Finite underidentification

I adapt the Generalised Method of Moments to deal with nonlinear models in which a finite number of isolated parameter values satisfy the moment conditions. I also study the closely related class of first-order underidentified models, whose expected Jacobian is rank deficient but not necessarily zero. In both cases, my proposed procedures exploit the underidentification structure to yield parameter estimators and underidentification tests within a standard asymptotically normal GMM framework. I study nonlinear models with and without separation of data and parameters. I also illustrate my proposed inference procedures with applications to production function estimation and dynamic panel data models.