Questions about using estimation and calibration at the same time

[jpfeifer]

SMM is not calibration, but a form of estimation. The problem with what you are describing is that you are mixing two types of estimation strategies: full information and limited information. Thus, there is an inherent tension:

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.