Nonparametric analyses of log-periodic precursors to financial crashes
We apply two nonpaxametric methods to further test the hypothesis that log-periodicity characterizes the detrended price trajectory of large financial indices prior to financial crashes or strong corrections. The term "parametric" refers here to the use of the log-periodic power law formula to fit the data; in contrast, "nonparametric" refers to the use of general tools such as Fourier transform, and in the present case the Hilbert transform and the so-called (H, q)-analysis. The analysis using the (H, q)-derivative is applied to seven time series ending with the October 1987 crash, the October 1997 correction and the April 2000 crash of the Dow Jones Industrial Average (DJIA), the Standard & Poor 500 and Nasdaq indices. The Hilbert transform is applied to two detrended price time series in terms of the In(t(c) - t) variable, where t(c) is the time of the crash. Taking all results together, we find strong evidence for a universal fundamental log-frequency f = 1.02 +/- 0.05 corresponding to the scaling ratio lambda = 2.67 +/- 0.12. These values axe in very good agreement with those obtained in earlier works with different parametric techniques. This note is extracted from a long unpublished report with 58 figures available at http://arXiv.org/abs/cond-mat/0205531, which extensively describes the evidence we have accumulated on these seven time series, in particular by presenting all relevant details so that the reader can judge for himself or herself the validity and robustness of the results.
log-periodicity financial crashes hilbert transform critical phenomena discrete scale invariance generalized q-analysis