Forecasting two part ARIMA-GARCH model

Question

I am conducting the ARIMA-GARCH model in two stages. First, I assess the ARIMA model and then apply GARCH model on the residuals from the ARIMA model. My model looks like this:

library(rugarch)    
arima_model <- arima(data, order = c(1,0,1), include.mean = FALSE)    
    garchSpec <- ugarchspec(variance.model = list(model = "sGARCH",
                                                        garchOrder = c(1,1)),
                                  mean.model = list(armaOrder = c(0,0), include.mean = FALSE),
                                  distribution.model = "sstd")
    garchFit <- ugarchfit(spec = garchSpec, data = arima_model$residuals)

I want to make a forecast the data 12 points ahead. I've only seen forecasting ARIMA-GARCH models when ARIMA is already included in mean.model. But I can't do that. So, how do I forecast if ARIMA is not included in mean.model?

Answer 1

You can obtain the point forecast from the ARIMA model alone, as long as the point forecast you are interested in targets the conditional mean of the time series. If you are estimating ARIMA and GARCH models separately, the GARCH part is irrelevant for point forecasts, as it does not affect the estimate of the conditional mean in any way.

(It would be relevant if you were estimating the two models simultaneously, as adding the GARCH part would affect the coefficient estimates of the ARIMA model. It would also be relevant if the point foreacast targeted something else than the conditional mean, e.g. some quantile such as the median; then these properties of the distribution would become relevant, and the GARCH part would be instrumental in calculating them. It would also be relevant for interval and density forecasting regardless of whether estimation is separate / stage-wise or simultaneous.)