An autoregressive moving-average model of orders 
and 
, denoted 
, combines an autoregressive
model with a finite moving-average term. Let 
be the shift operator defined
by 
.
For a centered time series, the model is
 |
(1)
|
where
 |  |  |
(2)
|
 |  |  |
(3)
|
The 
are uncorrelated innovations having population
mean zero and constant variance. A causal stationary
solution exists when every root of 
lies outside the unit circle,
and the model is invertible when every root of 
lies outside the unit circle.
For a causal model, the autocovariances 
at lags 
obey
 |
(4)
|
Dividing by 
gives the corresponding autocorrelation
values.
An autoregressive moving-average process can be represented in the Wolfram Language using ARMAProcess[a, b, v]. A seasonal autoregressive moving-average model adds autoregressive and moving-average terms at multiples of a seasonal period.
