Applications of distance correlation to time series

TY - JOUR

T1 - Applications of distance correlation to time series

AU - Davis, Richard

AU - Matsui, Munyea

AU - Mikosch, Thomas Valentin

AU - Wan, Phyllis

PY - 2018/11

Y1 - 2018/11

N2 - 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.

AB - 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.

U2 - 10.3150/17-bej955

DO - 10.3150/17-bej955

M3 - Journal article

SN - 1350-7265

VL - 24

SP - 3087

EP - 3116

JO - Bernoulli

JF - Bernoulli

IS - 4A

ER -