Adaptive Estimation of Functionals in Nonparametric Instrumental Regression

Breunig, Christoph; Johannes, Jan

doi:10.1017/s0266466614000966

Cited by 23 publications

(23 citation statements)

References 54 publications

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“…Recently, Breunig and Johannes (2013) also applied Lepski's method to study near L 2 -norm adaptive estimation of linear functionals of NPIV models. 10 Gautier and LePennec (2011) proposed a data-driven method for choosing the regularization parameter in a random coefficient binary choice 9 See Loubes and Marteau (2012) and Johannes and Schwarz (2013) for near L 2 -norm rate adaptivity of estimators similar to Horowitz (2014)'s when the eigenfunctions of the conditional expectation operator are known.…”

Section: Sup-norm Rate-adaptivity To the Oraclementioning

confidence: 99%

Optimal Sup-Norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation

Chen¹,

Christensen

2015

SSRN Journal

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Abstract 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. Terms of use: Documents in

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Section: Sup-norm Rate-adaptivity To the Oraclementioning

confidence: 99%

Optimal Sup-Norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation

Chen¹,

Christensen

2015

SSRN Journal

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“…Further, since n R kn 2 = O p (k 2 n ) (cf. Lemma A.1 of Breunig and Johannes [2011]) and n R kn 2 n −1/2 i e kn (W i ) ϕ kn (Z i ) − Y i…”

Section: From Lemmamentioning

confidence: 98%

“…In the following, we introduce the function ϕ kn (·) := e kn (·) t [T ] −1 kn [g] kn which belongs to L 2 Z . For all k 1 let us denote Ω k : Breunig and Johannes [2011]) and, hence…”

mentioning

confidence: 99%

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

Breunig

2015

Journal of Econometrics

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This paper proposes several tests of restricted specification in nonparametric instrumental regression. Based on series estimators, test statistics are established that allow for tests of the general model against a parametric or nonparametric specification as well as a test of exogeneity of the vector of regressors. The tests are asymptotically normally distributed under correct specification and consistent against any alternative model. Under a sequence of local alternative hypotheses, the asymptotic distribution of the tests is derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases.

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“…In the unconstrained problem, estimators of linear functionals do not necessarily converge at polynomial rates and may exhibit similarly slow, logarithmic rates as for estimation of the function g itself (e.g. Breunig and Johannes (2015)). Therefore, imposing monotonicity as we do in this paper may also improve statistical properties of estimators of such functionals.…”

Section: Non-asymptotic Risk Bounds Under Monotonicitymentioning

confidence: 99%

Nonparametric instrumental variable estimation under monotonicity

Chetverikov¹,

Wilhelm²

2015

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The ill-posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suffer from a very slow, logarithmic rate of convergence. In this paper, we show that restricting the problem to models with monotone regression functions and monotone instruments significantly weakens the ill-posedness of the problem. In stark contrast to the existing literature, the presence of a monotone instrument implies boundedness of our measure of ill-posedness when restricted to the space of monotone functions. Based on this result we derive a novel non-asymptotic error bound for the constrained estimator that imposes monotonicity of the regression function. For a given sample size, the bound is independent of the degree of ill-posedness as long as the regression function is not too steep. As an implication, the bound allows us to show that the constrained estimator converges at a fast, polynomial rate, independently of the degree of ill-posedness, in a large, but slowly shrinking neighborhood of constant functions. Our simulation study demonstrates significant finite-sample performance gains from imposing monotonicity even when the regression function is rather far from being a constant. We apply the constrained estimator to the problem of estimating gasoline demand functions from U.S. data. *

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Adaptive Estimation of Functionals in Nonparametric Instrumental Regression

Cited by 23 publications

References 54 publications

Optimal Sup-Norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation

Optimal Sup-Norm Rates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

Nonparametric instrumental variable estimation under monotonicity

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