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?