Optimal Change-point Testing for High-dimensional Linear Models with Temporal Dependence

In this paper, we study change-point testing for high-dimensional linear models, an important problem that has not been well explored in the literature. Specifically, we propose a quadratic-form cumulative sum (CUSUM) statistic to test the stability of regression coefficients in high-dimensional linear models. The test controls type-I error at any desired level and is robust to temporally dependent observations. We establish its asymptotic distribution under the null hypothesis, and demonstrate that it is asymptotically powerful against multiple change-point alternatives and achieves the optimal detection boundary for a wide class of high-dimensional models. We further develop an adaptive procedure to estimate the tuning parameters of the test, making our method practical in applications. Additionally, we extend our approach to localize change-points in the regression time series and establish sharp error bounds for our change-point estimator. Extensive numerical experiments and a real data application in macroeconomics are conducted to demonstrate the promising performance and practical utility of the proposed test.