CFDP 2238R

Adaptive, Rate-Optimal Hypothesis Testing in Nonparametric IV Models


Publication Date: June 2020

Revision Date: December 2021

Pages: 70


We propose a new adaptive hypothesis test for polyhedral cone (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 NPIV estimators. We provide computationally simple, data-driven choices of sieve tuning parameters and 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 L2. That is, the sum of its type I error uniformly over the composite null and its type II error uniformly over nonparametric alternative models cannot be improved by any other hypothesis test for NPIV models of unknown regularities. Data-driven confidence sets in L2 are obtained by inverting the adaptive test. Simulations con rm that our adaptive test controls size and its nite-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.

Keywords: Nonparametric instrumental variables, Shape restrictions, Nonparametric alternatives, Minimax rate of testing, Adaptive hypothesis testing, Random exponential scan, Sieve regularization, Quadratic functionals

See CFDP Version(s): CFDP 2238