On the difference in inference and prediction between the joint and independent t-error models for seemingly unrelated regressions

Abstract

We consider likelihood and Bayesian inferences for seemingly unrelated (linear) regressions for the joint multivariate t-error (e.g. Zellner, 1976) and the independent t-error (e.g. Maronna, 1976) models. For likelihood inference

the scale matrix and the shape parameter for the joint t-error model cannot be consistently estimated because of the lack of adequate information to identify the latter. The joint t-error model also yields the same MLEs for the regression coefficients and the scale matrix as for the independent normal error model, which are not robust against outliers. Further, linear hypotheses with respect to the regression coefficients Also give rise to the same null distributions as for the independent. normal error model, though the MLE has a non-normal limiting distribution. In contrast to the striking similarities between the joint terror and the independent normal error models, the independent t-error model yields MLEs that are robust against outliers. Since the MLE of the shape parameter reflects the tails of the data distributions, this model extends the independent normal error model for modeling data distributions with relatively thicker tails. These differences are also discussed with respect to the posterior and predictive distributions for Bayesian inference.