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

该项目导出了估计量的渐近风险度量和高水平假设下测试的渐近大小的显式公式，这些公式适用于非常广泛的一类具有不连续的模型。提供了一个非常普遍的统一理论，适用于以前在文献中只在个案基础上研究过的许多模型。许多计量经济学模型中的估计量和检验统计量具有依赖于干扰参数值的渐近分布。当干扰参数逼近某一不连续点时，渐近分布往往发生不连续变化。研究者和Donald Andrews以及研究者（2005a-e）研究了这种不连续模型中检验的渐近大小。他们过去的工作为测试的大小提供了一个明确的公式，并表明在实践中使用的许多测试是非常大小扭曲的。本课题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，可以使用风险公式根据它们的最大风险对备选估计器进行排序。风险公式适用于参数空间中某些点缺乏识别的模型，例如具有弱工具和阈值自回归模型的模型，根可能接近单位的标量和向量自回归（VAR）模型，参数可能接近边界的模型，参数由矩不等式定义的模型，基于超高效或收缩估计的模型，基于后模型选择估计的模型，具有近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。该项目的第二个目标是将测试大小的显式公式应用于需要大量计算工作的领域。这些检验包括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，有限支持随机变量不完全模型的片面Kolmogorov-Smirnov检验和后Hausman检验推理。更广泛的影响：本研究提供了在应用研究人员通常使用的模型中具有正确大小和具有有利的相对甚至最佳风险属性的估计器的测试程序。这里介绍的技术可以用于更多的应用经济学领域，如金融、产业组织或劳动经济学。这项工作的结果将提交给主要的经济学期刊，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.