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A multifractal detrended fluctuation analysis (MDFA) of the Chinese growth enterprise market (GEM)
Physica A 391 (2012) 3496–3502
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
A multifractal detrended fluctuation analysis (MDFA) of the Chinese growth enterprise market (GEM) Hui Wang a,b , Luojie Xiang c , R.B. Pandey d,∗ a
School of Management and Economics, UESTC, China
b
School of Physics and Electronics, UESTC, 610054, China
c
School of Computer Science, UESTC, 611054, China
d
Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, MS 39406-0001, USA
article
info
Article history: Received 2 November 2011 Received in revised form 29 January 2012 Available online 15 February 2012 Keywords: Multifractal detrended fluctuation approach Detrended cross-correlation analysis Generalized Hurst index Singularity exponent Singularity spectrum
abstract A multifractal, detrended fluctuation approach is used to analyze the growth enterprise market (GEM) in China involving a range of correlations in fluctuations of share prices (fat tail), persistent and anti-persistent states. Our analysis exhibits company-specific multifractal characteristics, which vary among the companies listed in the same industry, e.g., the power-law cross-correlations between computer and electronics sectors. These results may help reduce the risk in complex financial markets. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Modeling and forecasting has increasingly become very important in analysis of trends in stock markets [1–3], particularly in high frequency trades. It is difficult to predict the price return, i.e. profit or loss, due to many unknown variables including social and political unrest, catastrophic events, etc. Therefore, identifying the trends using empirical modeling based on historical data remains one of the main choices. Among many measures, the Hurst index, H(t) [3], is one of the frequently used quantities in data analyses. It is defined from the power-law exponent in the variation of the standard deviations (of the stock price) with the period (t) of the time series. Apart from analyzing the general trends in large mature markets (primarily due to availability of long time data), understanding the diverse markets is becoming increasingly important to assess multiscale correlations, local and global. For example, Carbone et al. [3] have evaluated the Hurst exponent for the return of high frequency series of the German market, while Zhou and Sornette have studied financial bubbles in the South African stock market [4]. There are numerous examples (too many to cite) on a variety of markets that involve power-law scaling, such as the Warsaw stock exchange [5], the Shanghai stock exchange [6], the Chinese stock market [7], and the Nordic spot electricity market [8]. It is important to analyze multi-scale patterns due to diverse variations in a highly connected global market. Contrary to simple universal scaling processes characterized by one fractal exponent, multifractal methods [9–21] may capture such characteristics that vary with time scale. Therefore, it may be more suitable to study irregularities in the stock market patterns. Kumar and Deo have analyzed the multifractal properties of the Indian financial market [22]. Norouzzadeh and Jafari have reported multifractal measures to the Tehran price index using the Hurst exponent [23].
∗
Corresponding author. E-mail addresses: [email protected] (H. Wang), [email protected] (L. Xiang), [email protected], [email protected] (R.B. Pandey).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.01.053
H. Wang et al. / Physica A 391 (2012) 3496–3502
3497
The multifractal detrended fluctuation analysis is believed to handle unstable financial time series, depict the distribution characteristics more clearly, and detect multifractal scaling. Onali and Goddard have analyzed the unifractality and multifractality in the Italian stock market [16]. Wang et al. [15] have examined the efficiency for the Shenzhen stock market based on multifractal detrended fluctuation analysis. Using an empirical analysis, Andreadis has shown the multifractal character of the Dow–Jones Industrial Average [21]. Detrended fluctuation analysis is recently generalized by Podobnik and Stanley [24] to investigate the cross-correlation between different quantities with simultaneous time series [25,26], e.g., volume change and price change in stock trades in the Standard and Poor 500 