A Bayesian analysis of log-periodic precursors to financial crashes

Abstract

A large number of papers have been written by physicists documenting an alleged signature of imminent financial crashes involving so-called log-periodic oscillations–oscillations which are periodic with respect to the logarithm of the time to the crash. In addition to the obvious practical implications of such a signature, log-periodicity has been taken as evidence that financial markets can be modelled as complex statistical-mechanics systems. However, while many log-periodic precursors have been identified, the statistical significance of these precursors and their predictive power remain controversial in part because log-periodicity is ill-suited for study with classical methods. This paper is the first effort to apply Bayesian methods in the testing of log-periodicity. Specifically, we focus on the Johansen–Ledoit–Sornette (JLS) model of log periodicity. Using data from the S&P 500 prior to the October 1987 stock market crash, we find that, if we do not consider crash probabilities, a null hypothesis model without log-periodicity outperforms the JLS model in terms of marginal likelihood. If we do account for crash probabilities, which has not been done in the previous literature, the JLS model outperforms the null hypothesis, but only if we ignore the information obtained by standard classical methods. If the JLS model is true, then parameter estimates obtained by curve fitting have small posterior probability. Furthermore, the data set contains negligible information about the oscillation parameters, such as the frequency parameter that has received the most attention in the previous literature.

Keywords:

• Financial crashes
• Bayesian inference
• Log-periodicity

Acknowledgments

The authors would like to thank Dave DeJong, John Geweke, Anders Johansen, Jean-Francois Richard, Gene Savin, Pedro Silos, and Chuck Whiteman for suggestions and discussions on this topic.

Notes

†For a review of log-periodic research and other examples of how physicists have applied their methods to problems of economic interest, see Feigenbaum (2003). For a more thorough discussion of the evidence in favour of log-periodicity, see Sornette (2003).

‡For responses to this criticism, see Sornette and Johansen (2001) and Johansen (2002a,b).

†Feigenbaum (2001a,b) did not impose the constraint |C|≤ B imposed in this paper, so the paper tested for log-periodicity primarily by testing for the significance of the oscillation term.

‡There have been a few applications of Bayesian methods in the log-periodic literature employing very rough estimates of the relevant likelihoods. See, for example, Johansen and Sornette (2000). However, to our knowledge, this is the first application of Bayesian methods that actually uses the likelihood function specified by the JLS model.

†For a more complete introduction to the subject of Bayesian inference, consult Berger (1985), Bernardo and Smith (1994), or Gelman et al. (2003).

†See Blanchard and Fischer (1989) for a review of bubble solutions.

‡Note that the analogy between the time to a crash and the distance in the phase space to the transition boundary is not exact, and this postulate has never been rigorously established.

†Sornette and Johansen (2001) also noted that B′>0 is necessary.

‡JLS (2000) disregard the contribution of the stochastic volatility σ on the expectation of ds that comes from Ito's Lemma.

§The beta distribution B(α,β) has support [0,1] with density proportional to . The mean is α/(α+β) and the variance is . The gamma distribution Γ(α,β) has support [0,∞) with density proportional to . The mean is α/β and the variance is α/β².

†While it is not possible for the maximum likelihood of a model to decrease after expanding the parameter space, it is possible for the marginal likelihood to decrease. Indeed, this is quite common when the larger parameter space has a higher dimension than the original space, as is the case here.

‡Note that calling this tighter distribution a ‘prior’ is actually an abuse of language since the prior should only reflect information available before we confront the data. Making inferences based on such a ‘prior’ would not be proper. We present computations based on the tight prior only as a robustness check.

†If a t _c was drawn that was too close to t _N , this could cause problems so we truncated the source distribution so t _c>t _N +0.5. To be consistent, we must truncate the prior distribution in the same way. This was not done in the marginal likelihood estimates reported above (and below). Nevertheless, truncating the priors had a negligible effect on marginal likelihoods.

†Strictly speaking, we should be comparing the market-time model to NLLS estimates from the market-time model. However, there is not much difference between the posteriors for the market- and calendar-time models with diffuse priors.

‡For φ and C, the difference between the posterior and prior means is not statistically significant.

†The parameters of this highest posterior draw were μ=9.87×10⁻⁵, τ=15 323, B=0.013, C=0.94, β=0.55, ω=5.70, φ=4.96 and t _c=10/20/87.

‡Note that there are classical methods that can take into account the crash probabilities. Maximum likelihood estimation (MLE) of the JLS model could incorporate the crash probabilities into the likelihood function as we have done, but, to our knowledge, no researcher has used MLE to estimate the model.

†If daily returns are normally distributed, the standard deviation would have to be on the order of 1% to fit most price data, in which case the probability of the market dropping by as much as 20% on any given day would be astronomically small. So we must introduce another mechanism to plausibly account for crashes in the null hypothesis.

†The posterior for ω in figure 27 does exhibit one abnormally high point, but this is still within two standard errors of the prior.

†The notion that psychological barriers might be involved in explaining log-periodicity has also been suggested by Johansen and Sornette (1999), who proposed this to explain the appearance of log-periodic antibubbles. The JLS model cannot account for such antibubbles since the critical event that triggers an antibubble occurs at the beginning rather than the end, so there is no uncertainty about when it will occur.