Inference in Nonlinear Models with Endogeneity

提出的研究包括为各种具有横截面或面板数据的非线性模型开发新的推理程序。所讨论的模型，如二元选择和罗伊模型已经在实证工作中广泛使用。建议的活动可分为三个部分。第一个与自我选择模型的面板数据版本有关。自我选择模型使计量经济学家能够控制经济主体的最优决策。例如，观察到的工资应反映出某一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，其中一个代理的结果在多个时间段内观察，在实证研究中越来越受欢迎。纵向面板数据集的可用性的增加为计量经济学家提供了新的机会来控制个体间未观察到的异质性。在非线性面板数据模型的重要工作进行了调查（Arellano和Honore(2001)）。然而，在自我选择模型的面板数据领域的工作很少，提出的研究旨在解决这个问题。提出了平稳和非平稳条件下的推理方法。前者指的是一种假设，即个体中未观察到的组成部分随着时间的推移具有相同的分布。后者放宽了这一假设，但要求截面上不同个体的未观察分量在同一时间段内具有相同的分布。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，例如劳动供给曲线的斜率。锐集是指当数据满足计量模型的假设条件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。这样的模型在治疗效果文献中经常出现，其中的内生变量往往是治疗状态，结果变量是二元的，如就业状态。在这些情况下，通常感兴趣的一个参数是回归框架中的处理系数。识别这种参数的两种方法在文献中被认为是控制函数和工具变量方法。这里建议的活动是建立两种方法之间的关系。特别地，对控制函数模型建立了一个定理，证明了对感兴趣的处理效果参数进行推理是多么困难。这类似于（Khan和Tamer(2010)）中关于工具变量模型的定理。因此，推理变得不规范，因此提出了新的推理方法。第三部分是关于为广泛的具有自我选择的横截面截尾回归模型（如Roy模型）建立最优性结果。首先，考虑了保证感兴趣参数点识别的条件，如独立性或支持条件，并推导了效率界限。点识别是指将尖锐集合缩小到单个值。效率界限是指在计量经济模型的假设下，估算过程中可达到的最小方差。这种界限的有用性是双重的——首先，它将使测量在实践中采用的方法的相对效率成为可能，其次，它将提出达到该界限的新估计程序。参考文献arellano， M.和B. Honore(2001)：“面板数据模型：一些最新发展”，《计量经济学手册》。第五卷，第3229-96页。Khan， S.和E. Tamer(2010)：“不规则识别、支持条件和逆权重估计”，《计量经济学》，即将出版。

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.