index. Detrended cross-correlation analysis (DCCA) is further extended for non-stationary time series with periodic trends [27] as well as cross-correlation with local and global detrending [28]. We have examined the multifractal detrended fluctuation and detrended cross-correlations of the Chinese GEM which exhibits multifractal characteristics that may be valuable in identifying the relative investment risk. In addition to application of cross-correlation between two time series (for stock price fluctuations considered here) there are numerous studies on the cross-correlations between many time series involving random matrix theory (RMT) [29,30]. Based on the random correlated matrix and the time series of different stocks of the finance market including the Standard and Poor 500, Laloux et al. [29] raised doubt on the blind use of empirical correlation matrices for risk management. Plerou et al. [30] identified the universal predictions of RMT of the cross-correlation matrix and the deviations of the empirical data on the stocks with cross correlations. A random matrix with a varying number of lags proposed by Potters et al. [31] has been implemented by Podobnik et al. in a range of collective phenomena, e.g. finance, physiology, and genomics; they find that the largest 500 singular values of the NYSE composite stocks follow a Zipf distribution. Wang et al. [32] have recently proposed a modified time lag random matrix theory and analyzed long-range cross correlations in multiple time series of price fluctuations between 48 world indices (one from each different countries). They find that ‘when a market shock is transmitted around the world, the risk decays very slowly’ due to strong cross-correlations between volatilities. These findings are in contrast to a common belief that the portfolio diversification is a necessary action in reducing the risk. Our detrended cross-correlation analysis in two time series (with empirical data) is consistent with these findings. 2. Method The multifractal detrended fluctuation analysis (MDFA) has been extensively used for many years (e.g. Ref. [11]). Therefore, we would like to provide a brief summary here. Use of the logarithmic unit for the price pt of stock share or index fund at time t is standard in such analysis. The price change from the previous close at a time step u is given by x(u) = ln(pt ) − ln(pt −1 ) = ln(pt /pt −1 )
(u = 1, 2, 3, . . . , N ).
(1)
The running deviation in price change from a mean x¯ over an interval u is y(u) =
u (x(u) − x¯ ) (u = 1, 2, 3, . . . , N ).
(2)
i
The length N of the time series is partitioned into n segments, each of length s, N = ns. A least squares method can be used to identify trends in running deviation over each segment k by a polynomial g (u). The average fluctuation Fk (s) in each subregion k is ks
1
[Fk (s)]2 =
[yk (u) − g (u)]2 .
s u=(k−1)s+1
(3)
The average moment of the fluctuation of order q over n segments of the time series is
F q ( s) =
n 1
n k=1
as q → 0
[Fk (s)]
1q q
ln F0 (s) =
(4) n 1
n k=1
Fk (s) ln Fk (s).
(5)
The power-law dependence of the q-th order moment of the fluctuation Fq (s) in interval s of the time series provides an estimate of the Hurst exponent h(q), i.e., Fq (s) ∝ sh(q) .
(6)
Thus, the Hurst exponent h(q) of the q-th order fluctuation moment can be evaluated from the slope of the ln Fq (s) versus ln s plot. The rate of change of the Hurst index with the order of the moment q defines the singularity exponent α , (using the Legendre transformation), i.e.,
α=
dh dq
(7)
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H. Wang et al. / Physica A 391 (2012) 3496–3502
Fig. 1. The return (x(t )) time series of GEM index and that of the computer and electronics sectors.
and the singularity spectrum f (α) f (α) = qα − α.
(8)
The width of fractal spectrum ∆α = αmax − αmin and the difference between maximum and minimum values of the singularity ∆f = f (αmin ) − f (αmax ) exponents, provide an estimate of the spread in changes in fractal patterns. The corresponding difference in the fractal spectrum provides an insight into the change in frequency. The detrended covariance fluctuation Fk (s) in two time series of y(u) and z (u) is calculated from the detrended covariance of each,
[Fk (s)] =
1
ks
s u=(k−1)s+1
[yk (u) − y˜ k (u)][zk (u) − z˜k (u)],
(9)
where y˜ k (u), z˜k (u) are least squares polynomials fit. The q-th order detrended covariance is,
Fyz (q, s) =
n 1
n k=1
Fyz (0, s) = exp
1/q q/ 2
Fk (s)
n 1
2n k=1
,
q ̸= 0
(10)
ln Fk (s) .
