Underidentification?

We develop methods for testing that an econometric model is underidentified and for estimating the nature of the failed identification. We adopt a generalized-method-of moments perspective in a possibly non-linear econometric specification. If, after attempting to replicate the structural relation, we find substantial evidence against the overidentifying restrictions of an augmented model, this is evidence against underidentification of the original model. To diagnose how identification might fail, we study the estimation of a one-dimensional curve that gives the parameter configurations that provide the greatest challenge to identification, and we illustrate this calculation in an empirical example.