We consider models defined by a set of moment restrictions that may be subject to weak identification. More specifically, we study the asymptotic properties of the standard GMM estimator and the Hansen J-test when additional moment restrictions that are weaker than the original ones are available. We show that the consistency of the GMM estimator is not affected by such restrictions even when they are invalid. We also provide conditions under which these restrictions may improve the efficiency of GMM estimator. Finally, we study the behavior of the Hansen J-test to assess the compatibility between existing restrictions and additional ones in order to detect ”spurious” identification that may rely on invalid moments. Our theoretical characterization of the J-test reveals that the J-test is approximately akin to checking whether the value of the parameter identified by the existing restrictions is conformable to the information provided by the additional ones. Our simulations confirm that the issue with the standard J-test is not its power but rather its size. We also show that the power of the J-test increases with the weakness of the additional restrictions and we provide some intuition for why that is.
