  We consider median regression and, more generally, a possibly infinite
collection of quantile regressions in high-dimensional sparse models. In these
models the overall number of regressors $p$ is very large, possibly larger than
the sample size $n$, but only $s$ of these regressors have non-zero impact on
the conditional quantile of the response variable, where $s$ grows slower than
$n$. We consider quantile regression penalized by the $\ell_1$-norm of
coefficients ($\ell_1$-QR). First, we show that $\ell_1$-QR is consistent at
the rate $\sqrt{s/n} \sqrt{\log p}$. The overall number of regressors $p$
affects the rate only through the $\log p$ factor, thus allowing nearly
exponential growth in the number of zero-impact regressors. The rate result
holds under relatively weak conditions, requiring that $s/n$ converges to zero
at a super-logarithmic speed and that regularization parameter satisfies
certain theoretical constraints. Second, we propose a pivotal, data-driven
choice of the regularization parameter and show that it satisfies these
theoretical constraints. Third, we show that $\ell_1$-QR correctly selects the
true minimal model as a valid submodel, when the non-zero coefficients of the
true model are well separated from zero. We also show that the number of
non-zero coefficients in $\ell_1$-QR is of same stochastic order as $s$.
Fourth, we analyze the rate of convergence of a two-step estimator that applies
ordinary quantile regression to the selected model. Fifth, we evaluate the
performance of $\ell_1$-QR in a Monte-Carlo experiment, and illustrate its use
on an international economic growth application.
