Abstract

Many applications record several correlated time series that respond to a common event, but not necessarily in the same way. We consider testing for a single common change point in the mean of a fixed-dimensional multivariate time series, allowing the size and direction of the shift to differ across coordinates. The proposed tests are based on a vector-valued two-sample U-statistic process and a weighted maximum of its quadratic form, where the weighting increases sensitivity to changes near the beginning or end of the observation period. Under weak dependence, we derive the asymptotic null distribution for bounded antisymmetric kernels, such as the Wilcoxon and Huber kernels, and separately for the classical CUSUM kernel. In each case, the limit is a weighted supremum of the squared norm of a standard multivariate Brownian bridge. Since the required long-run covariance can be unstable to estimate in finite samples, we also propose tests based on the standardized residuals of a fitted vector autoregression. For the CUSUM kernel, and for the Huber kernel under Gaussian innovations, the residual-based statistics are asymptotically equivalent to their infeasible counterparts, so critical values depend only on the dimension and the weight parameter.