Predictions Under a System of Linear Regression Models with Correlated Errors

Abstract

A system of linear regression models that consists of several regression equations is an extension of the linear regression models which allow correlated errors across the equations. In this study, we consider some fundamental problems on best linear unbiased predictors (BLUPs) of all unknown vectors under a system of linear regression models containing the set of m regression equations. We present analytical expressions of the predictors and establish some rank formulas to characterize covariance matrices of BLUPs under the system and its single models. We give the necessary and sufficient conditions for equalities of predictors to hold under considered models by using various rank formulas. As an application, some results are also presented for seemingly unrelated regression (SUR) models.