Definition:Moving Average Model/Moving Average Operator
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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 moving average model on $S$ of order $q$:
- $\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$
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:
- $\tilde z_t = \map \theta B a_t$
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
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: