When the number of moment conditions is greater than the dimension of the parameter vector θ, the model is said to be over-identified. Sargan (1958) proposed tests for over-identifying restrictions based on instrumental variables estimators that are distributed in large samples as Chi-square variables with degrees of freedom that depend on the number of over-identifying restrictions. Subsequently, Hansen (1982) applied this test to the mathematically equivalent formulation of GMM estimators. Note, however, that such statistics can be negative in empirical applications where the models are misspecified, and likelihood ratio tests can yield insights since the models are estimated under both null and alternative hypotheses (Bhargava and Sargan, 1983).


Conceptually we can check whether ${\displaystyle {\hat {m}}({\hat {\theta }})}$ is sufficiently close to zero to suggest that the model fits the data well. The GMM method has then replaced the problem of solving the equation ${\displaystyle {\hat {m}}(\theta )=0}$, which chooses ${\displaystyle \theta }$ to match the restrictions exactly, by a minimization calculation. The minimization can always be conducted even when no ${\displaystyle \theta _{0}}$ exists such that ${\displaystyle m(\theta _{0})=0}$. This is what J-test does. The J-test is also called a test for over-identifying restrictions.


Formally we consider two hypotheses:


Under hypothesis ${\displaystyle H_{0}}$, the following so-called J-statistic is asymptotically chi-squared distributed with k–l degrees of freedom. Define J to be:


where ${\displaystyle {\hat {\theta }}}$ is the GMM estimator of the parameter ${\displaystyle \theta _{0}}$, k is the number of moment conditions (dimension of vector g), and l is the number of estimated parameters (dimension of vector θ). Matrix ${\displaystyle {\hat {W}}_{T}}$ must converge in probability to ${\displaystyle \Omega ^{-1}}$, the efficient weighting matrix (note that previously we only required that W be proportional to ${\displaystyle \Omega ^{-1}}$ for estimator to be efficient; however in order to conduct the J-test W must be exactly equal to ${\displaystyle \Omega ^{-1}}$, not simply proportional).


Under the alternative hypothesis ${\displaystyle H_{1}}$, the J-statistic is asymptotically unbounded:


To conduct the test we compute the value of J from the data. It is a nonnegative number. We compare it with (for example) the 0.95 quantile of the ${\displaystyle \chi _{k-\ell }^{2}}$ distribution: