Nonlinear Models with Errors-in-Variables

利用工具变量对回归变量中具有测量误差的非线性模型进行辨识和根n相容估计是计量经济学和统计学中的一个长期问题。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，为这一问题提供了一个实用的解决方案。所研究的方法依赖于关于回归函数的参数假设来实现根n的一致性，但避免了对所有变量的分布的任何参数约束。在错误测量的协变量和工具上的假设的强度之间的两种替代权衡被考虑，其中一种直接适用于面板数据设置。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的实例说明了所提方法的有效性，大多数计量经济学和统计学教科书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存在的协变量测量误差造成的偏差。本项目提供了适用于非线性模型的这种方法的一般情况。在这项研究中设计的方法将是有帮助的，因为在经济学中的许多应用中，非线性和测量误差必然同时存在。例如，家庭在特定类型商品或服务上的支出通常是家庭收入的非线性函数，这个变量被误报是出了名的。对错误测量量的非线性反应在生物统计学和流行病学中也很常见，在生物统计学和流行病学中，暴露(对病原体或污染物)的暴露通常是有误差的，并且对暴露的生理反应通常是非线性的。更广泛地说，当涉及到人类受试者时，对错误测量的投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之于众。

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.