A Frequency Domain Bootstrap for General Stationary Processes

TL;DR

This paper introduces a new frequency domain bootstrap method that is consistent for a broad class of stationary processes, extending beyond linear models, and effectively applies to various spectral mean statistics.

Contribution

The paper proposes a novel bootstrap approach using convolved periodograms that accurately captures dependence structures for a wide range of stationary processes.

Findings

The new bootstrap method is consistent for general stationary processes.

It improves accuracy for spectral mean and ratio statistics.

Finite sample simulations demonstrate its effectiveness.

Abstract

Existing frequency domain methods for bootstrapping time series have a limited range. Consider for instance the class of spectral mean statistics (also called integrated periodograms) which includes many important statistics in time series analysis, such as sample autocovariances and autocorrelations among other things. Essentially, such frequency domain bootstrap procedures cover the case of linear time series with independent innovations, and some even require the time series to be Gaussian. In this paper we propose a new, frequency domain bootstrap method which is consistent for a much wider range of stationary processes and can be applied to a large class of periodogram-based statistics. It introduces a new concept of convolved periodograms of smaller samples which uses pseudo periodograms of subsamples generated in a way that correctly imitates the weak dependence structure of the periodogram. %The new bootstrap procedure %corrects for those aspects of the distribution of spectral means that cannot be mimicked by existing procedures. We show consistency for this procedure for a general class of stationary time series, ranging clearly beyond linear processes, and for general spectral means and ratio statistics. Furthermore, and for the class of spectral means, we also show, how, using this new approach, existing bootstrap methods, which replicate appropriately only the dominant part of the distribution of interest, can be corrected. The finite sample performance of the new bootstrap procedure is illustrated via simulations.