  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 data. By allowing
different covariates to be relevant for modeling conditional quantile functions
at different quantile levels, it provides a more complete picture of the
conditional distribution of a response variable than mean regression. Existing
work on penalized quantile regression has been mostly focused on point
estimation. Although bootstrap procedures have recently been shown to be
effective for inference for penalized mean regression, they are not directly
applicable to penalized quantile regression with heteroscedastic errors. We
prove that a wild residual bootstrap procedure for unpenalized quantile
regression is asymptotically valid for approximating the distribution of a
penalized quantile regression estimator with an adaptive $L_1$ penalty and that
a modified version can be used to approximate the distribution of
$L_1$-penalized quantile regression estimator. The new methods do not need to
estimate the unknown error density function. We establish consistency,
demonstrate finite sample performance, and illustrate the applications on a
real data example.

Methodology