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.

Estimating MA Parameters through Factorization of the Autocovariance Matrix and an MA‐Sieve Bootstrap

“…Alternatively, an arbitrary distribution with mean zero and standard deviation could be employed. For example, one may use a distribution that additionally achieves a desired kurtosis level as discussed in Krampe et al in the hope of covering more general statistics, e.g., the sample autocovariances, under a linear process setting.…”

“…Recently, Krampe et al proposed an alternative approach to estimating MA( ∞ ) models; their approach involves first estimating the spectral density f (·), then calculating the Fourier coefficients of , and finally using these Fourier coefficients to estimate the MA coefficients through a factorization given in Pourahmadi . Krampe et al were able to show the consistency of their approach for the MA( ∞ ) coefficients, as long as was uniformly consistent, although they did not give rates of convergence. Furthermore, they used their fitted MA model (of order say ) to devise a residual‐based bootstrap based on simulating innovations from as if they were i.i.d.…”

“…McMurry and Politis showed how to use banded and tapered autocovariance matrix estimates to obtain the aforementioned best one‐step‐ahead linear predictor of the unobserved X n +1 given the data X 1 ,…, X n without relying on the AR approximation . In this paper, we show that by employing banded and tapered autocovariances (McMurry and Politis, ), rather than the raw sample autocovariance function used in the innovations algorithm, fitting an MA( q ) or MA( ∞ ) model can be reframed as a factorization of a consistent estimate of the autocovariance matrix; this allows for easier selection of the model complexity parameter, i.e., , and provides an approach that is direct, easy to implement, and has more tractable theoretical properties than the one proposed by Krampe et al .…”

“…Remark Note that Krampe et al base their approach on an estimator of the spectral density, whereas we start with an estimator of the n × n autocovariance matrix. The MA coefficients are a feature of the underlying autocovariance structure that is encoded in an equivalent way in either the spectral density or the (infinite) autocovariance matrix.…”

“…It is also possible to construct an MA‐sieve bootstrap using the newly estimated MA() coefficients (with diverging); this should be compared to the MA‐sieve bootstrap of Krampe et al but also to the linear process bootstrap (LPB) of (McMurry and Politis, ). The latter is an alternative resampling procedure that is also based on resampling the ϵ t errors of as if they were i.i.d.…”