Estimated Wold Representation and Spectral-Density-Driven Bootstrap for Time Series

Abstract

Summary The second-order dependence structure of purely non-deterministic stationary processes is described by the coefficients of the famous Wold representation. These coefficients can be obtained by factorizing the spectral density of the process. This relationship together with some spectral density estimator is used to obtain consistent estimators of these coefficients. A spectral-density-driven bootstrap for time series is then developed which uses the entire sequence of estimated moving average coefficients together with appropriately generated pseudoinnovations to obtain a bootstrap pseudo-time-series. It is shown that if the underlying process is linear and if the pseudoinnovations are generated by means of an independent and identically distributed wild bootstrap which mimics, to the extent necessary, the moment structure of the true innovations, this bootstrap proposal asymptotically works for a wide range of statistics. The relationships of the proposed bootstrap procedure to some other bootstrap procedures, including the auto-regressive sieve bootstrap, are discussed. It is shown that the latter is a special case of the spectral-density-driven bootstrap, if a parametric auto-regressive spectral density estimator is used. Simulations investigate the performance of the new bootstrap procedure in finite sample situations. Furthermore, a real life data example is presented.