Generalized Wald Methods for Testing Nonlinear Implicit and Overidentifying Restrictions

The W- ld approach to testing direct explicit restrictions on a parameter vector is generalizetd to the case of nonlinear implicit constraints. When applied to subsystems of simultaneous equations models, the generalization enables the symmetric joinit testing of nonlinear overidentufying structural restrictions under very wide cornditions. By varying the choices of certain matrices used to construct the generalized Wald statistic, one produces a whole class of tests which have equal asymptotic power yet whose associated structural coefficient estim,-iators have different asymptotic efficiencies for any given rediuced-form estimator from which they are derived. restrictions on a parameter vector was originally formulated in terms of the unrestricted maximum-likelihood estimator of that vector. Stroud [13] showed clearly how Wald criteria can be defined in terms of consistent asymptotically normal estimators other than maximum likelihood. While this makes the Wald