Adaptive Estimation and Uniform Confidence Bands for Nonparametric Structural Functions and Elasticities

Abstract

We introduce computationally simple, data-driven procedures for estimation and inference on a structural function h 0 and its derivatives in nonparametric models using instrumental variables. Our first procedure is a bootstrap-based, data-driven choice of sieve dimension for sieve nonparametric instrumental variables (NPIV) estimators. When implemented with this data-driven choice, sieve NPIV estimators of h 0 and its derivatives are adaptive: they converge at the best possible (i.e., minimax) sup-norm rate, without having to know the smoothness of h 0 , degree of endogeneity of the regressors, or instrument strength. Our second procedure is a data-driven approach for constructing honest and adaptive uniform confidence bands (UCBs) for h 0 and its derivatives. Our data-driven UCBs guarantee coverage for h 0 and its derivatives uniformly over a generic class of data-generating processes (honesty) and contract at, or within a logarithmic factor of, the minimax sup-norm rate (adaptivity). As such, our data-driven UCBs deliver asymptotic efficiency gains relative to UCBs constructed via the usual approach of undersmoothing. In addition, both our procedures apply to nonparametric regression as a special case. We use our procedures to estimate and perform inference on a nonparametric gravity equation for the intensive margin of firm exports and find evidence against common parameterizations of the distribution of unobserved firm productivity.