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We consider varying coefficient models (with heteroscedastic error) allowing the coefficients to vary with time in a longitudinal data setting. Since important variables can influence various quantiles in different ways, the problem of variable selection in quantile regression is more challenging. We propose an easy way to check the influence of the covariates on the distribution of the response by investigating both the location and the scale. The functions are estimated with penalized B-splines.
Grouped adaptive Lasso and nonnegative garrote are considered for variable selection. We prove that the grouped adaptive Lasso is consistent in estimation as well as in variable selection. The simulation study confirms that both grouped adaptive Lasso and nonnegative garrote have good performance in selecting the important covariates as well as in estimating the (functional) coefficients with respect to both the location and the scale. The procedures are compared with grouped adaptive Lasso using B-splines estimation and grouped SCAD. Nonnegative garrote outperforms the other methods way far with respect to the computational time. The procedures are illustrated on two real-data examples.
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