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个体主体（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于所有agent都是相同的：agent在面对相同的选择集时会做出不同的选择。由于各代理的参数各不相同，这些模型的经验工作目标是估计未观察到的异质性的分布，或随机系数的分布。在计算一项政策的福利效应（比如税收政策带来的消费者剩余）或计算样本外新商品的需求时，需要这种分布。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。统计文献主要关注某些参数类的混合分布，如正态分布的混合。很少有人注意使用混合工具来研究更复杂的非线性统计模型，如结构经济选择模型。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限制信息一致。我们使用面对不同选择集的代理的横截面数据来显示识别。统计文献表明，混合模型类别的线性独立性对于识别是必要和充分的。对于经济选择模型来说，这个条件很难用代数方法来验证。这个项目引入了一个新的条件，可简化性，这是线性独立和识别的充分条件。可还原性是经济模型的一个属性，可以很容易地验证，正如经济学中广泛使用的一组模型所显示的那样。在显示识别后，未观察到的异质性的参数或非参数分布可以更自信地估计。估计混合模型最常用的工具是EM算法，它当然是适用的。然而，EM算法有数值问题，可能不适合本身需要复杂计算的经济选择模型，如动态规划模型。本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。估计量是非参数的。更广泛的影响：主要的更广泛的影响将是使未观察到的异质性（随机系数）分布的识别和估计更加简单。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。经济主体的选择模型每天都在成千上万的实证应用中使用。例如，已婚妇女参与劳动力市场的决定导致了一个选择问题：工资只观察到有工作的妇女。那些被观察到工作的人的偏好和工作机会并不代表那些不工作的人。为了解决这一选择问题，需要对参与率和工资决策进行联合估计。了解这一过程对于理解过去30年性别工资差距的变化是必要的。再举一个例子，考虑预测电器用电量的环境经济学问题。设备上唯一可用的数据是什么？S的用电量都是针对那些购买家电的消费者，选取的样本。改变电器或电力本身的价格（比如通过征税）将改变购买电器的消费者群体和购买电器所附带的用电量。估计异质性的分布对于计算税收影响的福利测量是必要的。其中一名研究人员使用这些方法来估计无线运营商合并对消费者福利的影响，印度教师出勤率对财政激励的反应，以及经验丰富的工程师对竞争公司工资报价的工作流动性反应。总的来说，我们认为显示身份可以使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，这样政策制定者和其他研究人员就能更有信心地对待这些模型的结果。我们还扩展了将要估计的模型的类别。由于其中一些方法的参数化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和相关领域的所有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为研究生开设了一门迷你课程，并建立了研究人员。两位研究者都在各自的研究机构向博士生教授这些技术，并将为获得这项资助的本科生和研究生更深入地教授这些技术。

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.