Simultaneous variable selection and parametric estimation for quantile regression

In this paper, variable selection techniques in the linear quantile regression model are mainly considered. Based on the penalized quantile regression model, a one-step procedure that can simultaneously perform variable selection and coefficient estimation is proposed. The proposed procedure has three distinctive features: (1) By considering quantile regression, the set of relevant variables can vary across quantiles, thus making it more flexible to model heterogeneous data; (2) The one-step estimator has nice properties in both theory and practice. By applying SCAD penalty (Fan and Li, 2001) and Adaptive-LASSO penalty (Zou, 2006), we establish the oracle property for the sparse quantile regression under mild conditions. Computationally, the one-step estimator is fast, dramatically reducing the computation cost; (3) We suggest a BIC-like tuning parameter selector for the penalized quantile regression and demonstrate the consistency of this criterion. That is to say the true model can be identified consistently based on the BIC-like criterion, making our one-step estimator more reliable practically. Monte Carlo simulation studies are conducted to examine the finite-sample performance of this procedure. Finally, we conclude with a real data analysis. The results are promising.