Quantile-Regression Inference With Adaptive Control of Size

ABSTRACT

Regression quantiles have asymptotic variances that depend on the conditional densities of the response variable given regressors. This article develops a new estimate of the asymptotic variance of regression quantiles that leads any resulting Wald-type test or confidence region to behave as well in large samples as its infeasible counterpart in which the true conditional response densities are embedded. We give explicit guidance on implementing the new variance estimator to control adaptively the size of any resulting Wald-type test. Monte Carlo evidence indicates the potential of our approach to deliver powerful tests of heterogeneity of quantile treatment effects in covariates with good size performance over different quantile levels, data-generating processes, and sample sizes. We also include an empirical example. Supplementary material is available online.

KEYWORDS:

• Asymptotic variance
• Conditional density estimation
• Regression quantile
• Standard error

Supplementary Materials

• Appendices: Appendix A contains precise statements of the assumptions used in Theorems 1 and 2; Appendix B contains proofs of Theorems 1 and 2; Appendix C shows that the estimators of G₀(α) proposed by Powell (1991) and Hendricks and Koenker (1992) cannot induce Wald-type tests that control size adaptively in large samples; Appendix D describes a data-driven, as opposed to a fixed, bandwidth to implement our proposed estimate of G₀(α); Appendix E reports further simulation evidence on the finite-sample performance of our proposed method relative to its competitors, while Appendix F contains further investigation of the empirical example presented in Section 5. (qdf61supp.pdf)

• R programs: We also include R code that enables reproduction of the simulation results in Section 4 and Appendix E and of the empirical analyses reported in Section 5 and Appendix F. (qdf61code.zip)

Acknowledgments

The authors thank Theory and Methods co-editors Nicholas Jewell and David Ruppert, an associate editor, and a referee for comments that greatly improved this article. They are also grateful to Victoria Zinde-Walsh for her comments and to Daniel Siercks for computing support.

Additional information

Funding

This work was partially supported by the Spanish Plan Nacional de I+D+I, reference number ECO2014-55858-P.