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