# What is fitted in a GARCH: residual or log-return?

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...

If you use the log returns, you're essentially making the assumption that there is no conditional variation in the mean. In some circumstances you may want to explicitly model both, but other times it may be sufficient to assume a constant mean and focus on the conditional variance. Depends on what you're trying to do.

In addition, if you fit a GARCH model with raw log returns, then you're also implicitly assuming the mean is zero. Centering the data may be important if the mean is large (i.e. especially in lower frequency data).

• I basically want to use an ARMA-GARCH model to model both my mean equation and variance equation. How should I proceed?Can you explain what you mean by no conditional variation in the mean?What would be the implication of that?
– ankc
Oct 5, 2013 at 11:48
• Can you explain mathematically what you meant? thanks
– ankc
Oct 5, 2013 at 11:57
• Basically if the mean varies over time, one way to model that is with ARMA. The residuals from a ARMA regression would (ideally) no longer have a time-varying mean. If it is important to do both, then you can take a pseudo-maximum likelihood approach, fit the ARMA model, get the residuals, then fit the Garch model to the residuals.
– John
Oct 5, 2013 at 18:16
• you mean if the series is not stationary in its mean?isn't there differencing for that?
– ankc
Oct 5, 2013 at 19:35
• If I fit the garch model to the log-returns, what would I get?the variance equation?is it possible to get the mean equation if I do that? If I first model the series using an ARMA,then model the residuals using a GARCH model, would'nt I need to re-estimate the parameters of the ARMA model?I saw in Statistics and Data Analysis for Financial Engineering talking about something like that but i'm not sure. thanks
– ankc
Oct 5, 2013 at 19:41