Bayesian treatment of the independent student-t linear model

Abstract

SUMMARY This article takes up methods for Bayesian inference in a linear model in which the disturbances are independent and have identical Student-t distributions. It exploits the equivalence of the Student-t distribution and an appropriate scale mixture of normals, and uses a Gibbs sampler to perform the computations. The new method is applied to some well-known macroeconomic time series. It is found that posterior odds ratios favour the independednt Student- linear model over the normal linear model inodl, and that the posterior odds ratio in favour of difference stationarity over trend stationarity is often substantially less in the favoured Student-t models. The possibility of leptokurtic disturbances is a common concern of econometricians and other users of the linear model. This article takes up methods for Bayesian inference in a linear model in which the disturbances are independent and have identical Student-t distributions. It exploits the equivalence of the Student-t distribution and an appropriate scale mixture of normals, and uses a Gibbs sampler to perform the computations. The main contribution is to provide a simple and stable computational method for full Bayesian inference in the independent Student-t linear model. The new method is applied to some well-known macroeconomic time series. It is found that posterior odds ratios favour the independent Student-t linear model over the normal linear model, and that the posterior odds ratio in favour of difference stationarity over trend stationarity is often substantially less in the favoured Student-t models. The work reported here builds on Bayesian treatments of heteroscedasticity, which began with hierarchical models for the analysis of variance (Lindley, 1965, 1971). With the adaptation of a prior distribution making cell means linear in cofactors (Lindley and Smith, 1972), this treatment was effectively extended to the linear regression model. Lindley (1971) took up the conjugate prior in which the inverses of the variances are x2(v), up to a factor of proportionality with an improper prior. It is shown in Section 2.1 that this is equivalent to the specification of an independent Student-t linear model with known degrees of freedom. In an important variant on this model, Leonard (1975) used a prior in which the log variances are linear functions of cofactors, and constructed an approximation to the posterior density. This article extends these developments in two ways. First, it shows how to construct the exact posterior distribution, whereas the earlier work was confined to obtaining the posterior mode of an approximate posterior. (Interestingly, however, the algorithm in the appendix of