Summary: In the general linear model with covariance structure, depending on an unknown parameter vector, it is shown that the greatest lower bound for the risk matrix of the generalized least squares estimator (GLSE) constructed with covariance structure estimated from the iterated residuals is that of the Gauss-Markov estimator.
A sufficient condition for the existence and the unbiasedness of the GLSE based on iterated residuals is given. It is shown that the use of the iterated residuals does not improve the risk matrix of GLSE through terms of order \(n^{-2}\) relative to that of the two step estimator.