I estimated a GARCH model to forecast the variance of a variable conditional on past information. I evaluated the forecast by comparing the squared forecast error, i.e. the squared value by which the conditional mean model forecast missed the actual value, with the forecasted conditional variance.
It turned out to be a terrible forecast on average (I did this for a multitude of time-series, i.e. I am working in a panel setting). Specifically, a small number of extremely bad forecasts in terms of the aforementioned evaluation appear to "make it bad" on average.
I then used the absolute values of the residuals as my dependent GARCH model variable instead of squared values, as the classic GARCH approach suggests. I compared the forecast values resulting from this model with the absolute value by which the conditional mean model forecast missed the actual value. It turned out to be much better on average.
Now my question is: Is it generally fine to use absolute values of the residuals instead of squared values in a GARCH model? From my understanding, instead of modelling the conditional variance one would then simply model the conditional standard deviation. Is this correct? Thank you in advance.