Abstract
The varying coefficient (VC) model is a generalization of ordinary linear model, which can not only retain strong interpretability but also has the flexibility of the nonparametric model. In this paper, we investigate a VC model with hierarchical structure. A unified variable selection method for VC model is proposed, which can simultaneously select the nonzero effects and estimate the unknown coefficient functions. Meanwhile, the selected model enforces the hierarchical structure, that is, interaction terms can be selected into the model only if the corresponding main effects are in the model. The kernel method is employed to estimate the varying coefficient functions, and a combined overlapped group Lasso regularization is introduced to implement variable selection to keep the hierarchical structure. It is proved that the proposed penalty estimators have oracle properties, that is, the coefficients are estimated as well as if the true model were known in advance. Simulation studies and a real data analysis are carried out to examine the performance of the proposed method in finite sample case.
Highlights
We aim to develop a unified variable selection method for varying coefficient (VC) model with hierarchical structure, which not merely can identify the significant variables with nonzero functional coefficients
The selected model keeps the hierarchical structure, that is, interaction terms can be selected into the model only if the corresponding main effects are in the model
It is proved that the proposed estimators have the oracle properties, that is, the functional coefficients are estimated as well as if the true model were known in advance
Summary
The varying coefficient model [1] is defined as Mathematics 2021, 9, 132. https://. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Combined the ideas of the local polynomial smoothing and Lasso (least absolute shrinkage and selection operator [8]) to estimate the coefficients and select variables simultaneously; Zhao and Xue [9] employed basis function approximations and SCAD (smoothly clipped absolute deviation [10]) penalty for the semiparametric varying coefficient partially linear model; Tang et al [11] developed a unified variable selection approach for varying coefficient models; Li et al [12] studied the model selection and structure specification for the generalized semi-varying coefficient models; and He et al [13] introduced a dimensionality reduction and variable selection method for multivariate varying-coefficient models with a large number of covariates, and so on. We aim to develop a unified variable selection method for VC model with hierarchical structure, which not merely can identify the significant variables with nonzero functional coefficients.
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