# Information criteria via different GARCH models

I have a question about comparison of different GARCH models via information criteria.

I use rugarch package. So, let's have 3 model types: "sGARCH", "eGARCH", "gjrGARCH". I fit all 3 type models for same time series, for example, for p=q=1. Could I use IC (BIC, for example) for best model selection among these 3 types? If not, what's the best method for this?

Thank you.

In short: Yes, you can use the $BIC$ (and $AIC$) information criteria, assuming the following:

• All models are applied on exactly the same data set.

• All model parameters are estimated via maximum likelihood estimation.

• Your sample size is much bigger, than the amount of parameters in the models (as GARCH models don't have many parameters, this will probably be no issue)

Even though it is often claimed, the compared models don't need to be nested for the $AIC$ and $BIC$ to be valid.

Lastly, note that there is no "best" way in model selection. It all depends on the purpose of the model that you are looking for - usually, in a real life setting, the true model isn't even one of the candidates. However, $AIC$ and $BIC$ can usually give you a good idea, which of the models is preferable.

• Yes, all models are estimated via MLE on same dataset. But main difference in GARCH part - there are sufficiently strong difference between models in formula. This is not a problem for IC, isn't it? – Dmitriy Jun 28 '17 at 9:54
• That is not an issue - as I said, the models don't have to be nested, which means in particular that their structure may vary. You only have to be careful, when comparing different likelihood functions, i.e. GARCH and GARCH-t. There is a nice summary about this topic (for the AIC) here. – Eldioo Jun 28 '17 at 11:57
• Thank you! So, I can't directly compare GARCH(1,1)-Normal and GARCH(1,1)-Student, can me? – Dmitriy Jun 28 '17 at 16:01
• I might have been a little unclear - you certainly can use it, but should note that the difference in the AIC/BIC will not only be due to better fitting volatility estimates, but also due to better fit of the data itself. Assuming that you're mostly interested in modelling the standard deviation, you should keep this in mind - the difference in AIC between GARCH and GARCH-t could be relatively large, but potentially have little impact on the actual volatility estimates, as the distribution doesn't directly affect the GARCH variance equation - only the ML estimates. – Eldioo Jun 28 '17 at 21:17