Analysis of Financial Time Series, Third Edition

2.6 Simple ARMA Models

In some applications, the AR or MA models discussed in the previous sections become cumbersome because one may need a high-order model with many parameters to adequately describe the dynamic structure of the data. To overcome this difficulty, the autoregressive moving-average (ARMA) models are introduced; see Box, Jenkins, and Reinsel (1994). Basically, an ARMA model combines the ideas of AR and MA models into a compact form so that the number of parameters used is kept small, achieving parsimony in parameterization. For the return series in finance, the chance of using ARMA models is low. However, the concept of ARMA models is highly relevant in volatility modeling. As a matter of fact, the generalized autoregressive conditional heteroscedastic (GARCH) model can be regarded as an ARMA model, albeit nonstandard, for the series; see Chapter 3 for details. In this section, we study the simplest ARMA(1,1) model.

A time series rt follows an ARMA(1,1) model if it satisfies

2.25

where {at} is a white noise series. The left-hand side of the Eq. (2.25) is the AR component of the model and the right-hand side gives the MA component. The constant term is ϕ0. For this model to be meaningful, we need ϕ1 ≠ θ1; otherwise, there is a cancellation in the equation and the process ...