  We develop a uniform test for detecting and dating explosive behavior of a
strictly stationary GARCH$(r,s)$ (generalized autoregressive conditional
heteroskedasticity) process. Namely, we test the null hypothesis of a globally
stable GARCH process with constant parameters against an alternative where
there is an 'abnormal' period with changed parameter values. During this
period, the change may lead to an explosive behavior of the volatility process.
It is assumed that both the magnitude and the timing of the breaks are unknown.
We develop a double supreme test for the existence of a break, and then provide
an algorithm to identify the period of change. Our theoretical results hold
under mild moment assumptions on the innovations of the GARCH process.
Technically, the existing properties for the QMLE in the GARCH model need to be
reinvestigated to hold uniformly over all possible periods of change. The key
results involve a uniform weak Bahadur representation for the estimated
parameters, which leads to weak convergence of the test statistic to the
supreme of a Gaussian Process. In simulations we show that the test has good
size and power for reasonably large time series lengths. We apply the test to
Apple asset returns and Bitcoin returns.
