Instrumental Variable Estimation of Nonlinear Errors-in-Variables Models

Abstract

In linear specifications, the bias due to the presence of measurement error in a regressor can be entirely avoided when either repeated measurements or instruments are available for the mismeasured regressor. The situation is more complex in nonlinear settings. While identification and root n consistent estimation of general nonlinear specifications have recently been proven in the presence of repeated measurements, similar results relying on instruments have so far only been available for polynomial specifications and absolutely integrable regression functions. This paper addresses two unresolved issues. First, it is shown that instruments indeed allow for the fully nonparametric identification of general nonlinear regression models in the presence of measurement error. Second, when the regression function is parametrically specified, a root n consistent and asymptotically normal estimator is provided. The starting point of the proposed approach is a system of two functional equations that relate conditional expectations of observed variables to the regression function of interest, as first proposed by Hausman, Ichimura, Newey and Powell (1991) for polynomial specifications. It is shown that these two equations have a unique solution, thus establishing identification. The proposed estimation procedure relies on the same functional equations, and the proof of asymptotic normality and root n consistency is based on standard results regarding the asymptotics of semiparametric estimatorserrors-in-variables, measurement error, Fourier transforms, nonlinear models, semiparametric estimation