(11)
The Hurst index hyz for the detrended cross-correlation is then estimated from the power-law scaling relation, Fyz (q, s) ∝ shyz (q) .
(12)
3. Data analysis The data for the GEM index, computer sector and electronics sector, are selected for one year, i.e., from September 2010 through August 2011. The return in share price in time series (which is the change in price in logarithmic units), x(t ) of the GEM index, computer sector and electronics sector, is presented in Fig. 1. Data for some of the listed companies in electronics and computer sectors in GEM market are selected from 2010 to 2011 (see Fig. 2) Obviously, it is difficult to identify any trend due to noise in data in both Figs. 1 and 2. A multifractal analysis is therefore needed to see if there is any trend. We have analyzed fluctuation moments Fq (s) for a range of order q and evaluated the Hurst index h(q) and the singularity spectrum f (α). Fig. 3 shows the variation of the Hurst index h(q) with q for the GEM index and the computer and electronics sectors. Corresponding values of the singularity spectrum is presented in Fig. 4. Different values of the Hurst index h(q) for different order q clearly exhibits multifractality. Differences in the multifractal patterns among the GEM index and the computer and electronics sectors are more pronounced at higher order moments (q ≥ 6) (see Fig. 3). The multifractal patterns of the computer and electronics sectors are somewhat similar and differ (with lower h(q)) slightly from that of the GEM index. Further, the magnitude of h(q) is close to 0.5, which shows short-range correlations in the fluctuations in the share price. Fig. 5 shows the variation of the Hurst exponent with the order (q) of the fluctuation moment for three representative companies listed in the computer sector. The corresponding singularity spectrum of these companies is presented in Fig. 6.
H. Wang et al. / Physica A 391 (2012) 3496–3502
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Fig. 2. The return (x(t )) time series of selected IT companies.
Fig. 3. Variation of the Hurst index h(q) with q for the GEM index and the computer and electronics sectors.
Fig. 4. Variation of the singularity spectrum f (α) with the singularity exponent α for the GEM index and the computer and electronics sectors.
The same value of the Hurst exponent h(q) at q = 2 shows similarity in the overall patterns in fluctuations in their share prices. The differences in h(q) of these companies increases with higher order (Fig. 5). For example, the correlation among the higher order fluctuations in share price of the Leshi Internet Information and Technology Corporation remains relatively long-range in time (with h(q) > 0.6) while that of the Beijing Ultrapower Software Company becomes short-range (with h(q) < 0.5) at higher q (Fig. 5). The spread of singularity spectrum (Fig. 6) does not seem to follow the trend in Hurst’s measures of the multifractality. For example, the long-range and short correlations lead to a relatively narrow width of the spectrum (lower risk) for the Leshi Internet Information and Technology Corporation and East Money Information Company.
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Fig. 5. Variation of the Hurst index h(q) with q for selected corporations in the computer sector.
Fig. 6. Variation of the singularity spectrum f (α) with the singularity exponent α for selected corporations in the computer sector.
Fig. 7. Variation of the Hurst index h(q) with q for selected corporations in the electronics sector.
Variations of the Hurst index h(q) with the order of the fluctuation moment q and that of the singularity spectrum with α are presented in Figs. 7 and 8, respectively. Dependence of h(q) on q of the Chengdu Galaxy Magnet Company is similar to that of the Zhuhai Orbital Control Engineering at q ≤ 0. The differences in the magnitude of h(q) of the two companies increases at higher order moments, i.e., at q ≥ 4 where the range of correlation in fluctuation in share price of the Chengdu Galaxy Magnet seems to remain long range while that of the Zhuhai Orbital Control Engineering seems to become short range. The corresponding range of correlation in the share price fluctuation of the EVE Energy Company remains short range at higher order (q) (see Fig. 7). The range of correlation in fluctuations of the lower order moments (0 ≤ q ≤ 3) of all of the companies remains larger than that of the higher order fluctuations.
