Definition:ARMA Model/ARMA Operator

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 an ARMA model on $S$ of order $p$:

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

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:

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

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: Mixed autoregressive -- moving average models