  We study a rational expectation model of bubbles and crashes. The model has
two components : (1) our key assumption is that a crash may be caused by local
self-reinforcing imitation between noise traders. If the tendency for noise
traders to imitate their nearest neighbors increases up to a certain point
called the ``critical'' point, all noise traders may place the same order
(sell) at the same time, thus causing a crash. The interplay between the
progressive strengthening of imitation and the ubiquity of noise is
characterized by the hazard rate, i.e. the probability per unit time that the
crash will happen in the next instant if it has not happened yet. (2) Since the
crash is not a certain deterministic outcome of the bubble, it remains rational
for traders to remain invested provided they are compensated by a higher rate
of growth of the bubble for taking the risk of a crash. Our model distinguishes
between the end of the bubble and the time of the crash,: the rational
expectation constraint has the specific implication that the date of the crash
must be random. The theoretical death of the bubble is not the time of the
crash because the crash could happen at any time before, even though this is
not very likely. The death of the bubble is the most probable time for the
crash. There also exists a finite probability of attaining the end of the
bubble without crash. Our model has specific predictions about the presence of
certain critical log-periodic patterns in pre-crash prices, associated with the
deterministic components of the bubble mechanism. We provide empirical evidence
showing that these patterns were indeed present before the crashes of 1929,
1962 and 1987 on Wall Street and the 1997 crash on the Hong Kong Stock
Exchange. These results are compared with statistical tests on synthetic data.
