@techreport{Belloni2009Penalised,
abstract = {We consider median regression and, more generally, quantile regression 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. Since in this case the ordinary quantile regression is not consistent, we consider quantile regression penalized by the L1-norm of coefficients (L1-QR). First, we show that L1-QR is consistent at the rate of the square root of (s/n) log p, which is close to the oracle rate of the square root of (s/n), achievable when the minimal true model is known. 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 L1-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 L1-QR is of same stochastic order as s, the number of non-zero coefficients in the minimal true model. 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 L1-QR in a Monte-Carlo experiment, and provide an application to the analysis of the international economic growth.},
address = {London},
author = {Alexandre Belloni and Victor Chernozhukov},
copyright = {http://www.econstor.eu/dspace/Nutzungsbedingungen},
doi = {10.1920/wp.cem.2009.1009},
keywords = {330; Regression; Theorie},
language = {eng},
number = {CWP10/09},
publisher = {Centre for Microdata Methods and Practice (cemmap)},
title = {L1-Penalised quantile regression in high-dimensional sparse models},
type = {cemmap working paper},
url = {http://hdl.handle.net/10419/64690},
year = {2009}
}