H. Wang et al. / Physica A 391 (2012) 3496–3502
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Fig. 8. Variation of the singularity spectrum f (α) with the singularity exponent α for selected corporations in the electronics sector.
Fig. 9. Variation of the scaling exponent hxy (q) for the cross-correlated share price between computer and electronics sectors with the order q of the fluctuation moment along with corresponding Hurst exponent hx (q) and hy (q) in these sectors.
The singularity spectrum of share price fluctuation of the Chengdu Galaxy Magnet is similar to that of the Zhuhai Orbital Control Engineering with their spread much narrower than that of the EVE Energy Company. Consequently, the investment in EVE Energy Company appears to be comparatively more risky. In order to study the cross-correlation in the share price between computer and electronics sectors, we have examined the Hurst exponent (hxy (q)) as a function of the fluctuation moment of order q using Eq. (12) with the detrended crosscorrelation analysis (DCCA) described above [24–28]. Fig. 9 shows the variation of hxy (q) with q. Corresponding Hurst exponents hx (q) and hy (q) for the share price in computer and electronics sectors respectively are also included for the comparison. At low moment (q = 0, 1, 2), the magnitude of hxy (q) is somewhat higher than the Hurst exponent in individual sectors but still less than 0.5. The value of the cross-correlated Hurst exponent hxy (q) decays slightly faster than hx (q) in the computer sector and slower than hy (q) in the electronics sector which show some correlated bias towards the computer sector. 4. Summary and conclusions Patterns of the fluctuation moments of the share prices of an index (GEM, computer and electronics sectors) and that of the individual companies for about a year exhibit multifractality. Our analysis consists of evaluating the Hurst exponent h(q) and its scaling with the order of the share price fluctuation moment q and the associated singularity spectrum f (α). We find that the trends in price return of each index and that of the individual company are described by a specific Hurst index h(q) which shows variations, though generally small but detectable, with the order q of the fluctuation moments at higher q ≥ 0. The spread in the singularity spectrum could be a measure of relative risk. Among the share price fluctuations of the indices, we find that the electronics sector is relatively more risky than the computer sector while the GEM appears to have the lowest risk. In the computer sector, trade and investment in Leshi Internet Information and Technology Corporation appears to have lower risk than that in Beijing Ultrapower Software Company and East Money Information Company. In the electronics sector, the risk of investment in Chengdu Galaxy Magnet and Zhuhai Orbital Control Engineering are similar and
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H. Wang et al. / Physica A 391 (2012) 3496–3502
appears to be lower than that in EVE Energy Company. Thus, the multifractal analysis of the trends in share prices in time series could be a useful tool to identify the relative risk of investments in both indices as well as in individual companies. The detrended cross-correlation analysis is consistent with the multi-fractal characteristics and shows strong correlation in share price between computer and electronics sectors. Finally, the results of our cross-correlation analysis are consistent with recent finding by Wang et al. [32] based on the analysis of 48 world stocks using a time lag random matrix theory. Acknowledgments We thank Diana Lovejoy for careful reading and corrections. We thank referee for pointing out the important development on cross-correlations in multiple time series based on random matrix theory. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
