This paper develops and applies tools to assess multivariate aspects of Bayesian Dynamic Stochastic General Equilibrium (DSGE) model forecasts and their ability to predict comovements among key macroeconomic variables. We construct posterior predictive checks to evaluate conditional and unconditional density forecasts, in addition to checks for root-mean-squared errors and event probabilities associated with these forecasts. The checks are implemented on a three-equation DSGE model as well as the Smets and Wouters (2007) model using real-time data. We find that the additional features incorporated into the Smets-Wouters model do not lead to a uniform improvement in the quality of density forecasts and prediction of comovements of output, inflation, and interest rates.