Definition of J Statistic: In a GMM context, when there are more moment conditions than parameters to be estimated, a chi-square test can be used to test the over-identifying restrictions. The test statistic can be called the J statistic.
In more detail: Say there are q moment conditions and p parameters to be estimated. Let the weighting matrix be the inverse of the asymptotic covariance matrix. Let T be the sample size. Then T times the minimized value of the objective function (TJT(bT)) is asymptotically distributed with a chi-square distribution with (q-p) degrees of freedom. (Econterms)