation problems (beyond the context of the linear model) where the estimator is not unbiased, and so the "optimal" estimator that minimises the mean-squared error is not BLUE. Additionally, even within the context of linear models, it is useful to separate these two concepts for pedagogical reasons ---i.e., because he has not yet shown that they are the same.
