Fourier Analysis of Serial Dependence Measures

Abstract

Classical spectral analysis is based on the discrete Fourier transform of the autocovariances. In this article we investigate the asymptotic properties of new frequency-domain methods where the autocovariances in the spectral density are replaced by alternative dependence measures that can be estimated by U-statistics. An interesting example is given by Kendall's , for which the limiting variance exhibits a surprising behavior.

Quantile-Frequency Analysis and Spectral Measures for Diagnostic Checks of Time Series With Nonlinear Dynamics

“…The former include conventional techniques such as the autocorrelation function and the periodogram (Priestley, 1981) and more recent ones such as that discussed by Hecke et al. (2018). The latter include the techniques based on the bivariate distribution function P ( X t ≤ x , X t + τ ≤ y ) or the corresponding bivariate characteristic function (Dette et al., 2015; Hong, 2000; Lee & Subba Rao, 2011; Skaug & Tjøstheim, 1993).…”

“…Among diagnostic tools for serial dependence, some are more narrowly focused but easier to use; others are more comprehensive but harder to compute, visualize, and interpret. The former include conventional techniques such as the autocorrelation function and the periodogram (Priestley 1981) and more recent ones such as that discussed by Hecke, Volgushev and Dette (2018). The latter include the techniques based on the bivariate distribution function P X t x X t y or the corresponding bivariate characteristic function (Skaug and Tjøstheim 1993;Hong 2000;Lee and Subba Rao 2011;Dette, Hallin, Kley and Volgushev 2015).…”