Bayesian model selection and prediction with empirical applications comments

Abstract

In the preceding article Peter Phillips has written a very stimulating paper in which he uses the posterior information criterium (PIC) put forward in joint work with Werner Ploberger to select forecast models. PIC is applied to the fourteen Nelson-Plosser and Schotman-van Dijk macro-economic time series for the U.S. economy. The selected models are found to be parsimonious and to perform well in forecasting over the period 197&1988. They generally outperform a fixed-format AR(3) model with linear trend. The question of whether a fixed-format model or an evolving-format model is more appropriate for forecasting the series at hand is of great practical importance. The issue whether the use of PIC leads to the selection of models which in forecasting outperform models selected using other criteria, such as full-fledged posterior odds, BIC, or AIC, is of even greater practical significance. It is not addressed in the paper but it should be considered before PIC can be advocated as a panacea when there is model uncertainty. For instance, Zellner (1978) has made the relationship between AIC and the posterior odds criterium (POC) explicit by showing that AIC is a truncated version of the POC. From a Bayesian point of view, the posterior odds ratio is the criterion to be used when the loss structure is proportional to the identity matrix. Now PIC can be interpreted as a posterior odds ratio under a specific prior distribution. Certainly, an attractive feature of PIC, in particular for automated decisionmaking, is that it can be easily computed. The prior probability distribution function (pdf) on the regression coefficients fl is uniform and centred at zero. The regression coefficients are assumed to be a priori independent. Furthermore,