On the range of validity of the autoregressive sieve bootstrap

TL;DR

This paper broadens the understanding of the AR-sieve bootstrap, demonstrating its validity for a wide class of stationary processes beyond linear models, and providing a practical criterion for its applicability.

Contribution

It extends the applicability of the AR-sieve bootstrap to all stationary, nondeterministic processes with positive spectral density, and offers a simple test for its validity in various situations.

Findings

AR-sieve bootstrap valid for a broad class of stationary processes

Main theorem provides a practical validity assessment tool

Counterexample shows limitations for autocovariance estimation

Abstract

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.