On Rational Bubbles and Fat Tails
利用乘法随机过程理论，证明理性泡沫模型产生的无条件分布尾部呈幂律分布且指数小于1，导致均值发散；但该预测与实证中指数在2到4之间的肥尾特征不符，表明理性泡沫难以解释金融数据的基本统计事实。
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.