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After having modelled conditional variance via an appropriate ARIMA(p,d,q) model I inspected the squared residuals ACF and noted that it does not look like a realisation of white noise.

Therefore (after an appropriate test on ACF) I decided to model an ARIMA(p,d,q)-GARCH(1,1), with p,d,q the same I got from the maximisation of the AIC of the ARIMA model.

The estimates of the garch parameters look very significant, however the AIC has dropped from the one I got with simple ARIMA. Moreover, Log-likelihood moved from positive to negative and many of the previously estimated ARIMA parameters now look non-significant.

These are the outputs from 3 similar series I modelled for comparison. As we can notice the problem i described is present for all the three:

  • ARIMA

ARIMA

  • ARIMA-GARCH

ARIMA-GARCH

Questions:

  • How can it be possibile?
  • Why it happened? Should I maybe redo the maximisation of the ARIMA(p,d,q) parameters jointly during ARMA-GARCH estimation? (i.e. calculate ARMA(1,0)-GARCH(1,1), ARMA(0,1)-GARCH(1,1), ARMA(1,1)-GARCH(1,1), and so on and keep the one with max AIC)
  • Should I allow higher orders GARCH (e.g. GARCH(2,1), GARCH(2,2))?
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  • $\begingroup$ How were you able to not do the maximization jointly and yet calculate the AIC of the ARIMA-GARCH model? Did you fit an ARIMA-GARCH model with restrictions on ARIMA parameters? $\endgroup$ – Richard Hardy May 3 '18 at 6:38
  • $\begingroup$ @RichardHardy thank you for your comment. I added further details on my question. What did i meant is that I didn't calculate ARMA(1,0)-GARCH(1,1), ARMA(1,1)-GARCH(1,1) and so on but simply took ARMA(5,5) as the one that maximised the AIC in the plain ARIMA model. $\endgroup$ – toyo10 May 3 '18 at 6:43

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