Overidentification and the Hansen J-Test

An instrumental-variables regression is just identified when you have exactly as many instruments as endogenous regressors — in that case, the instruments pin down a unique estimate and there's no way to check whether they're individually consistent with each other, because there's no leftover information to check against. A model is overidentified when you have more instruments than endogenous regressors, and this extra information is not wasted: GMM combines all the instruments into one efficient estimate, and the degree to which the different instruments' implied estimates disagree with each other becomes directly testable.

The Hansen J-test (also called the J-statistic or the overidentifying restrictions test) formalizes this. It takes the GMM objective function evaluated at the efficient estimate — a weighted sum of how far the sample moment conditions are from zero — and under the null hypothesis that all instruments are valid, this statistic is asymptotically chi-squared with degrees of freedom equal to the number of overidentifying restrictions (extra instruments beyond the minimum needed).

Worked illustration: suppose you have 2 endogenous regressors and 5 instruments, giving 3 overidentifying restrictions. Running GMM and computing the J-statistic yields, say, $J = 9.2$ against a $\chi^2_3$ critical value of 7.8 at the 5% level. Since $9.2 > 7.8$, you reject the null that all instruments are valid — at least one of your 5 instruments is likely correlated with the error term rather than being a genuinely exogenous source of variation, even though the test can't tell you which one.