R.N. Mantegna, H.E. Stanley, Nature 376 (1995) 46. R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics, Correlations and Complexity in Finance, Cambridge University Press, 2000. A. Carbone, G. Castelli, H.E. Stanley, Physica A 344 (2004) 267. W-X. Zhou, D. Sornette, Physica A 388 (2009) 869. D. Grech, G. Pamula, Physica A 387 (2008) 4299. Y. Ying, Z. Xin-tian, J. Xiu, Physica A 388 (2009) 2189. J. Jiang, W. Li, X. Cai, Q.A. Wang, Physica A 388 (2009) 1893. H. Erzgraber, et al., Physica A 387 (2008) 6567. F. Schmitt, D. Schertzer, S. Lovejoy, Appl. Stoch. Models Data Anal. 15 (1999) 29. F. Schmitt, D. Schertzer, S. Lovejoy, Int. J. Theor. Appl. Finance 3 (2000) 361. J.W. Kantehardt, S.A. Zschiegner, E. Koscelny-Bunde, S. Havlin, A. Bunde, H.E. Stanley, Physica A 316 (2002) 87. G. Mu, W. Chen, J. Kertesz, W-X. Zhou, arXiv:0904.1042v1 [q-fin.ST], 7 April, 2009. L. Borland, J-P. Bouchaud, J.F. Muzy, G. Zumbach, arXiv:cond-mat/0501292v1 [cond-mat.other], 12 Janm, 2005. Z-Q. Jiang, W. Chen, W-X. Zhou, Physica A 387 (2008) 5818. G. Lim, S. Kim, H. Lee, K. Kim, D.I. Lee, Physica A 386 (2007) 259. Y. Wang, L. Liu, R. Gu, Int. Rev. Financ. Anal. 18 (5) (2009) 271. E. Onali, J. Goddard, Int. Rev. Financ. Anal. 18 (2009) 154. M. Bacilar, Emerg. Mark. Financ. Tr. 39 (2003) 5. X.T. Zhuang, Y. Yuan, Physica A 387 (2008) 511. S. Bianchi, A. Pianese, Quant. Finance 7 (2007) 301. I. Andredis, A. Serltis, Chaos Solitons Fractals 13 (2002) 1309. S. Kumar, N. Deo, Physica A 388 (2009) 1593. P. Norouzzadeh, G.R. Jafari, Physica A 356 (2005) 609. B. Podobnik, H.E. Stanley, Phys. Rev. Lett. 100 (2008) 084102. B. Podobnik, D. Horvatic, A.M. Petersen, H.E. Stanley, PNAS 106 (2009) 22079. W-X. Zhou, Phys. Rev. E 77 (2008) 066211. D. Horvatic, H.E. Stanley, B. Podobnik, Eur. Phys. Lett. 94 (2011) 18007. B. Podobnik, I. Grosse, D. Horvatic, Eur. Phys. J. B 71 (2009) 243. L. Laloux, P. Cizeau, J.P. Bouchaud, M. Potters, Phys. Rev. Lett. 83 (1999) 1467. V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, H.E. Stanley, Phys. Rev. Lett. 83 (1999) 1471. B. Podobnik, D. Wang, D. Horvatic, I. Grosse, H.E. Stanley, Eur. Phys. Lett. 90 (2010) 68001. D. Wang, B. Podobnik, D. Horvatic, H.E. Stanley, Phys. Rev. E 83 (2011) 046121.
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
A multifractal detrended fluctuation analysis (MDFA) of the Chinese growth enterprise market (GEM) Hui Wang a,b , Luojie Xiang c , R.B. Pandey d,∗ a
School of Management and Economics, UESTC, China
b
School of Physics and Electronics, UESTC, 610054, China
c
School of Computer Science, UESTC, 611054, China
d
Department of Physics and Astronomy, University of Southern Mississippi, Hattiesburg, MS 39406-0001, USA
article
info
Article history: Received 2 November 2011 Received in revised form 29 January 2012 Available online 15 February 2012 Keywords: Multifractal detrended fluctuation approach Detrended cross-correlation analysis Generalized Hurst index Singularity exponent Singularity spectrum
abstract A multifractal, detrended fluctuation approach is used to analyze the growth enterprise market (GEM) in China involving a range of correlations in fluctuations of share prices (fat tail), persistent and anti-persistent states. Our analysis exhibits company-specific multifractal characteristics, which vary among the companies listed in the same industry, e.g., the power-law cross-correlations between computer and electronics sectors. These results may help reduce the risk in complex financial markets. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Modeling and forecasting has increasingly become very important in analysis of trends in stock markets [1–3], particularly in high frequency trades. It is difficult to predict the price return, i.e. profit or loss, due to many unknown variables including social and political unrest, catastrophic events, etc. Therefore, identifying the trends using empirical modeling based on historical data remains one of the main choices. Among many measures, the Hurst index, H(t) [3], is one of the frequently used quantities in data analyses. It is defined from the power-law exponent in the variation of the standard deviations (of the stock price) with the period (t) of the time series. Apart from analyzing the general trends in large mature markets (primarily due to availability of long time data), understanding the diverse markets is becoming increasingly important to assess multiscale correlations, local and global. For example, Carbone et al. [3] have evaluated the Hurst exponent for the return of high frequency series of the German market, while