Two-Stage Estimators of Seemingly Unrelated Regressions with Elliptical Distribution

Abstract

This article considers the problem of estimating the system of m seemingly unrelated regressions (SUR) with an elliptical class of distributions, as a generalization of the SUR model with normal distribution. It is shown that the dominance of the two-stage Aitken estimator (based on the unrestricted and restricted estimates of the dispersion matrix) introduced by Zellner (Zellner, A. (1962). An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. J. Amer. Statist. Assoc. 57:348–368) over the least squares estimator in terms of the mean square error matrix (MSEM) criterion remains unchanged, so does the two-stage covariance-improved estimator introduced by Liu and Wang (Liu, J., Wang, S. G. (1999). Two-stage estimate of the parameters in seemingly unrelated regression model. Progress in Natural Science 9:489–496). The exact MSEM of the two-stage covariance-improved estimator is obtained under a multivariate t-distribution, an important subclass of the elliptical class.

Keywords:

• Seemingly unrelated regressions
• Two-stage Aitken estimator
• Covariance-improved estimator
• Elliptical distribution
• Multivariate t-distribution
• MSC (1991): Primary: 62F11
• 62H12
• Secondary: 62J02
• 62H10

Acknowledgments

This research was practically supported by the National Natural Science Foundation of China. The author is grateful to the referee for his helpful comments.