تشخیص حباب و پیش بینی حباب سال های 2005-2007 و 2008-2009 بازار سهام چین

By combining (i) the economic theory of rational expectation bubbles, (ii) behavioral finance on imitation and herding of investors and traders and (iii) the mathematical and statistical physics of bifurcations and phase transitions, the log-periodic power law (LPPL) model has been developed as a flexible tool to detect bubbles. The LPPL model considers the faster-than-exponential (power law with finite-time singularity) increase in asset prices decorated by accelerating oscillations as the main diagnostic of bubbles. It embodies a positive feedback loop of higher return anticipations competing with negative feedback spirals of crash expectations. We use the LPPL model in one of its incarnations to analyze two bubbles and subsequent market crashes in two important indexes in the Chinese stock markets between May 2005 and July 2009. Both the Shanghai stock exchange composite index (US ticker symbol SSEC) and Shenzhen stock exchange component index (SZSC) exhibited such behavior in two distinct time periods: (1) from mid-2005, bursting in October 2007 and (2) from November 2008, bursting in the beginning of August 2009. We successfully predicted time windows for both crashes in advance (Sornette, 2007; Bastiaensen et al., 2009) with the same methods used to successfully predict the peak in mid-2006 of the US housing bubble (Zhou and Sornette, 2006b) and the peak in July 2008 of the global oil bubble (Sornette et al., 2009). The more recent bubble in the Chinese indexes was detected and its end or change of regime was predicted independently by two groups with similar results, showing that the model has been well-documented and can be replicated by industrial practitioners. Here we present a more detailed analysis of the individual Chinese index predictions and of the methods used to make and test them. We complement the detection of log-periodic behavior with Lomb spectral analysis of detrended residuals and (H,q)(H,q)-derivative of logarithmic indexes for both bubbles. We perform unit-root tests on the residuals from the log-periodic power law model to confirm the Ornstein–Uhlenbeck property of bounded residuals, in agreement with the consistent model of ‘explosive’ financial bubbles (Lin et al., 2009).

In the following two subsections, we present our analysis of the separate 2007 and 2009 bubbles using the methods described in Section 2. 3.1. Back test of Chinese bubble from 2005 to 2007 3.1.1. LPPL fitting with varying window sizes As discussed above, we test the stability of fit parameters for the two indexes, SSEC and SZSC, by varying the size of the fit intervals. Specifically, the logarithmic index is fit by the LPPL formula, Eq. (1): 1. in shrinking windows with a fixed end date t2=t2= 10 October 2007 with the start time t1t1 increasing from 1 October 2005 to 31 May 2007 in steps of 5 (trading) days and 2. in expanding windows with a fixed start date t1=t1= 1 December 2005 with the end date t2t2 increasing from 1 May 2007 to 1 October 2007 in steps of 5 (trading) days. In the above two fitting procedures, we fit the indexes 124 times in shrinking windows and 15 times in expanding windows. After filtering by the LPPL conditions, we finally observe 72 (78) results in the first step and 11 (15) results in the second step for SSEC (respectively, SZSC). Fig. 2(a) and (c) illustrates six chosen fitting results of the shrinking windows for SSEC and SZSC and Fig. 2 (b) and (d) illustrate six fitting results of the expanding time intervals for SSEC and SZSC. The dark and light shadow boxes in the figures indicate 20%/80% and 5%/95% quantile range of values of the crash dates for the fits that survived filtering. One can observe that the observed market peak dates (16 October 2007 for SSEC and 31 October 2007 for SZSC) lie in the quantile ranges of predicted crash dates tctc using only data from before the market crash (i.e., using t2<tc_obst2<tc_obs). Full-size image (100 K) Fig. 2. Daily trajectory of the logarithmic SSEC (a and b) and SZSC (c and d) index from 01 May 2005 to 18 October 2008 (dots) and fits to the LPPL formula (1). The dark and light shadow box indicate 20/80% and 5/95% quantile range of values of the crash dates for the fits, respectively. The two dashed lines correspond to the minimum date of t1t1 and the maximum date of t2t2. (a) Examples of fitting to shrinking windows with varied t1t1 and fixed t2=t2= 