Wild residual bootstrap inference for penalized quantile regression with heteroscedastic errors

We consider a heteroscedastic regression model in which some of the regression coefficients are zero but it is not known which ones. Penalized quantile regression is a useful approach for analyzing such heterogeneous data. By allowing different covariates to be relevant for model- 15 ing conditional quantile functions at different quantile levels, it permits a more realistic sparsity assumption and provides a more complete picture of the conditional distribution of a response variable. Existing work on penalized quantile regression has been mostly focused on point estimation. It is challenging to estimate the standard error. Although bootstrap procedures have recently been demonstrated effective for making inference for penalized mean regression, they 20 are not directly applicable to penalized quantile regression with heteroscedastic errors.We prove that a wild residual bootstrap procedure recently proposed by Feng et al. (2011) for unpenalized quantile regression is asymptotically valid for approximating the distribution of a penalized quantile regression estimator with an adaptive L1 penalty; and that a modified version of this wild residual bootstrap procedure can be used to approximate the distribution of L1 penalized 25 quantile regression. We establish the bootstrap consistency theory, demonstrate appropriate finite sample performance through a simulation study, and illustrate its application using an ozone effects data set. The new methods do not need to estimate the unknown error density function.