Definition:ARMA Model/Parameter
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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 $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations 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 ARMA model on $S$ of order $p$:
- $\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$
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 $\phi_1$ to $\phi_p$
- the coefficients $\theta_1$ to $\theta_q$.
In practice, each of these parameters needs to be estimated from the data.
It is often the case that an ARMA model can be effectively used in real-world applications where $p$ and $q$ are no greater than $2$, and often less.
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: Mixed autoregressive -- moving average models
- $1.2$ Stochastic and Deterministic Dynamic Mathematical Models
- $1$: Introduction: