Everything You Always Wanted to Know about Log Periodic Power Laws for Bubble Modelling but Were Afraid to Ask

Fantazzini, Dean and Geraskin, Petr (2011): Everything You Always Wanted to Know about Log Periodic Power Laws for Bubble Modelling but Were Afraid to Ask. Forthcoming in: European Journal of Finance

Abstract

Sornette et al. (1996), Sornette and Johansen (1997), Johansen et al. (2000) and Sornette (2003a) proposed that, prior to crashes, the mean function of a stock index price time series is characterized by a power law decorated with log-periodic oscillations, leading to a critical point that describes the beginning of the market crash. This paper reviews the original Log-Periodic Power Law (LPPL) model for financial bubble modelling, and discusses early criticism and recent generalizations proposed to answer these remarks. We show how to fit these models with alternative methodologies, together with diagnostic tests and graphical tools to diagnose financial bubbles in the making in real time. An application of this methodology to the Gold bubble which busted in December 2009 is then presented.

• Item Type: MPRA Paper
• Language: English
• Keywords: Log-periodic models, LPPL, Crash, Bubble, Anti-Bubble, GARCH, Forecasting, Gold, Bubble Burst, Bubble modelling, Bubble forecasting
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; G - Financial Economics > G1 - General Financial Markets > G17 - Financial Forecasting and Simulation

References

[1] Abreu, D., and M. K., Brunnermeier. 2003. Bubbles and crashes. Econometrica, 71, no. 1: 173204.

[2] Bastiaensen, K., Cauwels, P., Sornette, D., Woodard, R., and W.X. Zhou. 2009. The Chinese equity bubble: Ready to burst. http://arxiv.org/abs/0907.1827.

[3] Bauwens, L., Lubrano, M., and J.F. Richard. 2000. Bayesian Inference In Dynamic Econometric Models. New York: OUP.

[4] Bernardo, J.M. and A.F.M. Smith. 1994. Bayesian Theory. New York: Wiley.

[5] Blanchard, O. J. 1979. Speculative bubbles, crashes and rational ex- pectations. Economics Letters, 3:387-389.

[6] Cendrowski, S. 2009. Beware the gold bubble. http://money.cnn.com/2009/10/06/pf/gold investing bubble.fortune/index.htm.

[7] Chang , G. and J. Feigenbaum. 2006. A Bayesian analysis of log-periodic precursors to financial crashes, Quantitative Finance 6 (1), 15-36.

[8] Cvijovi´c, D. and J. Klinowski. (1995). Taboo search An approach to the multiple minima problem. Science, 267, no. 5: 664-666.

[9] Dennis, Jr., J.E., and R.B., Schnabel. 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs: Prentice-Hall.

[10] Derrida, B., De Seze, L., and Itzykson, C. 1983. Fractal structure of zeros in hierarchical models. Journal of Statistical Physics, 33: 559-569

[11] Fantazzini, D. 2010a. Modelling Bubbles And Anti-Bubbles In Bear Markets: A Medium-Term Trading Analysis. In Handbook of Trading, ed. G. Gregoriou, 365-388. New York: McGraw-Hill.

[12] Fantazzini, D. 2010b. Modelling and Forecasting the Global Financial Crisis: Initial Findings using Heterosckedastic Log-Periodic Models. Economics Bulletin, 30, no. 3: 1833-1841.

[13] Gazola,L., Fernandes, C., Pizzinga, A., and R. Riera. 2008. The log-periodic-AR(1)-GARCH(1,1) model for financial crashes, The European Physical Journal B, 61: 355-362.

[14] Goldenfeld, N. 1992. Lectures on phase transitions and the renormalization group. Reading: Addison-Wesley Publishing Company.

[15] Griffiths, W.E., Hill, R., and G.C. Lim. (2008). Using EViews for Principles of Econometrics. 3rd Edition, Hoboken, NJ: Wiley.

[16] Gulsten, M., Smith, E. A. and D.M. Tate. 1995. A Genetic Algorithm Approach to Curve Fitting. International Journal of Production Research, 33, no. 7: 1911-1923.

