Risk Properties of Estimators and the Size of Tests in Discontinuous Models

这个项目推导出了高水平假设下的估计量的渐近风险度量和检验的渐近大小的显式公式，这些假设适用于一类非常广泛的具有不连续性的模型。 一个非常普遍的统一理论，适用于许多模型，以前已经在文献中研究的情况下，在case-by-casesbasis.Estimators和许多计量经济模型的检验统计量有渐近分布，依赖于滋扰参数的值。 当干扰参数接近某个不连续点时，渐近分布常常不连续地变化。 研究者和Donald Andrews以及研究者（2005 a-e）研究了这种不连续模型中检验的渐近大小。 他们过去的工作提供了一个明确的公式的大小测试，并表明，许多测试在实践中使用的是非常大小扭曲。 本计画的第一个目标是研究具有干扰参数和损失函数的不连续模型中估计量的风险性质。 该项目推导出了渐近最大风险的显式公式。 在应用统计学中遇到的许多问题中，不存在最优估计量。 对于这些问题的风险公式可以用来排序的替代估计，根据他们的最大风险。 风险公式适用于在参数空间中的某些点处缺乏识别的模型，例如具有弱工具和阈值自回归模型的模型、具有可能接近于1的根的标量和向量自回归（VAR）模型、参数可能接近边界的模型、具有由矩不等式定义的参数的模型、基于超有效或收缩估计的模型，基于后模型选择估计量的模型、具有近似积分回归量的预测回归模型、具有参数不可微函数的模型以及具有一阶导数为零的参数可微函数的模型。 其中包括VAR模型中的格兰杰因果关系检验、有限支持随机变量的随机优势检验、有限支持随机变量的不完全模型的单侧Kolmogorov-Smirnov检验和后Hausman检验推断。这项研究提供了测试程序，有正确的大小和估计有利的相对甚至最佳的风险属性的模型，是常用的应用研究人员本文所介绍的技术可以应用于更实用的经济学领域，如金融学、产业组织学或劳动经济学。这项工作的结果将提交给领先的经济期刊，并在研究研讨会上发表。研究生将参与该项目。

This project derives explicit formulae for asymptotic risk measures for estimators and asymptotic size for tests under high level assumptions that are applicable to a very wide class of models with discontinuities. A very general unifying theory is provided that applies to many models that have previously been studied in the literature only on a case by case basis.Estimators and test statistics in many Econometric models have asymptotic distributions that depend on the values of nuisance parameters. The asymptotic distribution often changes discontinuously as the nuisance parameter approaches a certain discontinuity point. The investigator and Donald Andrews and the investigator (2005a-e) studied the asymptotic size of tests in such discontinuous models. Their past work provides an explicit formula for the size of tests and demonstrates that many tests used in practice are extremely size distorted. The first goal of this project is to investigate the risk properties of estimators in discontinuous models with nuisance parameters and a loss function. The project derives an explicit formula for the asymptotic maximal risk. In many problems encountered in applied statistics, there does not exist an optimal estimator. For these problems the risk formula can be used to rank alternative estimators according to their maximal risk. The risk formula is applied to models with lack of identification at some point(s) in the parameter space, such as models with weak instruments and threshold autoregressive models, scalar and vector autoregressive (VAR) models with roots that may be close to unity, models where a parameter may be near a boundary, models with parameters defined by moment inequalities, models based on super-efficient or shrinkage estimators, models based on post-model selection estimators, predictive regression models with nearly-integrated regressors, models with non-differentiable functions of parameters, and models with differentiable functions of parameters that have zero first-order derivatives.The second goal of this project is to apply the explicit formula for the size of tests to areas that require intensive computational effort. These include tests of Granger causality in VAR models, tests of stochastic dominance for random variables with finite support, and one-sided Kolmogorov-Smirnov tests of incomplete models for random variables with finite support and post Hausman test inference.Broader impact: This research provides testing procedures that have correct size and estimators with favorable relative or even optimal risk properties in models that are commonly used by applied researchers. The techniques introduced here can be used in more applied Economic fields, such as Finance, Industrial Organization or Labor Economics. The results of this work will be submitted to leading Economic journals and presented in research seminars. Graduate students will participate in the project.