When one introduces ARCH or GARCH errors into an ARIMA models, sometimes the "best" (lowest IC) will change using automated software (e.g. auto.arima).
In a theoretical sense I understand ARCH and GARCH errors to be something which is estimated "on top of" the ARIMA model. If that is true, it makes little sense that the model changes when using other errors. However in practice, I know that the two equations (one for the unconditional mean and one for the errors) are estimated simultaneously. Is it that estimation feature that makes the optimal ARIMA model change, or is it due to other factors?
