Testing temporal constancy of the spectral structure of a time series

Abstract

Statistical inference for stochastic processes with time-varying spectral characteristics has received considerable attention in recent decades. We develop a nonparametric test for stationarity against the alternative of a smoothly time-varying spectral structure. The test is based on a comparison between the sample spectral density calculated locally on a moving window of data and a global spectral density estimator based on the whole stretch of observations. Asymptotic properties of the nonparametric estimators involved and of the test statistic under the null hypothesis of stationarity are derived. Power properties under the alternative of a time-varying spectral structure are discussed and the behavior of the test for fixed alternatives belonging to the locally stationary processes class is investigated.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ179 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Costationarity of Locally Stationary Time Series

“…They then apply a two-dimensional wavelet transform which simultaneously smoothes across frequency (father wavelets) and identifies inhomogeneities using mother wavelets. Other recent stationarity tests are Stȃricȃ and Granger (2005) and Paparoditis (2009) that measure the difference between a periodogram and model spectrum on intervals. The typical stationarity test null hypothesis is:…”

A Measure of Stationarity in Locally Stationary Processes With Applications to Testing

“…Sakiyama and Taniguchi (2003) considered the problem of testing stationarity versus local stationarity in a parametric locally stationary model, while Lee et al (2003) investigated the constancy over time of a finite number of autocovariances. von Sachs and Neumann (2000) proposed a multiple testing procedure based on empirical wavelet coefficients estimated using localized versions of the periodogram, while Paparoditis (2010) used L 2 -distances between the local sample spectral density and an overall spectral density estimator [see also Paparoditis (2009)]. A common feature in many of these methods is the fact that the statistical inference depends on the choice of a regularization parameter.…”

“…A common feature in many of these methods is the fact that the statistical inference depends on the choice of a regularization parameter. For example, Paparoditis (2009) and Paparoditis (2010) compare nonparametric estimators of the spectral density of the stationary and locally stationary process, and as a consequence, the resulting statistical analysis depends sensitively on the choice of a smoothing parameter which is required for the density estimation. An alternative approach in this context is the application of the empirical spectral measure for inference in locally stationary time series [see Dahlhaus and Polonik (2009)].…”

Measuring stationarity in long-memory processes

“…Recently, Nason (2013) used the same principle and examined the constancy of a wavelet spectrum by examining its Haar wavelet coefficients over a finite set of wavelet scales. Paparoditis (2009Paparoditis ( , 2010 suggested an L 2 -distance between the estimated spectral densities under the assumptions of stationarity and of local stationarity. Many authors have used the localized periodogram to construct a test for stationarity.…”

“…the time varying spectral density does not depend on u) if and only if D 2 = 0, and D 2 can be considered as a measure of deviation from stationarity in the frequency domain. This quantity corresponds to the measure used in Paparoditis (2009), but unlike to this author, Dette, Preuß and Vetter (2011a) estimated D 2 directly via Riemann sums of the (squared) local periodogram instead of a smoothed local periodogram and thus avoided the choice of a smoothing parameter. Preuß and Vetter and Dette (2013) proposed an alternative measure for deviations from stationarity based on the Kolmogorov-Smirnov distance…”