中文摘要
针对非参数工具变量（NPIV）模型中结构函数上的不等式（如单调性、凸性）和等式（如参数、半参数）约束，我们提出了一种新的自适应假设检验。我们的检验统计量基于受限与无约束筛两阶段最小二乘估计量之间二次距离的修正留一法样本类似。我们提供了计算简便、数据驱动的筛调节参数选择和Bonferroni调整的卡方临界值。在存在未知内生性程度和工具变量未知强度的情况下，我们的检验能够自适应于备择函数的未知平滑度。该检验在L2中达到了自适应极小极大检验速率。也就是说，在复合原假设上的第一类错误上确界与在非参数备择模型上的第二类错误上确界之和，无法被任何其他针对未知正则性NPIV模型的检验所最小化。通过反演自适应检验，可以得到L2中的置信集。模拟证实，在不同的工具变量强度和样本量下，我们的自适应检验控制了检验水平，且其有限样本功效大大超过了现有的针对NPIV模型中单调性和参数约束的非自适应检验。文中展示了检验差异化产品需求与恩格尔曲线形状约束的实证应用。
Abstract
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.