Number of Parameters of Moving Average Model
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Theorem
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 an moving average model on $S$ of order $q$:
- $\tilde z_t = a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$
Then $M$ has $q + 2$ parameters.
Proof
By definition of the parameters of $M$:
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 $\theta_q$.
Thus:
- there are $q$ parameters of the form $\theta_j$
- $1$ parameter $\mu$
- $1$ parameter $\sigma_a^2$.
That is: $q + 1 + 1 = q + 2$ parameters.
$\blacksquare$
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: