Comparing time varying regression quantiles under shift invariance

This article investigates whether time-varying quantile regression curves are the same up to the horizontal shift or not. The errors and the covariates involved in the regression model are allowed to be locally stationary. We formalize this issue in a corresponding non-parametric hypothesis testing problem, and develop an integrated-squared-norm based test (SIT) as well as a simultaneous confidence band (SCB) approach. The asymptotic properties of SIT and SCB under null and local alternatives are derived. Moreover, the asymptotic properties of these tests are also studied when the compared data sets are dependent. We then propose valid wild bootstrap algorithms to implement SIT and SCB. Furthermore, the usefulness of the proposed methodology is illustrated via analysing simulated and real data related to COVID-19 outbreak and climate science. bootstrap, comparison of curves, confidence band, hypothesis testing, locally stationary process, nonparametric quantile regression, COVID-19. 1 ar X iv :2 01 1. 06 33 3v 2 [ st at .M E ] 2 4 D ec 2 02 1