Definition:ARIMA Model/Motivation

ARIMA Model: Motivation

Suppose $S$ is a stochastic process which is non-stationary, and in particular does not have a constant mean level.

$S$ may still have some sort of homogeneous behaviour.

Although the general level about which there are deviations may change over time, the overall behaviour of $S$ may be the same if these changes are taken into account.

It may be possible to model such behaviour using a variant of an autoregressive operator $\map \varphi B$ such that the polynomial $\map \varphi B$ has one or more of its roots actually lying on the unit circle.

(From Necessary Condition for Autoregressive Process to be Stationary, this means that $S$ is non-stationary.)

Suppose there are $d$ such roots, then the autoregressive operator $\map \varphi B$ can be written as:

$\map \varphi B = \map \phi B \paren {1 - B}^d$

where $\map \phi B$ is the autoregressive operator for a stationary stochastic process.

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

$1$: Introduction:

$1.2$ Stochastic and Deterministic Dynamic Mathematical Models

$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Nonstationary models