On Rate Optimality for Ill-Posed Inverse Problems in Econometrics

Chen, Xiaohong; Reiß, Markus

doi:10.1017/s0266466610000381

Cited by 107 publications

(134 citation statements)

References 27 publications

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“…Condition LB is standard in the optimal rate literature (see Hall and Horowitz (2005) and Chen and Reiss (2011)). The mildly ill-posed case corresponds to choosing ν(t) = t −ς , and says roughly that the conditional expectation operator T makes p-smooth functions of X into (ς + p)-smooth functions of W .…”

Section: Lower Boundsmentioning

confidence: 99%

“…As in Chen and Reiss (2011), Theorem 3.2 is proved by (i) noting that the risk (in sup-norm loss) for the NPIV model is at least as large as the risk (in sup-norm loss) for the NPIR model, and (ii) calculating a lower bound (in sup-norm loss) for the NPIR model. Theorem 3.2 therefore follows from a sup-norm analogue of Lemma 1 of Chen and Reiss (2011) and Theorem G.1, which establishes a lower bound on minimax risk over Hölder classes under sup-norm loss for the NPIR model.…”

Section: G2 Proofs For Section 32mentioning

confidence: 99%

“…6 In this paper, we derive the best possible (i.e., minimax) sup-norm convergence rates of any estimator of h 0 and its derivatives in mildly and severely ill-posed NPIV models. Surprisingly, the optimal sup-4 See e.g., Hall and Horowitz (2005); Blundell et al (2007); Chen and Reiss (2011);Darolles, Fan, Florens, and Renault (2011);Horowitz (2011);Chen and Pouzo (2012); Gagliardini and Scaillet (2012); Florens and Simoni (2012); Kato (2013) and references therein.…”

Section: Introductionmentioning

confidence: 99%

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

Chen¹,

Christensen²

2017

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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.

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Section: Lower Boundsmentioning

confidence: 99%

Section: G2 Proofs For Section 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Chen¹,

Christensen²

2017

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“…Condition LB(i)-(ii) is standard (see Hall and Horowitz (2005) and Chen and Reiss (2011)). Condition LB(iii) is a so-called link condition (Chen and Reiss, 2011).…”

Section: Lower Bounds On Sup-norm Ratesmentioning

confidence: 99%

Optimal sup-norm rates, adaptivity and inference in nonparametric instrumental variables estimation

Chen¹,

Christensen

2015

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This paper makes several contributions to the literature on the important yet difficult problem of estimating functions nonparametrically using instrumental variables. First, we derive the minimax optimal sup-norm convergence rates for nonparametric instrumental variables (NPIV) estimation of the structural function h 0 and its derivatives. Second, we show that a computationally simple sieve NPIV estimator can attain the optimal sup-norm rates for h 0 and its derivatives when h 0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal L 2 -norm rates for severely ill-posed problems, and are only up to a [log(n)] (with < 1/2) factor slower than the optimal L 2 -norm rates for mildly ill-posed problems. Third, we introduce a novel data-driven procedure for choosing the sieve dimension optimally. Our data-driven procedure is sup-norm rate-adaptive: the resulting estimator of h 0 and its derivatives converge at their optimal sup-norm rates even though the smoothness of h 0 and the degree of ill-posedness of the NPIV model are unknown. Finally, we present two non-trivial applications of the sup-norm rates to inference on nonlinear functionals of h 0 under low-level conditions. The first is to derive the asymptotic normality of sieve t-statistics for exact consumer surplus and deadweight loss functionals in nonparametric demand estimation when prices, and possibly incomes, are endogenous. The second is to establish the validity of a sieve score bootstrap for constructing asymptotically exact uniform confidence bands for collections of nonlinear functionals of h 0 . Both applications provide new and useful tools for empirical research on nonparametric models with endogeneity.

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“…If W were observed this would be the instrumental variables nonparametric regression model of, e.g., Newey and Powell (2003), Hall and Horowitz (2005), Darolles, Fan, Florens, and Renault, (2011) and Chen and Reiss (2011). Assume, as those authors do, that g .X / is identified from the conditional moments in equation (8), and consider how we might construct sample moment analogues to this equation in the case where W is not observed.…”

Section: Latent Nonparametric Instrumental Variablesmentioning

confidence: 99%