Estimating change-point latent factor models for high-dimensional time series

Abstract

We consider estimating a factor model for high-dimensional time series that contains structural breaks in the factor loading space at unknown time points. We first study the case when there is one change point in factor loadings, and propose a consistent estimator for the structural break location, whose convergence rate is shown to depend on an interplay between the dimension of the observed time series and the strength of the underlying factor structure. Our results reveal that the asymptotic behavior of the proposed estimator can be asymmetric in the sense that a larger estimation error can occur toward the regime with weaker factor strength. Based on the proposed estimator for the structural break location, we also consider the problem of estimating the factor loading spaces before and after the structural break. We show that the proposed estimators for change-point location and loading spaces are consistent when the numbers of factors are correctly estimated or overestimated. The algorithm for multiple change-point detection is also developed in the paper. Compared with existing results on change-point factor analyses of high-dimensional time series, a distinguished feature of the current paper is that the noise process is not necessarily assumed to be idiosyncratic and as a result we allow the noise process with potentially strong cross-sectional dependence. Another advantage for the proposed method is that it is specifically designed for the changes in the factor loading space and the stationarity assumption is not imposed on either the factor or noise process, while most existing methods for change-point detection of high-dimensional time series with/without a factor structure require the data to be stationary or ’close’ to a stationary process between two change points, which is rather restrictive. Numerical experiments including a Monte Carlo simulation and a real data application are presented to illustrate the proposed estimators perform well.