Estimation of Nonlinear Models with Measurement Error

This paper presents a solution to an important econometric problem, namely the root n consistent estimation of nonlinear models with measurement errors in the explanatory variables, when one repeated observation of each mismeasured regressor is available. While a root n consistent estimator has been derived for polynomial specifications (see Hausman, Ichimura, Newey and Powell, 1991), such an estimator for general nonlinear specifications has so far not been available. Using the additional information provided by the repeated observation, the suggested estimator separates the measurement error from the "true" value of the regressors thanks to a useful property of the Fourier transform: The Fourier transform converts the integral equations that relate the distribution of the unobserved "true" variables to the observed variables measured with error into algebraic equations. The solution to these equations yields enough information to identify arbitrary moments of the "true", unobserved variables. The value of these moments can then be used to construct any estimator that can be written in terms of moments, including traditional linear or nonlinear least squares estimators, general extremum estimators or even nonparametric kernel estimators. In the parametric case, a generalization of the delta method to functionals is used to show that the proposed estimator exhibits root n consistency as well as asymptotic normality. Monte-Carlo evidence and an application to Engel curve estimation illustrate the usefulness of this new approach