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

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)