Prediction and the efficiency of least squares

The general linear model y = Xβ + u is considered when the ranks of X and Γ, the error covariance matrix, are arbitrary. The least squares and best linear unbiased estimators of the part of β in the row space of X are shown to be unique. Necessary and sufficient conditions are given for these estimators to be identical. The treatment deliberately avoids the use of generalized inverses and estimable linear functions. The best linear predictor of y0 = x0’β+ u0 from the general linear model y = Xβ + u is studied. Necessary and sufficient conditions for the equality of the best linear predictor of y0 and the least squares estimator of X’q P are given. The conditions may be approximately satisfied in practice so that the simpler least squares estimator could then be used.