Some equalities for the covariance matrices of predictors in seemingly unrelated regression models

Seemingly unrelated regression (SUR) models are extensions of linear regression models by considering multiple regression equations with correlated errors among equations. In this study, two SUR models are considered with their general linear models which is obtained from combining the models by using block matrices. The prediction problem is examined under considered models. Several results are given on statistical properties of the best linear unbiased predictors (BLUPs) of all unknown vectors under two SUR models and under their general model. Especially, some results established on covariance matrices of BLUPs under two models by using some rank formulas of matrices. The study consists of five sections. The first part of the study is the introduction. Some information about the subject and its importance are given in this section. Some of the existing work related to SUR models considered before in the literature are also discussed. Some theorems, concepts and properties used throughout the study are given in the second section. In the third section of the study, definition, mathematical expression and properties of SUR models are given. The predictability of a general linear function under considered models is examined and in addition, estimators and predictors are described. The fourth section contains the main results. In this section, some equalities are obtained related to BLUPs of joint unknown vectors and their covariance matrices under considered models by using some basic properties related to ranks of matrices. The results corresponding to the special cases are also given. The last section is the conclusion and discussions section. Keywords: BLUE, BLUP, covariance matrix, rank, SUR model