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Statistical inference for high-dimensional spectral density matrix

Jinyuan Chang, Qing Jiang, Tucker S. McElroy, Xiaofeng Shao

TL;DR

This work tackles statistical inference for the high-dimensional spectral density matrix of multivariate time series by developing global and multiple testing procedures for the cross-spectral density at chosen frequencies and component pairs. It advances methodology by combining a novel Gaussian-approximation framework with a computationally feasible parametric bootstrap to approximate the distribution of a maximum-type test statistic, and it augments this with a false discovery rate controlled procedure for support recovery. Theoretical guarantees are established under weak temporal dependence and growing dimension, and practical performance is demonstrated via simulations and real-data analyses of county-level hires data. The results enable scalable, frequency-domain inference and decision-making for partitioned, batch-wise, or high-dimensional time series, with direct applicability to forecasting, seasonal adjustment evaluation, and connectivity analysis in complex systems.

Abstract

The spectral density matrix is a fundamental object of interest in time series analysis, and it encodes both contemporary and dynamic linear relationships between component processes of the multivariate system. In this paper we develop novel inference procedures for the spectral density matrix in the high-dimensional setting. Specifically, we introduce a new global testing procedure to test the nullity of the cross-spectral density for a given set of frequencies and across pairs of component indices. For the first time, both Gaussian approximation and parametric bootstrap methodologies are employed to conduct inference for a high-dimensional parameter formulated in the frequency domain, and new technical tools are developed to provide asymptotic guarantees of the size accuracy and power for global testing. We further propose a multiple testing procedure for simultaneously testing the nullity of the cross-spectral density at a given set of frequencies. The method is shown to control the false discovery rate. Both numerical simulations and a real data illustration demonstrate the usefulness of the proposed testing methods.

Statistical inference for high-dimensional spectral density matrix

TL;DR

This work tackles statistical inference for the high-dimensional spectral density matrix of multivariate time series by developing global and multiple testing procedures for the cross-spectral density at chosen frequencies and component pairs. It advances methodology by combining a novel Gaussian-approximation framework with a computationally feasible parametric bootstrap to approximate the distribution of a maximum-type test statistic, and it augments this with a false discovery rate controlled procedure for support recovery. Theoretical guarantees are established under weak temporal dependence and growing dimension, and practical performance is demonstrated via simulations and real-data analyses of county-level hires data. The results enable scalable, frequency-domain inference and decision-making for partitioned, batch-wise, or high-dimensional time series, with direct applicability to forecasting, seasonal adjustment evaluation, and connectivity analysis in complex systems.

Abstract

The spectral density matrix is a fundamental object of interest in time series analysis, and it encodes both contemporary and dynamic linear relationships between component processes of the multivariate system. In this paper we develop novel inference procedures for the spectral density matrix in the high-dimensional setting. Specifically, we introduce a new global testing procedure to test the nullity of the cross-spectral density for a given set of frequencies and across pairs of component indices. For the first time, both Gaussian approximation and parametric bootstrap methodologies are employed to conduct inference for a high-dimensional parameter formulated in the frequency domain, and new technical tools are developed to provide asymptotic guarantees of the size accuracy and power for global testing. We further propose a multiple testing procedure for simultaneously testing the nullity of the cross-spectral density at a given set of frequencies. The method is shown to control the false discovery rate. Both numerical simulations and a real data illustration demonstrate the usefulness of the proposed testing methods.
Paper Structure (57 sections, 18 theorems, 351 equations, 2 figures, 3 tables)

This paper contains 57 sections, 18 theorems, 351 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Some technical results
  3. Preliminary
  4. A general Gaussian approximation procedure
  5. Parametric bootstrap procedure
  6. Applications
  7. Global hypothesis testing
  8. Multiple testing with FDR control
  9. Numerical simulations
  10. Global hypothesis testing
  11. Multiple testing
  12. Real Data Analysis
  13. Batching county-level hires data
  14. Detecting seasonal over-adjustment in Texas hires
  15. Discussion
  16. ...and 42 more sections

Key Result

Proposition 1

Assume $r\geq n^\kappa$ for some sufficiently small constant $\kappa>0$, and let Conditions as:moment--as:eigen hold. As $n\rightarrow\infty$, the following two assertions are valid. (i) If $\mathcal{J}=\{\omega_1,\ldots,\omega_K\}$ and $\log(Kr)\ll n^{1/9}l_n^{-1}\log^{-8/3}(l_n)$, with the bandwi for a $(2Kr)$-dimensional normally distributed random vector ${\mathbf s}_{n,{\mathbf y}}=(s_{n,1},

Figures (2)

  • Figure 1: Heatmap of p-values for $51$ state pairs, testing whether the cross-spectrum of each pair is zero at seasonal frequencies $\mathcal{J}= \{-\pi, -\pi/2, 0, \pi/2\}$. A white box indicates a pair for which no test is computed; a star marks pairs that are not significant using FDR control, where $\alpha = 5\%$. Each series has been differenced.
  • Figure 2: Heatmap of p-values for $51$ state pairs, testing whether the cross-spectrum of each pair is zero at seasonal frequencies $\mathcal{J}= \{-\pi, -\pi/2, 0, \pi/2\}$. A white box indicates a pair for which no test is computed; a star marks pairs that are not significant using FDR control, where $\alpha = 5\%$. Each series has been seasonally differenced.

Theorems & Definitions (20)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Lemma L1
  • Lemma L2
  • Lemma L3
  • Lemma L4
  • ...and 10 more