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A Bootstrap Test for Independence of Time Series Based on the Distance Covariance

Annika Betken, Herold Dehling, Marius Kroll

Abstract

We present a test for independence of two strictly stationary time series based on a bootstrap procedure for the distance covariance. Our test detects any kind of dependence between the two time series within an arbitrary maximum lag $L$. In simulation studies, our test outperforms alternative testing procedures. In proving the validity of the underlying bootstrap procedure, we generalise bounds for the Wasserstein distance between an empirical measure and its marginal distribution under the assumption of $α$-mixing. Previous results of this kind only existed for i.i.d. processes.

A Bootstrap Test for Independence of Time Series Based on the Distance Covariance

Abstract

We present a test for independence of two strictly stationary time series based on a bootstrap procedure for the distance covariance. Our test detects any kind of dependence between the two time series within an arbitrary maximum lag . In simulation studies, our test outperforms alternative testing procedures. In proving the validity of the underlying bootstrap procedure, we generalise bounds for the Wasserstein distance between an empirical measure and its marginal distribution under the assumption of -mixing. Previous results of this kind only existed for i.i.d. processes.
Paper Structure (19 sections, 34 theorems, 325 equations, 8 figures, 11 tables, 2 algorithms)

This paper contains 19 sections, 34 theorems, 325 equations, 8 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Test Statistic and Bootstrap Technique
  3. Main Theoretical Results
  4. Summary
  5. Technical Preliminaries
  6. Results on the Test for Independence
  7. A Bound for the Wasserstein Distance
  8. The Condition of Bounded Densities
  9. Simulations and Data Analysis
  10. Simulations
  11. Data Analysis
  12. Outline of Proofs
  13. Outline for Theorem \ref{['thm:hyp_bootstrap']}
  14. Outline for Theorem \ref{['thm:wasserstein_gesamt']}
  15. Proofs and Auxiliary Results
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that $X_1$ and $Y_1$ both have finite $(4+\delta)$-th moments for some $\delta > 0$, and that the process $(Z_k)_{k \in \mathbb{N}}$ is strictly stationary and absolutely regular with $\beta(n) = \mathcal{O}\left(n^{-r}\right)$ for some $r > 18$. Furthermore, suppose that the joint distribut where $\zeta$ is the weak limit of $n\, \mathrm{dcov}(\theta_n)$ under the hypothesis

Figures (8)

  • Figure 1: Rejection rates of the test based on Pearson's correlation and the distance covariance for two independent AR(1) processes with parameter $\rho$ and length $n = 300$. As block length we choose $d\in \{2, n^{1/3}, \sqrt{n}\} = \{2, 6, 17\}$.
  • Figure 2: Rejection rates of the test based on Pearson's correlation and the distance covariance for $\mathrm{VAR}(1)$ processes with parameters $a=0.5$, $b=0$, correlation matrix $\Gamma(0)$ as in Eq. \ref{['eq:corr_matrix']}, and length $n = 300$. As block length we choose $d\in \{2, n^{1/3}, \sqrt{n}\} = \{2, 6, 17\}$.
  • Figure 3: Rejection rates of the test based on Pearson's correlation and the distance covariance for $\mathrm{VAR}(1)$ processes with parameters $a=0.5$, $b=0$, correlation matrix $\Gamma(0)$ as in Eq. \ref{['eq:corr_matrix']}, and length $n = 300$. We consider a quantile transformation that yields a $t$-distribution with $\nu = 3$ degrees of freedom. As block length we choose $d\in \{2, n^{1/3}, \sqrt{n}\} = \{2, 6, 17\}$.
  • Figure 4: Rejection rates of the test based on Pearson's correlation and the distance covariance for two independent AR(1) processes with parameter $\rho$ and length $n = 300$. We consider a quantile transformation that yields a $t$-distribution with $\nu = 3$ degrees of freedom. As block length we choose $d\in \{2, n^{1/3}, \sqrt{n}\} = \{2,6,17\}$.
  • Figure 5: Rejection rates of the test based on Pearson's correlation and the distance covariance for stochastic volatility time series of length $n = 300$, i.e. for $Y_i = X_{i+1}$, $X_j=(\xi_j-\mathbb{E}\xi_j)\exp(\eta_j)$, $j=1, \ldots, n$ where $\eta_j$, $j=1, \ldots, n$, stems from an AR(1) process with parameter $\rho$ and $\xi_j$, $j=1, \ldots, n$, from i.i.d. Pareto distributed random variables with shape parameter $\alpha = 5$. As block length we choose $d\in \{2, n^{1/3}, \sqrt{n}\} = \{2, 6, 17\}$.
  • ...and 3 more figures

Theorems & Definitions (70)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Remark 1
  • Corollary 3
  • Lemma 1
  • proof
  • Theorem 3
  • Proposition 1
  • ...and 60 more