Definition:ARIMA Model/ARIMA 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 $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ 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 an ARIMA process on $S$:

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

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:

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

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

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