Predictive Likelihood Comparisons with DSGE and DSGE-VAR Models

In this paper we treat the issue of forecasting with DSGE and DSGE-VAR models, with particular attention to Bayesian estimation of the predictive distribution and its mean and covariance. As a novel contribution to the forecasting literature, which extends beyond (log-linearized) DSGE models and DSGE-VARs, we show how the value of the h-step-ahead marginal and joint predictive likelihood for a subset of variables can be calculated. This is of interest when attempting to rank models in density forecasting exercises, but it is generally useful since the predictive likelihood is a natural model selection device under a Bayesian approach. The basic idea is to utilize well-known techniques for handling missing data when computing the likelihood function, while the predictive likelihood can thereafter be calculated using either exact (Monte Carlo integration) or approximate methods. This is particularly straigtforward for linear Gaussian models since the Kalman filter supports missing data with minor modifications and can thus be used to obtain the value of the likelihood, while the modified harmonic mean estimator and the Laplace approximation may be applied for computing the predictive likelihood. As an empirical illustration, we use euro area data and compare the forecasting performance of the New Area-Wide Model (NAWM)—a small-open-economy DSGE model developed zat the European Central Bank—to DSGE-VARs, which relax the cross-equation restrictions of the former, as well as to reduced-form forecasting models. In earlier work, we have shown that overall the NAWM performs well in comparison to reduced-form models over the forecast sample beginning with the introduction of the euro, but that it has difficulties when it comes to predicting, in particular, wages and private consumption. We therefore examine whether DSGE-VARs can account for the NAWM’s systematic overprediction of both variables and if they can compete with a large BVAR.