We explore the limits of the autoregressive (AR) sieve bootstrap, and show that its applicability extends well beyond the realm of linear time series as has been previously thought. In particular, for appropriate statistics, the AR-sieve bootstrap is valid for stationary processes possessing a general Wold-type autoregressive representation with respect to a white noise; in essence, this includes all stationary, purely nondeterministic processes, whose spectral density is everywhere positive. Our main theorem provides a simple and effective tool in assessing whether the AR-sieve bootstrap is asymptotically valid in any given situation. In effect, the large-sample distribution of the statistic in question must only depend on the first and second order moments of the process; prominent examples include the sample mean and the spectral density. As a counterexample, we show how the AR-sieve bootstrap is not always valid for the sample autocovariance even when the underlying process is linear.Comment: Published in at http://dx.doi.org/10.1214/11-AOS900 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A nonparametric, residual-based block bootstrap procedure is proposed in the context of testing for integrated (unit root) time series. The resampling procedure is based on weak assumptions on the dependence structure of the stationary process driving the random walk and successfully generates unit root integrated pseudo-series retaining the important characteristics of the data. It is more general than previous bootstrap approaches to the unit root problem in that it allows for a very wide class of weakly dependent processes and it is not based on any parametric assumption on the process generating the data. As a consequence the procedure can accurately capture the distribution of many unit root test statistics proposed in the literature. Large sample theory is developed and the asymptotic validity of the block bootstrap-based unit root testing is shown via a bootstrap functional limit theorem. Applications to some particular test statistics of the unit root hypothesis, i.e., least squares and Dickey-Fuller type statistics are given. The power properties of our procedure are investigated and compared to those of alternative bootstrap approaches to carry out the unit root test. Some simulations examine the finite sample performance of our procedure.

-The situation where the available data arise from a general linear process with a unit root is discussed. We propose a modi cation of the Block Bootstrap which generates replicates of the original data and which correctly imitates the unit root behavior and the weak dependence structure of the observed series. Validity of the proposed method for estimating the unit root distribution is shown.

A new goodness-of-®t test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quanti®cation of how well a parametric spectral density model ®ts the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justi®ed theoretically. The ®nite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.

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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

We consider the prediction problem of a time series on a whole time interval in terms of its past. The approach that we adopt is based on functional kernel nonparametric regression estimation techniques where observations are discrete recordings of segments of an underlying stochastic process considered as curves. These curves are assumed to lie within the space of continuous functions, and the discretized time series data set consists of a relatively small, compared with the number of segments, number of measurements made at regular times. We estimate conditional expectations by using appropriate wavelet decompositions of the segmented sample paths. A notion of similarity, based on wavelet decompositions, is used to calibrate the prediction. Asymptotic properties when the number of segments grows to 1 are investigated under mild conditions, and a nonparametric resampling procedure is used to generate, in a flexible way, valid asymptotic pointwise prediction intervals for the trajectories predicted. We illustrate the usefulness of the proposed functional wavelet-kernel methodology in finite sample situations by means of a simulated example and two real life data sets, and we compare the resulting predictions with those obtained by three other methods in the literature, in particular with a smoothing spline method, with an exponential smoothing procedure and with a seasonal autoregressive integrated moving average model.

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