Asymptotic validity of bootstrap methods for a structural break in trend
**Abstract:** The aim of this paper is to provide theoretical results for bootstrap methods related to estimating and forming confidence intervals for the structural break date. I consider a linear trend model in which the slope exhibits a structural change at an unknown time. As a standard method, we can construct confidence intervals by obtaining the quantiles of interest from the limiting distribution along with consistent estimates. Recently, some researchers criticize the standard method because the exact coverage rate of the confidence interval for the break date is far below the nominal coverage rate in small samples. Bootstrap methods are introduced to complement the standard method. The noise component in the model is allowed to be either a sequence of $i.i.d.$ random variables or a stationary process. Depending on the assumption on the errors, I consider two bootstrap schemes, residual and sieve bootstrap. Two important results are derived theoretically. First, I prove consistency of the bootstrap estimators and establish the asymptotic validity of the bootstrap distribution. Second, I show that those two bootstrap methods have higher order refinement. Simulation experiments are provided to confirm theoretical derivations. In particular, the bootstrap-based confidence intervals show better exact coverage rates than the confidence intervals constructed by the standard method. As an empirical application, I analyze the nominal exchange rates with respect to the US dollars for eight countries and construct the bootstrap percentile confidence intervals for the break date.