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.

In this paper we investigate the problem of testing the assumption of stationarity in locally stationary processes. The test is based on an estimate of a Kolmogorov-Smirnov type distance between the true time varying spectral density and its best approximation through a stationary spectral density. Convergence of a time varying empirical spectral process indexed by a class of certain functions is proved, and furthermore the consistency of a bootstrap procedure is shown which is used to approximate the limiting distribution of the test statistic. Compared to other methods proposed in the literature for the problem of testing for stationarity the new approach has at least two advantages: On one hand, the test can detect local alternatives converging to the null hypothesis at any rate $g_T\to0$ such that $g_TT^{1/2}\to \infty$, where $T$ denotes the sample size. On the other hand, the estimator is based on only one regularization parameter while most alternative procedures require two. Finite sample properties of the method are investigated by means of a simulation study, and a comparison with several other tests is provided which have been proposed in the literature.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ472 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

We propose a new nonparametric procedure for the detection and estimation of multiple structural breaks in the autocovariance function of a multivariate (secondorder) piecewise stationary process, which also identifies the components of the series where the breaks occur. The new method is based on a comparison of the estimated spectral distribution on different segments of the observed time series and consists of three steps: it starts with a consistent test, which allows to prove the existence of structural breaks at a controlled type I error. Secondly, it estimates sets containing possible break points and finally these sets are reduced to identify the relevant structural breaks and corresponding components which are responsible for the changes in the autocovariance structure. In contrast to all other methods which have been proposed in the literature, our approach does not make any parametric assumptions, is not especially designed for detecting one single change point and addresses the problem of multiple structural breaks in the autocovariance function directly with no use of the binary segmentation algorithm. We prove that the new procedure detects all components and the corresponding locations where structural breaks occur with probability converging to one as the sample size increases and provide data-driven rules for

In statistical research there usually exists a choice between structurally simpler or more complex models. We argue that, even if a more complex, locally stationary time series model were true, then a simple, stationary time series model may be advantageous to work with under parameter uncertainty. We present a new model choice methodology, where one of two competing approaches is chosen based on its empirical, finitesample performance with respect to prediction, in a manner that ensures interpretability. A rigorous, theoretical analysis of the procedure is provided. As an important side result we prove, for possibly diverging model order, that the localised Yule-Walker estimator is strongly, uniformly consistent under local stationarity. An R package, forecastSNSTS, is provided and used to apply the methodology to financial and meteorological data in empirical examples. We further provide an extensive simulation study and discuss when it is preferable to base forecasts on the more volatile timevarying estimates and when it is advantageous to forecast as if the data were from a stationary process, even though they might not be.