IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v136y2015icp41-55.html
   My bibliography  Save this article

Heteroscedasticity checks for single index models

Author

Listed:
  • Zhu, Xuehu
  • Guo, Xu
  • Lin, Lu
  • Zhu, Lixing

Abstract

To test heteroscedasticity in single index models, in this paper two test statistics are proposed via quadratic conditional moments. Without the use of dimension reduction structure, the first test has the usual convergence rate in nonparametric sense. Under the dimension reduction structure of mean and variance functions, the second one has faster convergence rate to its limit under the null hypothesis, and can detect local alternative hypotheses distinct from the null at a much faster rate than the one the first test can achieve. Numerical studies are also carried out to evaluate the performance of the developed tests. Interestingly, the second one works much better than the first one if the variance function does have a dimension reduction structure. However, it is not robust against the violation of dimension reduction structure, in other words, the power performance of the second test may not be encouraging if without the dimension reduction structure.

Suggested Citation

  • Zhu, Xuehu & Guo, Xu & Lin, Lu & Zhu, Lixing, 2015. "Heteroscedasticity checks for single index models," Journal of Multivariate Analysis, Elsevier, vol. 136(C), pages 41-55.
    • Handle: RePEc:eee:jmvana:v:136:y:2015:i:c:p:41-55
    DOI: 10.1016/j.jmva.2015.01.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X15000123
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2015.01.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. H. Dette & A. Munk, 1998. "Testing heteroscedasticity in nonparametric regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 693-708.
    2. Holger Dette & Natalie Neumeyer & Ingrid Van Keilegom, 2007. "A new test for the parametric form of the variance function in non‐parametric regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 903-917, November.
    3. Sheng, Wenhui & Yin, Xiangrong, 2013. "Direction estimation in single-index models via distance covariance," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 148-161.
    4. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    5. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    6. Zhang, Jun & Zhu, Li-Ping & Zhu, Li-Xing, 2012. "On a dimension reduction regression with covariate adjustment," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 39-55, February.
    7. Jeffrey S. Simonoff & Chih‐Ling Tsai, 1994. "Use of Modified Profile Likelihood for Improved Tests of Constancy of Variance in Regression," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 43(2), pages 357-370, June.
    8. John Xu Zheng, 1996. "A consistent test of functional form via nonparametric estimation techniques," Journal of Econometrics, Elsevier, vol. 75(2), pages 263-289, December.
    9. Xia, Yingcun, 2006. "Asymptotic Distributions For Two Estimators Of The Single-Index Model," Econometric Theory, Cambridge University Press, vol. 22(6), pages 1112-1137, December.
    10. You, Jinhong & Chen, Gemai, 2005. "Testing heteroscedasticity in partially linear regression models," Statistics & Probability Letters, Elsevier, vol. 73(1), pages 61-70, June.
    11. Dette, Holger & Marchlewski, Mareen, 2007. "A test for the parametric form of the variance function in apartial linear regression model," Technical Reports 2007,26, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    12. Liping Zhu & Tao Wang & Lixing Zhu & Louis Ferré, 2010. "Sufficient dimension reduction through discretization-expectation estimation," Biometrika, Biometrika Trust, vol. 97(2), pages 295-304.
    13. Yanyuan Ma & Liping Zhu, 2013. "A Review on Dimension Reduction," International Statistical Review, International Statistical Institute, vol. 81(1), pages 134-150, April.
    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


    Cited by:

    1. Zhu, Xuehu & Chen, Fei & Guo, Xu & Zhu, Lixing, 2016. "Heteroscedasticity testing for regression models: A dimension reduction-based model adaptive approach," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 263-283.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhu, Xuehu & Chen, Fei & Guo, Xu & Zhu, Lixing, 2016. "Heteroscedasticity testing for regression models: A dimension reduction-based model adaptive approach," Computational Statistics & Data Analysis, Elsevier, vol. 103(C), pages 263-283.
    2. Jun Zhang & Zhenghui Feng & Xiaoguang Wang, 2018. "A constructive hypothesis test for the single-index models with two groups," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1077-1114, October.
    3. Xie, Chuanlong & Zhu, Lixing, 2019. "A goodness-of-fit test for variable-adjusted models," Computational Statistics & Data Analysis, Elsevier, vol. 138(C), pages 27-48.
    4. Xia, Yingcun & Härdle, Wolfgang Karl & Linton, Oliver, 2009. "Optimal smoothing for a computationally and statistically efficient single index estimator," SFB 649 Discussion Papers 2009-028, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Lu, Xuewen, 2010. "Asymptotic distributions of two "synthetic data" estimators for censored single-index models," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 999-1015, April.
    6. Jing Sun, 2016. "Composite quantile regression for single-index models with asymmetric errors," Computational Statistics, Springer, vol. 31(1), pages 329-351, March.
    7. Xu Guo & Tao Wang & Lixing Zhu, 2016. "Model checking for parametric single-index models: a dimension reduction model-adaptive approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1013-1035, November.
    8. Yang, Jing & Tian, Guoliang & Lu, Fang & Lu, Xuewen, 2020. "Single-index modal regression via outer product gradients," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    9. Liu, Jicai & Xu, Peirong & Lian, Heng, 2019. "Estimation for single-index models via martingale difference divergence," Computational Statistics & Data Analysis, Elsevier, vol. 137(C), pages 271-284.
    10. Cui, Xia & Härdle, Wolfgang Karl & Zhu, Lixing, 2009. "Generalized single-index models: The EFM approach," SFB 649 Discussion Papers 2009-050, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    11. Huybrechts F. Bindele & Ash Abebe & Karlene N. Meyer, 2018. "General rank-based estimation for regression single index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(5), pages 1115-1146, October.
    12. Wong, Heung & Liu, Feng & Chen, Min & Ip, Wai Cheung, 2009. "Empirical likelihood based diagnostics for heteroscedasticity in partial linear models," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3466-3477, July.
    13. Lei Wang, 2019. "Dimension reduction for kernel-assisted M-estimators with missing response at random," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(4), pages 889-910, August.
    14. repec:hum:wpaper:sfb649dp2009-028 is not listed on IDEAS
    15. Estate V. Khmaladze, 2021. "Distribution-free testing in linear and parametric regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(6), pages 1063-1087, December.
    16. Wang, Lei & Zhao, Puying & Shao, Jun, 2021. "Dimension-reduced semiparametric estimation of distribution functions and quantiles with nonignorable nonresponse," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    17. Hongxia Wang & Zihan Zhao & Hongxia Hao & Chao Huang, 2023. "Estimation and Inference for Spatio-Temporal Single-Index Models," Mathematics, MDPI, vol. 11(20), pages 1-32, October.
    18. Feng, Long & Zou, Changliang & Wang, Zhaojun, 2012. "Rank-based inference for the single-index model," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 535-541.
    19. Rothe, Christoph, 2009. "Semiparametric estimation of binary response models with endogenous regressors," Journal of Econometrics, Elsevier, vol. 153(1), pages 51-64, November.
    20. Feng, Sanying & Kong, Kaidi & Kong, Yinfei & Li, Gaorong & Wang, Zhaoliang, 2022. "Statistical inference of heterogeneous treatment effect based on single-index model," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    21. Sadikoglu, Serhan, 2019. "Essays in econometric theory," Other publications TiSEM 99d83644-f9dc-49e3-a4e1-5, Tilburg University, School of Economics and Management.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:136:y:2015:i:c:p:41-55. See general information about how to correct material in RePEc.

    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 CitEc recognized a bibliographic reference but did not link an item in RePEc 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 RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.