ACF of a MA(2) process

Question

Does the ACF of a MA(2) process or generally other MA(q) processes show signs of slow decay? I have difficulties in pairing ACF and PACF graphs to AR or MA realizations.

Could anyone give me a short guidance?

Answer 1

The ACF for a MA process has a very drastic decay. Take a MA(2) for instance, $$x_t = \epsilon_t + \theta_1\epsilon_{t-1} + \theta_2\epsilon_{t-2}.$$

Then its autocovariances are given by \begin{aligned} \gamma(1) &= \operatorname{Cov}[x_t, x_{t-1}] = \left(\theta_1 + \theta_1\theta_2\right)\sigma^2\\ \gamma(2) &= \operatorname{Cov}[x_t, x_{t-2}] = \theta_2\sigma^2\\ \gamma(k) &= \operatorname{Cov}[x_t, x_{t-3}] = 0, \qquad k\ge 3. \end{aligned}