Attribution of Spurious Factors from High-Dimensional Functional Time Series

2026-03-27

AI Summary

This study addresses the challenge that empirical eigenanalysis of high-dimensional nonstationary functional time series is prone to spurious factors, which distort the sample covariance operator and impede accurate recovery of the true latent factor structure. The authors develop a general functional factor model and, leveraging operator spectral analysis and asymptotic theory for covariance operators, establish a novel effective rank condition tailored to functional data. This condition provides sufficient criteria for the emergence of spurious factors, revealing that even a small number of strongly nonstationary factors can induce spurious limiting behavior in the functional setting—thereby extending beyond the applicability of conventional results in high-dimensional time series. The theoretical findings are corroborated through simulations and further validated by empirical analysis of age-specific mortality data across multiple regions, where spurious factors induced by empirical eigenanalysis are successfully identified.

Abstract

This article explores a general factor structure for high-dimensional nonstationary functional time series, encompassing a wide range of factor models studied in the existing literature. We investigate the asymptotic spectral behaviors of the sample covariance operator under this general data structure. A novel fundamental sufficient condition, formulated in terms of a newly introduced effective rank tailored to this setup, is established under which empirical eigen-analysis yields spurious results, rendering sample eigenvalues and eigenvectors unreliable for accurately recovering the underlying factor structure. This generalizes the results of Onatski and Wang [2021] from typical high-dimensional time series (HDTS) to the more intricate functional framework. The newly defined effective rank is rigorously analyzed through a decomposition of the effects attributable to functional factor loadings and functional factors. Contrary to the findings in the HDTS setting, empirical eigen-analysis of models with only a small number of strong non-stationary factors may still produce spurious limits in the functional framework. Therefore, additional caution is warranted when applying covariance-based statistical methods to potentially nonstationary functional data. Simulation studies are performed to determine conditions under which spurious limits occur. Real data analysis on age-specific mortality rate data from multiple locations is conducted for evidence of spurious factors induced by empirical eigen-analysis.

Problem

• spurious factors
• functional time series
• high-dimensional data
• nonstationarity
• eigen-analysis

Innovation

• functional time series
• spurious factors
• effective rank
• nonstationarity
• eigen-analysis