Measuring stationarity in long-memory processes

TL;DR

This paper introduces a method to measure stationarity in long-memory processes using an $L_2$-distance based on spectral density, providing a test for stationarity and a bootstrap procedure for improved accuracy.

Contribution

It proposes a novel $L_2$-distance measure for stationarity in long-memory processes and develops a bootstrap method for better hypothesis testing accuracy.

Findings

Asymptotic normality of the spectral distance estimate is established.

The bootstrap procedure is proven to be consistent.

Differences between Riemann sums and integrals in spectral analysis are demonstrated.

Abstract

In this paper we consider the problem of measuring stationarity in locally stationary long-memory processes. We introduce an $L_2$-distance between the spectral density of the locally stationary process and its best approximation under the assumption of stationarity. The distance is estimated by a numerical approximation of the integrated spectral periodogram and asymptotic normality of the resulting estimate is established. The results can be used to construct a simple test for the hypothesis of stationarity in locally stationary long-range dependent processes. We also propose a bootstrap procedure to improve the approximation of the nominal level and prove its consistency. Throughout the paper, we will work with Riemann sums of a squared periodogram instead of integrals (as it is usually done in the literature) and as a by-product of independent interest it is demonstrated that the two approaches behave differently in the limit.