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.

翻译：本文提出了一种新的自适应假设检验方法，用于检验非参数工具变量（NPIV）模型中结构函数的不等式约束（如单调性、凸性）与等式约束（如参数化、半参数化）。我们的检验统计量基于限制性与无限制性筛分两阶段最小二乘估计量之间二次距离的修正留一法样本模拟。我们提供了计算简便、数据驱动的筛分调参选择及Bonferroni校正卡方临界值。该检验能够在内生性程度未知与工具变量强度未知的情况下，自适应地适应备择函数的未知光滑度。该检验在$L^{2}$范数下达到了自适应极小极大检验速率。这意味着，在未知正则性的NPIV模型中，任何其他检验都无法使复合零假设上的第一类错误上确界与非参数备择模型上的第二类错误上确界之和更小。通过逆自适应检验可获得$L^{2}$置信集。模拟实验证实，在不同工具变量强度与样本量下，我们的自适应检验能有效控制检验水平，且其有限样本功效显著超越NPIV模型中现有非自适应单调性与参数约束检验方法。文中还展示了该方法在检验差异化产品需求与恩格尔曲线形态约束中的实证应用。