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.

stemp.arima<- auto.arima(stemp.ts)

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.


1 Answer 1


To gain efficiency, it would make sense to fit a SARIMA-GARCH model in one step rather than starting with SARIMA and following it up with GARCH. Unfortunately, I am not aware of any R package that would have the functionality for one-step fitting of SARIMA-GARCH models. Restricted ARIMA-GARCH could be used as a substitute, 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 example, the one-step-ahead 80% forecast interval would be the point forecast from SARIMA plus the interval between 10% and 90% quantiles of the forecasted error distribution from GARCH.
  • 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 $h=1$.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.