Mixed autoregressive moving-average process
autoregressive moving-average process, ARMA process
A wide-sense stationary stochastic process with discrete time
the values of which satisfy a difference equation
![]() | (1) |
![]() |
where ,
,
being the Kronecker delta (i.e.
is a white noise process with spectral density
),
and
are non-negative integers, and
,
are constant coefficients. If all roots of the equation
![]() |
are of modulus distinct from 1, then the stationary autoregressive moving-average process exists and has spectral density
![]() |
where . However, for the solution of equation (1) with given initial values
to tend to the stationary process
as
, it is necessary that all roots of the equation
be situated outside the unit disc
(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 , cf. Auto-regressive process) and moving-average processes (when
, 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 |
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 [a2], [a4] for references on this problem.
Autoregressive moving-average processes are used by statisticians [a3], econometricians [a1] and engineers [a5].
References
[a1] | M. Aoki, "Notes on economic time series analysis: system theory perspectives" , Lect. notes in econom. and math. systems , 220 , Springer (1983) |
[a2] | P. Faurre, M. Clerget, F. Germain, "Opérateurs rationnels positifs" , Dunod (1979) |
[a3] | E.J. Hannan, "Multiple time series" , Wiley (1970) |
[a4] | A. Lindquist, G. Picci, "Realization theory for multivariate stationary Gaussian processes" SIAM J. Control Optim. , 23 (1985) pp. 809–857 |
[a5] | L. Ljung, T. Söderström, "Theory and practice of recursive identification" , M.I.T. (1983) |
Mixed autoregressive moving-average process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed_autoregressive_moving-average_process&oldid=47861