Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models
This work addresses the problem of testing structural function restrictions in econometrics and statistics, offering a rate-optimal solution that adapts to unknown smoothness and instrument strength, which is incremental but provides strong specific gains.
The authors developed an adaptive hypothesis test for inequality and equality restrictions in nonparametric instrumental variable models, achieving the adaptive minimax rate of testing in L² and demonstrating in simulations that it controls size and significantly outperforms existing non-adaptive tests in power across various instrument strengths and sample sizes.
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.