Consistent Testing for Pairwise Dependence in Time Series

ABSTRACT

We consider the problem of testing pairwise dependence for stationary time series. For this, we suggest the use of a Box–Ljung-type test statistic that is formed after calculating the distance covariance function among pairs of observations. The distance covariance function is a suitable measure for detecting dependencies between observations as it is based on the distance between the characteristic function of the joint distribution of the random variables and the product of the marginals. We show that, under the null hypothesis of independence and under mild regularity conditions, the test statistic converges to a normal random variable. The results are complemented by several examples. This article has supplementary material online.

KEYWORDS:

• Distance covariance
• Empirical characteristic function
• Generalized spectral density
• Kernel
• U-statistic
• V-statistic

Supplementary Materials

The online supplementary materials include some further results concerning applications (Section 1) and the proofs of the main theorems (Section 2).

Acknowledgments

The authors thank the editor, associate editor, and two anonymous reviewers for constructive criticism. In addition, the authors acknowledge the project eMammoth - Compute and Store on Grids and Clouds infrastructure(ANABATHMISI/06609/09), which is co-funded by the Republic of Cyprus and the European Regional Development Fund of the EU, for computational support.