Number of Parameters of Moving Average 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 $\tilde z_t$ 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 shock 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 $M$:

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.