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Attribution of Spurious Factors from High-Dimensional Functional Time Series

Adam Nie, Yanrong Yang, Han Lin Shang, Yi He

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.

Attribution of Spurious Factors from High-Dimensional Functional Time Series

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.

Paper Structure

This paper contains 18 sections, 9 theorems, 122 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup: Model and Assumptions
  3. A General Functional Factor Model
  4. Model Generality
  5. The Sample Covariance Operator Matrix
  6. Asymptotic Theory for Spurious Phenomenon
  7. Assumptions
  8. Asymptotic Theory
  9. Separable Effects from Factors and Factor Loadings
  10. Simulation Study
  11. Setup
  12. Results
  13. Empirical applications
  14. Conclusion and Future Work
  15. Proofs
  16. ...and 3 more sections

Key Result

Proposition 2.6

Let $C_1$ and $C_2$ be given as in Definition definition - C Omega, and $\Omega$ is defined in (eq- def of Omega). Then

Figures (8)

  • Figure 1: Setting 1 - $C_\epsilon$ delocalized, loading matrices are of full-column rank
  • Figure 2: Setting 3 - $C_\epsilon$ localized, loading matrices are of full-column rank
  • Figure 3: Setting 4 - $C_\epsilon$ localized, loading matrices are of low-effective rank
  • Figure 4: Setting 5 - $C_\epsilon$ is of rank 2, loading matrices are of full-column rank
  • Figure 5: Setting 6 - $C_\epsilon$ is of rank 2, loading matrices are of low-effective rank
  • ...and 3 more figures

Theorems & Definitions (32)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • Remark 2.7
  • Remark 3.2
  • Remark 3.4
  • Remark 3.5
  • ...and 22 more