Cluster-Robust Bootstrap Inference in Quantile Regression Models

ABSTRACT

In this article I develop a wild bootstrap procedure for cluster-robust inference in linear quantile regression models. I show that the bootstrap leads to asymptotically valid inference on the entire quantile regression process in a setting with a large number of small, heterogeneous clusters and provides consistent estimates of the asymptotic covariance function of that process. The proposed bootstrap procedure is easy to implement and performs well even when the number of clusters is much smaller than the sample size. An application to Project STAR data is provided. Supplementary materials for this article are available online.

KEYWORDS:

• Cluster-robust standard errors
• Correlated data
• Grouped data
• Quantile
• Resampling

Acknowledgments

The author would like to thank Roger Koenker for his continued support and encouragement. The author would also like to thank the co-editor, an associate editor, two referees, Matias Cattaneo, Sílvia Gonçalves, Carlos Lamarche, Sarah Miller, Stephen Portnoy, João Santos Silva, Ke-Li Xu, and Zhou Zhou for comments and discussions.

Notes

1 By construction, the correlation coefficient of X²_ikU_ik and X²_ilU_il is corr(U_ik, U_il)(2ϱ² + 1)/3. I generate data such that corr(U_ik, U_il) = min {1, 3ϱ/(2ϱ² + 1)}. The within-cluster correlation coefficient of X²_ikU_ik is then exactly ϱ for ϱ ∈ [0, 0.5] and has a value slightly below ϱ for ϱ ∈ (0.5, 1). This choice for corr(U_ik, U_il) ensures that the other restrictions on the DGP hold for all values of ϱ used in the experiments.

2 This bandwidth choice required a robust estimate of scale. Koenker (2013) used the minimum of the standard deviation of the QR residuals and their normalized interquartile range. Parente and Santos Silva (2016) suggested the median absolute deviation of the QR residuals with a scaling constant of 1. I chose Koenker’s implementation because it yielded better results in nearly all cases.

3 The sample also contains a large number of students who identify as white and a very small number of students who identify as Hispanic, Asian, American Indian, or other.