Multivariate student-t regression models:: Pitfalls and inference

We consider likelihood-based inference from multivariate regression models with independent Student-t errors with unknown degrees of freedom. Some pitfalls are revealed of both Bayesian and maximum likelihood methods. Under a commonly used non-informative prior, Bayesian inference is precluded for certain samples, even though a well-defined conditional distribution exists of the parameters given the observables. We also find that adding new observations can destroy the possibility of conducting posterior inference. Global maximisation of the likelihood function is a vacuous exercise since the latter is unbounded as we tend to the boundary of the parameter space. The unboundedness of the likelihood function also implies that the problems mentioned above for Bayesian analysis can even occur under proper priors. These pitfalls arise as a consequence of the fact that the recorded data have zero probability under the assumed sampling model. Therefore, a Bayesian analysis on the basis of set observations, which takes into account the precision with which the data were originally recorded, i.e. the 'rounding', is proposed and illustrated by several examples.