![]() ![]() ![]() ![]() ![]() The J-test is also called test for overidentifying restrictions - i.e., if you have more instruments than you need, you can exploit that overidenfication to test the joint validity of all instruments. Thus, under exact identification, the criterion function is zero for any admissible weighting matrix $\hat W$, and thus, the $J$-statistic is identically zero. Sample moment vectors and matrices are denoted by $s_\\ The notation follows Hayashi, i.e., instruments $x$, regressors $z$. Here is a proof for linear GMM estimators. In that case, the J-statistic is by construction zero. You have as many instruments in Rh as endogenous variables, i.e., exact identification. ![]()
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