Statistical inference for high-dimensional spectral density matrix

TL;DR

This paper introduces novel statistical inference methods for high-dimensional spectral density matrices in multivariate time series, enabling accurate testing of relationships across frequencies and components.

Contribution

It develops the first Gaussian approximation and bootstrap-based global testing procedures for high-dimensional spectral density matrices in the frequency domain.

Findings

New global testing procedure for cross-spectral density nullity.

Bootstrap methods provide accurate size and power guarantees.

Method controls false discovery rate in multiple testing.

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.