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I was modelling a time series with an ARIMA(1,1,1) model which had an AIC of -4782.96. However, after checking squared residuals and performing ARCH tests (Engle's and McLeod-Li) I detected the presence of ARCH effects up to lag 7. Therefore, it seemed like ARIMA(1,1,1)-GARCH(1,1) would be a good idea. However, the AIC of the ARIMA-GARCH model dropped significantly; it is only -2.36. Does this indicate the ARIMA-GARCH model is worse? What could have caused this and how should I interpret this? My rcode for ARIMA-GARCH is:

tryg3<- garchFit(formula= ~arma(1,1) + garch(1,1), data = na.omit(xts.data$ldVIXCLS))

Where ldVIXCLS is differenced log series of the VIX index.

I would greatly appreciate any input!:)

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It is highly implausible that the AICs of such relatively similar models differ as much. Most likely they are not directly comparable due to quirks of definitions of AIC (and the likelihood on which AIC is based) across different software implementations. Different packages and functions may make different simplifications, so AICs/likelihoods may be shifted by a constant. Also, some pieces of software (probably rugarch in R, if I remember correctly, probably also fGarch that you seem to be using) report AIC/likelihood per observation instead of AIC/likelihood for the entire sample.

Try fitting your ARIMA model using the same software as you did for fitting ARIMA-GARCH. If you cannot do it directly, consider fitting a restricted ARIMA-GARCH with the GARCH parameters set to, say, 0.001 for the lagged squared error term and 0.999 for the lagged conditional variance. This would approximate constant variance, so you would get an approximate AIC/likelihood (perhaps per observation) of the pure ARIMA model that is comparable to the AIC/likelihood of the ARIMA-GARCH model.

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  • $\begingroup$ Thank you so much! I checked out the packages and it appears that the information criteria are indeed reported per observation. Your input was really helpful so thank you again! $\endgroup$
    – Ula
    May 7, 2022 at 9:07
  • $\begingroup$ @Ula, you may consider multiplying the per-observation AIC/likelihood by the number of observations to make it comparable. However, likelihoods are sometimes shifted by a constant in different packages, so there is no guarantee that will work. It would be safer to use the same package for obtaining AIC/likelihood for different models. $\endgroup$ May 7, 2022 at 9:18

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