View source for Mixed autoregressive moving-average process

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''autoregressive moving-average process, ARMA process''

A wide-sense stationary [[Stochastic process|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|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 [[#References|[1]]] and [[#References|[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 [[#References|[3]]]). Special cases of autoregressive moving-average processes are auto-regressive processes (when  $  q = 0 $, 
cf. [[Auto-regressive process|Auto-regressive process]]) and moving-average processes (when  $  p = 0 $, 
cf. [[Moving-average process|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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.E.P. Box,   G.M. Jenkins,   "Time series analysis. Forecasting and control" , '''1–2''' , Holden-Day  (1976)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,   "The statistical analysis of time series" , Wiley  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Doob,   "The elementary Gaussian processes"  ''Ann. Math. Stat.'' , '''15'''  (1944)  pp. 229–282</TD></TR></table>

====Comments====
The class of autoregressive moving-average processes is of interest because they represent stationary processes with a rational spectral density.

The problem of representing a stationary process as an autoregressive moving-average process is known in the Western literature as the stochastic realization problem; see [[#References|[a2]]], [[#References|[a4]]] for references on this problem.

Autoregressive moving-average processes are used by statisticians [[#References|[a3]]], econometricians [[#References|[a1]]] and engineers [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Aoki,   "Notes on economic time series analysis: system theory perspectives" , ''Lect. notes in econom. and math. systems'' , '''220''' , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Faurre,   M. Clerget,   F. Germain,   "Opérateurs rationnels positifs" , Dunod  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.J. Hannan,   "Multiple time series" , Wiley  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Lindquist,   G. Picci,   "Realization theory for multivariate stationary Gaussian processes"  ''SIAM J. Control Optim.'' , '''23'''  (1985)  pp. 809–857</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Ljung,   T. Söderström,   "Theory and practice of recursive identification" , M.I.T.  (1983)</TD></TR></table>