Quantile spectral analysis for locally stationary time series
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Date
2014-04-25
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Abstract
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, 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 time-varying versions of the
copula spectra and periodograms that have been recently proposed in the literature, along with
a new defi nition 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 the consistency of our methods, and illustrate their power 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 di fferent 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.
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Keywords
time series, local stationarity, ranks, copulas, quantile regression, periodogram, spectral analysis