Definition:Moving Average Model

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Definition

Let $S$ be a stochastic process based on an equispaced time series.

Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$

Let $\tilde z_t$ be the deviation from a constant mean level $\mu$:

$\tilde z_t = z_t - \mu$

Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$


Let $M$ be a model where the current value of $\tilde z_t$ is expressed as a finite linear aggregate of the shocks:

$\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$

$M$ is known as a moving average (MA) process of order $q$.


Moving Average Operator

Let $\map \theta B$ be defined as:

$\map \theta B = 1 - \theta_1 B - \theta_2 B^2 - \dotsb - \theta_q B^q$

where $B$ denotes the backward shift operator.


Then $\map \theta B$ is referred to as the moving average operator.

Hence the moving average model can be written in the following compact manner:

$\tilde z_t = \map \theta B a_t$


Parameter

The parameters of $M$ consist of:

the constant mean level $\mu$
the variance $\sigma_a^2$ of the underlying (usually white noise) process of the independent shocks $a_t$
the coefficients $\theta_1$ to $\theta_q$.


Nomenclature

The name moving average model is a bit of a misnomer because:

$\text{(a)}: \quad$ the weights $1, -\theta_1, -\theta_2, \ldots$ need not all add up to $1$ (and in fact will generally not do so)
$\text{(b)}: \quad$ the weights need not in fact all be positive.

But the term is in common use, and so will be used on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also see

  • Results about moving average models can be found here.


Sources

$1$: Introduction:
$1.2$ Stochastic and Deterministic Dynamic Mathematical Models
$1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Moving average models $(1.2.3)$