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为在有效的政策制定中利用这些模型带来了重大障碍。该项目通过开发一种新的统计推断方法来协助制定经济政策，该方法可用于有效评估与经济模型估计参数的政策相关功能有关的统计不确定性。这种新方法结合了单边数值微分和Bootstrap重采样技术，允许策略函数的不可微性，并且计算简单，易于实现。它可以用来分析产业组织中的寡头垄断竞争、教育政策的效果，如小班规模的影响，以及许多其他领域的应用经济学分析。这些工具使模型能够评估经济政策的有效性，从而为社会福利作出贡献。该项目研究了一种用于推断正则参数的方向可微函数的数值Delta方法。该方法计算效率高，不需要对感兴趣函数的结构的分析知识，并为检验参数的凸函数的单边假设提供了一致有效的推理。在限制分布的一阶Delta方法退化的情况下，二阶(或更高)Delta方法可以提供必要的非退化大样本近似。研究人员进一步将数值Delta方法推广到一种新的重采样技术，称为数值自举，它可以在许多情况下一致地估计极限分布--在许多情况下，传统的自举是无效的，二次抽样一直是最常用的推理方法，并且参数在方向上是未知的。应用包括以规则和非标准速率收敛的约束和无约束M-估计量，例如最大得分模型、部分识别模型、错误指定的模拟GMM模型和许多依赖于样本大小的统计量。

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.