  This paper is devoted to testing for the explosive bubble under time-varying
non-stationary volatility. Because the limiting distribution of the seminal
Phillips et al. (2011) test depends on the variance function and usually
requires a bootstrap implementation under heteroskedasticity, we construct the
test based on a deformation of the time domain. The proposed test is
asymptotically pivotal under the null hypothesis and its limiting distribution
coincides with that of the standard test under homoskedasticity, so that the
test does not require computationally extensive methods for inference.
Appealing finite sample properties are demonstrated through Monte-Carlo
simulations. An empirical application demonstrates that the upsurge behavior of
cryptocurrency time series in the middle of the sample is partially explained
by the volatility change.
