Significance of Log - Periodic Signatures in Cumulative Noise

In the last years there has been a heated discussion about the presence of log-periodic signatures in financial data [Fei01b], [SJ01], [Fei01a]. One tool to detect these signatures has been the Lomb periodogram introduced by Lomb [Lom76] and improved by Scargle in [Sca82]. An important property of Scargle’s periodogram is that the individual Lomb powers of independently normal distributed noise follow approximately an exponential distribution. Unfortunately this property is lost, when one calculates Scargle’s periodogram for cumulative noise, where the differences between two observations are independently normal distributed. In this Brownian-Motion case, the expected Lomb powers are much greater for small frequencies than for large ones. Therefore the significance of large Lomb powers at small frequencies is difficult to estimate. Huang, Saleur, Sornette and Zhou tackle this problem and several related ones with extensive Monte Carlo simulations in [HSS00] and [ZS02]. Here we present an analytic approach. In the first part of this paper we introduce a small correction to Scargle’s Lomb periodogram, to make the distribution of Lomb powers exactly exponential for independently normal distributed noise. In the second part we use the same methods to derive a normalisation of the Lomb periodogram that assures an frequency independent exponential distribution of Lomb powers for cumulative noise. In the last section we apply these new methods to estimate the significance of log-periodic signatures in so called S&P 500-anti-bubble after the crash of 2000. We show how our methods greatly simplify the whole analysis and derive that there is about a 6% chance that a signature like the one detected by Sornette and Zhou in [SZ02] arises by chance if one only considers frequencies smaller than 10.0. If one searches all frequencies up to the Nyquist frequency peaks of this hight become much more common. Furthermore we detect equally significant peaks at harmonics of the fundamental frequency of Sornette and Zhou. This complements evidence for a