  We extend the model of rational bubbles of Blanchard and of Blanchard and
Watson to arbitrary dimensions d: a number d of market time series are made
linearly interdependent via d times d stochastic coupling coefficients. We
first show that the no-arbitrage condition imposes that the non-diagonal
impacts of any asset i on any other asset j different from i has to vanish on
average, i.e., must exhibit random alternative regimes of reinforcement and
contrarian feedbacks. In contrast, the diagonal terms must be positive and
equal on average to the inverse of the discount factor. Applying the results of
renewal theory for products of random matrices to stochastic recurrence
equations (SRE), we extend the theorem of Lux and Sornette (cond-mat/9910141)
and demonstrate that the tails of the unconditional distributions associated
with such d-dimensional bubble processes follow power laws (i.e., exhibit
hyperbolic decline), with the same asymptotic tail exponent mu<1 for all
assets. The distribution of price differences and of returns is dominated by
the same power-law over an extended range of large returns. This small value
mu<1 of the tail exponent has far-reaching consequences in the non-existence of
the means and variances. Although power-law tails are a pervasive feature of
empirical data, the numerical value mu<1 is in disagreement with the usual
empirical estimates mu approximately equal to 3. It, therefore, appears that
generalizing the model of rational bubbles to arbitrary dimensions does not
allow us to reconcile the model with these stylized facts of financial data.
The non-stationary growth rational bubble model seems at present the only
viable solution (see cond-mat/0010112).
