Comparing AIC between ARMA and GARCH

Question

This is a basic theoretical doubt that I'd like to clarify with you.

Suppose that I want to compare AIC between ARMA, GARCH and ARMA-GARCH models which I fitted to the same data sample.

From this post Can AIC be used to compare an ARMA model to an ARMA-GARCH model?, I understand that ARMA models and ARMA-GARCH models fitted to the same sample can be compared in terms of AIC because specifying a GARCH model on top of an ARMA model does not change the resulting hybrid model dependent variable.

My question is:

Is it possible to compare goodness of fit between ARMA models and GARCH models that are fitted to the same sample even though they feature a different dependent variable? If so, can AIC be used for this purpose?

Answer 1

Actually, the dependent variable remains the same in ARMA and GARCH models, even though the models focus on different parameters of the underlying random process. That is, both models specify a distribution of the same dependent variable, but ARMA introduces a nonconstant conditional mean while GARCH introduces a nonconstant conditional variance. Still, the likelihood is calculated for the same dependent variable. So yes, you can use AIC to compare ARMA and GARCH.

On the other hand, be careful about losing initial observations while fitting the models. You need to make sure the likelihood is calculated for the same sample in both cases. For example, you cannot directly compare the likelihood of AR(1) and AR(2) estimated on the same sample if the models are fitted by OLS -- because AR(1) effectively loses the first observation while AR(2) loses the first two observations while fitting, and the likelihood is for observations 2 through $T$ in case of AR(1) but for 3 through $T$ in case of AR(2).