Variable Selection in Single Index Quantile Regression for Heteroscedastic Data

Quantile regression (QR) has become a popular method of data analysis, especially when the error term is heteroscedastic, due to its relevance in many scientific studies. The ubiquity of high dimensional data has led to a number of variable selection methods for linear/nonlinear QR models and, recently, for the single index quantile regression (SIQR) model. We propose a new algorithm for simultaneous variable selection and parameter estimation applicable also for heteroscedastic data. The proposed algorithm, which is noniterative, consists of two steps. Step 1 performs an initial variable selection method. Step 2 uses the results of Step 1 to obtain better estimation of the conditional quantiles and, using them, to perform simultaneous variable selection and estimation of the parametric component of the SIQR model. It is shown that the initial variable selection method of Step 1 consistently estimates the relevant variables, and that the estimated parametric component derived in Step 2 satisfies the oracle property.