When I read this group questions about lag selection for ARMA part of ARMA-GARCH models I found 2 different answers from moderator: The use of GARCH and ARMA GARCH estimation process in practice

I can't understand - could I try to select AR and MA part order by information criterion ignoring that the errors have a GARCH structure, get lags for AR and MA parts (p and q for example) and after that select best model from ARMA(p, q)-GARCH(s, t), where p and q are constants, s and t could be in 1:N range, using, for example, AIC?



1 Answer 1


The answers are different in that they highlight different aspects of the subject. However, their implications do not conflict. Sequential modelling of the conditional mean and variance can be justified when the conditional mean part can be estimated consistently even in presence of GARCH errors. AR-GARCH model would be one such example. Meanwhile, it is more difficult to justify sequential modelling when neglected GARCH errors ruin the consistency of the conditional mean estimation. ARMA-GARCH with non-empty MA part is one such example.

In practice, you may of course try selecting the conditional mean model ignoring the GARCH structure in the errors and see what happens. This is an easier setup and requires less computing. However, you have no guarantee over how much the selected model will differ from a model selected incorporating the information on the GARCH structure. It is only fair to admit that even considering the conditional mean and variance models simultaneously does not guarantee "perfect" results. It is just that this approach is easier to justify based on properties of the resulting estimators.

Putting diplomacy aside, why don't you try it your way and see if the results are satisfactory. You could hide some data from yourself in a hold-out sample when selecting the model. When the "optimal" model has been selected, you would test the model performance on it.

  • $\begingroup$ Thank you. I asked this question, because I have strange problem. For example, I simulated ARMA(2,1)-GARCH(1,1) series. When I use AIC/BIC criterion for ARMA-GARCH model fitting in general, I have ARMA part overfitting each time I tried to use information criterion (I make models pool from ARMA(0,0)-GARCH(1,0) till ARMA(5,5)-GARCH(3,3) models) - I have estimated ARMA part as ARMA(4,1), ARMA(5,0) and so on. But when I try to fit series as pure ARMA, I get ARMA(1,1), ARMA(2,1), ARMA(2,2) models as optimal by AIC, for example (fArma doesn't have BIC criterion). Main question - why? $\endgroup$
    – Dmitriy
    Apr 19, 2016 at 13:20
  • $\begingroup$ Please, could you give articles/books, where I could read about this more (about difference in fitting ARMA and ARMA-GARCH models) and so on? $\endgroup$
    – Dmitriy
    Apr 19, 2016 at 13:25
  • $\begingroup$ I will try to come back to you a little later. $\endgroup$ Apr 19, 2016 at 13:50
  • $\begingroup$ Regarding the first comment, I don't have an answer as to "why". But keep in mind that ARMA processes with different orders may generate similar behaviour. If you look at impulse-response functions you may occasionally notice that. Also, if you are selecting from a large pool of models, there is little chance of selecting the excactly correct model. However, you could see whether the exactly correct model is among top 5% of the models when ranked by AIC; I guess it should be. Also, perhaps the selected models generate similar behaviour to the true model; again, use impulse-responses for that. $\endgroup$ Apr 19, 2016 at 18:04
  • $\begingroup$ Regarding references, there are quite a few time series textbooks, none of them clearly better than the rest. Some of the textbooks are available free of charge. Also, there are lecture notes and other resources. So just try a few and see if you find what you need. What I have on my shelf is Zivot "Modelling Financial Time Series with S-PLUS", Tsay "Analysis of Financial Time Series" and Hamilton "Time Series Analysis". $\endgroup$ Apr 19, 2016 at 18:10

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