On Bayesian Modeling of Fat Tails and Skewness

Abstract

We consider a Bayesian analysis of linear regression models that can account for skewed error distributions with fat tails. The latter two features are often observed characteristics of empirical datasets, and we formally incorporate them in the inferential process. A general procedure for introducing skewness into symmetric distributions is first proposed. Even though this allows for a great deal of flexibility in distributional shape, tail behavior is not affected. Applying this skewness procedure to a Student t distribution, we generate a “skewed Student” distribution, which displays both flexible tails and possible skewness, each entirely controlled by a separate scalar parameter. The linear regression model with a skewed Student error term is the main focus of the article. We first characterize existence of the posterior distribution and its moments, using standard improper priors and allowing for inference on skewness and tail parameters. For posterior inference with this model, we suggest a numerical procedure using Gibbs sampling. The latter proves very easy to implement and renders the analysis of quite challenging problems a practical possibility. Some examples illustrate the use of this model in empirical data analysis.

Key Words:

• Gibbs sampling
• Improper prior
• Linear regression model
• Posterior moments
• Stable distribution
• Student t sampling