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Applications of distance correlation to time series. (English) Zbl 1414.62357

Summary: 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 G. J. Székely et al. [Ann. Stat. 35, No. 6, 2769–2794 (2007; Zbl 1129.62059)] and G. J. Székely and M. L. Rizzo [Ann. Appl. Stat. 3, No. 4, 1236–1265 (2009; Zbl 1196.62077)] 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.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60E10 Characteristic functions; other transforms
60G10 Stationary stochastic processes
62E20 Asymptotic distribution theory in statistics
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