Definition:Box-Jenkins Model/ARMA

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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 $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:

$\tilde z_t = z_t - \mu$

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

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

$M$ is known as a mixed autoregressive (order $p$), moving average (order $q$) process, usually referred as an ARMA process.


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

$\map \phi B \tilde z_t = \map \theta B a_t$

where $B$ denotes the backward shift operator.

Hence:

$\tilde z_t = \map {\phi^{-1} } B \map \theta B a_t$


Parameter

The parameters of $M$ consist of:

the constant mean level $\mu$
the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$
the coefficients $\phi_1$ to $\phi_p$
the coefficients $\theta_1$ to $\theta_q$.


Also see

  • Results about ARMA models can be found here.


Sources

$1$: Introduction:
$1.2$ Stochastic and Deterministic Dynamic Mathematical Models
$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Mixed autoregressive -- moving average models: $(1.2.4)$
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