  Using a recently introduced rational expectation model of bubbles, based on
the interplay between stochasticity and positive feedbacks of prices on returns
and volatility, we develop a new methodology to test how this model classifies
9 time series that have been previously considered as bubbles ending in
crashes. The model predicts the existence of two anomalous behaviors occurring
simultaneously: (i) super-exponential price growth and (ii) volatility growth,
that we refer to as the ``fearful singular bubble'' regime. Out of the 9 time
series, we find that 5 pass our tests and can be characterized as ``fearful
singular bubbles.'' The 4 other cases are the information technology Nasdaq
bubble and three bubbles of the Hang Seng index ending in crashes in 1987, 1994
and 1997. According to our analysis, these four bubbles have developed with
essentially no significant increase of their volatility. This paper thus
proposes that speculative bubbles ending in crashes form two groups hitherto
unrecognized, namely those accompanied by increasing volatility (reflecting
increasing risk perception) and those without change of volatility (reflecting
an absence of risk perception).
