  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.
