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Varying Coefficient Regression Models: A Review and New Developments

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  • Byeong U. Park
  • Enno Mammen
  • Young K. Lee
  • Eun Ryung Lee

Abstract

type="main" xml:id="insr12029-abs-0001"> Varying coefficient regression models are known to be very useful tools for analysing the relation between a response and a group of covariates. Their structure and interpretability are similar to those for the traditional linear regression model, but they are more flexible because of the infinite dimensionality of the corresponding parameter spaces. The aims of this paper are to give an overview on the existing methodological and theoretical developments for varying coefficient models and to discuss their extensions with some new developments. The new developments enable us to use different amount of smoothing for estimating different component functions in the models. They are for a flexible form of varying coefficient models that requires smoothing across different covariates' spaces and are based on the smooth backfitting technique that is admitted as a powerful technique for fitting structural regression models and is also known to free us from the curse of dimensionality.

Suggested Citation

  • Byeong U. Park & Enno Mammen & Young K. Lee & Eun Ryung Lee, 2015. "Varying Coefficient Regression Models: A Review and New Developments," International Statistical Review, International Statistical Institute, vol. 83(1), pages 36-64, April.
    • Handle: RePEc:bla:istatr:v:83:y:2015:i:1:p:36-64
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    1. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    2. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    3. Cai, Zongwu, 2002. "Two-Step Likelihood Estimation Procedure for Varying-Coefficient Models," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 189-209, July.
    4. Ip, Wai-Cheung & Wong, Heung & Zhang, Riquan, 2007. "Generalized likelihood ratio test for varying-coefficient models with different smoothing variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4543-4561, May.
    5. Oliver Linton & E. Mammen & J. Nielsen, 1997. "The Existence and Asymptotic Properties of a Backfitting Projection Algorithm Under Weak Conditions," Cowles Foundation Discussion Papers 1160, Cowles Foundation for Research in Economics, Yale University.
    6. Lam, Clifford & Fan, Jianqing, 2008. "Profile-kernel likelihood inference with diverging number of parameters," LSE Research Online Documents on Economics 31548, London School of Economics and Political Science, LSE Library.
    7. Wang, Hansheng & Xia, Yingcun, 2009. "Shrinkage Estimation of the Varying Coefficient Model," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 747-757.
    8. Zhang, Riquan & Li, Guoying, 2007. "Averaged estimation of functional-coefficient regression models with different smoothing variables," Statistics & Probability Letters, Elsevier, vol. 77(4), pages 455-461, February.
    9. Zhang, Wenyang & Lee, Sik-Yum, 2000. "Variable Bandwidth Selection in Varying-Coefficient Models," Journal of Multivariate Analysis, Elsevier, vol. 74(1), pages 116-134, July.
    10. Colin Wu & Kai Yu & Chin-Tsang Chiang, 2000. "A Two-Step Smoothing Method for Varying-Coefficient Models with Repeated Measurements," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 519-543, September.
    11. Cai, Zongwu & Fan, Jianqing & Yao, Qiwei, 2000. "Functional-coefficient regression models for nonlinear time series," LSE Research Online Documents on Economics 6314, London School of Economics and Political Science, LSE Library.
    12. Jianhua Z. Huang & Haipeng Shen, 2004. "Functional Coefficient Regression Models for Non‐linear Time Series: A Polynomial Spline Approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(4), pages 515-534, December.
    13. Jianhua Z. Huang, 2002. "Varying-coefficient models and basis function approximations for the analysis of repeated measurements," Biometrika, Biometrika Trust, vol. 89(1), pages 111-128, March.
    14. Wang, Lifeng & Li, Hongzhe & Huang, Jianhua Z., 2008. "Variable Selection in Nonparametric Varying-Coefficient Models for Analysis of Repeated Measurements," Journal of the American Statistical Association, American Statistical Association, vol. 103(484), pages 1556-1569.
