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

TL;DR

This paper develops data-driven methods for optimal nonparametric estimation and inference of structural functions and elasticities using instrumental variables, achieving minimax rates and constructing efficient uniform confidence bands.

Contribution

It introduces a novel data-driven sieve dimension selection method and constructs uniform confidence bands that are minimax optimal and more efficient than traditional undersmoothing approaches.

Findings

Estimators achieve minimax convergence rates in sup-norm.

Uniform confidence bands provide guaranteed coverage and contract at minimax rates.

Application to international trade elasticity estimation demonstrates practical effectiveness.

Abstract

We introduce two data-driven procedures for optimal estimation and inference in nonparametric models using instrumental variables. The first is a data-driven choice of sieve dimension for a popular class of sieve two-stage least squares estimators. When implemented with this choice, estimators of both the structural function $h_0$ and its derivatives (such as elasticities) converge at the fastest possible (i.e., minimax) rates in sup-norm. The second is for constructing uniform confidence bands (UCBs) for $h_0$ and its derivatives. Our UCBs guarantee coverage over a generic class of data-generating processes and contract at the minimax rate, possibly up to a logarithmic factor. As such, our UCBs are asymptotically more efficient than UCBs based on the usual approach of undersmoothing. As an application, we estimate the elasticity of the intensive margin of firm 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.