Bootstrapping Structural Change Tests

This paper analyses the use of bootstrap methods to test for parameter change in linear models estimated via Two Stage Least Squares (2SLS). Two types of test are considered: one where the null hypothesis is of no change and the alternative hypothesis involves discrete change at k unknown break-points in the sample; and a second test where the null hypothesis is that there is discrete parameter change at l break-points in the sample against an alternative in which the parameters change at l + 1 break-points. In both cases, we consider inferences based on a sup-Wald-type statistic using either the wild recursive bootstrap or the wild fixed bootstrap. We establish the asymptotic validity of these bootstrap tests under a set of general conditions that allow the errors to exhibit conditional and/or unconditional heteroskedasticity, and report results from a simulation study that indicate the tests yield reliable inferences in the sample sizes often encountered in macroeconomics. The analysis covers the cases where the first-stage estimation of 2SLS involves a model whose parameters are either constant or themselves subject to discrete parameter change. If the errors exhibit unconditional heteroscedasticity and/or the reduced form is unstable then the bootstrap methods are particularly attractive because the limiting distributions of the test statistics are not pivotal.

Bootstrap methods have been applied extensively in testing for structural breaks in the past few decades, but the conditions under which they are valid are, for the most part, unknown. In this paper, we fill this gap for the empirically important scenario in which supremum-type tests are used to test for discrete parameter change in linear models estimated by least squares methods. Our analysis covers models with exogenous regressors estimated by Ordinary Least Squares (OLS), and models with endogenous regressors estimated by Two Stage Least Squares (2SLS). Specifically, we show the asymptotic validity of the (IID and wild) recursive and fixed-regressors bootstraps for inference based on sup-F and sup-Wald statistics for testing both the null hypothesis of no parameter change versus an alternative of parameter change at k>0 unknown break points, and also the null hypothesis of parameter change at L break points versus an alternative of parameter change at L+1 break points. For the case of exogenous regressors, Bai & Perron (1998) derive and tabulate the limiting distributions of the test statistics based on OLS under the appropriate null hypothesis; for the case of endogenous regressors, Hall, Han & Boldea (2012) show that the same limiting distributions hold for the analogous test statistics based on 2SLS when the first stage model is stable. As part of our analysis, we derive the limiting distribution of the test statistics based on 2SLS when the regressors are endogenous and the first stage regression exhibits discrete parameter change. We show that the asymptotic distributions of the second-stage break-point tests are non-pivotal, and as a consequence the usual Bai & Perron (1998) critical values cannot be used. Thus, our bootstrap-based methods represent the most practically feasible approach to testing for multiple discrete parameter changes in the empirically relevant scenario of endogenous regressors and an unstable first stage regression. Our simulation results show very good finite sample properties with all the versions of the bootstrap considered here, and indicate that the bootstrap tests are preferred over the asymptotic tests, especially in the presence of conditional heteroskedasticity of unknown form.