Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression

Xiaohong Chen, Timothy Christensen

    Research output: Contribution to journalArticle

    Abstract

    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.

    Original languageEnglish (US)
    Pages (from-to)39-84
    Number of pages46
    JournalQuantitative Economics
    Volume9
    Issue number1
    DOIs
    StatePublished - Mar 1 2018

    Fingerprint

    Inference
    Minimax
    Deadweight loss
    Confidence
    Derivatives
    Instrumental variables
    Consumer surplus
    Gasoline demand
    Approximation
    Statistics
    Instrumental variable estimation
    Splines
    Wavelets
    Bootstrap
    Least squares estimator
    Factors
    Instrumental variables estimator
    Two-stage least squares
    Lower bounds
    Convergence rate

    Keywords

    • nonlinear welfare functionals
    • nonparametric demand with endogeneity
    • optimal sup-norm convergence rates
    • score bootstrap uniform confidence bands
    • Series two-stage least squares
    • uniform Gaussian process strong approximation

    ASJC Scopus subject areas

    • Economics and Econometrics

    Cite this

    Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression. / Chen, Xiaohong; Christensen, Timothy.

    In: Quantitative Economics, Vol. 9, No. 1, 01.03.2018, p. 39-84.

    Research output: Contribution to journalArticle

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