The integrated copula spectrum

TL;DR

This paper introduces the integrated copula spectrum, a new frequency domain method for time series that captures joint distribution dynamics without smoothing parameters, enabling novel hypothesis testing.

Contribution

It proposes the integrated copula spectrum, which can be estimated without smoothing parameters, and provides theoretical analysis and hypothesis testing tools.

Findings

Estimation of the integrated copula spectrum without smoothing parameters.

Asymptotic properties established via a functional central limit theorem.

New tests for time-reversibility and tail asymmetry in time series.

Abstract

Frequency domain methods form a ubiquitous part of the statistical toolbox for time series analysis. In recent years, considerable interest has been given to the development of new spectral methodology and tools capturing dynamics in the entire joint distributions and thus avoiding the limitations of classical, $L^2$-based spectral methods. Most of the spectral concepts proposed in that literature suffer from one major drawback, though: their estimation requires the choice of a smoothing parameter, which has a considerable impact on estimation quality and poses challenges for statistical inference. In this paper, associated with the concept of copula-based spectrum, we introduce the notion of copula spectral distribution function or integrated copula spectrum. This integrated copula spectrum retains the advantages of copula-based spectra but can be estimated without the need for smoothing parameters. We provide such estimators, along with a thorough theoretical analysis, based on a functional central limit theorem, of their asymptotic properties. We leverage these results to test various hypotheses that cannot be addressed by classical spectral methods, such as the lack of time-reversibility or asymmetry in tail dynamics.