Quantile Spectral Analysis for Locally Stationary Time Series

Classical spectral methods are subject to two fundamental limitations: they only can account for covariance-related serial dependencies, and they require second-order stationarity. Much attention has been devoted recently to quantile-based spectral methods that go beyond covariance-based serial dependence features. At the same time, covariance-based methods relaxing stationarity into much weaker local stationarity conditions have been developed for a variety of time-series models. Here, we are combining those two approaches by proposing quantile-based spectral methods for locally stationary processes. We therefore introduce a time-varying version of the copula spectra that have been recently proposed in the literature, along with a suitable local lag-window estimator. We propose a new definition of strict local stationarity that allows us to handle completely general non-linear processes without any moment assumptions, thus accommodating our quantile-based concepts and methods. We establish a central limit theorem for our estimator, and illustrate the power of the new methodology by means of simulations and an empirical study of the Standard & Poor's 500 series. This empirical study brings evidence of important variations in serial dependence structures both across time (crises and quiet periods exhibit quite different dependence structures) and across quantiles (dependencies between extreme quantiles are not the same as in the median range of the series). Such variations remain completely undetected, and are actually undetectable, via classical covariance-based spectral methods.