Long Memory in Nonlinear Processes

TL;DR

This paper reviews nonlinear long memory models for time series exhibiting strong dependence, discussing their properties and estimation methods, motivated by applications in finance and other fields.

Contribution

It introduces various nonlinear long memory models, analyzing their properties and discussing both parametric and semiparametric estimation techniques.

Findings

Nonlinear models can capture long memory in series where linear models are insufficient.

Properties of different nonlinear long memory models are analyzed.

Estimation methods for these models are discussed in detail.

Abstract

It is generally accepted that many time series of practical interest exhibit strong dependence, i.e., long memory. For such series, the sample autocorrelations decay slowly and log-log periodogram plots indicate a straight-line relationship. This necessitates a class of models for describing such behavior. A popular class of such models is the autoregressive fractionally integrated moving average (ARFIMA) which is a linear process. However, there is also a need for nonlinear long memory models. For example, series of returns on financial assets typically tend to show zero correlation, whereas their squares or absolute values exhibit long memory. Furthermore, the search for a realistic mechanism for generating long memory has led to the development of other nonlinear long memory models. In this chapter, we will present several nonlinear long memory models, and discuss the properties of the models, as well as associated parametric andsemiparametric estimators.