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全计量经济模型的识别和推断的文献。例如，当样本实现不能完全观察到，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关系时，计量经济学模型可能是不完整的。在这些情况下，抽样过程和维持的假设与表征模型的参数向量（或统计函数）的一组值一致。这组值是模型参数的尖锐识别区域。当尖锐识别区域不是单一时，模型被部分识别。研究者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，因为它们提供了条件和无条件。随机集的概率分布和期望，使研究人员能够在集合空间中表征模型的已识别特征，其方式与向量空间中点识别模型的通常执行方式完全类似。研究人员旨在开发的方法侧重于一类特定的不完整模型，为此它提供了一个计算上易于处理的尖锐识别区域的表征。如果一个不完全模型预测给定协变量的结果的条件概率分布的凸集，而不是一个单一的条件概率分布，那么它就属于本文所讨论的这类模型。本课程中的模型示例包括：存在多个混合策略纳什均衡的静态，同时移动的完全信息有限博弈；以及区间回归数据的多分选择模型。在建议中对这些例子进行了明确的分析。在相关文献中，这类模型参数的尖锐识别区域的计算可处理的表征被认为是无法实现的。部分识别模型在最近的经济学理论和实证文献中无处不在。虽然有时很容易明确地表征它们的识别区域，但存在许多难以获得可处理表征的重要问题。建立清晰度可能特别困难，也就是说，要表明一个推测区域只包含可行参数值而不包含其他参数值。在不清晰的推测区域上进行推理可能会大大削弱研究人员做出有用预测和检验模型错误规范的能力。该建议的智力价值是双重的：(1)提供了一种方法框架，以获得模型的尖锐识别区域的计算易于处理的特征；(2)为实践者提供随时可用的软件来应用该方法，并在无法进行点识别时进行估计和推断。更广泛的影响：提出的尖锐识别区域的表征和计算方法使从业者能够评估现有政策研究的可信度，并通过解决问题的识别方面和统计推断方面，比较不同政策研究方法的结果。本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指导，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.