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 and nonlinear least squares estimators, or general extremum estimators. The proposed estimator is shown to admit a representation in terms of an influence function, thus establishing its root n consistency and asymptotic normality. Monte Carlo evidence and an application to Engel curve estimation illustrate the usefulness of this new approach. Copyright Econometric Society 2004.

The purpose of this note is to show how semiparametric estimators with a small bias property can be constructed. The small bias property (SBP) of a semiparametric estimator is that its bias converges to zero faster than the pointwise and integrated bias of the nonparametric estimator on which it is based. We show that semiparametric estimators based on twicing kernels have the SBP. We also show that semiparametric estimators where nonparametric kernel estimation does not affect the asymptotic variance have the SBP. In addition we discuss an interpretation of series and sieve estimators as idempotent transformations of the empirical distribution that helps explain the known result that they lead to the SBP. In Monte Carlo experiments we find that estimators with the SBP have mean-square error that is smaller and less sensitive to bandwidth than those that do not have the SBP.