Predictive likelihood for Bayesian model selection and averaging

This paper investigates the performance of the predictive distributions of Bayesian models. To overcome the difficulty of evaluating the predictive likelihood, we introduce the concept of expected log-predictive likelihoods for Bayesian models, and propose an estimator of the expected log-predictive likelihood. The estimator is derived by correcting the asymptotic bias of the log-likelihood of the predictive distribution as an estimate of its expected value. We investigate the relationship between the proposed criterion and the traditional information criteria and show that the proposed criterion is a natural extension of the traditional ones. A new model selection criterion and a new model averaging method are then developed, with the weights for the individual models being dependent on their expected log-predictive likelihoods. We examine the performance of the proposed method using Monte Carlo experiments and a real example, which concerns the prediction of quarterly growth rates of real gross domestic product in the G7 countries. Out-of-sample forecasts show that the proposed methodology outperforms other methods available in the literature.