Bayesian skew selection for multivariate models

We develop a Bayesian approach for the selection of skew in multivariate skew tt distributions constructed through hidden conditioning in the manners suggested by either Azzalini and Capitanio (2003) or Sahu et al. (2003). We show that the skew coefficients for each margin are the same for the standardized versions of both distributions. We introduce binary indicators to denote whether there is symmetry, or skew, in each dimension. We adopt a proper beta prior on each non-zero skew coefficient, and derive the corresponding prior on the skew parameters. In both distributions we show that as the degrees of freedom increases, the prior smoothly bounds the non-zero skew parameters away from zero and identifies the posterior. We estimate the model using Markov chain Monte Carlo (MCMC) methods by exploiting the conditionally Gaussian representation of the skew tt distributions. This allows for the search through the posterior space of all possible combinations of skew and symmetry in each dimension. We show that the proposed method works well in a simulation setting, and employ it in two multivariate econometric examples. The first involves the modeling of foreign exchange rates and the second is a vector autoregression for intra-day electricity spot prices. The approach selects skew along the original coordinates of the data, which proves insightful in both examples.