Bootstrapping tests for breaks in mean or variance based on U-statistics

Bootstrapping tests (Formula presented.) are implemented for the tests U by Schmidt [Detecting changes in the trend function of heteroscedastic time series; 2021. Preprint: arXiv:2108.09206 [math.ST]] for mean break and Schmidt et al. [An asymptotic test for constancy of the variance under short-range dependence. Ann Stat. 2021;49:3460–3481.] for variance break based on U-statistics. The tests U have good powers against epidemic breaks that are common in practice. The test U for variance break is proved to have the nice property of consistency against a general class of alternatives. However, the tests U have non-ignorable finite sample size distortion under serial correlation and/or conditional heteroscedasticity. Our implementation (Formula presented.) based on autoregressive residual bootstrapping and moving block bootstrapping are shown to remedy the size distortion problems of U for mean break and for variance break, respectively, in a Monte-Carlo experiment. The experiment also demonstrates the power advantages of bootstrapping tests over the original tests and other standard break tests against epidemic breaks, which, however, are accompanied by disadvantages against simple single breaks. The proposed bootstrapping tests are well illustrated by break analyses of means and variances of the log-return and realized the volatility of the US S&P 500.