ARIMA Models

A small, yet comprehensive, class of stationary time-series models consists of the nonseasonal ARMA (Autoregressive Moving Average) processes defined by

\[\phi(B) (W_t - \mu) = \theta(B)A_t,\; t \in Z,\]

where \(Z = \{\ldots,-2,-1,0,1,2,\ldots\}\) denotes the set of integers, B is the backward shift operator defined by \(B^kW_t=W_{t-k}\), \(\mu\) is the mean of \(W_t\), and the following equations are true:

\[\begin{split}\phi(B) = 1 -\phi_1 B-\phi_2 B^2-\ldots-\phi_p B^p, p \ge 0 \\ \theta(B) = 1 -\theta_1 B-\theta_2 B^2-\ldots-\theta_q B^q, q \ge 0 \,.\end{split}\]

The model is of order (p, q) and is referred to as an ARMA(p, q) model.

An equivalent version of the ARMA(p, q) model is given by

\[\phi(B) W_t = \theta_0 + \theta (B) A_t, \; t \in Z,\]

where \(\theta_0\) is an overall constant defined by the following:

\[\theta_0 = \mu \left( 1- \sum_{i=1}^p \phi_i \right) \,.\]

See [1], p. 97, for a discussion of the meaning and usefulness of the overall constant.

If the “raw” data, \(\{Z_t\}\), are homogeneous and nonstationary, then differencing induces stationarity, and the model is called ARIMA (AutoRegressive Integrated Moving Average). Parameter estimation is performed on the stationary time series \(W_t=\nabla^d Z_t\), where \(\nabla^d=(1-B)^d\) is the backward difference operator with period 1 and order d, d>0.

[1]	Box, G., G. Jenkins and G. Reinsel (1994), Time Series Analysis : Forecasting and Control, Prentice Hall, New Jersey.