Optimizing robust conditional moment tests an estimating function approach

Robust conditional moment (RCM) tests for partial specifications are derived without a full specification assumption. Yet, researchers usually claim the optimality of these RCM tests by reinterpreting them as score tests under certain full specifications. This argument is in fact incompatible with the rationale of RCM tests. In this study, we consider a generalized RCM test based on the estimating function (EF) approach and explore a semi-parametric optimality criterion that does not require full specifications. Specifically, we derive the upper bound of the noncentrality parameter of the generalized RCM test and propose a method to optimize RCM tests so as to achieve this upper bound. The optimized RCM test is associated with the optimal EF method, and it is useful for improving the asymptotic local power of existing RCM tests. The proposed method thus permits researchers to pursue optimality without sacrificing robustness in estimating and testing partial specifications. We illustrate our method using various partial specifications and demonstrate the improved power property of the optimized tests by simulations.