Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

Abstract

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.

Keywords:

• Bayesian inference
• Exchange rates
• Generalized error distribution
• Generalized t distribution
• Indirect inference
• Linear model
• Monte Carlo methods

Mathematics Subject Classification:

• 62F15
• 65C60

Acknowledgments

The author wishes to thank, but in no way implicate, two anonymous referees for comments on an earlier version.

Notes

Note: Posterior means are reported with posterior standard deviations in parentheses.

Note: Figures in parentheses represent approximate Bayes factors.

¹This model could be explored in further research. The presence of three shape parameters, however, would naturally require large sample sizes.

²The results for regression parameters and the scale parameter were insensitive as well. These results are not reported to save space.