Maximum likelihood estimation of the GLS model with unknown parameters in the disturbance covariance matrix

Abstract

This paper considers the regression model y = Xβ+ε with all the classical assumptions (including normality) but one, viz. it is assumed that the covariance matrix of the disturbances depends upon a finite number of unknown parameters θ1 … θm. The paper gives a method to derive simultaneously the maximum likelihood estimates of β and θ. Also the information matrix is presented. It is proved that β̂ is unbiased if its mean exists. Conditions are given under which the maximum likelihood estimates are consistent, asymptotically normal, and asymptotically efficient. Finally, applications are given to the autocorrelated errors model and to Zellner-type regressions.