This paper investigates the potentials of the bootstrap as a tool for inference on the parameters of macroeconometric models which admit a state space representation. We consider a bootstrap estimator of the parameters of state space models and show that the bootstrap realizations of this estimator, usually employed to approximate asymptotic confidence intervals, p-values, and critical values of tests, can be also constructively used to build a test for forms of misspecifications which invalidate asymptotic normality. The test evaluates how close or distant the estimated state space model is from the case where asymptotic inference based on the Gaussian distribution applies. We derive sufficient conditions on the number of bootstrap repetitions, B, relative to the number of sample observations, T, for the test statistic to have a well-defined asymptotic distribution under the null. Throughout the paper, we focus on the state space form of small-scale monetary dynamic stochastic general equilibrium (DSGE) models and investigate the usefulness of our approach through Monte Carlo experiments and empirical illustrations based on US quarterly data. Results show that (i) bootstrapping the state space form provides highly reliable inference and (ii) the suggested test detects weakly identified parameters reasonably well in finite samples.
