Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 3087-3116.

Applications of distance correlation to time series

Richard A. Davis, Muneya Matsui, Thomas Mikosch, and Phyllis Wan

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Abstract

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.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 3087-3116.

Dates
Received: July 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051234

Digital Object Identifier
doi:10.3150/17-BEJ955

Mathematical Reviews number (MathSciNet)
MR3779711

Zentralblatt MATH identifier
06853274

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

Citation

Davis, Richard A.; Matsui, Muneya; Mikosch, Thomas; Wan, Phyllis. Applications of distance correlation to time series. Bernoulli 24 (2018), no. 4A, 3087--3116. doi:10.3150/17-BEJ955. https://projecteuclid.org/euclid.bj/1522051234


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Supplemental materials

  • Supplement to “Applications of distance correlation to time series”. Complementary results to the proofs of Theorem 4.2 and Lemma 4.1 are provided in the supplement.