Inference in Nonlinear Models with Endogeneity

拟议的研究涉及开发新的推理程序的各种非线性模型的横截面或面板数据。所讨论的模型，如二元选择和罗伊模型，在实证工作中得到了广泛的应用。第一个是关于具有自我选择的模型的面板数据版本。自我选择模型使计量经济学家能够控制经济主体的最佳决策。例如，观察到的工资应反映一个部门提供给个人的工资超过所有其他部门提供的工资。面板数据模型，其中代理人的结果在多个时间段内观察，已成为越来越受欢迎的实证研究。纵向面板数据集的增加提供了新的机会，计量经济学家控制个体未观察到的异质性跨代理。Arellano和Honore（2001）综述了非线性面板数据模型的重要工作。然而，有很少的工作在面板数据领域的模型与自我选择，所提出的研究旨在解决这个问题。推断方法提出了平稳和非平稳条件下。前者是指一种假设，即个体的未观察成分随时间具有相同的分布。后者放宽了这一假设，但规定，未观察到的组件，为不同的个人在横截面具有相同的分布在同一时间段。在这两种情况下，新方法都能够估计感兴趣的参数的尖锐集合，例如劳动力供给曲线的斜率。锐集是指当数据满足计量经济学模型的假设时所能得到的最小集合。本建议的第二部分涉及具有离散内生协变量的横截面二元选择模型。这种模型经常出现在治疗效果文献中，其中内生变量通常是治疗状态，结果变量是二元的，如就业状态。在这些情况下，通常感兴趣的参数是回归框架中的处理系数。两种方法来确定这样一个参数，已被认为是在文献中的控制功能和工具变量的方法。这里建议的活动是建立两种方法之间的关系。特别是，一个定理，建立了一个控制函数模型，这表明它是多么困难，进行推理的治疗效果参数的利益。这类似于（Khan和Tamer（2010））中工具变量模型的定理。第三部分研究了一类具有自选择的截尾回归模型（如Roy模型）的最优性问题。首先，条件，确保点识别的参数的兴趣被认为是，如独立性，或支持条件，并推导出效率界。点识别是指将锐集归约为单个值。效率界限是指在计量经济学模型的假设下，估计程序可达到的最小方差。这种界限的有用性是双重的-一方面，它将能够测量在实践中采用的方法的相对效率，第二，它将建议达到界限的新的估计程序。和B。Honore（2001）：“Panel Data Models：Some Recent Developments，”Handbook of Econometrics.第五卷，第100页。3229- 96. Khan，S.，和E. Tamer（2010）：“Irregular Identification，Support Conditions and Inverse Weight Estimation，”Econometrica，forthcoming.

The proposed research involves developing new inference procedures for a variety of non-linear models with cross sectional or panel data. The models discussed, such as the binary choice and Roy model have seen widespread use in empirical work.The proposed activity can be divided into three parts. The first pertains to panel data versions of models with self selection. Self selection models enable the econometrician to control for optimal decisions of the economic agent. For example, observed wages shouldreflect that the wage offered to an individual in one sector exceeds the wage offered in all other sectors. Panel data models, where an agent's outcomes are observed over multiple time periods, have become increasingly popular in empirical research. The increased availability of longitudinal panel data sets has presented new opportunities for econometricians to control for individual unobserved heterogeneity across agents. Important work in nonlinear panel data models is surveyed in (Arellano and Honore (2001)). However, there is very little work in the area of panel data for models with self selection, and the proposed research aims to address this.Inference methods are proposed under both stationary and nonstationary conditions. The former refers to an assumption that unobserved components of individuals have the same distribution over time. The latter relaxes this assumption but imposes that unobserved components for different individuals in the cross section have the same distribution in the same time period. In both cases the new methods are able to estimate sharp sets for parameter of interest, such as the slope of a labor supply curve. A sharp set refers to the smallest set that can be obtained when the data satisfies the assumptions of the econometric model.The second part of this proposal pertains to cross sectional binary choice models with discrete endogenous covariates. Such models arise frequently in the treatment effect literature, where the endogenous variable is often the treatment status, and the outcome variable is binary, such as employment status. A parameter that is often of interest in these situations is the coefficient on treatment in a regression framework. Two approaches to identifying such a parameter that have been considered in the literature are the control function and the instrumental variable methods. The proposed activity here is to establish a relation between the two methods. In particular, a theorem is established for a control function model which demonstrates how difficult it is to conduct inference on the treatment effect parameter of interest. This is analogous to the theorem in (Khan and Tamer (2010)) for the instrumental variable model. Consequently, inference becomes nonstandard and so new inference methods are proposed.The third part is about establishing optimality results for a wide class of cross sectional censored regression models with self selection, such as the Roy model. First, conditions that ensure point identification of the parameters of interest are considered, such as independence, or support conditions, and efficiency bounds are derived. Point identification refers to the sharp set reducing to a single value. Efficiency bounds refer to the smallest attainable variance for an estimation procedure under the assumptions of the econometric model. The usefulness of such bounds is twofold - for one it will enable measuring the relative efficiency of methods that are adopted in practice, and second it will suggest new estimation procedures which attain the bound.ReferencesArellano, M., and B. Honore (2001): "Panel Data Models: Some Recent Developments," Handbook of econometrics. Volume 5, pp. 3229-96.Khan, S., and E. Tamer (2010): "Irregular Identification, Support Conditions and Inverse Weight Estimation," Econometrica, forthcoming.