  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.
