(Solved): Calculate MA(3) Moving Average Process MA(1) \[ y_{t}=\mu+e_{t}+\theta e_{t-1} \] \( e_{t} \) is w ...

Calculate MA(3)

Moving Average Process MA(1) \[ y_{t}=\mu+e_{t}+\theta e_{t-1} \] \( e_{t} \) is white noise with \( V\left[e_{t}\right]=\sigma^{2} \) \( y_{t} \) is autocorrelated \[ \begin{array}{l} E\left[y_{t}\right]=\mu \\ \gamma_{0}=V\left[y_{t}\right]=E\left[\left(y_{t}-\mu\right)^{2}\right]=\left(1+\theta^{2}\right) \sigma^{2} \\ \gamma_{1}=\operatorname{cov}\left(\mathrm{y}_{\mathrm{t}}, \mathrm{y}_{\mathrm{t}-1}\right)=\mathrm{E}\left[\left(y_{t}-\mu\right)\left(y_{t-1}-\mu\right)\right]=\theta \sigma^{2} \\ \rho_{1}=\operatorname{corr}\left(y_{t}, y_{t+h}\right)=\frac{\theta}{1+\theta^{2}} \\ \rho_{h}=\gamma_{h}=0 \text { for } h \geq 2 \\ M A(1) \text { is covariance stationary } \end{array} \] Moving Average Process MA(2) \[ y_{t}=\mu+e_{t}+\theta_{1} e_{t-1}+\theta_{2} e_{t-2} \] \( e_{t} \) is white noise with \( V\left[e_{t}\right]=\sigma^{2} \) \( y_{t} \) is autocorrelated \( E\left[y_{t}\right]=\mu \) \( \gamma_{0}=V\left[y_{t}\right]=E\left[\left(y_{t}-\mu\right)^{2}\right]=\left(1+\theta_{1}^{2}+\theta_{2}^{2}\right) \sigma^{2} \) \( \gamma_{1}=\operatorname{cov}\left(\mathrm{y}_{\mathrm{t}}, \mathrm{y}_{\mathrm{t}-1}\right)=\left(\theta_{1}+\theta_{1} \theta_{2}\right) \sigma^{2} \) \( \gamma_{1}=\operatorname{cov}\left(\mathrm{y}_{\mathrm{t}}, \mathrm{y}_{\mathrm{t}-2}\right)=\theta_{2} \sigma^{2} \) \( \gamma_{h}=0 \) for \( h \geq 3 \) \( M A(2) \) is covariance stationary