Open Access
November 2018 Applications of distance correlation to time series
Richard A. Davis, Muneya Matsui, Thomas Mikosch, Phyllis Wan
Bernoulli 24(4A): 3087-3116 (November 2018). DOI: 10.3150/17-BEJ955


The use of empirical characteristic functions for inference problems, including estimation in some special parametric settings and testing for goodness of fit, has a long history dating back to the 70s. More recently, there has been renewed interest in using empirical characteristic functions in other inference settings. The distance covariance and correlation, developed by Székely et al. (Ann. Statist. 35 (2007) 2769–2794) and Székely and Rizzo (Ann. Appl. Stat. 3 (2009) 1236–1265) for measuring dependence and testing independence between two random vectors, are perhaps the best known illustrations of this. We apply these ideas to stationary univariate and multivariate time series to measure lagged auto- and cross-dependence in a time series. Assuming strong mixing, we establish the relevant asymptotic theory for the sample auto- and cross-distance correlation functions. We also apply the auto-distance correlation function (ADCF) to the residuals of an autoregressive processes as a test of goodness of fit. Under the null that an autoregressive model is true, the limit distribution of the empirical ADCF can differ markedly from the corresponding one based on an i.i.d. sequence. We illustrate the use of the empirical auto- and cross-distance correlation functions for testing dependence and cross-dependence of time series in a variety of contexts.


Download Citation

Richard A. Davis. Muneya Matsui. Thomas Mikosch. Phyllis Wan. "Applications of distance correlation to time series." Bernoulli 24 (4A) 3087 - 3116, November 2018.


Received: 1 July 2016; Revised: 1 February 2017; Published: November 2018
First available in Project Euclid: 26 March 2018

zbMATH: 06853274
MathSciNet: MR3779711
Digital Object Identifier: 10.3150/17-BEJ955

Keywords: $U$-statistics , AR process , auto- and cross-distance correlation function , ergodicity , Fourier analysis , residuals , Strong mixing , testing independence , time series

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 4A • November 2018
Back to Top