  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.
