Optimal Sup-Norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation

Abstract

This paper makes several contributions to the literature on the important yet difficult problem of estimating functions nonparametrically using instrumental variables. First, we derive the minimax optimal sup-norm convergence rates for nonparametric instrumental variables (NPIV) estimation of the structural function h_0 and its derivatives. Second, we show that a computationally simple sieve NPIV estimator can attain the optimal sup-norm rates for h_0 and its derivatives when h_0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal L^2-norm rates for severely ill-posed problems, and are only up to a [log(n)]^epsilon (with epsilon < 1/2) factor slower than the optimal L^2-norm rates for mildly ill-posed problems. Third, we introduce a novel data-driven procedure for choosing the sieve dimension optimally. Our data-driven procedure is sup-norm rate-adaptive: the resulting estimator of h_0 and its derivatives converge at their optimal sup-norm rates even though the smoothness of h_0 and the degree of ill-posedness of the NPIV model are unknown. Finally, we present two non-trivial applications of the sup-norm rates to inference on nonlinear functionals of h_0 under low-level conditions. The first is to derive the asymptotic normality of sieve t-statistics for exact consumer surplus and deadweight loss functionals in nonparametric demand estimation when prices, and possibly incomes, are endogenous. The second is to establish the validity of a sieve score bootstrap for constructing asymptotically exact uniform confidence bands for collections of nonlinear functionals of h_0. Both applications provide new and useful tools for empirical research on nonparametric models with endogeneity.

Keywords: Ill-posed inverse problems, Series 2SLS, Optimal sup-norm convergence rates, Adaptive estimation, Random matrices, Bootstrap uniform confidence bands, Nonlinear welfare functionals, Nonparametric demand analysis with endogeneity

JEL Classification: C13, C14, C32