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Varying Index Coefficient Models

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  • Shujie Ma
  • Peter X.-K. Song

Abstract

It has been a long history of using interactions in regression analysis to investigate alterations in covariate-effects on response variables. In this article, we aim to address two kinds of new challenges arising from the inclusion of such high-order effects in the regression model for complex data. The first kind concerns a situation where interaction effects of individual covariates are weak but those of combined covariates are strong, and the other kind pertains to the presence of nonlinear interactive effects directed by low-effect covariates. We propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable. As a result, most of the existing semiparametric regression models are special cases of our proposed models. We develop a numerically stable and computationally fast estimation procedure using both profile least squares method and local fitting. We establish both estimation consistency and asymptotic normality for the proposed estimators of index coefficients as well as the oracle property for the nonparametric function estimator. In addition, a generalized likelihood ratio test is provided to test for the existence of interaction effects or the existence of nonlinear interaction effects. Our models and estimation methods are illustrated by simulation studies, and by an analysis of child growth data to evaluate alterations in growth rates incurred by mother's exposures to endocrine disrupting compounds during pregnancy. Supplementary materials for this article are available online.

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  • Shujie Ma & Peter X.-K. Song, 2015. "Varying Index Coefficient Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 341-356, March.
    • Handle: RePEc:taf:jnlasa:v:110:y:2015:i:509:p:341-356
    DOI: 10.1080/01621459.2014.903185
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    References listed on IDEAS

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    3. Christoph Breunig, 2018. "Varying Random Coefficient Models," Papers 1804.03110, arXiv.org, revised Aug 2020.
    4. Li, Kangqiang & Tang, Songqiao & Zhang, Lixin, 2022. "Robust parameter estimation of regression models under weakened moment assumptions," Statistics & Probability Letters, Elsevier, vol. 191(C).
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    7. Hongyu An & Boping Tian, 2024. "Varying Index Coefficient Model for Tail Index Regression," Mathematics, MDPI, vol. 12(13), pages 1-34, June.
    8. Zhou, Ling & Lin, Huazhen & Chen, Kani & Liang, Hua, 2019. "Efficient estimation and computation of parameters and nonparametric functions in generalized semi/non-parametric regression models," Journal of Econometrics, Elsevier, vol. 213(2), pages 593-607.
    9. Chaohua Dong & Jiti Gao & Bin Peng & Yayi Yan, 2023. "Estimation and Inference for a Class of Generalized Hierarchical Models," Papers 2311.02789, arXiv.org, revised Apr 2024.
    10. Zhou, Fei & Ren, Jie & Ma, Shuangge & Wu, Cen, 2023. "The Bayesian regularized quantile varying coefficient model," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    11. Gao, Zhikun & Tang, Yanlin & Wang, Huixia Judy & Wu, Guangying K. & Lin, Jeff, 2020. "Automatic identification of curve shapes with applications to ultrasonic vocalization," Computational Statistics & Data Analysis, Elsevier, vol. 148(C).
    12. Breunig, Christoph, 2021. "Varying random coefficient models," Journal of Econometrics, Elsevier, vol. 221(2), pages 381-408.
    13. Jing Lv & Chaohui Guo, 2019. "Quantile estimations via modified Cholesky decomposition for longitudinal single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1163-1199, October.
    14. Wang, Taining & Henderson, Daniel J., 2022. "Estimation of a varying coefficient, fixed-effects Cobb–Douglas production function in levels," Economics Letters, Elsevier, vol. 213(C).
    15. Chaohua Dong & Jiti Gao & Bin Peng & Yayi Yan, 2023. "Estimation of Semiparametric Multi-Index Models Using Deep Neural Networks," Monash Econometrics and Business Statistics Working Papers 21/23, Monash University, Department of Econometrics and Business Statistics.
    16. Liu, Hefei & Song, Xinyuan & Zhang, Baoxue, 2022. "Varying-coefficient hidden Markov models with zero-effect regions," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).

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