Adaptive, Rate‐Optimal Hypothesis Testing in Nonparametric IV Models
提出一种自适应假设检验方法，用于检验非参数工具变量模型中结构函数的形状约束（如单调性、凸性）和参数约束，通过数据驱动选择调谐参数和Bonferroni调整的卡方临界值，实现自适应最优检验速率，并应用于差异化产品需求和恩格尔曲线的形状检验。
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.