Estimation in the restricted general linear model with a positive semidefinite covariance matrix

Abstract

It is well known that the linear model y=XY+∊, with Y restricted by a set of linear constraints, may be expressed as an unrestricted model with a singular covariance matrix. The problem of estimation in the general linear model y = XY + ∊ has received recent attention in the literature when the covariance matrix of ∊ is assumed to be of arbitrary rank. One approach to this problem is to perform a linear transformation to obtain a linear model of smaller dimensions with full rank covariance matrix and then estimate the parameters using the “smaller” model. More recently, Ahlers and Lewis (1972) showed that in the unrestricted general linear model, estimators may be calculated directly without the need of this linear transformation. In this paper, their result is extended to obtain a minimum variance estimate of γ with γ restricted by linear restrictions which, at most, are assumed to be consistent, and with the covariance matrix of ∊ assumed to be of arbitrary rank.