Bayes and Empirical Bayes Estimators with Their Unique Simpler Forms and Their Superiorities over BLUE in Two Seemingly Unrelated Regressions

This paper considers Bayes and empirical Bayes estimation of the regression parameters in a system of two seemingly unrelated regression (SUR) models with Gaussian disturbances. Employing the covariance-adjusted technique, we obtain a sequence of Bayes estimators and show that this is the best Bayes estimator (BE) in the sense of having least covariance matrix. We establish the superiority of this best BE over the best linear unbiased estimator (BLUE) in terms of the mean square error matrix (MSEM) criterion. When the covariance matrix of disturbances is unknown, we obtain a sequence of corresponding empirical Bayes (EB) estimators and show that this too is superior to BLUE in MSEM criterion. In addition, we establish an interesting fact which shows that both Bayes and empirical Bayes estimators have only unique simpler forms, which too are superior to BLUE, and further allow us to study them in more details. This paper shows that the proposed Bayes and EB estimators, which combine the Bayesian method with the covariance-adjusted technique, are very efficient even for small sample size. Finally, we generalize our results to the system of two SURs with unequal numbers of observations and to the system of more than two SURs.