# Fitting GARCH(1,1) to log returns instead of residuals - centering crucial?

For a project I need to fit a GARCH(1,1) model to the log returns of an index. When using the residuals of an ARMA or ARIMA model it is clear that the (conditional) mean is 0. When using the log returns, do I have to do centering first? I know that log returns tend to be stationary, but that is only for the unconditional moments. And will centering in that case only make the unconditional or also the conditional mean zero? I feel like I'm mixing things up a bit.

I found a similar question here, but that didn't fully help me understand.

Thanks in advance! Any help is appreciated!

• As written in the answer to the linked question above, centering the log-returns may be important if the constant conditional mean is substantially large (assuming that by centering, you mean demeaning the log-returns). If you just insert raw log-returns you make the implicit assumption that the conditional mean is zero and we are in a model-setup where the discretized log-returns are on the form $r_t = \sigma_t z_t$. You can do an ad-hoc analysis on whether it makes a difference by estimating the GARCH model on centered vs uncentered log-returns and observe the deviation in the parameters.
– Pleb
Jun 15 at 9:20
• Thanks a lot! Now I'm understanding it.
– Toni
Jun 15 at 11:18