<p>Penalized quantile regression can detect heteroscedasticity, and is robust to heavy-tailed errors. However, it could be less efficient than the penalized least squared regression when the error follows a normal distribution. To solve this problem, we propose a robust and efficient variable selection procedure via a convex combination of the penalized least squared regression and penalized quantile regression with data-driven weights. The proposed method can adapt to different error structures and automatically chooses the weight to achieve both robustness and high efficiency. Under some conditions, the asymptotic properties of our proposed estimators are established. Besides, we apply a minorization–maximization algorithm to solve the proposed optimization problem. Extensive numerical studies are carried out to compare the performances of our method and other existing methods, and the results reveal that the newly proposed method is equivalent to the penalized least squared regression when the error follows a normal distribution, can detect heteroscedasticity, and is robust to heavy-tailed errors including the infinite variance case. Finally, we present a real example for demonstration.</p>