Stationary ARMA Processes

Abstract

In this chapter we introduce an extremely important class of time series {X _t , t = 0, ± 1, ± 2,...} defined in terms of linear difference equations with constant coefficients. The imposition of this additional structure defines a parametric family of stationary processes, the autoregressive moving average or ARMA processes. For any autocovariance function γ(·) such that lim _h→∞ γ(h) = 0, and for any integer k > 0, it is possible to find an ARMA process with autocovariance function γ _X (·) such that γ _X (h) = γ(h), h = 0, 1,...., k. For this (and other) reasons the family of ARMA processes plays a key role in the modelling of time-series data. The linear structure of ARMA processes leads also to a very simple theory of linear prediction which is discussed in detail in Chapter 5.