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学模型中识别和推断的文献。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部变量之间的关系是对应而不是函数时，计量经济学模型可能是不完整的。在这些情况下，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点时，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备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，因为它们为随机集提供有条件和无条件的概率分布和期望，这允许研究人员以与通常在向量空间中的点识别模型如何执行该任务的精确模拟的方式来表征集合空间中的模型的识别特征。研究人员的目标是开发一种方法，重点放在一类特定的不完整模型上，它为这些模型提供了一个易于计算的清晰识别区域的特征。如果一个不完整的模型预测的是给定协变量的结果的条件概率分布的凸集，而不是单一的条件概率分布，那么它就属于拟议研究中处理的那类模型。这类模型的例子包括：在存在多个混合策略纳什均衡的情况下，完全信息的静态、同时移动有限博弈；以及具有区间回归数据的多目标选择模型。提案中对这些例子进行了明确的分析。在相关文献中，对于这类模型参数的精确辨识域的精确刻画被认为是不容易得到的。在最近的经济学理论和经验文献中，部分辨认模型是普遍存在的。虽然有时很容易明确地描述它们的识别区域，但存在许多重要的问题，其中很难获得易于处理的描述。要建立清晰度可能特别困难，也就是说，要证明一个猜测区域正好包含可行的参数值，而不包含其他参数值。将推理建立在一个不清晰的推测区域上，可能会显著削弱研究人员做出有用预测和测试模型错误说明的能力。这一建议的学术价值有两个方面：(1)提供一个方法框架，以获得一个模型的尖锐识别区域的易于计算的表征；(2)为从业者提供随时可以使用软件来应用这种方法，并在无法获得点识别时进行估计和推理。广泛的影响：拟议的尖锐识别区域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，并通过处理识别方面以及问题的统计推断方面的问题，比较不同政策研究方法的结果。这项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对使用随机集理论的部分识别模型的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.