The predictive likelihood is useful for ranking models in forecast comparison exercises using Bayesian inference. We discuss how it can be estimated, by means of marzginalization, for any subset of the observables in linear Gaussian state-space models. We compare macroeconomic density forecasts for the euro area of a DSGE model to those of a DSGE-VAR, a BVAR and a multivariate random walk over 1999:Q1-2011:Q4. While the BVAR generally provides superior forecasts, its performance deteriorates substantially with the onset of the Great Recession. This is particularly notable for longer-horizon real GDP forecasts, where the DSGE and DSGE-VAR models perform better.
