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Conditional quantile processes based on series or many regressors

Author

Listed:
• Alexandre Belloni

(Institute for Fiscal Studies)

• Victor Chernozhukov

() (Institute for Fiscal Studies and MIT)

• Denis Chetverikov

() (Institute for Fiscal Studies and UCLA)

• Ivan Fernandez-Val

(Institute for Fiscal Studies and Boston University)

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-speci fic coefficients and estimate the function-valued coefficients from the data. We develop large sample theory for the QR-series coefficient process, namely we obtain uniform strong approximations to the QR-series coefficient process by conditionally pivotal and Gaussian processes. Based on these two strong approximations, or couplings, we develop four resampling methods (pivotal, gradient bootstrap, Gaussian, and weighted bootstrap) that can be used for inference on the entire QR-series coefficient function. 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. Speci fically, we obtain uniform rates of convergence and show how to use the four resampling methods mentioned above for inference on the functionals. All of the above results are for function-valued parameters, holding uniformly in both the quantile index and the covariate value, and covering the pointwise case as a by-product. We demonstrate the practical utility of these results with an empirical example, where we estimate the price elasticity function and test the Slutsky condition of the individual demand for gasoline, as indexed by the individual unobserved propensity for gasoline consumption.

Suggested Citation

• Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Ivan Fernandez-Val, 2016. "Conditional quantile processes based on series or many regressors," CeMMAP working papers CWP46/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
• Handle: RePEc:ifs:cemmap:46/16
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References listed on IDEAS

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Citations

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Cited by:

1. Shujie Ma & Oliver Linton & Jiti Gao, 2017. "Estimation and inference in semiparametric quantile factor models," Monash Econometrics and Business Statistics Working Papers 8/17, Monash University, Department of Econometrics and Business Statistics.
2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP44/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
3. Manuel Arellano & Stéphane Bonhomme, 2016. "Nonlinear panel data estimation via quantile regressions," Econometrics Journal, Royal Economic Society, vol. 19(3), pages 61-94, October.
4. Kong, Efang & Linton, Oliver & Xia, Yingcun, 2013. "Global Bahadur Representation For Nonparametric Censored Regression Quantiles And Its Applications," Econometric Theory, Cambridge University Press, vol. 29(05), pages 941-968, October.
5. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
6. Manuel Arellano & Richard Blundell & Stéphane Bonhomme, 2017. "Earnings and Consumption Dynamics: A Nonlinear Panel Data Framework," Econometrica, Econometric Society, vol. 85, pages 693-734, May.
7. Victor Chernozhukov & Vira Semenova, 2018. "Simultaneous inference for Best Linear Predictor of the Conditional Average Treatment Effect and other structural functions," CeMMAP working papers CWP40/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
8. Charlier, Isabelle & Paindaveine, Davy & Saracco, Jérôme, 2015. "Conditional quantile estimation based on optimal quantization: From theory to practice," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 20-39.
9. R H Spady & S Stouli, 2018. "Dual regression," Biometrika, Biometrika Trust, vol. 105(1), pages 1-18.
10. Belloni, Alexandre & Chernozhukov, Victor & Chetverikov, Denis & Kato, Kengo, 2015. "Some new asymptotic theory for least squares series: Pointwise and uniform results," Journal of Econometrics, Elsevier, vol. 186(2), pages 345-366.
11. Rosen, Adam M., 2012. "Set identification via quantile restrictions in short panels," Journal of Econometrics, Elsevier, vol. 166(1), pages 127-137.
12. Manuel Arellano & Stéphane Bonhomme, 2017. "Nonlinear Panel Data Methods for Dynamic Heterogeneous Agent Models," Annual Review of Economics, Annual Reviews, vol. 9(1), pages 471-496, September.
13. Alexandre Belloni & Mingli Chen & Victor Chernozhukov, 2016. "Quantile Graphical Models: Prediction and Conditional Independence with Applications to Systemic Risk," Papers 1607.00286, arXiv.org, revised Dec 2017.
14. Chernozhukov, Victor & Fernández-Val, Iván & Hoderlein, Stefan & Holzmann, Hajo & Newey, Whitney, 2015. "Nonparametric identification in panels using quantiles," Journal of Econometrics, Elsevier, vol. 188(2), pages 378-392.
15. Victor Chernozhukov & Ivan Fernandez-Val & Blaise Melly & Kaspar Wüthrich, 2016. "Generic inference on quantile and quantile effect functions for discrete outcomes," CeMMAP working papers CWP35/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
16. Alexandre Belloni & Victor Chernozhukov, 2009. "L1-Penalized Quantile Regression in High-Dimensional Sparse Models," Papers 0904.2931, arXiv.org, revised Nov 2011.
17. Shih-Kang Chao & Wolfgang K. Härdle & Ming Yuan, 2016. "Factorisable Multi-Task Quantile Regression," SFB 649 Discussion Papers SFB649DP2016-057, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
18. Ferreira, Francisco H. G. & Firpo, Sergio & Galvao, Antonio F., 2017. "Estimation and Inference for Actual and Counterfactual Growth Incidence Curves," IZA Discussion Papers 10473, Institute of Labor Economics (IZA).
19. Bonaccolto, G. & Caporin, M. & Gupta, R., 2018. "The dynamic impact of uncertainty in causing and forecasting the distribution of oil returns and risk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 446-469.
20. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2013. "Valid Post-Selection Inference in High-Dimensional Approximately Sparse Quantile Regression Models," Papers 1312.7186, arXiv.org, revised Jun 2016.
21. Mehmet Balcilar & Seyi Saint Akadiri & Rangan Gupta & Stephen M. Miller, 2017. "Partisan Conflict and Income Distribution in the United States: A Nonparametric Causality-in-Quantiles Approach," Working Papers 201741, University of Pretoria, Department of Economics.
22. repec:spr:soinre:v:142:y:2019:i:1:d:10.1007_s11205-018-1906-3 is not listed on IDEAS

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