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Unified specification tests in partially linear quantile regression models

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  • Song, Xiaojun
  • Yang, Zixin

Abstract

We propose specification tests for parametric quantile regression models versus semiparametric alternatives over a continuum of quantile levels. The test statistics are constructed as continuous functionals of a quantile-marked residual process. We show that using an orthogonal projection on the tangent space of nuisance parameters at each quantile index delivers unified asymptotic properties for tests based on different estimators. Consistency of the tests and asymptotic power under a sequence of local alternatives converging to the null at a parametric rate are also discussed. We propose a simple multiplier bootstrap procedure to carry out the tests, whose nominal levels are well approximated in our simulation study for modest sample sizes.

Suggested Citation

  • Song, Xiaojun & Yang, Zixin, 2025. "Unified specification tests in partially linear quantile regression models," Statistics & Probability Letters, Elsevier, vol. 216(C).
    • Handle: RePEc:eee:stapro:v:216:y:2025:i:c:s0167715224002128
    DOI: 10.1016/j.spl.2024.110243
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    References listed on IDEAS

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    1. Zhongjun Qu & Jungmo Yoon & Pierre Perron, 2024. "Inference on Conditional Quantile Processes in Partially Linear Models with Applications to the Impact of Unemployment Benefits," The Review of Economics and Statistics, MIT Press, vol. 106(2), pages 521-541, March.
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    4. Escanciano, J.C. & Goh, S.C., 2014. "Specification analysis of linear quantile models," Journal of Econometrics, Elsevier, vol. 178(P3), pages 495-507.
    5. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    6. Wangli Xu & Xu Guo & Lixing Zhu, 2012. "Goodness-of-fitting for partial linear model with missing response at random," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(1), pages 103-118.
    7. Escanciano, Juan Carlos & Velasco, Carlos, 2010. "Specification tests of parametric dynamic conditional quantiles," Journal of Econometrics, Elsevier, vol. 159(1), pages 209-221, November.
    8. Feng, Xingdong & Liu, Qiaochu & Wang, Caixing, 2023. "A lack-of-fit test for quantile regression process models," Statistics & Probability Letters, Elsevier, vol. 192(C).
    9. Matthew Harding & Carlos Lamarche, 2017. "Penalized Quantile Regression with Semiparametric Correlated Effects: An Application with Heterogeneous Preferences," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 32(2), pages 342-358, March.
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    Cited by:

    1. Xiaojun Song & Jichao Yuan, 2025. "Specification tests for regression models with measurement errors," Papers 2511.04127, arXiv.org.

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