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Suppose I have one time series and two competing models that describe it. Model 1 is ARMA$(p_1,q_1)$, model 2 is ARMA$(p_2,q_2)$-GARCH$(r,s)$.

I obtain AIC values of model 1 and model 2. I would like to conclude that the model with the lower AIC value is the prefered model. However, I am not sure whether I can compare AIC values coming from two different classes of models such as ARMA and ARMA-GARCH.

My intuition is the following: if we find a model that nests both model 1 and model 2, we should be able to use AIC to compare the former two. Indeed, ARMA$(p_1,q_1)$ and ARMA$(p_2,q_2)$-GARCH$(r,s)$ are both nested within ARMA$(\max\{p_1,p_2\},\max\{q_1,q_2\})$-GARCH$(r,s)$ and may be obtained from the latter model by applying zero restrictions on coefficients.

However, I am not sure whether my approach makes sense. Could someone please elaborate on this issue and answer the question: can we, or can we not, use AIC to compare ARMA models with ARMA-GARCH models?

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Yes. AIC can be used to compare an ARMA model to an ARMA-GARCH model.

GARCH part is adding the variance equation. Specifying GARCH does not change the dependent variable and the loglikelihood is comparable. Hence AIC.

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