  In this paper, we develop two families of sequential monitoring procedure to
(timely) detect changes in a GARCH(1,1) model. Whilst our methodologies can be
applied for the general analysis of changepoints in GARCH(1,1) sequences, they
are in particular designed to detect changes from stationarity to explosivity
or vice versa, thus allowing to check for volatility bubbles. Our statistics
can be applied irrespective of whether the historical sample is stationary or
not, and indeed without prior knowledge of the regime of the observations
before and after the break. In particular, we construct our detectors as the
CUSUM process of the quasi-Fisher scores of the log likelihood function. In
order to ensure timely detection, we then construct our boundary function
(exceeding which would indicate a break) by including a weighting sequence
which is designed to shorten the detection delay in the presence of a
changepoint. We consider two types of weights: a lighter set of weights, which
ensures timely detection in the presence of changes occurring early, but not
too early after the end of the historical sample; and a heavier set of weights,
called Renyi weights which is designed to ensure timely detection in the
presence of changepoints occurring very early in the monitoring horizon. In
both cases, we derive the limiting distribution of the detection delays,
indicating the expected delay for each set of weights. Our theoretical results
are validated via a comprehensive set of simulations, and an empirical
application to daily returns of individual stocks.
