# 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.

• Anyone able to clarify on this please?
– Albe
Mar 19, 2017 at 0:35

I believe your question is closely related to this question : How should we select efficiently orders parameters in time series modelling? . (ps : the accepted answer did not really answer it...)

When modelling both the mean and variance process together, we must choose either to determine the order of lags in the mean process first and the conditional variance in a second step (what I called a two step approach) or to determine orders of lag for the mean and variance process together (direct approach). And you are right, depending of the method you use, you'll end up with different combinations. I believe the current literature is not clear about this issue.

If you don't have a solid economical reason to choose one method over the other one, and with a pure econometric point of view I would recommand you to employ the direct approach as it covers a much larger parameter space (i.e you test more combinations with the direct approach than with the two step approach).

• Thanks for replying. Just to clarify, the two step approach would be similar to what I described in my original post from paragraphs 1-5 correct?.....Whereas the direct approach would be what I describe in my second last paragraph, correct?.......Could you kindly provide literature/links to references where both of these methodologies are employed, as I'm doing a paper, and would like to read, analyse and then cite. Thank you.
– Albe
Mar 20, 2017 at 21:20
• In the other link you provided with an example "Example: I fit all ARMA(p,q) to the series with (p,q)=0:2 and select the most parsimonious one. Let’s say the best model is p=1 and q=2. Second step : if fit all ARMA(1,2)-GARCH(s,t) models to the serie with (s,t)=0:2 and I select the "best" s,t parameters using rule A again.".......The difference with what I'm doing and the example you provided is that I keep s,t fixed as I wish to do a GARCH(1,1) rather than using rule A. However using the same AR and MA terms for my GARCH estimation leads to some of the AR and MA terms not being significant.
– Albe
Mar 20, 2017 at 21:30
• First comment , yes 1-5 paragraphs are the two step approach. I have no references, I haven't read papers that explicitly mention this methodological issue, usually researches just employ one method without more details. Note that the "terminology" direct vs two step methods is my own and is not recognized by the community. Mar 21, 2017 at 14:37
• Second comment: If you fit the ARMA part alone and then you include the GARCH term (two steps methods), you'll indeed see that ARMA coefficients changes, that's why I think it is more efficient to fit directly the model with the GARCH term included (direct approach). At the end you still may get some coefficient not being significant must it is, per se, not a serious concern. The important parts are : that you are able to justify you choice of lags (AIC method for instance) and that model's hypothesis are fulfilled ( IID standardized errors ) Mar 21, 2017 at 14:43
• Malick, from a statistical point of view, direct approach is nicer as it is more efficient, while the two-step approach can be even inconsistent (at least I have not seen a proof it would always be consistent). However, direct approach will be more computationally demanding. @Albe, "model hypothesis" should be "model assumptions" in this case. Apr 3, 2017 at 12:15