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:
- 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))?