Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula

TL;DR

This paper proposes an objective Bayesian method to estimate the discrete degrees of freedom for multivariate t distributions and t-copulas, addressing a longstanding inference challenge with practical testing on simulated and real financial data.

Contribution

It introduces a novel objective prior for the degrees of freedom parameter, overcoming previous difficulties in inference for multivariate t and t-copula models.

Findings

Prior performs well in simulated scenarios

Method effectively estimates degrees of freedom in real financial data

Provides open-source R code for replication

Abstract

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.