[17] G¨urkaynak, R. 2008. Econometric tests of asset price bubbles: taking stock. Journal of Economic Surveys, 22, 1:166186.

[18] Hayashi, F. 2000. Econometrics. Princeton: PUP.

[19] Jacobsson, E. 2009. How to predict crashes in financial markets with the Log-Periodic Power Law. Master diss., Department of Mathematical Statistics, Stockholm University.

[20] Jiang, Z.Q., Zhou, W.H., Sornette, D., Woodard, R., Bastiaensen, K., and P. Cauwels. 2009. Bubble Diagnosis and Prediction of the 2005-2007 and 2008-2009 Chinese stock market bubbles (2009). Journal of Economic Behavior and Organization 74, 149-162 (2010). Available at http://arxiv.org/abs/0909.1007

[21] Johansen, A. 2002. Comment on “Are financial crashes predictable?”, Europhysics Letters, 60, no. 5: 809-810.

[22] Johansen, A. 2003. Characterization of large price variations in financial markets. Physica A, 324: 157-166.

[23] Johansen, A., and D., Sornette. 1999. Financial anti-bubbles: Log-periodicity in Gold and Nikkei collapses. International Journal of Modern Physics C, 10, no. 4: 563-575.

[24] Johansen, A., and D., Sornette. 2000. Evaluation of the quantitative prediction of a trend reversal on the Japanese stock market in 1999. International Journal of Modern Physics C, 11, no. 2: 359-364.

[25] Johansen, A., and D., Sornette. 2004. Endogenous versus Exogenous Crashes in Financial Markets, in Contemporary Issues in International Finance, Nova Science Publishers. Reprinted as ’Shocks, Crashes and Bubbles in Financial Markets’, Brussels Economic Review (Cahiers economiques de Bruxelles), 49 (3/4), Special Issue on Nonlinear Analysis, 2006.

[26] Johansen, A., Ledoit, O. and D., Sornette. 2000. Crashes as critical points, International Journal of Theoretical and Applied Finance, 3, no. 2: 219-255.

[27] Kaizoji, T., and D. Sornette., Market Bubbles and Crashes, in press in the Encyclopedia of Quantitative Finance (Wiley) http://www.wiley.com//legacy/wileychi/eqf/ (long version available at http://arXiv.org/abs/0812.2449)

[28] Koop, G. 2003. Bayesian Econometrics. Chichester: Wiley.

[29] Krugman, P. 1998. The Confidence Game, How Washington worsened Asia’s crash. The New Republic, October 5.

[30] Kwiatkowski, D., Phillips, P., Schmidt, P., and Y., Shin. 1992. Testing the null hypothesis of stationary against the alternative of a unit root. Journal of Econometrics, 54: 159-178.

[31] Laloux, L., Potters, M., Cont,R., Aguilar, J.P and J.P. Bouchaud. 1999. Are financial crashes predictable?, Europhysics Letters, 45, no. 1: 1-5.

[32] Landau, L.D. and E.M. Lifshitz. (1980). Statistical Physics, Part 1 and Part 2. 3rd ediion, Butterworth-Heinemann.

[33] Lin, L. and D. Sornette. 2009. Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-Reverting Termination Times. ETH working paper, http://arxiv.org/abs/0911.1921 .

[34] Lin, L., Ren., R.E. and D. Sornette. 2009. A Consistent Model of Explosive Financial Bubbles With Mean-Reversing Residuals, Quantitative Finance, in press. Available at http://arxiv.org/abs/0905.0128 and http://papers.ssrn.com/abstract=1407574.

[35] Mogi, C. 2009. Gold hits record high as dollar sets new lows. Reuters, November 26 2009, http://www.reuters.com/article/goldMktRpt/idUSSP7486520091126

[36] Ng, S. and P. Perron. 2001. Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69, no. 6: 1519-1554.

[37] Onsager, L. 1944. Crystal statistics. A two-dimensional model with an order-disorder transition. Physics Review, 65: 117-149.

[38] Pfaff, B. (2008). Analysis of Integrated and Cointegrated Time Series with R. New York: Wiley.

