Correct procedure for modelling GARCH for forecasting volatility of stock Index returns

I will be using Eviews and am looking to forecast volatility of stock index returns using ARCH/GARCH models. I've generated the logarithmic returns and done the unit root tests. I then proceeded to plot the ACF and PAC functions of returns and squared returns to get an indication of the lags to include in my mean equation to remove autocorrelation.

Eviews has a way to to do ARIMA forecasting using multiple combinations to get the appropriate AR and MA terms for my mean equation using a pre-selected criteria such as lowest AIC. When you run this, it shows all the combinations and the respective AICS's. Therefore I do this, and obtain the relevant AR and MA terms.

Then I ran a Least Squares regression using these AR and MA terms from the automatic ARIMA forecasting, and after this I was able to see if there were ARCH effects using the residual diagnostics ARCH-LM test. I also run some other diagnostics.

Having seen that there are ARCH effects I proceed to estimate a GARCH(1,1). In my mean equation I used the same AR and MA terms generated in the Automatic ARIMA forecasting. After estimating, I check for significance, and run some residual diagnostics as well as checking fit for my model.

My question then is: Is there a need to do this ARIMA forecasting to find the most suitable AR and MA terms to use in the GARCH estimation? This procedure obviously will allow me to do a Least Squares regression for the AR and MA terms I get prior to estimating GARCH. It allows me to run Engle's ARCH-LM test for presence of ARCH effects, thus acting as a trigger for me to use GARCH models. Would the above procedure of running a Least Squares regression first be satisfactory, even if perhaps not optimal?

Alternatively, however, I have read a bit on this site (correct me if I'm wrong please) that one could skip the Automatic ARIMA forecasting and jump straight to GARCH estimation after plotting the ACF and PAC (i.e. joint estimation)? I believe this would entail entering different AR and MA combinations in the mean equation when estimating GARCH, recording the AIC's and log-likelihood function of every combination run and then selecting the GARCH estimation with the lowest AIC. After this one would then run the standard residual diagnostic tests on the optimal model, and proceed to the forecasting stage.

Any detailed explanation of this would be most appreciated, as I'm struggling to get my head round the concept. Preferably if anyone could provide an Eviews-centric step-by-step guide to modelling and forecasting volatility that would be even better.

• What I understood from this long post is that you need to select an appropriate ARMA-GARCH model and then forecast from it. Your questions seem to be, (1) How to select an ARMA-GARCH model? and (2) Is sequential model selection (first the ARMA part, then the GARCH part) satisfactory? Am I right? – Richard Hardy Mar 22 '17 at 8:53
• The answer to (2) is: stepwise model building is problematic because getting the ARMA order right depends on getting the GARCH order right, and vice versa. An alternative to consider for (1) is: check all possible combinations of ARMA(p,q)-GARCH(s,r) within, say, $0<p,q<4$ and $0<s,r<2$ (with $r=0$ whenever $s=0$), and pick those with the lowest AIC. Since there will be many models that have relatively similar and low AIC, it could make sense to use a bunch of them for forecasting and just average their forecasts. If you want to reduce the computational burden, you could fix $s=r=1$. – Richard Hardy Mar 22 '17 at 8:59
• However, stock index is unlikely to have genuine ARMA patterns, so you could set $p=q=0$ and focus entirely on the conditional variance. You could try something beyond the standard GARCH, e.g. GJR-GARCH (but there are many other alternatives, too). Now, If I got your questions right, I could probably post an answer afterwards. Let me know. – Richard Hardy Mar 22 '17 at 9:00
• Apologies for the late reply, had some commitments, but I should be fairly quick in responding now. Appreciate the help. In terms of how you've broken down my two questions.....yes you have it right......I wish to fix s,r to 1, as I'm intending to use a GARCH(1,1). The issue i'm having is because I'm using Eviews, if I don't do sequential (i.e. the ARMA part first) but opt for the joint estimation method, I can't do an ARCH LM test to reject the null hypothesis that there are no ARCH effects, this seems to be quite important to my supervisor. – Albe Mar 25 '17 at 20:51
• In any case since you seem to understand this subject in great detail and are willing to elaborate further, please do proceed. I'd also appreciate if you could elaborate further on: 1) Why stock index is unlikely to have ARMA patterns (with journal references). 2) Journal references to those who employ the sequential method and those who employ the joint estimation method you've described above......Thanks for the assistance :) NOTE: Please note I also do plan to use GJR, EGARCH, GARCH-M as you allude to, so any reply with this in mind might help you recommend the best approach. – Albe Mar 25 '17 at 20:51