We clarify the status of log-periodicity associated with speculative bubbles
preceding financial crashes. In particular, we address Feigenbaum's [2001]
criticism and show how it can be rebuked. Feigenbaum's main result is as
follows: ``the hypothesis that the log-periodic component is present in the
data cannot be rejected at the 95% confidence level when using all the data
prior to the 1987 crash; however, it can be rejected by removing the last year
of data.'' (e.g., by removing 15% of the data closest to the critical point).
We stress that it is naive to analyze a critical point phenomenon, i.e., a
power law divergence, reliably by removing the most important part of the data
closest to the critical point. We also present the history of log-periodicity
in the present context explaining its essential features and why it may be
important. We offer an extension of the rational expectation bubble model for
general and arbitrary risk-aversion within the general stochastic discount
factor theory. We suggest guidelines for using log-periodicity and explain how
to develop and interpret statistical tests of log-periodicity. We discuss the
issue of prediction based on our results and the evidence of outliers in the
distribution of drawdowns. New statistical tests demonstrate that the 1% to 10%
quantile of the largest events of the population of drawdowns of the Nasdaq
composite index and of the Dow Jones Industrial Average index belong to a
distribution significantly different from the rest of the population. This
suggests that very large drawdowns result from an amplification mechanism that
may make them more predictable than smaller market moves.Comment: Latex document of 38 pages including 16 eps figures and 3 tables, in
press in Quantitative Financ