Quantile regression with interval data

Abstract

This paper investigates the identification of quantiles and quantile regression parameters when observations are set valued. We define the identification set of quantiles of random sets in a way that extends the definition of quantiles for regular random variables. We then give sharp characterization of this set by extending concepts from random set theory. Applying the identification set of quantiles and its sharpness to parametric quantile regression models yields the identification set of the parameters and its sharpness. We apply our methods to data on localized environmental benefits and their impact on house values.

Keywords:

• Partial identification
• quantile regression
• quantile sets
• random sets

JEL Code:

• C21

Acknowledgment

We have benefited from discussions with Tong Li, Francesca Molinari, Ilya Molchanov, Tatsushi Oka, Roee Teper, and Adam Rosen and very useful comments by Esfandiar Maasoumi (the editor), the editors of this special issue in honor of Cheng Hsiao, an anonymous referee, and participants in NASM 2016, New York Camp Econometrics XIII, CMES 2018, and IAAE 2018. All remaining errors are ours.

Notes

1 This implication corresponds to the necessity part of Artstein’s Lemma, which does not restrict to compact sets. See discussion in Beresteanu, Molchanov, and Molinari (2012).

2 See for example Belloni, Bugni, and Chernozhukov (2018) and Bugni, Canay, and Shi (2017). Also see Kaido, Molinari and Stoye (2016).

3 We used R = 2, ${ϵ}_n=0.05,\hspace{0.5em}{\kappa }_n={(0.3\hspace{0.17em}\text{ln}\hspace{0.17em}(n))}^{\frac{1}{2}}$ and $B_n={(0.4\hspace{0.17em}\text{ln}\hspace{0.17em}(n)/\hspace{0.17em}\text{ln}\hspace{0.17em}\hspace{0.17em}\text{ln}\hspace{0.17em}(n))}^{\frac{1}{2}}.$

4 We remark that the sample used by GRT and the sample used by us are slightly different likely due to different data selection procedures. Furthermore, they use sampling weights, while we do not.

5 For two sets, A and B, in a finite dimensional Euclidean space $({\mathbb{R}}^k,\Vert \Vert ),$ the directed Hausdorff distance from A to B is $d_H(A,B)=\underset{a\in A}{\sup }\underset{b\in B}{\inf }\Vert a-b\Vert$ and the Hausdorff distance between A and B is $H(A,B)=\max \left\{d_H(A,B),d_H(B,A)\right\}.$

Additional information

Funding

This research was sponsored in part by NSF grant SES-0922373.