We take as a starting point the existence of a joint distribution implied by different dynamic stochastic general equilibrium (DSGE) models, all of which are potentially misspecified. Our objective is to compare "true" joint distributions with ones generated by given DSGEs. This is accomplished via the construction of a new tool for comparing the empirical joint distribution of historical time series with the empirical distribution of simulated time series. The tool draws on recent advances in the theory of the bootstrap, Kolmogorov type testing, and other work on the evaluation of DSGEs, aimed at comparing the second order properties of historical and simulated time series. We begin by fixing a given model as the "benchmark" model, against which all "alternative" models are to be compared. Our comparison is done using a distributional generalization of White's (2000) reality check. In particular, we test whether at least one of the alternative models provides a more "accurate" approximation to the true cumulative distribution than does the benchmark model. Accuracy is measured in terms of distributional square error. As the data are simulated using estimated parameters (as well as previously calibrated parameters), the limiting distribution of the test statistic is a Gaussian process with a covariance kernel that reflects the contribution of parameter estimation error. Thus, the limiting distribution is not nuisance parameter free, and critical values cannot be tabulated. In order to address this issue, we show the validity of two versions of the block bootstrap in our context. An illustrative example is also given, in which the testing approach is applied to a real business cycle model. It is shown that alternative versions of the model in which calibrated parameters are allowed to vary slightly perform equally well. On the other hand, there are stark differences between models when the shocks driving the models are assigned non-plausible variances and/or distributional assumptions.
Real Business Cycles Output empirical Distribution Simulated Models Model Selection