Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER TEN

Nonlinear and Long Memory Models

In previous chapters, the properties and uses of linear autoregressive moving average models have been extensively examined and illustrated for representing linear time series processes. Time series encountered in practice may not always exhibit characteristics of a linear process or for which a linear ARMA model will provide a good representation. Thus, in this chapter we explore some additional topics of special interest in relation to modeling that extend beyond the linear ARMA class. In Section 10.1 we consider models for conditional heteroscedastic time series, which are particularly relevant for economic and financial time series. Such series exhibit periods of differing degrees of volatility or (conditional) variability depending on the past history of the series. In Section 10.2 we introduce several classes of nonlinear time series models, which are capable of capturing some distinctive features in the behavior of processes that deviate from linear Gaussian time series. Section 10.3 looks at models for long memory processes, which are characterized by the much slower convergence to zero of their autocorrelation function ρ_k as k → ∞ compared to (short memory) ARMA processes.

10.1 AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTIC (ARCH) MODELS

In the ARMA(p, q) model ϕ(B)z_t = θ₀ + θ(B)a_t for time series z_t, when the errors a_t are independent random variables (with the usual assumptions of zero mean and constant variance ), an implication ...