THE PREDICTIVE DISTRIBUTION FOR THE HETEROSCEDASTIC MULTIVARIATE LINEAR MODELS WITH ELLIPTICALLY CONTOURED ERROR DISTRIBUTIONS

Abstract

This paper considers the heteroscedastic multivariate linear model with errors
following elliptically contoured distributions. The marginal likelihood function of
the unknown covariance parameters and the predictive distribution of future responses
have been derived. The predictive distribution obtained is a product of m
multivariate Student’s t distributions. It is interesting to note that when the models
are assumed to have elliptically contoured distributions the marginal likelihood
function of the parameters as well as the predictive distribution are identical to
those obtained under independently distributed normal errors or dependent but
uncorrelated Student’s t errors. Therefore, the distribution of future responses
is unaffected by a change in the error distribution from the multivariate normal
and multivariate t distributions to elliptically contoured distributions. This gives
inference robustness with respect to departure from the reference case of independent
sampling from the multivariate normal or dependent but uncorrelated
sampling from multivariate t distributions to elliptically contoured distributions.