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

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Abstract
This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h_0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h_0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h_0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h_0 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 h_0 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.

Suggested Citation

  • Xiaohong Chen & Timothy Christensen, 2013. "Optimal Sup-Norm Rates and Uniform Inference on Nonlinear Functionals of Nonparametric IV Regression," Cowles Foundation Discussion Papers 1923R2, Cowles Foundation for Research in Economics, Yale University, revised Jan 2017.
    • Handle: RePEc:cwl:cwldpp:1923r2
    Note: Includes Supplimental Material
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    References listed on IDEAS

    as
    1. Xiaohong Chen & Demian Pouzo, 2015. "Sieve Wald and QLR Inferences on Semi/Nonparametric Conditional Moment Models," Econometrica, Econometric Society, vol. 83(3), pages 1013-1079, May.
    2. Chen, Xiaohong & Christensen, Timothy M., 2015. "Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions," Journal of Econometrics, Elsevier, vol. 188(2), pages 447-465.
    3. Chen, Xiaohong & Reiss, Markus, 2011. "On Rate Optimality For Ill-Posed Inverse Problems In Econometrics," Econometric Theory, Cambridge University Press, vol. 27(3), pages 497-521, June.
    4. Hausman, Jerry A & Newey, Whitney K, 1995. "Nonparametric Estimation of Exact Consumers Surplus and Deadweight Loss," Econometrica, Econometric Society, vol. 63(6), pages 1445-1476, November.
    5. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    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;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation

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