My intention is to calculate the MAE for different (G)ARCH-models (comparing the one-step-ahead forecast for $\sigma$ with the absolute return that day).
The formula for MAE is actually clear, but I'm not quite sure which two series to use, when I do a rolling forecast in
R for a (G)ARCH-model including mean.
Some Output I can extract after the
roll.forecast is a series of "$\mu$" as well as a series of "$\sigma$". Is it okay to compare this mentioned $\sigma$ with the absReturn [MAE=Sigma-absReturn] or do I have to consider the $\mu$ as well and compare $\sigma$ and (absReturn-$\mu$) [MAE=$\sigma$-(absReturn-$\mu$)]?
Because according to theory return = conditional mean + cond. volatility and if this $\sigma$ is identical to cond. volatility I have to adjust the return?