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Conditional quantile processes based on series or many regressors

  • Alexandre Belloni

    (Institute for Fiscal Studies)

  • Victor Chernozhukov


    (Institute for Fiscal Studies and MIT)

  • Ivan Fernandez-Val

    (Institute for Fiscal Studies and University of Boston)

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.

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Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP19/11.

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Date of creation: 27 May 2011
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Handle: RePEc:ifs:cemmap:19/11
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  1. Roger Koenker & Kevin F. Hallock, 2001. "Quantile Regression," Journal of Economic Perspectives, American Economic Association, vol. 15(4), pages 143-156, Fall.
  2. Victor Chernozhukov & Sokbae Lee & Adam Rosen, 2011. "Intersection bounds: estimation and inference," CeMMAP working papers CWP34/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  3. Victor Chernozhukov & Ivan Fernandez-Val & Alfred Galichon, 2008. "Improving point and interval estimates of monotone functions by rearrangement," CeMMAP working papers CWP17/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  4. Adonis Yatchew & Joungyeo Angela No, 2001. "Household Gasoline Demand in Canada," Econometrica, Econometric Society, vol. 69(6), pages 1697-1709, November.
  5. Kong, Efang & Linton, Oliver & Xia, Yingcun, 2010. "Uniform Bahadur Representation For Local Polynomial Estimates Of M-Regression And Its Application To The Additive Model," Econometric Theory, Cambridge University Press, vol. 26(05), pages 1529-1564, October.
  6. Omar Arias & Walter Sosa-Escudero & Kevin F. Hallock, 2001. "Individual heterogeneity in the returns to schooling: instrumental variables quantile regression using twins data," Empirical Economics, Springer, vol. 26(1), pages 7-40.
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