Linear Models and Time-Series Analysis

7ARMA Processes

Parsimoniously parameterized time‐series models were developed as aids to short‐term forecasting, where the fiction that the analyst has discovered the ‘true’ model is innocuous. Such fiction, however, is far from innocuous when attempting to base inference about long‐run behavior on these fitted models.

(Paul Newbold et al., 1993)

The AR() and MA() time‐series models seen in the previous chapter are straightforward to combine, resulting in the so‐called ARMA() model—a very flexible model class capable of producing a variety of autocorrelation structures. The infinite AR and MA expansions of this model, as developed in Section 7.2 , will be seen to play an important role in estimation and forecasting, these being discussed in Sections 7.3 , 7.4 , and 7.5 . This is done within the ARMAX model, which augments the ARMA error structure with a set of regressors, as in Chapter 5. Section 7.6 builds on the material in Section 5.4 for obtaining an improved estimator of the AR(1) parameter. Finally, Section 7.7 briefly introduces ARMA‐type models that embody certain forms of nonlinearity, and/or can serve as an alternative to a near or exact unit root process.

7.1 Basics ...