Measuring stationarity in long-memory processes
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Date
2013-03-14
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Abstract
In this paper we consider the problem of measuring stationarity in locally stationary longmemory
processes. We introduce an L2-distance between the spectral density of the locally
stationary process and its best approximation under the assumption of stationarity. The distance
is estimated by a numerical approximation of the integrated spectral periodogram and
asymptotic normality of the resulting estimate is established. The results can be used to construct
a simple test for the hypothesis of stationarity in locally stationary long-range dependent
processes. We also propose a bootstrap procedure to improve the approximation of the nominal
level and prove its consistency. Throughout the paper, we will work with Riemann sums of a
squared periodogram instead of integrals (as it is usually done in the literature) and as a byproduct
of independent interest it is demonstrated that the two approaches behave differently
in the limit.
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Keywords
bootstrap, empirical spectral measure, goodness-of-fit tests, integrated periodogram, locally stationary process, long-memory, non-stationary processes, spectral density