Collaborative Research: Identification in incomplete econometric models using random set theory

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目将有助于在不完整的计量经济模型的识别和推理的文献。例如，当样本实现不完全可观察时，或者当模型断言感兴趣的结果变量与外生变量之间的关系是对应关系而不是函数时，计量经济学模型可能是不完整的。 在这些情况下，采样过程和保持的假设与表征模型的参数向量（或统计泛函）的一组值一致。这组值是模型参数的尖锐识别区域。当尖锐的识别区域不是单点时，模型被部分识别。研究人员使用随机集理论的工具来研究不完全计量经济模型中的识别问题。这些工具特别适合于部分识别分析，因为它们提供了随机集的条件和无条件概率分布和期望，允许研究人员以一种精确模拟的方式来描述集合空间中模型的识别特征，这种方式与矢量空间中的点识别模型通常执行的任务完全相同。 研究人员旨在开发的方法专注于一类特定的不完整模型，它提供了一个计算上易于处理的尖锐识别区域的特征。一个不完整的模型属于所提出的研究中处理的类，如果它预测的结果给定协变量的条件概率分布的凸集，而不是一个单一的条件概率分布。这类模型的例子包括：静态的，同时移动有限的完全信息博弈中存在多个混合策略纳什均衡;和多元选择模型与区间回归数据。提案中明确分析了这些例子。部分可辨识模型在最近的经济学理论和经验文献中普遍存在，但在相关文献中，这类模型的参数的精确可辨识区域的计算刻画是难以实现的。虽然有时很容易明确地描述其识别区域，但存在许多重要问题，其中难以获得易于处理的特征。建立清晰度可能特别困难，也就是说，要显示一个精确的区域包含可行的参数值，而不包含其他参数值。基于一个不尖锐的约束区域的推断可能会大大削弱研究人员做出有用预测的能力，并测试模型的错误设定。这一建议的智力价值是双重的：（1）提供一个方法框架，以获得模型的尖锐识别区域的计算上易于处理的特征;（2）为从业人员提供现成的软件，以应用这一方法，并在点识别不可用时进行估计和推断。所提出的尖锐识别区域的表征和计算方法使从业人员能够评估现有政策研究的可信度，并比较不同政策研究方法的结果，通过解决识别方面，以及问题的统计推断方面。该研究计划旨在通过本科生的研究经验，研究生助理的使用，以及使用随机集理论在部分识别模型中进行推理的研究生课程的教学，将教学和研究结合起来。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project would contribute to the literature on identification and inference in incomplete econometric models. An econometric model may be incomplete when, for example, sample realizations are not fully observable, or when the model asserts that the relationship between the outcome variable of interest and the exogenous variables is a correspondence rather than a function. In these cases, the sampling process and the maintained assumptions are consistent with a set of values for the parameter vectors (or statistical functionals) characterizing the model. This set of values is the sharp identification region of the models parameters. When the sharp identification region is not a singleton, the model is partially identified. The investigators use the tools of random sets theory to study identification in incomplete econometric models. These tools are especially suited for partial identification analysis, because they provide conditional and unconditional .probability distributions and expectations for random sets, that allow researchers to characterize the identified features of a model in the space of sets, in a manner which is the exact analog of how this task is commonly performed for point identified models in the space of vectors. The methodology that the investigators aim to develop focuses on a specific class of incomplete models, for which it provides a computationally tractable characterization of the sharp identification region. An incomplete model belongs to the class treated in the proposed research, if it predicts a convex set of conditional probability distributions of outcomes given covariates, rather than a single conditional probability distribution. Examples of models in this class include: static, simultaneous move finite games of complete information in the presence of multiple mixed strategy Nash equilibria; and polychotomous choice models with interval regressor data. These examples are explicitly analyzed in the proposal. A computationally tractable characterization of the sharp identification region of the parameters of models in this class was considered unattainable in the related literature.Partially identified models are ubiquitous in the recent theoretical and empirical literature in economics. Although it sometimes is easy to characterize their identification region explicitly, there exist many important problems in which a tractable characterization is difficult to obtain. It may be particularly difficult to establish sharpness, that is, to show that a conjectured region contains exactly the feasible parameter values and no others. Basing inference on a conjectured region which is not sharp may significantly weaken the ability of the researcher to make useful predictions, and to test for model misspecification. The intellectual merit of this proposal is twofold: (1) To provide a methodological framework to obtain a computationally tractable characterization of the sharp identification region of a model; (2) To provide practitioners with ready to use software to apply this methodology and conduct estimation and inference when point identification is not available.Broader impacts: The proposed methodology for characterization and computation of the sharp identification region enables practitioners to evaluate the credibility of existing policy studies, and compare the results of different approaches to policy research, by addressing both the identification aspects, as well as the statistical inference aspects of the problem. This research program aims to integrate teaching and research through research experience for undergraduates, the use of graduate assistants, and the instruction of a graduate course on inference in partially identified models using random sets theory.