cemmap working paper CWP59/15 Constrained Conditional Moment Restriction Models

This paper examines a general class of inferential problems in semiparametric and nonparametric models defined by conditional moment restrictions. We con-struct tests for the hypothesis that at least one element of the identified set satisfies a conjectured (Banach space) “equality ” and/or (a Banach lattice) “inequality ” con-straint. Our procedure is applicable to identified and partially identified models, and is shown to control the level, and under some conditions the size, asymptotically uni-formly in an appropriate class of distributions. The critical values are obtained by building a strong approximation to the statistic and then bootstrapping a (conser-vatively) relaxed form of the statistic. Sufficient conditions are provided, including strong approximations using Koltchinskii’s coupling. Leading important special cases encompassed by the framework we study in-clude: (i) Tests of shape restrictions for infinite dimensional parameters; (ii) Confi-dence regions for functionals that impose shape restrictions on the underlying pa-rameter; (iii) Inference for functionals in semiparametric and nonparametric models defined by conditional moment (in)equalities; and (iv) Uniform inference in possibly nonlinear and severely ill-posed problems