Robust and optimal estimation for partially linear instrumental variables models with partial identification

This paper studies robust and optimal estimation of the slope coefficients in a partially linear instrumental variables model with nonparametric partial identification. We establish the root-n asymptotic normality of a penalized sieve minimum distance estimator of the slope coefficients. We show that the asymptotic normality holds regardless of whether the nonparametric function is point identified or only partially identified. However, in the presence of nonparametric partial identification, the slope coefficients may not be continuous in the underlying distribution and the asymptotic variance matrix may depend on the penalty, so classical efficiency analysis does not apply. We instead develop an optimally penalized estimator that minimizes the asymptotic variance of a linear functional of the slope coefficients estimator by employing an optimal penalty for a given weight, and propose a feasible two-step procedure. We also propose an iterated procedure to address how to choose both penalty and weight optimally and further improve efficiency. To conduct inference, we provide a consistent variance matrix estimator. Monte Carlo simulations examine the finite sample performance of our estimators.