Model checking for parametric single-index quantile models

In this work, we construct a lack-of-fit test for testing parametric single-index quantile regression models. We apply the kernel smoothing technique for the multivariate nonparametric estimation involved in this task. To avoid the "curse of dimensionality" in multivariate nonparametric estimation and to fully utilize the information contained in the model, we employ a sufficient dimension reduction technique to identify the corresponding dimensionally reduced subspace and then construct our test statistic in this subspace. At different quantile levels, the test statistics given in this paper can quickly detect local alternative hypotheses, which are different from the null hypothesis for small and moderate sample sizes. A new wild bootstrap method is applied to approximate the critical values of the quantile regression model test. The effectiveness of the method is verified by simulation experiments and a real data application.