Identifying the Structure of High-Dimensional Time Series via Eigen-Analysis

Abstract

Cross-sectional structures and temporal tendency are important features of high-dimensional time series. Based on eigen-analysis on sample covariance matrices, we propose a novel approach to identifying four popular structures of high-dimensional time series, which are grouped in terms of factor structures and stationarity. The proposed three-step method includes a ratio statistic of empirical eigenvalues;a projected Augmented Dickey-Fuller Test;a new unit-root test based on the largest empirical eigenvalues.We develop asymptotic properties for these three statistics to ensure the feasibility of the whole identifying procedure. Finite sample performances are illustrated via various simulations. We also analyze U.S. mortality data, U.S. house prices and income, and U.S. sectoral employment, all of which possess cross-sectional dependence and nonstationary temporal dependence. It is worth mentioning that we also contribute to statistical justification for the benchmark paper by Lee and Carter in mortality forecasting. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.