A statistical analysis of log-periodic precursors to financial crashes^*

Abstract

Motivated by the hypothesis that financial crashes are macroscopic examples of critical phenomena associated with a discrete scaling symmetry, we reconsider the evidence of log-periodic precursors to financial crashes and test the prediction that log-periodic oscillations in a financial index are embedded in the mean function of this index. In particular, we examine the first differences of the logarithm of the S&P 500 prior to the October 87 crash and find the log-periodic component of this time series is not statistically significant if we exclude the last year of data before the crash. We also examine the claim that two separate mechanisms are needed to explain the frequency distribution of draw downs in the S&P 500 and find the evidence supporting this claim to be unconvincing.

Prediction accuracy and sloppiness of log-periodic functions

“…the predictive power of log-periodic functions in this context. The question is still unsettled as of yet (Laloux et al 1999, Feigenbaum 2001, van Bothmer 2003, Chang and Feigenbaum 2006, Gazola et al 2008. Problems are indeed numerous: what definition of a bubble and a crash to adopt (Jacobsen 2009, Lin et al 2009),?…”

“…Problems are indeed numerous: what definition of a bubble and a crash to adopt (Jacobsen 2009, Lin et al 2009),? should the price in a bubble always be increasing (Bothmer and Meister 2003), should one impose constraints on the fitted parameters , where to start a fit of a bubble,y what test of goodness of fit to use,z why having different lengths of the data window greatly affects the parameters of the best fit of the LPPL to the data (Feigenbaum 2001), why leaving out a few data points can alter the parameters of the best fit sufficiently to change a no/bubble decision (Lin et al 2009), and why is the fitting error very sensitive to small (but not large!) changes in one of the parameters of the model (Bre´e and Joseph 2010)?…”

A Consistent Model of 'Explosive' Financial Bubbles with Mean-Reversing Residuals

“…This so-called log-periodic power law (LPPL) dynamics given by (2) has been previously proposed in different forms in various papers (see for instance Sornette et al [1996], Feigenbaum and Freund [1996], , 2001], Feigenbaum [2001, Zhou and Sornette [2003a], Drozdz et al [2003], Sornette [2004b]). The power law A−B(t c −t i ) β expresses the super-exponential acceleration of prices due to positive feedback mechanisms, alluded to above.…”

Nonparametric Analyses of Log-Periodic Precursors to Financial Crashes

“…Associated with this "king" effect, log-periodic patterns [26] in financial price time series have been found to be precursory signatures of large crashes or large drawdowns [32,6,31,33,34,2,35,7,15,5,17,1] (see also [27,29] and references therein). Most of the analyses have been of a parametric nature, based on formulas involving combinations of powers laws and of log-periodic functions of the type cos[ω ln(t c − t)] [31,13,14], where t c is a "critical" time at or close to the crash.…”