  We present a self-consistent model for explosive financial bubbles, which
combines a mean-reverting volatility process and a stochastic conditional
return which reflects nonlinear positive feedbacks and continuous updates of
the investors' beliefs and sentiments. The conditional expected returns exhibit
faster-than-exponential acceleration decorated by accelerating oscillations,
called "log-periodic power law." Tests on residuals show a remarkable low rate
(0.2%) of false positives when applied to a GARCH benchmark. When tested on the
S&P500 US index from Jan. 3, 1950 to Nov. 21, 2008, the model correctly
identifies the bubbles ending in Oct. 1987, in Oct. 1997, in Aug. 1998 and the
ITC bubble ending on the first quarter of 2000. Different unit-root tests
confirm the high relevance of the model specification. Our model also provides
a diagnostic for the duration of bubbles: applied to the period before Oct.
1987 crash, there is clear evidence that the bubble started at least 4 years
earlier. We confirm the validity and universality of the volatility-confined
LPPL model on seven other major bubbles that have occurred in the World in the
last two decades. Using Bayesian inference, we find a very strong statistical
preference for our model compared with a standard benchmark, in contradiction
with Chang and Feigenbaum (2006) which used a unit-root model for residuals.
