I estimated a couple of GARCH models on the basis of of different estimators for a given sample of data.
Now I have two more data sets, that I want to use to evaluate which of these estimated models gives me the best forecasting accuracy.
For the realized volatility I use the Garman and Klass estimator.
Now my question is:
In the literature almost always the ML estimator is used to estimate the parameters of a GARCH model. However when it comes to the evaluation of the forecast accuracy of GARCH models it seems like Maximum Likelihood is never used as evaluation criterion. I always see the typical loss functions like MSE or MAE as evaluation/ranking criterion. Why is that?
The Maximum Likelihood tells use what the likelihood of the forecasted value being the actual value is, given a particular density function.
To me looking at the likelihood of the forecast beeing the actual value is a much better measure then looking at the MSE. I understand it depends on the use case, but I have NEVER seen it beeing used in a paper on that topic.
So why ML isn't used as evaluation criterion, when it is almost always used to estimate the models?