How does the GARCH part affect the ACF/PACF of an ARMA-GARCH process? I need some help with fitting an ARMA-GARCH model.
I'm analyzing a daily time series. I don't understand how the order of the GARCH impacts ACF or PACF plots. I mean: what is the difference I should notice in the ACF plot when I'm using GARCH(1,1)+ARMA(1,1) instead of GARCH(2,1)+ARMA(1,1)? What 'part' of the ACF plot should change when I modify the GARCH or/and ARMA orders?
 A: An ARMA(p,q)-GARCH(r,s) model looks like this:
\begin{aligned}
x_t &= \mu_t + u_t, \\
\mu_t &= c + \varphi_1 x_{t-1} + \dots + \varphi_p x_{t-p} + \theta_1 u_{t-1} + \dots + \theta_q u_{t-q} \quad \text{("ARMA")}, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dots + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dots + \beta_r \sigma_{t-r}^2 \quad \text{("GARCH")}, \\
\varepsilon_t &\sim i.i.D(0,1),
\end{aligned}
where $D$ is some probability distribution with zero mean and unit variance.
The ACF/PACF of the original series $x_t$ would mostly be determined by the ARMA part, not the GARCH part. The GARCH part implies nonconstant variance of $u_t$, so you have an ARMA model with the noise part being a bit ill-behaved (not i.i.d.). This might blur the ACF/PACF a little bit, but the general patterns would be the same.
If you are interested in selecting and fitting an ARMA-GARCH model, then you would probably also look at the ACF/PACF plots for the model's residuals with the aim of model diagnostics. You may look at the raw standardized residuals $\hat\varepsilon_t$. If you change something in the equation for $\sigma_t^2$ (the GARCH part of ARMA-GARCH), you may not notice much difference in their ACF/PACF plots. However, you should notice some changes in the ACF/PACF plots of squared standardized residuals $\hat\varepsilon_t^2$.
