I believe your question is closely related to this question : How should we select efficiently orders parameters in time series modelling? . (ps : the accepted answer did not really answer it...)

When modelling both the mean and variance process together, we must choose either to determine the order of lags in the mean process first and the conditional variance in a second step (what I called a two step approach) or to determine orders of lag for the mean and variance process together (direct approach). And you are right, depending of the method you use, you'll end up with different combinations. I believe the current literature is not clear about this issue.

If you don't have a solid economical reason to choose one method over the other one, and with a pure econometric point of view I would recommand you to employ the direct approach as it covers a much larger parameter space (i.e you test more combinations with the direct approach than with the two step approach).

I believe your question is closely related to this question : How should we select efficiently orders parameters in time series modelling? . (ps : the accepted answer did not really answer it...)

When modelling both the mean and variance process together, we must choose either to determine the order of lags in the mean process first and the conditional variance in a second step (what I called a two step approach) or to determine orders of lag for the mean and variance process together (direct approach). And you are right, depending of the method you use, you'll end up with different combinations. I believe the current literature is not clear about this issue.

If you don't have a solid economical reason to choose one method over the other one, and with a pure econometric point of view I would recommand you to employ the direct approach as it covers a much larger parameter space (i.e you test more combinations with the direct approach than with the two step approach).