Optimal Sup-Norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression

94 Pages Posted: 15 Feb 2017  

Xiaohong Chen

Yale University - Cowles Foundation

Timothy Christensen

New York University (NYU) - Department of Economics

Date Written: January 25, 2017

Abstract

This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Empiricists could read our real data application of UCBs for exact CS and DL functionals of gasoline demand that reveals interesting patterns and is applicable to other markets.

Keywords: Series 2SLS, Optimal Sup-Norm Convergence Rates, Uniform Gaussian Process Strong Approximation, Score Bootstrap Uniform Confidence Bands, Nonlinear Welfare Functionals, Nonparametric Demand with Endogeneity

JEL Classification: C13, C14, C36

Suggested Citation

Chen, Xiaohong and Christensen, Timothy, Optimal Sup-Norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression (January 25, 2017). Cowles Foundation Discussion Paper No. 1923R2. Available at SSRN: https://ssrn.com/abstract=2916740 or http://dx.doi.org/10.2139/ssrn.2916740

Xiaohong Chen (Contact Author)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States

Timothy Christensen

New York University (NYU) - Department of Economics ( email )

269 Mercer Street, 7th Floor
New York, NY 10011
United States

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