  We propose a new adaptive hypothesis test for inequality (e.g., monotonicity,
convexity) and equality (e.g., parametric, semiparametric) restrictions on a
structural function in a nonparametric instrumental variables (NPIV) model. Our
test statistic is based on a modified leave-one-out sample analog of a
quadratic distance between the restricted and unrestricted sieve two-stage
least squares estimators. We provide computationally simple, data-driven
choices of sieve tuning parameters and Bonferroni adjusted chi-squared critical
values. Our test adapts to the unknown smoothness of alternative functions in
the presence of unknown degree of endogeneity and unknown strength of the
instruments. It attains the adaptive minimax rate of testing in $L^2$. That is,
the sum of the supremum of type I error over the composite null and the
supremum of type II error over nonparametric alternative models cannot be
minimized by any other tests for NPIV models of unknown regularities.
Confidence sets in $L^2$ are obtained by inverting the adaptive test.
Simulations confirm that, across different strength of instruments and sample
sizes, our adaptive test controls size and its finite-sample power greatly
exceeds existing non-adaptive tests for monotonicity and parametric
restrictions in NPIV models. Empirical applications to test for shape
restrictions of differentiated products demand and of Engel curves are
presented.
