TY - JOUR
T1 - Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression
AU - Chen, Xiaohong
AU - Christensen, Timothy M.
N1 - Funding Information:
Xiaohong Chen: xiaohong.chen@yale.edu Timothy M. Christensen: timothy.christensen@nyu.edu This paper is a revised version of the preprint arXiv:1508:03365v1 (Chen and Christensen (2015a)) which was submitted to Econometrica in March 2015 and was a major extension of Sections 2 and 3 of the preprint arXiv:1311.0412 (Chen and Christensen (2013)). We are grateful to Y. Sun for careful proofreading and useful comments, and to M. Parey for sharing the gasoline demand data set. We thank L. P. Hansen, R. Matzkin, W. Newey, J. Powell, A. Tsybakov, and participants of SETA2013, AMES2013, SETA2014, the 2014 International Symposium in Honor of Jerry Hausman, the 2014 Cowles Summer Conference, the 2014 SJTU-SMU Econometrics Conference, the 2014 CEMMAP Celebration Conference, the 2015 NSF Conference—Statistics for Complex Systems, the 2015 International Workshop for Enno Mammen’s 60th birthday, and the 2015 ES World Congress for comments. Support from the Cowles Foundation is gratefully acknowledged.
Publisher Copyright:
Copyright © 2018 The Authors.
PY - 2018/3
Y1 - 2018/3
N2 - This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h0 and functionals of h0. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series two-stage least squares) 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. Our real data application of UCBs for exact CS and DL functionals of gasoline demand reveals interesting patterns and is applicable to other goods markets.
AB - This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h0 and functionals of h0. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series two-stage least squares) 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. Our real data application of UCBs for exact CS and DL functionals of gasoline demand reveals interesting patterns and is applicable to other goods markets.
KW - Series two-stage least squares
KW - nonlinear welfare functionals
KW - nonparametric demand with endogeneity
KW - optimal sup-norm convergence rates
KW - score bootstrap uniform confidence bands
KW - uniform Gaussian process strong approximation
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U2 - 10.3982/QE722
DO - 10.3982/QE722
M3 - Article
AN - SCOPUS:85045444580
VL - 9
SP - 39
EP - 84
JO - Quantitative Economics
JF - Quantitative Economics
SN - 1759-7323
IS - 1
ER -