自相关(ACF)与偏自相关(PACF)（2）

§3.The Autocorrelation Function for MA(1) Process
Consider the MA(1) process：
$$y_t={\epsilon }_t+\beta {\epsilon }_{t-1}$$

We can apply the Yule-Walker equations to derive(注意：$\{y_t\}$为白噪声序列的线性组合，是平稳的):
${\gamma }_0=E{y}_t{y}_t=E[({\epsilon }_t+\beta {\epsilon }_{t-1})({\epsilon }_t+\beta {\epsilon }_{t-1})]=(1+{\beta }^2){\sigma }^2$
${\gamma }_1=E{y}_t{y}_{t-1}=E[({\epsilon }_t+\beta {\epsilon }_{t-1})({\epsilon }_{t-1}+\beta {\epsilon }_{t-2})]=\beta {\sigma }^2$
and
${\gamma }_s=E[({\epsilon }_t+\beta {\epsilon }_{t-1})({\epsilon }_{t-s}+\beta {\epsilon }_{t-s-1})]=0,foralls>1$
Hence,${\rho }_0=1,{\rho }_1=\beta /(1+{\beta }^2)$,${\rho }_s=0,foralls>1$