Definition:Box-Jenkins Model/ARIMA

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Definition

Let $S$ be a stochastic process based on an equispaced time series.

Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$

Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$

Let:

$w_t = \nabla^d z_t$

where $\nabla^d$ denotes the $d$th iteration of the backward difference operator.


Let $M$ be a model where the current value of $w_t$ is expressed as a combination of a finite linear aggregate of the past values along with a finite linear aggregate of the shocks:

$w_t = \phi_1 w_{t - 1} + \phi_2 w_{t - 2} + \dotsb + \phi_p w_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$

$M$ is known as an autoregressive integrated moving average (ARIMA) process of order $p$, $d$, $q$.


In practice, $d$ is usually $0$ or $1$, or at most $2$.


ARIMA Operator

Using the autoregressive operator:

$\map \phi B = 1 - \phi_1 B - \phi_2 B^2 - \dotsb - \phi_p B^p$

and the moving average operator:

$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$

the ARIMA model can be written in the following compact manner:

$\map \phi B w_t = \map \theta B a_t$

where $B$ denotes the backward shift operator.


Hence:

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

where:

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


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.


Also see

  • Results about ARIMA models can be found here.


Linguistic Note

The I in the acronym ARIMA stands for integrated.

This arises because the relationship which is the inverse of the backward difference operator:

$w_t = \map {\nabla^d} {z_t}$

is:

$z_t = S^d w_T$

where $S$ is the summation operator, defined as:

$S = \nabla^{-1} = \paren {1 - B}^{-1}$

so that:

$\map S {w_t} = \ds \sum_{j \mathop = 0}^\infty w_{t - j} = w_t + w_{t - 1} + w_{t - 2} + \dotsb$

Hence the autoregressive integrated moving average (ARIMA) process can be generated by summing or integrating the ARMA process a total of $d$ times.


Sources

$1$: Introduction:
$1.2$ Stochastic and Deterministic Dynamic Mathematical Models
$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Nonstationary models