Comparing models using log data density vs using moments

[jpfeifer]

We had a similar discussion in

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

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