I am having difficulty determining how to forecast values for an ARIMA-GARCH model manually (by hand).
I understand that for an ARIMA model, the inputs for the MA terms are the residuals (i.e. the fitted value - true value). However, since in an ARIMA - GARCH, the GARCH model is fit on the residuals produced by the ARIMA model, how would I now interpret the MA terms? Are the inputs for the MA terms now the residuals produced by the GARCH model? Or is it still the residuals from the ARIMA model?
To give a hypothetical example, say the ARIMA model is a (0,0,1), and the first two residuals (εt and εt-1) from the ARIMA model are 0.62 and 0.52 respectively. A GARCH model is then fit on the residuals and the first two residuals of the GARCH model are 0.23 and 0.25. When I am predicting the Yt+1 of the ARIMA model, would it be 0.62+θ0.52 or 0.23+θ0.25?
If it is the former (i.e. 0.62+θ0.52), would that imply that the GARCH model has no effect on the first forecast of the ARIMA(0,0,1)?