A Unified Variable Selection Approach for Varying Coefficient Models

In varying coefficient models, three types of variable selection problems are of practical interests: separation of varying and constant effects, selection of variables with nonzero varying effects, and selection of variables with nonzero constant effects. Existing variable selection methods in the literature often focus on only one of the three types. In this paper, we develop a unified variable selection approach for both least squares regression and quantile regression models with possibly varying coefficients. The developed method is carried out by using a two-step iterative procedure based on basis expansion and a double adaptive-LASSO-type penalty. Under some regularity conditions, we show that the proposed procedure is consistent in both variable selection and the separation of varying and constant coefficients. In addition, the estimated varying coefficients possess the optimal convergence rate under the same smoothness assumption, and the estimated constant coefficients have the same asymptotic distribution as their counterparts obtained when the true model is known. Finally, we investigate the finite sample performance of the proposed method through a simulation study and the analysis of the Childhood Malnutrition Data in India.