Numerical Bootstrap and Constrained Estimation

许多用于政策评估的经济模型是高度非线性的，并且通常受到参数的非线性约束。估计这些模型的计算挑战已经构成了利用这些模型在有效的政策制定的重大障碍。本研究课题开发了一种新的统计推断方法，可以有效地评估与经济模型的参数估计值的政策相关功能相关的统计不确定性，从而有助于经济政策的制定。这种新方法结合了单边数值微分与自举响应技术，以允许在政策功能的不可微性，是计算简单，易于实现。它可以应用于分析产业组织中的寡头竞争，教育政策的影响，如小班教学的影响，以及许多其他应用经济分析领域。这些工具通过使模型能够评估经济政策的有效性，为社会福利做出了贡献。本计画研究一个数值Delta方法，用来推论一个正则参数的方向可微函数。该方法计算效率高，不需要感兴趣的函数的结构的分析知识，并提供了一致有效的推理，用于测试参数的凸函数的单侧假设。在一阶Delta方法极限分布退化的情况下，二阶（或更高阶）Delta方法可以提供必要的非退化大样本近似。研究者进一步将数值Delta方法推广到一种称为数值引导的新的rescue技术，该技术可以在许多情况下一致地估计极限分布-其中传统的引导是无效的，子采样是最常用的推断方法，并且参数不知道是方向可微的。应用程序包括约束和无约束M-估计收敛在经常和非标准的速度，如最大得分模型，部分识别模型，误指定的模拟GMM模型，和许多样本量相关的统计。

Many economic models used for policy evaluation are highly nonlinear, and are typically subject to nonlinear constraints on the parameters. The computational challenge in estimating these models has posed a significant obstacle for utilizing these models in effective policy making. This project assists in economic policy making by developing a new method of statistical inference that can be used to provide valid evaluation of the statistical uncertainty associated with policy-related functions of estimated parameters of economic models. This new method combines one-sided numerical differentiation with bootstrap resampling techniques to allow for non-differentiability in the policy function, and is both computationally simple and easy to implement. It can be applied to analyze oligopolistic competition in industrial organization, the effect of education policy such as the impact of smaller class sizes, and many other areas of applied economic analysis. These tools contribute to the welfare of the society by enabling models to evaluate the effectiveness of economic policies. This project studies a numerical Delta method for inference on a directionally differentiable function of regular parameters. This method is computationally efficient, does not require analytic knowledge of the structure of the function of interest, and provides uniformly valid inference for testing a one-sided hypothesis of a convex function of the parameters. In situations where the first order Delta method limiting distribution is degenerate, the second (or higher) order Delta method may provide the necessary nondegenerate large sample approximation. The investigator further generalizes the numerical Delta method to a new resampling technique called the numerical bootstrap that can consistently estimate the limit distribution in many cases -- where the conventional bootstrap is not valid and subsampling has been the most commonly used inference approach, and where the parameters are not known to be directionally differentiable. Applications include constrained and unconstrained M-estimators converging at both regular and nonstandard rates such as the maximum score model, partially identified models, misspecified simulated GMM models, and many sample size dependent statistics.