# Mixed autoregressive moving-average process

autoregressive moving-average process, ARMA process

A wide-sense stationary stochastic process $X ( t)$ with discrete time $t = 0 , \pm 1 \dots$ the values of which satisfy a difference equation

$$\tag{1 } X ( t) + a _ {1} X ( t - 1 ) + \dots + a _ {p} X ( t - p ) =$$

$$= \ Y ( t) + b _ {1} Y ( t - 1 ) + \dots + b _ {q} Y ( t - q ) ,$$

where ${\mathsf E} Y ( t) = 0$, ${\mathsf E} Y ( t) Y ( s) = \sigma ^ {2} \delta _ {ts}$, $\delta _ {ts}$ being the Kronecker delta (i.e. $Y ( t)$ is a white noise process with spectral density $\sigma ^ {2} / 2 \pi$), $p$ and $q$ are non-negative integers, and $a _ {1} \dots a _ {p}$, $b _ {1} \dots b _ {q}$ are constant coefficients. If all roots of the equation

$$\phi ( z) = 1 + a _ {1} z + \dots + a _ {p} z ^ {p} = 0$$

are of modulus distinct from 1, then the stationary autoregressive moving-average process $X ( t)$ exists and has spectral density

$$f ( \lambda ) = \frac{\sigma ^ {2} }{2 \pi } \frac{| \psi ( e ^ {i \lambda } ) | ^ {2} }{| \phi ( e ^ {i \lambda } ) | ^ {2} } ,$$

where $\psi ( z) = 1 + b _ {1} z + \dots + b _ {q} z ^ {q}$. However, for the solution of equation (1) with given initial values $X ( t _ {0} - 1 ) \dots X ( t _ {0} - p )$ to tend to the stationary process $X ( t)$ as $t - t _ {0} \rightarrow \infty$, it is necessary that all roots of the equation $\phi ( z) = 0$ be situated outside the unit disc $| z | \leq 1$( see [1] and [2], for example).

The class of Gaussian autoregressive moving-average processes coincides with the class of stationary processes that have a spectral density and are one-dimensional components of multi-dimensional Markov processes (see [3]). Special cases of autoregressive moving-average processes are auto-regressive processes (when $q = 0$, cf. Auto-regressive process) and moving-average processes (when $p = 0$, cf. Moving-average process).

Generalizations of autoregressive moving-average processes are the autoregressive integrated moving-average processes introduced by G.E.P. Box and G.M. Jenkins (see [1]) and often used in applied problems. These are non-stationary processes with stationary increments such that the increments of some fixed order form an autoregressive moving-average process.

#### References

 [1] G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , 1–2 , Holden-Day (1976) [2] T.W. Anderson, "The statistical analysis of time series" , Wiley (1971) [3] J.L. Doob, "The elementary Gaussian processes" Ann. Math. Stat. , 15 (1944) pp. 229–282