A lack-of-fit test for quantile regression models with high-dimensional covariates

TL;DR

This paper introduces a new lack-of-fit test for high-dimensional quantile regression models that effectively handles many covariates and heteroscedasticity, validated through simulations and real data analysis.

Contribution

It adapts existing projection-based tests to high-dimensional settings and employs a wild bootstrap for critical value approximation.

Findings

The test performs well with high-dimensional covariates.

It is effective under heteroscedastic regression models.

Simulation and real data demonstrate its practical utility.

Abstract

We propose a new lack-of-fit test for quantile regression models that is suitable even with high-dimensional covariates. The test is based on the cumulative sum of residuals with respect to unidimensional linear projections of the covariates. The test adapts concepts proposed by Escanciano (Econometric Theory, 22, 2006) to cope with many covariates to the test proposed by He and Zhu (Journal of the American Statistical Association, 98, 2003). To approximate the critical values of the test, a wild bootstrap mechanism is used, similar to that proposed by Feng et al. (Biometrika, 98, 2011). An extensive simulation study was undertaken that shows the good performance of the new test, particularly when the dimension of the covariate is high. The test can also be applied and performs well under heteroscedastic regression models. The test is illustrated with real data about the economic growth of 161 countries.