Loss based prior for the degrees of freedom of the Wishart distribution

In this paper we propose a novel method to deal with Vector Autoregressive models, when the Normal-Wishart prior is considered. In particular, we depart from the current approach of setting $\nu=m+1$ by setting a loss-based prior on $\nu$. Doing so, we have been able to exploit any information about $\nu$ in the data and achieve better predictive performances than the method currently used in the literature. We show how this works both on simulated and real data sets where, in the latter case, we used data of macroeconometric fashion as well as viral data. In addition, we show the reason why we believe we achieve a better performance by showing that the data appears to suggest a value of $\nu$ far from the canonical $m+1$ value.