Demeand MA(1):

$$

MA(1): \quad x_t = \epsilon_t + \theta \times \epsilon_{t-1}, \epsilon_t \sim iid(0,\sigma^2_{\epsilon}); \quad x_0 = 0, \; \theta \in \mathcal{R}, \; \sigma^2_{\epsilon} \in \mathcal{R}_+ .

$$

Demeand MA(q):

$$

MA(q): \quad x_t = \epsilon_t + \theta_1 \times \epsilon_{t-1} + \dotsc +\theta_q \times \epsilon_{t-q}, \quad x_0 = 0, \; \theta_1, ..., \theta_q \in \mathcal{R}

$$

How to define q with ACF and PACF

To define a MA process the ACF should show a sharp drop after a certain q number of lags while PACF should show a geometric or gradual decreasing trend.

Short: use ACF, q is amount of the significant lags before the drop

Example

The ACF and PACF plots indicate that an MA (1) model would be appropriate for the time series because the ACF cuts after 1 lag while the PACF shows a slowly decreasing trend.