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The log-periodic power law (LPPL) is a model of asset prices duringendogenous bubbles. A major open issue is to verify the presence of LPPL inprice sequences and to estimate the LPPL parameters. Estimation is complicatedby the fact that daily LPPL returns are typically orders of magnitude smallerthan measured price returns, suggesting that noise obscures the underlying LPPLdynamics. However, if noise is mean-reverting, it would quickly cancel out oversubsequent measurements. In this paper, we attempt to reject mean-revertingnoise from price sequences by exploiting frequency-domain properties of LPPLand of mean reversion. First, we calculate the spectrum of mean-reverting \ounoise and devise estimators for the noise's parameters. Then, we derive theLPPL spectrum by breaking it down into its two main characteristics of powerlaw and of log-periodicity. We compare price spectra with noise spectra duringhistorical bubbles. In general, noise was strong also at low frequencies and,even if LPPL underlied price dynamics, LPPL would be obscured by noise.

Financial LPPL Bubbles with Mean-Reverting Noise in the Frequency Domain