Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER THREE

Linear Stationary Models

A general linear stochastic model is described that supposes a time series to be generated by a linear aggregation of random shocks. For practical representation it is desirable to employ models that use parameters parsimoniously. Parsimony may often be achieved by representation of the linear process in terms of a small number of autoregressive and moving average terms. The properties of the resulting autoregressive–moving average (ARMA) models are discussed in preparation for their use in model building.

3.1 GENERAL LINEAR PROCESS

3.1.1 Two Equivalent Forms for the Linear Process

In Section 1.2.1 we discussed the representation of a stochastic process as the output from a linear filter, whose input is white noise a_t, that is,

where is the deviation of the process from some origin, or from its mean, if the process is stationary. The general linear process (3.1.1) allows us to represent as a weighted sum of present and past values of the “white noise” process a_t. The literature provides many important references on the development of linear stochastic models [32, 106, 130, 135, 136, 178, 239, 240, 251, 265, 298, 311, 317]. The white noise process ...