Single-index thresholding in quantile regressio

Threshold models have a wide variety of applications in statistics and economics. We generalize the threshold quantile regression in literature to single-index thresholding quantile regression which depends on a linear combination of regressors as threshold variable. We first propose smoothed estimator and study its asymptotic properties for a single quantile as well as for the quantile process. To perform inference for index parameters, we propose a Wald-type method and a Mixed-bootstrap method which is more stable. We also extend the Mixed-bootstrap to test constancy of index parameters across different quantile indexes. The finite sample performance of the proposed method is assessed through simulation and the analysis of a real data.