Estimators of Long-Memory: Fourier versus Wavelets

TL;DR

This paper compares Fourier and wavelet methods for estimating long-memory in time series, analyzing their theoretical properties, practical performance, and providing simulation results to guide method selection.

Contribution

It offers a comprehensive comparison of Fourier and wavelet estimators, highlighting their strengths, weaknesses, and conditions for optimal use, supported by simulations and software tools.

Findings

Wavelet methods perform better with non-stationary data.

Fourier methods are more effective for stationary processes.

Simulation results demonstrate the practical differences between methods.

Abstract

There have been a number of papers written on semi-parametric estimation methods of the long-memory exponent of a time series, some applied, others theoretical. Some using Fourier methods, others using a wavelet-based technique. In this paper, we compare the Fourier and wavelet approaches to the local regression method and to the local Whittle method. We provide an overview of these methods, describe what has been done, indicate the available results and the conditions under which they hold. We discuss their relative strengths and weaknesses both from a practical and a theoretical perspective. We also include a simulation-based comparison. The software written to support this work is available on demand and we illustrate its use at the end of the paper.