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Conditional Quantile Processes based on Series or Many Regressors

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  • Alexandre Belloni
  • Victor Chernozhukov
  • Denis Chetverikov
  • Iv'an Fern'andez-Val

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 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 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. Specifically, 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 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.

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  • Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Iv'an Fern'andez-Val, 2011. "Conditional Quantile Processes based on Series or Many Regressors," Papers 1105.6154, arXiv.org, revised Aug 2018.
  • Handle: RePEc:arx:papers:1105.6154
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. R H Spady & S Stouli, 2018. "Dual regression," Biometrika, Biometrika Trust, vol. 105(1), pages 1-18.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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 for the Study of Labor (IZA).
    12. Manuel Arellano & Stéphane Bonhomme, 2015. "Nonlinear Panel Data Estimation via Quantile Regression," Working Papers wp2015_1505, CEMFI.
    13. repec:eee:phsmap:v:507:y:2018:i:c:p:446-469 is not listed on IDEAS
    14. Victor Chernozhukov & Iv'an Fern'andez-Val & Blaise Melly & Kaspar Wuthrich, 2016. "Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes," Papers 1608.05142, arXiv.org, revised Aug 2018.
    15. 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.
    16. 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.
    17. Alexandre Belloni & Victor Chernozhukov, 2009. "L1-Penalized Quantile Regression in High-Dimensional Sparse Models," Papers 0904.2931, arXiv.org, revised Nov 2011.
    18. Rosen, Adam M., 2012. "Set identification via quantile restrictions in short panels," Journal of Econometrics, Elsevier, vol. 166(1), pages 127-137.
    19. 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.
    20. Qu, Zhongjun & Yoon, Jungmo, 2015. "Nonparametric estimation and inference on conditional quantile processes," Journal of Econometrics, Elsevier, vol. 185(1), pages 1-19.
    21. 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.

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