[39] Reinhart, C. and K. Rogoff. 2009. This Time is Different: Eight Centuries of Financial Folly. Princeton: PUP.

[40] Shefrin. H. 2005. A Behavioral Approach to Asset Pricing. Academic Press Advanced Finance Series. Burlington: Academic Press.

[41] Sornette, D. 2003a. Why Stock Markets Crash (Critical Events in Complex Financial Systems), Princeton: PUP.

[42] Sornette, D. 2003b. Critical market crashes. Physics Reports, 378, no. 1: 1-98.

[43] Sornette, D. 2009. Dragon-Kings, Black Swans and the Prediction of Crises, International Journal of Terraspace Science and Engineering, 2, no. 1: 1-18.

[44] Sornette, D. and A., Johansen. 1997. Large financial crashes, Physica A, 245: 411-422.

[45] Sornette, D. and A., Johansen. 2001. Significance of log-periodic precursors to financial crashes, Quantitative Finance, 1, no. 4: 452-471.

[46] Sornette, D., and R.Woodard. 2010. Financial Bubbles, Real Estate bubbles, Derivative Bubbles, and the Financial and Economic Crisis, in “Proceedings of APFA7 (Applications of Physics in Financial Analysis)”, New Approaches to the Analysis of Large-Scale Business and Economic Data, Misako Takayasu, Tsutomu Watanabe and Hideki Takayasu, eds., Springer. Available at http://www.thic-apfa7.com/en/htm/index.html , http://arxiv.org/abs/0905.0220, http://web.sg.ethz.ch/wps/CCSS-09-00003/, and http://papers.ssrn.com/sol3/papers.cfm?abstract id=1407608

[47] Sornette, D. and W.X. Zhou. 2006. Predictability of Large Future Changes in major financial indices. International Journal of Forecasting, 22: 153-168.

[48] Sornette, D., Johansen, A. and J.P., Bouchaud. 1996. Stock market crashes, Precursors and Replicas, Journal de Physique I, 6: 167-175.

[49] Sornette, D., Woodard, R. and W.X. Zhou. 2008. The 2006-2008 Oil Bubble and Beyond. ETH Zurich preprint. http://arXiv.org/abs/0806.1170

[50] Sornette, D., Woodard, R. and W.X. Zhou. 2009. The 2006-2008 Oil Bubble: Evidence of Speculation, and Prediction. Physica A, 388: 1571-1576.

[51] Stanley, H. E. 1971. Introduction to Phase Transitions and Critical Phenomena. New York: OUP.

[52] Stock J.H. 1994. Unit roots, structural breaks and trends. In Handbook of econometrics, ed. R.F. Engle, D.L. McFadden, vol. 4, ch. 47. Elsevier.

[53] Stock J.H., and M.W. Watson. (2002). Introduction to Econometrics. Addison Wesley.

[54] Tsay, R. 2005. Analysis of Financial Time Series. Chicago: Wiley.

[55] Tung, R. 2009. China c.Bank: Gold Prices High, Warns of Bubble. Reuters, December 02 2009, http://www.reuters.com/article/idUSTPU00193020091202 .

[56] White, G. 2009. Golden times for precious metals but pick carefully. The Telegraph, September 08 2009, http://www.telegraph.co.uk/finance/markets/questor/6156983/Golden-times-for-precious-metals-but-pick-carefully.html .

[57] Zhou, W.X., and D., Sornette. 2003. 2000-2003 Real Estate Bubble in the UK but not in the USA. Physica A, 329: 249-263.

[58] Zhou, W.X., and D., Sornette. 2005. Testing the Stability of the 2000-2003 US Stock Market Antibubble. Physica A, 348: 428-452.

[59] Zhou, W.X., and D., Sornette. 2006. Is There a Real-Estate Bubble in the US?. Physica A, 361: 297-308.

[60] Zhou, W.X., and D., Sornette. 2008. Analysis of the real estate market in Las Vegas: Bubble, seasonal patterns, and prediction of the CSW indexes. Physica A, 387: 243-260.

[61] Zhou, W.X., and D., Sornette. 2009. A case study of speculative financial bubbles in the South African stock market 2003-2006. Physica A, 388: 869-880.