Definition:ARMA Model/Parameter

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 shock $a_t$
 * 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.