10 October 2007 for SSEC. The six fitting illustrations are corresponding to t1=t1= 30 September 2005, 5 December 2005, 13 February 2006, 24 April 2006, 15 January 2007, and 12 March 2007. (b) Examples of fitting to expanding windows with fixed t1=t1= 01 December 2005 and varied t2t2 for SSEC. The six fitting illustrations are associated with t2=t2= 20 August 2007, 29 August 2007, 7 September 2007, 17 September 2007, 26 September 2007, 5 October 2007. (c) Examples of fitting to shrinking windows with varied t1t1 and fixed t2=t2= 10 October 2007 for SZSC. The six fitting illustrations are corresponding to t1=t1= 30 September 2005, 12 December 2006, 24 February 2006, 12 May 2006, 9 January 2007, and 13 April 2007. (d) Examples of fitting to expanding windows with fixed t1=t1= 1 December 2005 and varied t2t2 for SZSC. The six fitting illustrations are associated with t2=t2= 1 August 2007, 10 August 2007, 24 August 2007, 7 September 2007, 21 September 2007, and 8 October 2007. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Figure options 3.1.2. Lomb analysis, parametric approach Fig. 3 summarizes the results of our Lomb analysis on the detrended residuals r(t)r(t). The Lomb periodograms (PNPN with respect to ωLombωLomb) are plotted in Fig. 3(a) for four typical examples, which are (t1,t2t1,t2)=(13 March 2006, 10 October 2007) and (12 December 2005, 7 September 2007) for SSEC and (4 April 2006, 10 October 2007) and (1 December 2005, 9 September 2007) for SZSC. The inset illustrates the corresponding detrended residuals r(t)r(t) as a function of ln⁡(tc−t)ln⁡(tc−t). We select the highest peak with its associated ωLombωLomb. Full-size image (54 K) Fig. 3. Lomb tests of the detrending residuals r(t)r(t) for SSEC and SZSC. The residuals are obtained from Eq. (4) by substituting different survival LPPL calibrating windows with the corresponding fitting results including tctc, m, and A. (a) Lomb periodograms for four typical examples, which are presented in the legend. The time periods followed the index names represent the LPPL calibrating windows. The inset illustrates the corresponding residuals r(t)r(t) as a function of ln⁡(tc−t)ln⁡(tc−t). (b) Bivariate distribution of pairs View the MathML source(ωLomb,PNmax⁡) for different LPPL calibrating intervals. Each point in the figure stands for the highest peak and its associated angular log-frequency in the Lomb periodogram of a given detrended residual series. The inset shows ωfitωfit as a function of ωLombωLomb. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.) Figure options The values of ωLombωLomb must be consistent with the values of ωfitωfit obtained from the fitting. We plot the bivariate distribution of pairs View the MathML source(ωLomb,PNmax⁡) for different LPPL calibrating windows in Fig. 3(b) and find that the minimum value of View the MathML sourcePNmax⁡ is approximately 54 for SSEC and approximately 30 for SZSC. These peaks are linked to a false alarm probability, which is defined as the chance that we falsely detect log-periodicity in a signal without true log-periodicity. To calculate this false alarm probability, a model of the distribution of the residuals must be used. We ‘bracket’ a range of models, from uncorrelated white noise to long-range correlated noise. For white noise, we find the false alarm probability to be Pr≪10−5Pr≪10−5 (Press et al., 1996). If the residuals have power-law behaviors with exponent in the range 2–4 and long-range correlations characterized by a Hurst index H≤0.7H≤0.7, we have Pr<10−2Pr<10−2 (Zhou and Sornette, 2002c). The inset of Fig. 3(b) plots ωfitωfit with respect to ωLombωLomb. One can see that most pairs of (ωLomb,ωfit)(ωLomb,ωfit) are overlapping on the line y=xy=x, which indicates the consistency between ωfitωfit and ωLombωLomb. The other pairs are located on the line y=2xy=2x. We can interpret these as a fundamental log-periodic component at ωω and its harmonic component at 2ω2ω. The existence of harmonics of log-periodic components can be expected generically in log-periodic signals (Sornette, 1998, Gluzman and Sornette, 2002, Zhou and Sornette, 2002a and Zhou and Sornette, 2009) and has been documented in earlier studies both of financial time series and for other systems (Zhou et al., 2007). When the harmonics are well defined with close-to-integer ratios to a common fundamental frequency as is the case here, this is in general a diagnostic of a very significant log-periodic component.