Calculate the autocorr. function of ARMA process I'm new to time series. I would like to calculate this.. but I really don't know how to begin... 
ARMA(1,2)
$X_{t}=\underbrace{\phi X_{t-1}}_{AR(1)}+\underbrace{\epsilon_{t}-\theta_{1}\epsilon_{t-1}-\theta_{2}\epsilon_{t-2}}_{MA(2)}$
AUTOCOV. FUNCTION: 
$\gamma_{t, t+h}=\operatorname{Cov}\left(X_{t}, X_{t+h})=E\left[\left(X_{t}-\mu_{t}\right)\left(X_{t+h}-\mu_{t+h}\right)\right]\right.$
AUTOCORR.FUNCTION OF $X$:
$\rho_{t, t+h}=\frac{\gamma_{t, t+h}}{\sqrt{\gamma_{t, t} \gamma_{t+h, t+h}}}=\frac{\gamma_{t, t+h}}{\sigma_{t} \sigma_{t+h}}$
All useful comments will be rewarded.
 A: In general the autocovariance function satisfies
\begin{align}
\gamma_k
  &=E(X_{t-k} X_t)
\\&=E(X_{t-k} (\phi X_{t-1}+\epsilon_t-\theta_1\epsilon_{t-1}-\theta_2\epsilon_{t-2}))
\\&=\phi\gamma_{k-1} + E(X_{t-k}\epsilon_t)-\theta_1 E(X_{t-k}\epsilon_{t-1})-\theta_2 E(X_{t-k}\epsilon_{t-2}). \tag{1}
\end{align}
Setting $k=0$, (1) simplifies to 
$$
\gamma_0 = \phi\gamma_1 + \sigma_\epsilon^2(1-\theta_1(\phi-\theta_1)-\theta_2(\phi(\phi-\theta_1)-\theta_2)), \tag{2}
$$
since 
\begin{align}
E(X_{t}\epsilon_t)
  &=E((\phi X_{t-1}+\epsilon_t-\theta_1\epsilon_{t-1}-\theta_2\epsilon_{t-2})\epsilon_{t}) 
\\&=E\epsilon_t^2=\sigma_\epsilon^2, \tag{2a}
\\E(X_{t}\epsilon_{t-1})
  &=E((\phi X_{t-1}+\epsilon_t-\theta_1\epsilon_{t-1}- 
  \theta_2\epsilon_{t-2})\epsilon_{t-1}) 
\\&=\phi E(X_{t-1}\epsilon_{t-1}) -\theta_1 E\epsilon_{t-1}^2, 
\\&=\phi\sigma_\epsilon^2 - \theta_1\sigma_\epsilon^2, \tag{2b}
\\E(X_{t}\epsilon_{t-2})
  &=E((\phi X_{t-1}+\epsilon_t-\theta_1\epsilon_{t-1}-\theta_2\epsilon_{t-2})\epsilon_{t-2})
\\&=\phi E(X_{t-1}\epsilon_{t-2})-\theta_2E\epsilon_{t-2}^2, 
\\&=\phi(\phi\sigma_\epsilon^2 - \theta_1\sigma_\epsilon^2)-\theta_2\sigma_\epsilon^2. \tag{2c}
\end{align}
Note how (2c) follows from (2b) which follows from (2a).
Similarly, by setting $k=1$ and $k=2$, you obtain two more equations in addition to (2) (not included since this is self-study) that you can solve for the three unknowns $\gamma_0$, $\gamma_1$ and $\gamma_2$.
For $k>2$ (the order of the MA part), since $X_{t-k}$ is then uncorrelated with $\epsilon_t$, $\epsilon_{t-1}$, and $\epsilon_{t-2}$, (1) simplifies to
$$
\gamma_k=\phi \gamma_{k-1}.
$$
More generally, for an ARMA$(p,q)$ model, one can show that $\gamma_k$ for lags $k>q$ satisfies the linear difference equation $\phi(B)\gamma_k=0$, where $\phi(B)$ is the AR operator polynomial of the model, see e.g. Wei (2007), ch. 3
