Spiked Eigenvalues of High-Dimensional Separable Sample Covariance Matrices

Abstract

This paper establishes asymptotic properties for spiked empirical eigenvalues of sample co- variance matrices for high-dimensional data with both cross-sectional dependence and a dependent sample structure. A new finding from the established theoretical results is that spiked empirical eigenvalues will reflect the dependent sample structure instead of the cross-sectional structure under some scenarios, which indicates that principal component analysis (PCA) may provide inaccurate inference for cross-sectional structures. An illustrated example is provided to show that some commonly used statistics based on spiked empirical eigenvalues mis-estimate the true number of common factors. As an application of high-dimensional time series, we propose a test statistic to distinguish the unit root from the factor structure and demonstrate its effective finite sample performance on simulated data. Our results are then applied to analyze OECD healthcare expenditure data and U.S. mortality data, both of which possess cross-sectional dependence as well as non-stationary temporal dependence. It is worth mentioning that we contribute to statistical justification for the benchmark paper by Lee and Carter in mortality forecasting.