Conditional quantile processes based on series or many regressors
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.
|Date of creation:||May 2011|
|Date of revision:|
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- 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.
- V. Chernozhukov & I. Fernández-Val & A. Galichon, 2009. "Improving point and interval estimators of monotone functions by rearrangement," Biometrika, Biometrika Trust, vol. 96(3), pages 559-575.
- Omar Arias & Walter Sosa-Escudero & Kevin F. Hallock, 2001.
"Individual heterogeneity in the returns to schooling: instrumental variables quantile regression using twins data,"
Springer, vol. 26(1), pages 7-40.
- Omar Arias & Kevin F. Hallock & Walter Sosa Escudero, 1999. "Individual Heterogeneity in the Returns to Schooling: Instrumental Variables Quantile Regression using Twins Data," Department of Economics, Working Papers 016, Departamento de Economía, Facultad de Ciencias Económicas, Universidad Nacional de La Plata.
- Kong, Efang & Linton, Oliver & Xia, Yingcun, 2010.
"Uniform Bahadur Representation For Local Polynomial Estimates Of M-Regression And Its Application To The Additive Model,"
Cambridge University Press, vol. 26(05), pages 1529-1564, October.
- Efang Kong & Oliver Linton & Yingcun Xia, 2009. "Uniform Bahadur Representation for LocalPolynomial Estimates of M-Regressionand Its Application to The Additive Model," STICERD - Econometrics Paper Series 535, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
- Adonis Yatchew & Joungyeo Angela No, 2001. "Household Gasoline Demand in Canada," Econometrica, Econometric Society, vol. 69(6), pages 1697-1709, November.
- Koenker,Roger, 2005.
Cambridge University Press, number 9780521608275, 1.
- Victor Chernozhukov & Sokbae (Simon) Lee & Adam Rosen, 2009.
"Intersection Bounds: estimation and inference,"
CeMMAP working papers
CWP19/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Victor Chernozhukov & Sokbae (Simon) Lee & Adam Rosen, 2012. "Intersection bounds: estimation and inference," CeMMAP working papers CWP33/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Victor Chernozhukov & Sokbae (Simon) Lee & Adam Rosen, 2011. "Intersection bounds: estimation and inference," CeMMAP working papers CWP34/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
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