GMM with Nearly-Weak Identification

A unified framework for the asymptotic distributional theory of GMM with nearly-weak instruments is provided. It generalizes a previously proposed framework in two main directions: first, by allowing instruments’ weakness to be less severe in the sense that some GMM estimators remain consistent, while featuring low precision; and second, by relaxing the so-called ”separability assumption” and considering generalized versions of local-to-zero asymptotics without partitioning a priori the vector of parameters in two subvectors converging at different rates. It is shown how to define directions in the parameter space whose estimators come with different rates of convergence characterized by the Moore-Penrose inverse of the Jacobian matrix of the moments. Furthermore, regularity conditions are provided to ensure standard asymptotic inference for these estimated directions.