Definition:Box-Jenkins Model/ARMA
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
- 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: $(1.2.4)$
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: