Consistent model identification of varying coefficient quantile regression with BIC tuning parameter selection

Quantile regression provides a flexible platform for evaluating covariate effects on different segments of the conditional distribution of response. As the effects of covariates may change with quantile level, contemporaneously examining a spectrum of quantiles is expected to have a better capacity to identify variables with either partial or full effects on the response distribution, as compared to focusing on a single quantile. Under this motivation, we study a general adaptively weighted LASSO penalization strategy in the quantile regression setting, where a continuum of quantile index is considered and coefficients are allowed to vary with quantile index. We establish the oracle properties of the resulting estimator of coefficient function. Furthermore, we formally investigate a Bayesian information criterion (BIC)-type uniform tuning parameter selector and show that it can ensure consistent model selection. Our numerical studies confirm the theoretical findings and illustrate an application of the new variable selection procedure.