Bootstrapping Structural Change Tests

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 and 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, and 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 and 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.