Test for independence of long-range dependent time series using distance covariance

TL;DR

This paper introduces a new statistical test based on distance covariance to determine independence between long-range dependent time series, with a novel asymptotic theory and a subsampling method for critical values.

Contribution

It develops a new test statistic for independence, derives its asymptotic distribution using a novel non-central limit theorem, and proposes a subsampling approach for practical implementation.

Findings

The test performs well in finite samples.

It effectively detects dependence in long-range dependent data.

The subsampling method provides valid critical values.

Abstract

We apply the concept of distance covariance for testing independence of two long-range dependent time series. As test statistic we propose a linear combination of empirical distance cross-covariances. We derive the asymptotic distribution of the test statistic, and we show consistency against a very general class of alternatives. The asymptotic theory developed in this paper is based on a novel non-central limit theorem for stochastic processes with values in an $L^2$-Hilbert space. This limit theorem is of general theoretical interest which goes beyond the context of this article. Subject to the dependence in the data, the standardization and the limit distributions of the proposed test statistic vary. Since the limit distributions are unknown, we propose a subsampling procedure to determine the critical values for the proposed test, and we provide a proof for the validity of subsampling. In a simulation study, we investigate the finite-sample behavior of our test, and we compare its performance to tests based on the empirical cross-covariances. As an application of our results we analyze the cross-dependencies between mean monthly discharges of three rivers.