2.5 Simple MA Models
We now turn to another class of simple models that are also useful in modeling return series in finance. These models are the moving-average (MA) models. As is shown in Chapter 5, the bid–ask bounce in stock trading may introduce an MA(1) structure in a return series. There are several ways to introduce MA models. One approach is to treat the model as a simple extension of white noise series. Another approach is to treat the model as an infinite-order AR model with some parameter constraints. We adopt the second approach.
There is no particular reason, but simplicity, to assume a priori that the order of an AR model is finite. We may entertain, at least in theory, an AR model with infinite order as
However, such an AR model is not realistic because it has infinite many parameters. One way to make the model practical is to assume that the coefficients ϕi's satisfy some constraints so that they are determined by a finite number of parameters. A special case of this idea is
where the coefficients depend on a single parameter θ1 via for i ≥ 1. For the model in Eq. (2.17) to be stationary, θ1 must be less than 1 in absolute value; otherwise, and the series will ...