A Comparison of Two Criteria for Ordinary-Least-Squares Estimators to Be Best Linear Unbiased Estimators

The problem of the equality between ordinary-least-squares estimators and best linear unbiased estimators is discussed in the literature in two versions: in the context of a fixed model (design) matrix and in the context of all model (design) matrices having a fixed common linear part.Unfortunately, one of the criteria derived in the latter case (McElroy 1967) is incorrectly quoted by some authors as a solution to the former version of the problem.This article clarifies the difference between the two versions and shows that there is no practical situation where the corresponding solutions can coincide.