Nonparametric instrumental variables estimation of a quantile regression model

Horowitz, Joel L.; Lee, Sokbae

doi:10.1920/wp.cem.2006.0906

Cited by 22 publications

(26 citation statements)

References 16 publications

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“…The resulting convergence rate h n − h 0 s = O p (n −α/(2α+d) ) coincides with the known optimal rate for the additive quantile regression model: Y 3 = h 01 (X 1 ) + h 02 (X 2 ) + U Pr(U ≤ 0|X 1 X 2 ) = γ; see, for example, Horowitz and Lee (2005) and Horowitz and Mammen (2007). The resulting convergence rate h n − h 0 s = O p (n −α/(2α+d) ) coincides with the known optimal rate for the additive quantile regression model: Y 3 = h 01 (X 1 ) + h 02 (X 2 ) + U Pr(U ≤ 0|X 1 X 2 ) = γ; see, for example, Horowitz and Lee (2005) and Horowitz and Mammen (2007).…”

Section: Application To Nonparametric Additive Quantile IV Regressionsupporting

confidence: 67%

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Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals

2012

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This paper studies nonparametric estimation of conditional moment restrictions in which the generalized residual functions can be nonsmooth in the unknown functions of endogenous variables. This is a nonparametric nonlinear instrumental variables (IV) problem. We propose a class of penalized sieve minimum distance (PSMD) estimators, which are minimizers of a penalized empirical minimum distance criterion over a collection of sieve spaces that are dense in the infinite-dimensional function parameter space. Some of the PSMD procedures use slowly growing finite-dimensional sieves with flexible penalties or without any penalty; others use large dimensional sieves with lower semicompact and/or convex penalties. We establish their consistency and the convergence rates in Banach space norms (such as a sup-norm or a root mean squared norm), allowing for possibly noncompact infinite-dimensional parameter spaces. For both mildly and severely ill-posed nonlinear inverse problems, our convergence rates in Hilbert space norms (such as a root mean squared norm) achieve the known minimax optimal rate for the nonparametric mean IV regression. We illustrate the theory with a nonparametric additive quantile IV regression. We present a simulation study and an empirical application of estimating nonparametric quantile IV Engel curves.KEYWORDS: Nonlinear ill-posed inverse, penalized sieve minimum distance, modulus of continuity, convergence rate, nonparametric additive quantile IV, quantile IV Engel curves. 1 This paper is a substantially revised version of Cowles Foundation Discussion Paper 1650. We are grateful to two co-editors, three anonymous referees, R. Blundell, V. Chernozhukov, T.

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Section: Application To Nonparametric Additive Quantile IV Regressionsupporting

confidence: 67%

“…Model (1) also nests the quantile instrumental variables (IV) treatment effect model of Chernozhukov and Hansen (2005) (CH), and the nonparametric quantile instrumental variables regression (NPQIV) of Chernozhukov, Imbens, and Newey (2007) (CIN) and Horowitz and Lee (2007) (HL):…”

Section: Introductionmentioning

confidence: 99%

Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals

2012

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“…Theorem 6 corrects Theorem 3.2 of Chernozhukov, Imbens, and Newey (2007) by adding the bound on ∞ j=1 b 2 j /μ 2 j . Horowitz and Lee (2007) imposed analogous conditions in their paper on convergence rates of nonparametric endogenous quantile estimators, but assumed identification. Horowitz and Lee (2007) imposed analogous conditions in their paper on convergence rates of nonparametric endogenous quantile estimators, but assumed identification.…”

Section: A Quantile IV Examplementioning

confidence: 99%

Local Identification of Nonparametric and Semiparametric Models

et al. 2014

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In parametric, nonlinear structural models, a classical sufficient condition for local identification, like Fisher (1966) and Rothenberg (1971), is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix. We derive an analogous result for the nonparametric, nonlinear structural models, establishing conditions under which an infinite dimensional analog of the full rank condition is sufficient for local identification. Importantly, we show that additional conditions are often needed in nonlinear, nonparametric models to avoid nonlinearities overwhelming linear effects. We give restrictions on a neighborhood of the true value that are sufficient for local identification. We apply these results to obtain new, primitive identification conditions in several important models, including nonseparable quantile instrumental variable (IV) models and semiparametric consumption-based asset pricing models.

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“…These include Newey andPowell (1989, 2003), Darolles, Florens, and Renault (2002), Ai and Chen (2003), Hall and Horowitz (2005), Blundell, Chen, and Kristensen (2007), Darolles, Fan, Florens, and Renault (2011), and for models with nonadditive random terms, Chernozhukov and Hansen (2005), Chernozhukov, Imbens, and Newey (2007), Horowitz and Lee (2007), Chen and Pouzo (2012), and Chen, Chernozhukov, Lee, and Newey (2014). Several nonparametric estimators have been developed for models with simultaneity, based on conditional moment restrictions.…”

Section: Introductionmentioning

confidence: 99%

Estimation of Nonparametric Models With Simultaneity

Matzkin

2015

Econometrica

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We introduce methods for estimating nonparametric, nonadditive models with simultaneity. The methods are developed by directly connecting the elements of the structural system to be estimated with features of the density of the observable variables, such as ratios of derivatives or averages of products of derivatives of this density. The estimators are therefore easily computed functionals of a nonparametric estimator of the density of the observable variables. We consider in detail a model where to each structural equation there corresponds an exclusive regressor and a model with one equation of interest and one instrument that is included in a second equation. For both models, we provide new characterizations of observational equivalence on a set, in terms of the density of the observable variables and derivatives of the structural functions. Based on those characterizations, we develop two estimation methods. In the first method, the estimators of the structural derivatives are calculated by a simple matrix inversion and matrix multiplication, analogous to a standard least squares estimator, but with the elements of the matrices being averages of products of derivatives of nonparametric density estimators. In the second method, the estimators of the structural derivatives are calculated in two steps. In a first step, values of the instrument are found at which the density of the observable variables satisfies some properties. In the second step, the estimators are calculated directly from the values of derivatives of the density of the observable variables evaluated at the found values of the instrument. We show that both pointwise estimators are consistent and asymptotically normal.

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Nonparametric instrumental variables estimation of a quantile regression model

Cited by 22 publications

References 16 publications

Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals

Estimation of Nonparametric Conditional Moment Models With Possibly Nonsmooth Generalized Residuals

Local Identification of Nonparametric and Semiparametric Models

Estimation of Nonparametric Models With Simultaneity

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