Definition:Moving Average Model
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
- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):
- $1$: Introduction:
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1.2.1$ Stationary and Nonstationary Stochastic Models for Forecasting and Control: Moving average models $(1.2.3)$
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: