SUMMARY The second-order expansion of the mean squared error matrix of the generalized least squares estimators for the regression parameters is obtained when the samples and the estimators for the structural parameters contained in the variance-covariance matrix are independent. In this paper we treat the second-order expansion of the mean squared error matrix of the generalized least squares estimator fJ in the statistical linear model, y = X: + u. We consider the case where the variance-covariance matrix V(00) of u is specified by finite parameters 00 and where a consistent estimator 0 of 00 is available which is independent of the observation y. Williams (1975) gave a sufficient condition concerning the order of the weak consistency of #fJ and commented on the asymptotic efficiency of #fJ. But he did not obtain a result for the latter. Fuller & Rao (1978) dealt with the case where the observations fell into k groups with constant error variance for each group. They obtained the asymptotic distribution of an estimator fJ which used a weight matrix when the number of groups was infinite, assuming the numbers of elements {niJ in the groups formed a fixed sequence. Our model corresponds to fixing k and letting ni increase.