How to do model testing in Indirect Inference/Simulated/General Method of Moments I have a model estimated via indirect inference, so I have a set of auxiliary moment conditions.  What I am doing is very similar to Method of Simulated Moments.  I want to do a couple of specification tests.  In one specification, I restrict one parameter to be zero.  In another, I change the model in a more fundamental way, but estimate the same set of parameters and the same auxiliary model as in the baseline. 
I think I can do a simple likelihood ratio like test when I restrict the parameter to be zero.  I can look at the ratio of goodness of fit metrics, and the test statistic will be asymptotically distributed chi squared with the correct number of degrees of freedom.  In the second model, in which I make a more fundamental change, I believe I need to employ some sort of non-nested test, but I am having trouble with my google-fu.  Can anyone point me in the right direction?
 A: For your first problem (restricting a parameter to be zero) you want what I would call a D-test, but this source calls a likelihood ratio test (which seems odd for GMM, but it's Newey - who am I to argue?). D-tests test:
$$
H_0:R\theta-q = 0
$$
where $\theta$ is a vector of parameters you're estimating, and $q$ has dimension $r\times 1$ It relies on the J statistic for over-identifying restrictions. It goes like this:


*

*Estimate unrestricted model, calculate J-statistic, $J_u$.

*Use same weight matrix and estimate model with restriction. Calculate J-stat, $J_r$.

*Under the null, $J_R-J_u$ is asymptotically $\chi^2$ with $r$ degress of freedom.


A similar test can be done for non-linear restrictions as detailed in the first source, and there is an equivalent of a Lagrange multiplier test. All these tests work for indirect inference as well as they do for GMM because they are ultimately derived from the properties of GMM as an M estimator, properties which indirect inference shares.
Non-nested model comparison is more tricky. Here's two papers which use it: Rivers and Vuong (2002) and Nikolov and Whited (2013).
