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In order to model time series with GARCH models in R, you first determine the AR order and the MA order using ACF and PACF plots. But then how do you determine the order of the actual GARCH model?

Ie. say you find ARMA(0,1) fits your model then you use:

garchFit(formula=~arma(0,1)+garch(1,1),data=XX,trace=FALSE,include.mean=FALSE)

I know GARCH(1,1) is the most widely used, but what's the best way to determine the order? AIC?

Richard Hardy
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user3384794
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  • @Richard Hardy Thanks for that response,however another quaestion, as you choose the candidate models,is there a logic behind the top most models you choose just to avoid the problem of having many models that would lead to the problem of overfitting earlier stated? secondly,can the plot of the histogram or Q-Q plot help you detrmine the best distribution to use in the model fitting or its just a matter of cross validating the distributions? – Christine Kibui Apr 06 '22 at 21:27
  • @ChristineKibui, please post your questions as two separate new threads. – Richard Hardy Apr 07 '22 at 05:37

1 Answers1

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You should determine both the ARMA and the GARCH orders simultaneously.

If the process is indeed well approximated by an ARMA-GARCH model, considering the conditional mean model (ARMA) while neglecting the conditional variance model (GARCH) -- and this way (implicitly) assuming the conditional variance to be constant -- will lead to trouble. Similarly, when considering the conditional variance model you should not neglect the model for the conditional mean.

This is because neither the conditional mean model nor the conditional variance model can be estimated consistently if taken separately, unless under special conditions (e.g. if the MA part of the ARMA model is empty, the AR part can be estimated consistently even if the GARCH model is neglected). However, joint estimation will typically be more efficient and that is why it is preferred.

Unfortunately, the task of jointly determining the ARMA-GARCH orders is difficult, as I understand it. You may experiment with a few different models and compare their AICs or BICs. The larger the pool of candidate models, the more likely you are to overfit. If you have a large enough sample, you may try cross validation. That is,
(1) define a pool of candidate models,
(2) estimate the models on part of the sample,
(3) use the estimated models to predict the remainder of the sample,
(4) pick the model that has the lowest prediction error.
Still, this will not prevent overfitting if the pool of candidate models is quite large as compared with the sample size.

You can also examine the properties of the residuals of the different models. You would prefer a model with well-behaved residuals (no remaining autocorrelations, no remaining ARCH patterns, etc.). Again, this is subject to overfitting. It is easy to imagine data dredging or model mining when one is not stopping until one gets all the model coefficients to be significant. But can this result be trusted if achieved via data dredging? (Of course, not.)

I am sorry I do not have a better solution.

Richard Hardy
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  • +1: Richard, I've been going through a number of your answers on ARCH/GARCH and have been learning quite a bit. Did you say that if the true model is AR(p) + GARCH and we estimate the AR(p) model assuming constant variance, the estimators of the AR coefficients are still consistent? Did I get that right? If so, why is that the case? – ColorStatistics Sep 22 '21 at 22:11
  • @ColorStatistics, yes, this is so. Given a correct specification of the conditional mean, OLS is consistent under heteroskedasticity. This is a classic result available in many econometrics textbooks. – Richard Hardy Sep 23 '21 at 05:05
  • Is that then the core of the answer to my question here? Meaning that assuming the true processes for the mean has no MA terms, fitting a model with constant variance still gives us consistent estimators of the parameters. https://stats.stackexchange.com/q/545016/198058 Can you please provide a reference for the fact that if the model for the mean contains no MA terms and we estimate a model with constant variance the AR coefficients are still consistent? I am trying to understand why the "if the mean process contains no MA terms". Thank you. – ColorStatistics Sep 23 '21 at 10:21
  • @ColorStatistics, yes. – Richard Hardy Sep 23 '21 at 10:30
  • @richard-hardy this a brilliant answer and I've been looking for this for my project,do you have an idea of the code/plot used to check for the Garch order? – Kagwiria kibui Apr 03 '22 at 04:15
  • @Kagwiriakibui, you could estimate a bunch of candidate models and pick the one with the lowest AIC value. I do not have code for that, but it would be quite simple and based on loops. Regarding plots, you could look at ACF and PACF of squared residuals from the conditional mean model, but I find them more useful for model diagnostics than for model selection. – Richard Hardy Apr 03 '22 at 06:45