Nonparametric Testing of an Exclusion Restriction in Quantile Regression

Using the framework proposed by Bickel et al. (2006 Bickel , P. J. , Ritov , Y. , Stoker , T. ( 2006 ). Tailor-made tests for goodness-of-fit to semiparametric hypotheses . Ann. Stat. 34 ( 2 ): 721 – 741 . Google Scholar]), we provide a score-based testing method to check the exclusion restriction in quantile regression, i.e., H: ν_α(Y|U, V) = ν_α(Y|U) w.p.1, where ν_α denotes the αth (0 < α < 1) quantile. A subsampling method is suggested to acquire the critical values and justified. The tests are all found to be consistent against fixed alternatives and have discriminating power against local alternatives at root-n scale. We address this particular problem as a representative among a wide family of semiparametric model checking problems. The methodology can be carried over to other goodness-of-fit testing of semiparametric models, possibly involve non smooth functions.