Tikhonov Regularization for Functional Minimum Distance Estimators

We study the asymptotic properties of a Tikhonov Regularized (TiR) estimator of a functional parameter based on a minimum distance principle for nonparametric conditional moment restrictions. The estimator is computationally tractable and takes a closed form in the linear case. We derive its asymptotic Mean Integrated Squared Error (MISE), its rate of convergence and its pointwise asymptotic normality under a regularization parameter depending on sample size. The optimal value of the regularization parameter is characterized. We illustrate our theoretical findings and the small sample properties with simulation results for two numerical examples. We also discuss two data driven selection procedures of the regularization parameter via a spectral representation and a subsampling approximation of the MISE. Finally, we provide an empirical application to nonparametric estimation of an Engel curve.