  This paper addresses the statistical properties of time series driven by
rational bubbles a la Blanchard and Watson (1982), corresponding to
multiplicative maps, whose study has recently be revived recently in physics as
a mechanism of intermittent dynamics generating power law distributions. Using
insights on the behavior of multiplicative stochastic processes, we demonstrate
that the tails of the unconditional distribution emerging from such bubble
processes follow power-laws (exhibit hyperbolic decline). More precisely, we
find that rational bubbles predict a 'fat' power tail for both the bubble
component and price differences with an exponent smaller than 1, implying
absence of convergence of the mean. The distribution of returns is dominated by
the same power-law over an extended range of large returns. Although power-law
tails are a pervasive feature of empirical data, these numerical predictions
are in disagreement with the usual empirical estimates of an exponent between 2
and 4. It, therefore, appears that exogenous rational bubbles are hardly
reconcilable with some of the stylized facts of financial data at a very
elementary level.
