19.8 MA(2)

When a stationary process \(\left(Y_{t}\right)\) follows a \(\operatorname{MA}(2),\)

\[ Y_{t}=\mu+\omega_{t}-\theta_{1} \omega_{t-1}-\theta_{2} \omega_{t-2} \]

with \(\theta_{2}+\theta_{1}<1, \theta_{2}-\theta_{1}<1,\left|\theta_{2}\right|<1\) (see 2.3.1) y \(\left(\omega_{t}\right) \sim \operatorname{IID}\left(0, \sigma_{\omega}^{2}\right)\), it can be seen that:

• Mean:

\[ \mu_{Y}=\mu \]

• ACF:

\[ \rho_{k}=\left\{\begin{array}{ll} -\left[\theta_{1}\left(1-\theta_{2}\right)\right] /\left(1+\theta_{1}^{2}+\theta_{2}^{2}\right) & \text { si } k=1 \\ -\theta_{2} /\left(1+\theta_{1}^{2}+\theta_{2}^{2}\right) & \text { si } k=2 \\ 0 & \text { para todo } k>2 . \end{array}\right. \]

• Variance: \[ \sigma_{Y}^{2}=\left(1+\theta_{1}^{2}+\theta_{2}^{2}\right) \sigma_{\omega}^{2} \text { . } \]