Variable-selection consistency of linear quantile regression by validation set approach

We consider the problem of variable selection in the quantile regression model by cross-validation. Although cross-validation is commonly used in quantile regression for model selection, its theoretical justification has not yet been verified. In this work, we prove that cross-validation with the check loss function can lead to variable-selection consistency in quantile regression. Specifically, we investigate its asymptotic properties in linear quantile regression and its penalized version under both fixed and diverging number of parameters. For penalized models, penalties with the oracle property combined with cross-validation are shown to provide variable-selection consistency. In general, one of the crucial requirements for this consistency to hold is that the validation set size should be asymptotically equivalent to the total number of observations, which is also required in the conditional mean linear regression.