ENH: score/Lagrange Multiplier test for GMM

[josef-pkt]

This seems to be missing in current GMM code, I don't remember if or what I read about lagrange multiplier tests in GMM.

At the end here
https://github.com/josef-pkt/misc/blob/gmm_glm/notebooks/ex_gmm_glm.ipynb
I compare score test in GLM with test of overidentifying restriction in GMM, but that's not the same as a score test since we don't keep the parameters fixed. Or if we put zeros in the weight matrix, then we don't get the unrestricted values (i.e. with full weight matrix).

related: maxiter=0 changes the parameters from the starting values, AFAICS in an example
(I thought that I have an issue that maxiter=0 keeps params unchanged, e.g. #1728)

related issues:
#2041 score tests, conditional moment tests
#2096 PR for score tests, lagrange multiplier tests and conditional moment tests (PR for translating theory into code)
#1096 review and more fixes for GMM (issue sounds outdated)

[josef-pkt]

A good read on testing in GMM including LM
https://eml.berkeley.edu/~mcfadden/e240b_f01/ch3.pdf from McFadden's course
https://eml.berkeley.edu/~mcfadden/e240b_sp03/e240b.html

It has more intuition than the handbook article (AFAIR that one)
I remember now reading a lot about this but with the context of QMLE in mind in #2096 and didn't look at implications for my GMM implementation.

McFadden has some good examples for the reformulation of other models (extremum estimators) as GMM.

[josef-pkt]

maxiter=0, I need to check unit tests whether maxiter=0 does the same as maxiter=1, unittests distinguish between onestep (maxiter=0 OLS equivalent) and oneiter (maxiter=1)

maxiter=0 might mean not to update weights (cov of moment conditions) for cov_params

need documentation since I don't remember