Robust inference in high- dimensional approximately sparse quantile regression models

This work proposes new inference methods for the estimation of a regression coefficientof interest in quantile regression models. We consider high-dimensional models where the number ofregressors potentially exceeds the sample size but a subset of them suffice to construct a reasonableapproximation of the unknown quantile regression function in the model. The proposed methods areprotected against moderate model selection mistakes, which are often inevitable in the approximatelysparse model considered here. The methods construct (implicitly or explicitly) an optimal instrumentas a residual from a density-weighted projection of the regressor of interest on other regressors. Underregularity conditions, the proposed estimators of the quantile regression coefficient are asymptoticallyroot-n normal, with variance equal to the semi-parametric efficiency bound of the partially linear quan-tile regression model. In addition, the performance of the technique is illustrated through Monte-carloexperiments and an empirical example, dealing with risk factors in childhood malnutrition. The nu-merical results confirm the theoretical findings that the proposed methods should outperform the naivepost-model selection methods in non-parametric settings. Moreover, the empirical results demonstratesoundness of the proposed methods.