    15. Li, Gaorong & Xue, Liugen & Lian, Heng, 2011. "Semi-varying coefficient models with a diverging number of components," Journal of Multivariate Analysis, Elsevier, vol. 102(7), pages 1166-1174, August.
    16. You, Jinhong & Chen, Gemai, 2006. "Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 324-341, February.
    17. Jianqing Fan & Wenyang Zhang, 2000. "Simultaneous Confidence Bands and Hypothesis Testing in Varying‐coefficient Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 715-731, December.
    18. Lan Xue & Hua Liang, 2010. "Polynomial Spline Estimation for a Generalized Additive Coefficient Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(1), pages 26-46, March.
    19. Zhang, Wenyang & Peng, Heng, 2010. "Simultaneous confidence band and hypothesis test in generalised varying-coefficient models," Journal of Multivariate Analysis, Elsevier, vol. 101(7), pages 1656-1680, August.
    20. Colin O. Wu & Kai F. Yu, 2002. "Nonparametric Varying-Coefficient Models for the Analysis of Longitudinal Data," International Statistical Review, International Statistical Institute, vol. 70(3), pages 373-393, December.
    21. Wang, Lan & Kai, Bo & Li, Runze, 2009. "Local Rank Inference for Varying Coefficient Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1631-1645.
    22. Yang, Lijian & Park, Byeong U. & Xue, Lan & Hardle, Wolfgang, 2006. "Estimation and Testing for Varying Coefficients in Additive Models With Marginal Integration," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1212-1227, September.
    23. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    24. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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    3. Samuele CENTORRINO & Jeffrey S. RACINE, 2017. "Semiparametric Varying Coefficient Models with Endogenous Covariates," Annals of Economics and Statistics, GENES, issue 128, pages 261-295.
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    5. Yongsheng Jiang & Dong Zhao & Andrew Sanderford & Jing Du, 2018. "Effects of Bank Lending on Urban Housing Prices for Sustainable Development: A Panel Analysis of Chinese Cities," Sustainability, MDPI, vol. 10(3), pages 1-16, February.
    6. Zhang, Xiaoke & Zhong, Qixian & Wang, Jane-Ling, 2020. "A new approach to varying-coefficient additive models with longitudinal covariates," Computational Statistics & Data Analysis, Elsevier, vol. 145(C).
    7. Christoph Breunig, 2018. "Varying Random Coefficient Models," Papers 1804.03110, arXiv.org, revised Aug 2020.
    8. Djeundje, Viani Biatat & Crook, Jonathan, 2019. "Dynamic survival models with varying coefficients for credit risks," European Journal of Operational Research, Elsevier, vol. 275(1), pages 319-333.
    9. Olarreaga, Marcelo & Sperlich, Stefan & Trachsel, Virginie, 2016. "Export Promotion: what works?," CEPR Discussion Papers 11270, C.E.P.R. Discussion Papers.
    10. Chen, Jia & Li, Degui & Xia, Yingcun, 2019. "Estimation of a rank-reduced functional-coefficient panel data model with serial correlation," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 456-479.
    11. Koo, Chao, 2018. "Essays on functional coefficient models," Other publications TiSEM ba87b8a5-3c55-40ec-967d-9, Tilburg University, School of Economics and Management.
    12. Lee, Kyeongeun & Lee, Young K. & Park, Byeong U. & Yang, Seong J., 2018. "Time-dynamic varying coefficient models for longitudinal data," Computational Statistics & Data Analysis, Elsevier, vol. 123(C), pages 50-65.
    13. Loann David Denis Desboulets, 2018. "A Review on Variable Selection in Regression Analysis," Econometrics, MDPI, vol. 6(4), pages 1-27, November.
    14. Sunil Kanwar & Stefan Sperlich, 2023. "Direct foreign investment and intellectual property reform in the South," Journal of International Development, John Wiley & Sons, Ltd., vol. 35(6), pages 1456-1477, August.
    15. Breunig, Christoph, 2021. "Varying random coefficient models," Journal of Econometrics, Elsevier, vol. 221(2), pages 381-408.
    16. Jooyong Shim & Changha Hwang & Kyungha Seok, 2016. "Support vector quantile regression with varying coefficients," Computational Statistics, Springer, vol. 31(3), pages 1015-1030, September.
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    19. Ingrid Mauerer & Gerhard Tutz, 2023. "Heterogeneity in general multinomial choice models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 32(1), pages 129-148, March.

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