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

Abstract

July 28, 2010 AbstractIn this paper we investigate the problem of measuring deviations from stationarity in locally stationary time series. Our approach is based on a direct estimate of the L 2 -distance between the spectral density of the locally stationary process and its best approximation by a spectral density of a stationary process. An explicit expression of the minimal distance is derived, which depends only on integrals of the spectral density of the stationary process and its square. These integrals can be estimated directly without estimating the spectral density, and as a consequence, the estimation of the measure of stationarity does not require the specification of smoothing parameters. We show weak convergence of an appropriately standardized version of the statistic to a standard normal distribution. The results are used to construct confidence intervals for the measure of stationarity and to develop a new test for the hypothesis of stationarity which does not require regularization. Finally, we investigate the finite sample properties of the resulting confidence intervals and tests by means of a small simulation study and illustrate the methodology in three data examples.

Measuring stationarity in long-memory processes

“…A similar result for the short-memory situation has been derived by Dette, Preuß and Vetter (2011a) under the condition √ T /N → 0. A comparison with (3.4) shows that a larger size of the window length N is necessary in the case of long-range dependence, which corresponds to intuition.…”

“…By combining the arguments from the proof of part a) and from Dette, Preuß and Vetter (2011b), we obtain that…”

“…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…”

“…As all procedures which have been suggested so far for discriminating between stationarity and non-stationarity, the tests proposed by Dette, Preuß and Vetter (2011a) and Preuß and Vetter and Dette (2013) are only applicable to short-memory processes, and the development of a corresponding methodology in the context of long-range dependence is missing. In fact, although stationary long-memory models are often employed in practice, there do not exist many tests for the hypothesis of stationarity which include these processes.…”

“…All these methods require the choice of at least one smoothing parameter, such as the order of the wavelet expansion, a bandwidth for the estimation of the spectral density or the order in a Portmanteau-type test. Dette, Preuß and Vetter (2011a) developed tests for the hypothesis of stationarity in the framework of locally stationary processes which is based on an estimate of the L 2 -distance…”