Title: Adaptive Estimation and Uniform Confidence Bands for Nonparametric Structural Functions and Elasticities
Abstract: We introduce two practical methods for estimation and inference on a nonparametric structural function h0 and its derivatives such as elasticities or other marginal effects - using instrumental variables. The first is a data-driven choice of sieve dimension. The second is a data-driven approach for constructing uniform confidence bands (UCBs) for h0 and its derivatives. Both procedures are simple to implement, have strong theoretical justification, and do not require prior information about the smoothness of h0 or instrument strength. Our first procedure leads to estimators of h0 and its derivatives that converge at the fastest possible (i.e., minimax) rate in sup-norm. Our second procedure yields UCBs for h0 and its derivatives that have correct asymptotic coverage and contract at, or within a logarithmic factor of, the minimax rate. Our UCBs are asymptotically more efficient (i.e., narrower) than UCBs based on the usual approach of undersmoothing. As an application, we estimate the elasticity of the intensive margin of rm exports in a monopolistic competition model of international trade. Simulations illustrate the good performance of our procedures in empirically calibrated designs. Our results provide evidence against common parameterizations of the distribution of unobserved firm heterogeneity.