In this paper the problem of computing maximum likelihood estimates of the parameters of linear statistical models is considered. The proposed approach relies on the prediction error decomposition of the likelihood function. A distinctive feature is that the required prediction errors are obtained using conventional linear least squares methods rather than the more usual Kalman filter. More specifically, it is shown that the orthogonalization procedure based on fast Givens transformations, used to obtain the triangular representation of the normal equations, automatically yields the one-step ahead prediction errors and their mean squared errors without additional side calculations.