这是一份liverpool利物浦大学ECON311的成功案例

as a $p$ th order autoregressive process
$$
\phi(B) \bar{z}{t}=e{t}
$$
with $e_{t}$ following the qth order moving average process
$$
e_{t}=\theta(B) a_{t}
$$
as a $q$ th order moving average process
$$
z_{t}=\theta(B) b_{t}
$$
with $b_{t}$ following the $p$ th order autoregressive process
$$
\phi(B) b_{t}=a_{t}
$$
so that
$$
\phi(B) z_{\mathrm{r}}=\theta(B) \phi(B) b_{t}=\theta(B) a_{t}
$$

ECON311 COURSE NOTES ：

$$
\gamma(B)=\sum_{k=-\infty}^{\infty} \gamma_{k} B^{k}
$$
to give
$$
\begin{aligned}
\gamma(B) &=\sigma_{a}^{2} \sum_{k=-\infty}^{\infty} \sum_{j=0}^{\infty} \psi_{j} \psi_{j+k} B^{k} \
&=\sigma_{a}^{2} \sum_{j=0}^{\infty} \sum_{k=-j}^{\infty} \psi_{j} \psi_{j+k} B^{k}
\end{aligned}
$$
since $\psi_{h}=0$ for $h<0$. Writing $j+k=h$, so that $k=h-j$,
$$
\begin{aligned}
\gamma(B) &=\sigma_{a}^{2} \sum_{j=0}^{\infty} \sum_{h=0}^{\infty} \psi_{j} \psi_{k} B^{k-j} \
&=\sigma_{a}^{2} \sum_{h=0}^{\infty} \psi_{h} B^{h} \sum_{j=0}^{\infty} \psi_{j} B^{-j}
\end{aligned}
$$
that is
$$
\gamma(B)=\sigma_{a}^{2} \psi(B) \psi\left(B^{-1}\right)=\sigma_{a}^{2} \psi(B) \psi(F)
$$