How to see whether the model fit the real data well?

[jpfeifer]

You are confusing fit in the context of full information and limited information estimation techniques. They imply a different weighting of moments. See above:

How to see whether the model fit the real data well?

Full information estimation works essentially via minimizing the one-step ahead forecast errors. So checking the forecasting performance within the estimation sample does not make sense. Comparing the standard deviations of simulated and actual data in contrast is a valid and worthwhile exercise, because the ML estimator will weight all moments according to their precision (and there are covariances at all leads and lags) while economists care particularly about a few select second moments.

The marginal data density will take the moments into account, but in a not obvious way. Also, you may not care about the moments that the MDD assigns the highest weight.