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

This paper treats the estimation of an unknown function θ0 which is identified via the conditional moment restriction E[ρ(Z, θ0)|W ] = 0 where W is a continuously distributed instrument and ρ is a known mapping. Using the principle of Tikhonov regularization, a new estimator minimizing some penalized objective is shown to be MSE consistent under a relatively strong Hilbert norm. This Tikhonov estimator achieves a rate of convergence in probability equal to n−1/(4+γ), where γ is a positive constant. Furthermore, γ vanishes under adequate smoothness conditions. This methodology does not require the unknown function to be continuous. Some Monte Carlo experiments are conducted highlighting the good finite sample properties of this estimator.