In the general linear model with observations not necessarily uncorrelated or homoscedastic, Gauss-Markov regression coefficients are superior to ordinary unweighted least squares in the well known BLU sense if the model is correct. However, it is shown that there is a weaker, but always applicable, minimum overall mean squared error sense in which Gauss-Markov residuals and biased residuals are inferior to ordinary least squares residuals as estimators of possible lack-of-fit in the model. This optimality of ordinary least squares is further illustrated by three other types of results about residuals.