Testing for jumps in the presence of smooth changes in trends of nonstationary time series

Nonparametric smoothing methods have been widely used in trend analysis. However, the inference procedure usually requires the crucial assumption that the underlying trend function is smooth. This paper considers the situation where the trend function has potential jumps in addition to smooth changes. In order to determine the existence of jumps, we propose a nonparametric test that can survive under dependent and nonstationary errors, where existing tests assuming independence or stationarity can fail. When the existence of jumps is affirmative, we further consider the problem of estimating the number, location and size of jumps. The results are illustrated via both Monte Carlo simulations and a real data example.