Change-point inference in high-dimensional regression models under temporal dependence

Abstract

This paper concerns the limiting distributions of change-point estimators, in a high-dimensional linear regression time-series context, where a regression object $({\mathit{y}}_{\mathit{t}},{\mathit{X}}_{\mathit{t}})\in \mathbb{R}\times {\mathbb{R}}^{\mathit{p}}$ is observed at every time point $\mathit{t}\in \{1,\dots ,\mathit{n}\}$. At unknown time points, called change points, the regression coefficients change, with the jump sizes measured in ${\ell }_2$-norm. We provide limiting distributions of the change-point estimators in the regimes where the minimal jump size vanishes and where it remains a constant. We allow for both the covariate and noise sequences to be temporally dependent, in the functional dependence framework, which is the first time seen in the change-point inference literature. We show that a block-type long-run variance estimator is consistent under the functional dependence, which facilitates the practical implementation of our derived limiting distributions. We also present a few important byproducts of our analysis, which are of their own interest. These include a novel variant of the dynamic programming algorithm to boost the computational efficiency, consistent change-point localization rates under temporal dependence and a new Bernstein inequality for data possessing functional dependence. Extensive numerical results are provided to support our theoretical results. The proposed methods are implemented in the R package $\mathtt{changepoints}$ (Xu et al. (2022)).