Instrumental Nonparametric Estimation Under Conditional Moment Restrictions : A Nonlinear Tikhonov Approach

This paper is concerned about the estimation of an infinite-dimensional parameter θ0 which is identified via E[ρ(Z, θ0)|W ] = 0 where W is a continuous instrument and ρ(.) a known mapping. Using the principle of Tikhonov regularization developed in Bissantz, Hohage and Munk (2004), a first stage estimator θ̂ minimizing some penalized objective achieves ||θ̂ − θ0|| = Op(n 1 4+γ ) for some positive constant γ which vanishes under adequate conditions with ||.|| being a relatively strong Hilbert norm. Under extra regularity conditions, a second stage estimation, inspired from the Levenberg-Marquardt method for solving iteratively nonlinear integral equations, permits to construct a CAN estimator for < θ0, f > whenever f meets a certain type of smoothness. Finally, a consistent estimator for the asymptotic variance is suggested in order to conduct inferences from data.