Nonmonotonic power for tests of a mean shift in a time series

The null hypothesis-based statistics CUSUM and QS are widely used for testing parameter stability. We provide examples, extensive simulation studies and theoretical results showing that these statistics fail to detect obvious shifts in the mean of a time series. Moreover, the detection probability can decrease when the magnitude of the shift in mean increases. Estimation of nuisance parameters under the null is identified as an important cause of this counterintuitive behavior of the power function. Results indicate that tests designed for the specific alternative of a shift in mean of a time series should be preferred.