# ARMA/GARCH estimation in sequence

I have a time series that shows a nonstationary seasonal autoregressive component as well as known heteroshedasticity. In order to model the series, I have fit a seasonal ARIMA model for the mean with the auto.arima model in the forecast package in R and a GARCH model on the residuals of the ARIMA model.

Is the procedure of sequentially estimating ARIMA and GARCH model correct or would it have been better to jointly model the mean and the variance of the series? If this were correct, is there a (possibly R) function to do it?

Functions ugarchspec and ugarchfit in package rugarch (see here for a vignette) allow specifying and estimating ARMA+GARCH models simultaneously for a variety of GARCH model classes. Unfortunately, seasonal ARMA models do not seem to be implemented there. Perhaps you could try seasonally adjusting your series before fitting an ARMA+GARCH model (although this would be suboptimal if the "true" model is seasonal ARIMA with conditionally heteroskedastic errors).
• @user6472523, trial and error is a possibility. auto.arima is another one, just be aware that auto.arima will work better under no conditional heteroskedasticity since it was not designed to handle ARMA-GARCH processes. But it might work nevertheless. Yes, GARCH(1,1) could be a good starting point for the conditional variance part. – Richard Hardy May 4 '18 at 19:40