# Conditional quantile processes based on series or many regressors

## Author Info

• Alexandre Belloni
• Victor Chernozhukov

()
(Institute for Fiscal Studies and Massachusetts Institute of Technology)

• Ivan Fernandez-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 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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
File URL: http://cemmap.ifs.org.uk/wps/cwp1911.pdf

## Bibliographic Info

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

as in new window
Length:
Date of revision:
Handle: RePEc:ifs:cemmap:19/11

Contact details of provider:
Postal: The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE
Phone: (+44) 020 7291 4800
Fax: (+44) 020 7323 4780
Email:
Web page: http://cemmap.ifs.org.uk

Order Information:
Postal: The Institute for Fiscal Studies 7 Ridgmount Street LONDON WC1E 7AE
Email:

## Related research

Keywords:

This paper has been announced in the following NEP Reports:

## References

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
as in new window
1. 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.
2. 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.
3. Roger Koenker & Kevin F. Hallock, 2001. "Quantile Regression," Journal of Economic Perspectives, American Economic Association, vol. 15(4), pages 143-156, Fall.
4. 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.
5. Adonis Yatchew & Joungyeo Angela No, 2001. "Household Gasoline Demand in Canada," Econometrica, Econometric Society, vol. 69(6), pages 1697-1709, November.
6. 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.
Full references (including those not matched with items on IDEAS)

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as in new window

Cited by:
1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Gaussian approximation of suprema of empirical processes," CeMMAP working papers CWP75/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
2. Efang Kong & Oliver Linton & Yingcun Xia, 2011. "Global Bahadur representation for nonparametric censored regression quantiles and its applications," CeMMAP working papers CWP33/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
3. Adam Rosen, 2009. "Set identification via quantile restrictions in short panels," CeMMAP working papers CWP26/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

## Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

## Corrections

When requesting a correction, please mention this item's handle: RePEc:ifs:cemmap:19/11. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Stephanie Seavers).

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.