# Applying GARCH Model to Forecasts

I'm reading up on GARCH models in Springer Introductory Time Series, and had a question on how we actually apply the model to forecasts.

The data can be grabbed like so:

stemp<- scan("http://www.maths.adelaide.edu.au/andrew.metcalfe/Data/stemp.dat")
stemp.ts<- ts(stemp, start = 1850, frequency = 12)


I originally fit a seasonal ARIMA to the data using auto.arima. The squared residuals appear to be correlated so I thought I'd fit a GARCH(1, 1) model to the residuals.

library(forecast)
stemp.arima<- auto.arima(stemp.ts)
acf(stemp.arima$residuals^2) library(tseries) stemp.garch<- garch(stemp.arima$residuals, trace = F)


The correlogram of the squared stemp.garch residuals appear to be white noise. Now that I have that knowledge, though, I don't understand how it would impact my predictions. I was thinking I could simulate a GARCH process using the coefficients from the GARCH output and the distribution of the seasonal ARIMA residuals. I could then take that and add it to my seasonal ARIMA predictions. Not sure if that's a good method, though.

In the case of a two-step solution, having a volatility forecast from GARCH allows you to replace the constant $h$-step-ahead forecast intervals with time-varying ones. You would derive them from the predicted conditional variance from the GARCH model.
• For $h$-step-ahead forecast intervals when $h>1$, you would estimate the variance as the sum of conditional variances over the $h$ periods and use that to determine the expected forecast error distribution from GARCH. Then take the point forecast and add quantiles as in the case of $h=1$.