Quantile regression provides a parsimonious model for the conditional quantile function of the response variable Y given the vector of covariates X, and describes the whole conditional distribution of the response, yielding estimators that are more robust to the presence of outliers. Quantile regression models specify, for each quantile level τ , the functional form for the conditional τ -th quantile of the response, which brings complexity to perform variable selection using regularization techniques, such as LASSO or adaptive LASSO (adaLASSO), as one might obtain a different set of selected variables for each quantile level. In this work, we propose a method for global variable selection and coefficient estimation in the linear quantile regression framework, imposing few restrictions on the functional form of β(·), and applying group adaLASSO penalization for variable selection. We set up a Monte Carlo study comparing six different proposed estimators based on LASSO, adaLASSO and group LASSO in six scenarios that diversify sample and quantile levels grid sizes. The findings demonstrate that the selection of the tuning parameter λ for penalization is critical for model selection and coefficient estimation. It was observed that the methods using traditional LASSO are more prone to include the true model as compared to adaLASSO, but renouncing model shrinkage and not removing irrelevant covariates, while the grouped approaches are more effective in zeroing coefficients that are less relevant.
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