Nonparametric Identification and Estimation of Distributions of Unobserved Heterogeneity in Economic Choice Models using Mixtures

结构性经济选择模型显示了个体代理人(企业、消费者、工人)如何在不同的选择集下做出选择。没有理由相信这些模型的参数对所有智能体都是相同的：当面对相同的选择集时，智能体会做出不同的选择。由于参数在不同的代理人之间不同，这些模型的经验工作的目标是估计未观察到的异质性的分布，即随机系数的分布。这种分布需要用来计算一项政策的福利效应(比如税收政策带来的消费者盈余)，或者用来计算对一种新商品的需求。在统计理论方面，具有未观察到的异质性的模型称为混合模型。统计学文献关注的是某些参数类中分布的混合，例如正态分布的混合。使用混合模型的工具来研究更复杂的非线性统计模型，例如结构性经济选择模型，受到的关注要少得多。非参数识别表明，未观察到的异质性的一个特定分布与条件结果概率的有限信息是一致的。我们使用横截面数据对面临不同选择集的代理进行识别。统计学文献证明，混合模型的线性独立性对于识别是必要的，也是充分的。对于经济选择模型来说，这一条件很难从代数上得到验证。这个项目引入了一个新的条件，即可约性，它足以使线性无关，从而识别。简约性是经济模型的一种属性，很容易得到验证，这一点在经济学中广泛使用的一组经验模型中得到了证明。在显示识别之后，未观察到的异质性的参数或非参数分布可以更有把握地估计。估计混合模型最常用的工具是EM算法，这当然是适用的。然而，EM算法存在数值问题，可能不适合于本身需要复杂计算的经济选择模型，如动态规划模型。该项目引入了一种新的、计算简单的混合估计器来解决这些问题。BROADER影响：主要的更广泛的影响将使未观察到的异质性(随机系数)的分布的识别和估计变得更加简单。这是在两个方面完成的：使显示新模型识别的条件更容易验证，并引入计算简单的估计器。经济主体选择的模型每天都在数以千计的实证应用中使用。例如，已婚妇女参加劳动力市场的决定引发了一个选择问题：工资只对工作的妇女遵守。观察到有工作的人的偏好和工作机会并不能代表那些不工作的人。为了解决这一选择问题，联合估计参与和工资决定是必要的。了解这一过程对于了解过去30年来性别工资差距的变化是必要的。再举一个例子，考虑一下预测电器用电量的环境经济学问题。电器上唯一可用的数据？S的用电量是那些购买电器的消费者的数据，是经过挑选的样本。改变家用电器或电力本身的价格(比方说从税收)将改变购买家用电器的消费者群体和以购买为条件的电力使用。估计异质性的分布对于计算税收效应的福利衡量是必要的。其中一名研究人员使用这些方法估计了无线运营商合并对消费者福利的影响，印度教师出勤率对经济激励的反应，以及经验丰富的工程师对竞争对手公司提供的工资的工作流动性反应。总体而言，我们认为，展示认同感可以让经济学家和统计学家更容易估计复杂的模型。我们希望将这些模型建立在坚实的理论基础上，以便政策制定者和其他研究人员更有信心地对待这些模型的结果。我们还扩展了将被估计的模型的类别。由于其中一些方法的参数版本在实证工作中不断使用，我们认为这项工作对应用经济学的所有领域和相关领域都有巨大的影响。其中一名研究人员在法国巴黎的INSEE-CREST/ENSAE为研究生和知名研究人员教授了一门小型课程。这两位研究人员都在自己的机构向博士生教授这些技术，并将更深入地为由这笔资金资助的本科生和研究生传授这些技术。

Structural economic choice models show how individual agents (firms, consumers, workers) make choices under different choice sets. There is no reason to believe that the parameters of these models are the same for all agents: agents will make different choices when faced with the same choice set. As the parameters vary across agents, the goal of empirical work with these models is to estimate the distribution of unobserved heterogeneity, or the distribution of random coefficients. This distribution is needed to calculate the welfare effects of a policy (say consumer surplus from a tax policy) or to compute demand for a new good, out of sample. In terms of statistical theory, a model with unobserved heterogeneity is called a mixtures model. The statistics literature has focused attention on mixtures of distributions in some parametric class, such as mixtures of normals. Much less attention has been paid to using the tools of mixtures to study more complex nonlinear statistical models, such as structural economic choice models. Nonparametric identification shows that one particular distribution of unobserved heterogeneity is consistent with limiting information on conditional outcome probabilities. We show identification using cross-sectional data on agents facing different choice sets. The statistics literature establishes that linear independence of the class of models being mixed over is necessary and sufficient for identification. This condition is difficult to algebraically verify for economic choice models. This project introduces a new condition, reducibility, that is sufficient for linear independence and hence identification. Reducibility is a property of economic models that can be easily verified, as is shown for a group of models of wide empirical use in economics. After showing identification, parametric or nonparametric distributions of unobserved Heterogeneity can be more confidently estimated. The most common tool for estimating mixtures models is the EM algorithm, which is certainly applicable. However, the EM algorithm has numerical issues and may be inappropriate for economic choice models that themselves require complex calculations, such as dynamic programming models. This project introduces a new, computationally simple mixtures estimator to resolve these issues. The estimator is nonparametric.BROADER IMPACT: The main broader impact will be to make the identification and estimation of distributions of unobserved heterogeneity (random coefficients) much simpler. This is done on two fronts: making the conditions for showing identification of new models easier to verify and introducing a computationally simple estimator. Models of choice by economic agents are used every day in thousands of empirical applications. For example, the decision of a married woman to participate in the labor market induces a selection problem: wages are observed only for the women who do work. The preferences and job opportunities of those observed to work are not representative of those who do not work. Jointly estimating the participation and wages decisions is necessary to resolve this selection problem. Understanding this process is necessary for understanding changes in the gender-wage gap over the last 30 years. For another example, consider the environmental economics problem of forecasting electricity use for appliances. The only data available on an appliance?s electricity use are for those consumers who buy the appliance, a selected sample. Changing the prices (say from a tax) of the appliances or of electricity itself will shift both the set of consumers who buy the appliance and electricity use conditional on buying. Estimating the distribution of heterogeneity is necessary for computing welfare measurements of the effect of the tax. One of the investigators has used these methods to estimate the consumer welfare implications of mergers of wireless carriers, the response of teacher attendance in India to financial incentives, and the job mobility response of experienced engineers to wage offers from competing firms. Overall, we see showing identification as making economists and statisticians more comfortable with estimating complex models. We hope to place these models on firm theoretical ground, so policymakers and other researchers treat results from these models with more confidence. We also expand the class of models that will be estimated. As parametric versions of some of these methods are in constant use in empirical work, we see this work as having tremendous impact of all areas of applied economics and related fields. One of the investigators has taught a mini course to graduate students and established researchers on these techniques at INSEE-CREST / ENSAE in Paris, France. Both investigators are teaching these techniques to PhD students at their home institutions and will do so more intensively for the undergraduate and graduate students funded by this grant.