Conditional Quantile Processes Based on Series or Many Regressors

71 Pages Posted: 11 Aug 2011

See all articles by Alexandre Belloni

Alexandre Belloni

Massachusetts Institute of Technology (MIT) - Operations Research Center

Victor Chernozhukov

Massachusetts Institute of Technology (MIT) - Department of Economics; New Economic School

Iván Fernández‐Val

Boston University - Department of Economics

Date Written: August 11, 2011

Abstract

Quantile regression (QR) is a principal regression method for analyzing the impact of covariates on outcomes. The impact is described by the conditional quantile function and its functionals. In this paper we develop the nonparametric QR series framework, covering many regressors as a special case, for performing inference on the entire conditional quantile function and its linear functionals. In this framework, we approximate the entire conditional quantile function by a linear combination of series terms with quantile-specific coefficients and estimate the function-valued coefficients from the data. We develop large sample theory for the empirical QR coefficient process, namely we obtain uniform strong approximations to the empirical QR coefficient process by conditionally pivotal and Gaussian processes, as well as by gradient and weighted bootstrap processes.

We apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function, such as the conditional quantile function itself, its partial derivatives, average partial derivatives, and conditional average partial derivatives. Specifically, we obtain uniform rates of convergence, large sample distributions, and inference methods based on strong pivotal and Gaussian approximations and on gradient and weighted bootstraps. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and in the covariate value, and covering the pointwise case as a by-product. If the function of interest is monotone, we show how to use monotonization procedures to improve estimation and inference. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption.

Keywords: quantile regression series processes, uniform inference

JEL Classification: C12, C13, C14

Suggested Citation

Belloni, Alexandre and Chernozhukov, Victor and Fernandez-Val, Ivan, Conditional Quantile Processes Based on Series or Many Regressors (August 11, 2011). MIT Department of Economics Working Paper No. 11-15, Available at SSRN: https://ssrn.com/abstract=1908413 or http://dx.doi.org/10.2139/ssrn.1908413

Alexandre Belloni

Massachusetts Institute of Technology (MIT) - Operations Research Center ( email )

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Victor Chernozhukov (Contact Author)

Massachusetts Institute of Technology (MIT) - Department of Economics ( email )

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New Economic School

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Ivan Fernandez-Val

Boston University - Department of Economics ( email )

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Boston, MA 02215
United States

HOME PAGE: http://people.mit.edu/ivanf

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