Given a time-series of log-return of SP500, then to obtain the volatility process what should we do? Some people say that we need using the ARMA model to withdraw the residual series, then plug this residual series into the GARCH model to obtain the conditional variance process? Or directly plug the log-return process of SP500 into the GARCH model to obtain the conditional variance?
I saw that in the book Introduction to Time Series in R, the author fits simulated residuals in GARCH model then he fits SP500 log returns in the GARCH model.
I got confused...