Notwithstanding the software one uses, an ideal way to go about GARCH estimations is to let the data speak for itself. After prewhitening the data using an apropriate AR(p)-filter (let's say using AIC or SBC), test for ARCH effects using Engle's LM test. Since ARCH is a stylized fact, your dataset is more likely to possess ARCH efect. Then proceed with AR(p)-GARCH(1,1) filter of normal distribution (the default option in many statistical packages). Then subject the standardized residuals of the AR-GARCH(1,1) process to Engle's LM test, Ljung Box test and Engle& Ng test to test for residual traces of ARCH, serial correlation and asymmetry effects. Before doing so you may see if the model is stable. You may then keep on iterating this process with higher orders of GARCH processes and/or GARCH processes of different distributions (GED, student-T etc.) until you derive standardized residuals that are devoid of residual ARCH effects, residual autocorrelation and residual asymmetry effects. In essence, ideally speaking, you should end up getting standardized residuals that are closer to white noise. At that point, one stops.

Two researchers having access to the same datset may use different prewhitening filters (one may use an AR filter while the other may use an MA filter) and then proceed with GARCH modeling of the variance. It may also happen that the final mean-variance model in both the cases (in one case AR(p)-GARCH(p,q) and in the other case MA(q)-GARCH(p,q)) may be such that the standardized residuals are devoid of serial autocorrelation, asymmetry and ARCH effects. In such a scenario, both the models are equally robust from residual diagnostics point of view. Having said so, two or more models thatare equally robust from residual diagnostics point of view are likely to possess unequal forecasting capabilities.

Dear Vinodh Madhavan , thank you for your answer, which is proving to be very helpful.

I'm am experienced with econometrics so i was wondering if you can explain me how to " prewhitening the data using an apropriate AR(p)-filter". I understand that we can also add MA(q) or even both (AR(q) and MA(q)) but I have no idea how to rightly choose those terms.

The best way to choose values of p & q is to find ACF and PACF.

The best way to choose values of p & q is to find ACF and PACF.

Dear Muhammad Ali can you be more specific?

use the attached presentation!

Dear Mustafa Özer thank you for your useful answer.

Nonetheless, I was wondering if the process is the same if we have more than one explanatory variables. To be more specific, when testing for the right AR and\or MA values, and the right ARCH\GARCH type, should we use all our explanatory variables or only use our dependent variable followed by the constant? And what about the trend? Also, when should we add our explanatory variables?

Use them as exogeneous variables!