Consistent nonparametric test for structural change in time series models via Cusum and Fourier transform

In this study, we address the problem of testing for structural changes in time series models that allow for time-varying distributions of regressors. We propose a novel stochastic process within a nonparametric framework, which combines the CUSUM method with Fourier-transformed data, and based on this process, develop a Cramér–von Mises (CvM) type statistic. We show that the constructed process weakly converges to a centered complex-valued Gaussian process and derive the asymptotic distribution of the test under the null hypothesis, while also examining its properties under alternative hypotheses. The test is consistent against a wide range of global alternatives and can detect Pitman sequences of local alternatives at the parametric rate T−1/2, outperforming most existing nonparametric tests. Due to the nonstandard asymptotic distribution of the test statistic, we propose a bootstrap method to obtain critical values and establish its validity. Monte Carlo simulations and an empirical application to major Chinese stock indices demonstrate that the proposed test achieves accurate size and strong power in finite samples, while the empirical results reveal recurrent structural breaks with strong co-movement across indices, highlighting the role of common market-wide factors in driving regime shifts.