  An objective Bayesian approach to estimate the number of degrees of freedom
$(\nu)$ for the multivariate $t$ distribution and for the $t$-copula, when the
parameter is considered discrete, is proposed. Inference on this parameter has
been problematic for the multivariate $t$ and, for the absence of any method,
for the $t$-copula. An objective criterion based on loss functions which allows
to overcome the issue of defining objective probabilities directly is employed.
The support of the prior for $\nu$ is truncated, which derives from the
property of both the multivariate $t$ and the $t$-copula of convergence to
normality for a sufficiently large number of degrees of freedom. The
performance of the priors is tested on simulated scenarios. The R codes and the
replication material are available as a supplementary material of the
electronic version of the paper and on real data: daily logarithmic returns of
IBM and of the Center for Research in Security Prices Database.
