We propose a new class of tests to evaluate the correct specification of quantile regression models over a continuum of quantiles. A cumulative sum (cusum) process is established by using all components in the gradient of the check function, which is further modified by (1) replacing the weighting functions with dimension reduction ability, and (2) incorporating an orthogonal projection onto the tangent space of nuisance parameters in order to eliminate the uncertainty of preliminary parameter estimation. Besides, this projection is also able to facilitate an attractive multiplier bootstrap procedure for the computation of critical values. We then introduce several Cramér-von Mises-type test statistics with convenient closed-form expressions. The asymptotic properties of the proposed test statistics are investigated under the null, the alternative, and a sequence of local alternatives converging to the null at the rate, respectively. Simulation studies show that our tests have good finite sample performance. A real data example is also introduced to illustrate the usefulness of our tests in practice.
