The Wald approach to testing direct explicit restrictions on a parameter vector is generalized to the case of nonlinear implicit constraints. When applied to subsystems of simultaneous equations models, the generalization enables the symmetric joint testing of nonlinear overidentifying structural restrictions under very wide conditions. 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 estimators have different asymptotic efficiencies for any given reduced-form estimator from which they are derived.