Attribution of Spurious Factors from High-Dimensional Functional Time Series

TL;DR

This paper investigates when empirical eigen-analysis of high-dimensional nonstationary functional time series produces spurious factors, introducing a new effective rank condition and analyzing its implications through simulations and real data.

Contribution

It generalizes existing results to functional data, introduces a new effective rank criterion, and highlights conditions leading to spurious factors in eigen-analysis.

Findings

Empirical eigen-analysis can produce spurious factors in functional time series.

A new effective rank condition predicts when spurious results occur.

Simulation and real data confirm the theoretical findings.

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.