Moving-average process

A stochastic process which is stationary in the wide sense and which can be obtained by applying some linear transformation to a process with non-correlated values (that is, to a white noise process). The term is often applied to the more special case of a process $X ( t)$ in discrete time $t = 0 , \pm 1 \dots$ that is representable in the form

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

where ${\mathsf E} Y ( t) = 0$, ${\mathsf E} Y ( t) Y ( s) = \sigma ^ {2} \delta _ {ts}$, with $\delta _ {ts}$ the Kronecker delta (so that $Y ( t)$ is a white noise process with spectral density $\sigma ^ {2} / 2 \pi$), $q$ is a positive integer, and $b _ {1} \dots b _ {q}$ are constant coefficients. The spectral density $f ( \lambda )$ of such a process is given by

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

$$\psi ( z) = b _ {0} + b _ {1} z + \dots + b _ {q} z ^ {q} ,\ b _ {0} = 1 ,$$

and its correlation function $r ( k) = {\mathsf E} X ( t) X ( t - k )$ has the form

$$r ( k) = \sigma ^ {2} \sum _ { j= } 0 ^ { {q } - | k | } b _ {j} b _ {j + | k | } \ \ \textrm{ if } | k | \leq q ,$$

$$r ( k) = 0 \ \textrm{ if } | k | > q .$$

Conversely, if the correlation function $r ( k)$ of a stationary process $X ( t)$ in discrete time $t$ has the property that $r ( k) = 0$ when $| k | > q$ for some positive integer $q$, then $X ( t)$ is a moving-average process of order $q$, that is, it has a representation of the form (1) where $Y ( t)$ is a white noise (see, for example, [1]).

Along with the moving-average process of finite order $q$, which is representable in the form (1), there are two types of moving-average processes in discrete time of infinite order, namely: one-sided moving-average processes, having a representation of the form

$$\tag{2 } X ( t) = \ \sum _ { j= } 0 ^ \infty b _ {j} Y ( t - j ) ,$$

where $Y ( t)$ denotes white noise and the series on the right-hand side converges in mean-square (so that $\sum _ {j=} 0 ^ \infty | b _ {j} | ^ {2} < \infty$), and also more general two-sided moving-average processes, of the form

$$\tag{3 } X ( t) = \ \sum _ {j = - \infty } ^ \infty b _ {j} Y ( t - j ) ,$$

where $Y ( t)$ denotes white noise and $\sum _ {j = - \infty } ^ \infty | b _ {j} | ^ {2} < \infty$. The class of two-sided moving-average processes coincides with that of stationary processes $X ( t)$ having spectral density $f ( \lambda )$, while the class of one-sided moving-average processes coincides with that of processes having spectral density $f ( \lambda )$ such that

$$\int\limits _ {- \pi } ^ \pi \mathop{\rm log} f ( \lambda ) \ d \lambda > - \infty$$

(see [2], [1], [3]).

A continuous-time stationary process $X ( t)$, $- \infty < t < \infty$, is called a one-sided or two-sided moving-average process if it has the form

$$X ( t) = \int\limits _ { 0 } ^ \infty b ( s) d Y ( t - s ) ,\ \ \int\limits _ { 0 } ^ \infty | b ( s) | ^ {2} d s < \infty ,$$

or

$$X ( t) = \int\limits _ {- \infty } ^ \infty b ( s) d Y ( t - s ) ,\ \ \int\limits _ {- \infty } ^ \infty | b ( s) | ^ {2} d s < \infty ,$$

respectively, where ${\mathsf E} [ d Y ( t) ] ^ {2} = \sigma ^ {2} d t$, that is, $Y ^ \prime ( t)$ is a generalized white noise process. The class of two-sided moving-average processes in continuous time coincides with that of stationary processes $X ( t)$ having spectral density $f ( \lambda )$, while the class of one-sided moving-average processes in continuous time coincides with that of processes having spectral density $f ( \lambda )$ such that

$$\int\limits _ {- \infty } ^ \infty \mathop{\rm log} f ( \lambda ) ( 1 + \lambda ^ {2} ) ^ {-} 1 \ d \lambda > - \infty$$

(see [4], [3], [5]).

References

[1]	T.M. Anderson, "The statistical analysis of time series" , Wiley (1971)
[2]	A.N. Kolmogorov, "Stationary sequences in Hilbert space" T. Kailath (ed.) , Linear Least-Squares Estimation , Benchmark Papers in Electric Engin. Computer Sci. , 17 , Dowden, Hutchington & Ross (1977) pp. 66–89 (Translated from Russian)
[3]	J.L. Doob, "Stochastic processes" , Wiley (1953)
[4]	K. Karhunun, "Ueber lineare Methoden in der Wahrscheinlichkeitsrechnung" Ann. Acad. Sci. Fennicae Ser. A. Math. Phys. , 37 (1947)
[5]	Yu.A. Rozanov, "Stationary random processes" , Holden-Day (1967) (Translated from Russian)

Comments

Both auto-regressive processes (cf. Auto-regressive process) and moving-average processes are special cases of so-called ARMA processes, i.e. auto-regressive moving-average processes (cf. Mixed autoregressive moving-average process), which are of great importance in the study of time series.