Number of Parameters of ARMA Model

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 $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ 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 $p$:

$\tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$

Then $M$ has $p + q + 2$ 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 $\phi_1$ to $\phi_p$

the coefficients $\theta_1$ to $\theta_q$.

Thus:

there are $p$ parameters of the form $\phi_i$

there are $q$ parameters of the form $\theta_j$

$1$ parameter $\mu$

$1$ parameter $\sigma_a^2$.

That is: $p + q + 1 + 1 = p + q + 2$ 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: Mixed autoregressive -- moving average models