Nonlinear Models with Errors-in-Variables

利用工具变量对回归变量具有测量误差的非线性模型进行辨识和根n相合估计是计量经济学和统计学中的一个长期难题。 该项目通过广泛使用傅立叶分析和广义函数理论，结合半参数估计方法，为这个问题提供了一个实际的解决方案。 研究的方法依赖于参数假设的回归函数，以实现根n的一致性，但避免任何参数约束的所有变量的分布。 被认为是两个替代权衡的强度的假设上的误测协变量和工具，其中之一是直接适用于面板数据设置。 本文通过生产函数分析、恩格尔曲线估计、流行病学和营养学研究中的实例说明了所提出方法的实用性，这些研究的公开数据是现成的，大多数计量经济学和统计学教科书都描述了如何通过使用所谓的工具变量来消除线性回归分析中由于协变量测量误差而产生的偏差。 这个项目提供了适用于非线性模型的这种方法的推广。 在这项研究中设计的方法将是有帮助的，因为非线性和测量误差必然会同时存在于经济学中的许多应用中。 例如，家庭在某种商品或服务上的支出通常是家庭收入的非线性函数，而家庭收入是一个众所周知的误报变量。 对误测量的非线性反应在生物统计学和流行病学中也很常见，其中暴露（病原体或污染物）通常被错误测量，并且对暴露的生理反应通常是非线性的。 更一般地说，只要涉及到人类主体，对错误测量的输入的非线性反应是规则而不是例外，这对经济行为和对疾病或药物的生理反应同样适用。 实施这些方法的计算机程序将公开提供。

The identification and the root n consistent estimation of nonlinear models with measurement error in the regressors using instrumental variables is a long-standing problem in econometrics and statistics. This project provides a practical solution to this problem through extensive use of Fourier analysis and the theory of generalized functions, combined with semiparametric estimation methods. The methods investigated rely on parametric assumptions regarding the regression function to achieve root n consistency, but avoid any parametric constraints on the distribution of all the variables. Two alternative trade-offs between the strength of the assumptions on the mismeasured covariates and on the instruments are considered, one of which is directly applicable to panel data settings. The usefulness of the proposed approaches is illustrated through examples drawn from production function analysis, Engel curve estimation, epidemiology and nutrition studies, for which public data are readily available.Most econometrics and statistics textbooks describe how to eliminate the bias due to the presence of covariate measurement error in linear regression analysis through the use of so-called instrumental variables. This project provides generalizations of this approach that are applicable to nonlinear models. The methods devised during this study will be helpful, because nonlinearity and measurement error are bound to be simultaneously present in a number of applications in economics. For instance, household expenditure on a given type of goods or services is typically a nonlinear function of household income, a variable that is notoriously misreported. Nonlinear responses to mismeasured quantities are also common in biostatistics and epidemiology, where exposures (to pathogens or contaminants) are typically measured with error and where the physiological response to the exposure is typically nonlinear. More generally, whenever human subjects are involved, a nonlinear response to mismeasured inputs is the rule rather than the exception, and this holds equally for economic behavior as for physiological responses to diseases or medications. A computer program implementing these methods will be made publicly available.