Zhou and Sornette have studied financial bubbles in the South African stock market [4]. There are numerous examples (too many to cite) on a variety of markets that involve power-law scaling, such as the Warsaw stock exchange [5], the Shanghai stock exchange [6], the Chinese stock market [7], and the Nordic spot electricity market [8]. It is important to analyze multi-scale patterns due to diverse variations in a highly connected global market. Contrary to simple universal scaling processes characterized by one fractal exponent, multifractal methods [9–21] may capture such characteristics that vary with time scale. Therefore, it may be more suitable to study irregularities in the stock market patterns. Kumar and Deo have analyzed the multifractal properties of the Indian financial market [22]. Norouzzadeh and Jafari have reported multifractal measures to the Tehran price index using the Hurst exponent [23].
∗
Corresponding author. E-mail addresses: [email protected] (H. Wang), [email protected] (L. Xiang), [email protected], [email protected] (R.B. Pandey).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.01.053
H. Wang et al. / Physica A 391 (2012) 3496–3502
3497
The multifractal detrended fluctuation analysis is believed to handle unstable financial time series, depict the distribution characteristics more clearly, and detect multifractal scaling. Onali and Goddard have analyzed the unifractality and multifractality in the Italian stock market [16]. Wang et al. [15] have examined the efficiency for the Shenzhen stock market based on multifractal detrended fluctuation analysis. Using an empirical analysis, Andreadis has shown the multifractal character of the Dow–Jones Industrial Average [21]. Detrended fluctuation analysis is recently generalized by Podobnik and Stanley [24] to investigate the cross-correlation between different quantities with simultaneous time series [25,26], e.g., volume change and price change in stock trades in the Standard and Poor 500 index. Detrended cross-correlation analysis (DCCA) is further extended for non-stationary time series with periodic trends [27] as well as cross-correlation with local and global detrending [28]. We have examined the multifractal detrended fluctuation and detrended cross-correlations of the Chinese GEM which exhibits multifractal characteristics that may be valuable in identifying the relative investment risk. In addition to application of cross-correlation between two time series (for stock price fluctuations considered here) there are numerous studies on the cross-correlations between many time series involving random matrix theory (RMT) [29,30]. Based on the random correlated matrix and the time series of different stocks of the finance market including the Standard and Poor 500, Laloux et al. [29] raised doubt on the blind use of empirical correlation matrices for risk management. Plerou et al. [30] identified the universal predictions of RMT of the cross-correlation matrix and the deviations of the empirical data on the stocks with cross correlations. A random matrix with a varying number of lags proposed by Potters et al. [31] has been implemented by Podobnik et al. in a range of collective phenomena, e.g. finance, physiology, and genomics; they find that the largest 500 singular values of the NYSE composite stocks follow a Zipf distribution. Wang et al. [32] have recently proposed a modified time lag random matrix theory and analyzed long-range cross correlations in multiple time series of price fluctuations between 48 world indices (one from each different countries). They find that ‘when a market shock is transmitted around the world, the risk decays very slowly’ due to strong cross-correlations between volatilities. These findings are in contrast to a common belief that the portfolio diversification is a necessary action in reducing the risk. Our detrended cross-correlation analysis in two time series (with empirical data) is consistent with these findings. 2. Method The multifractal detrended fluctuation analysis (MDFA) has been extensively used for many years (e.g. Ref. [11]). Therefore, we would like to provide a brief summary here. Use of the logarithmic unit for the price pt of stock share or index fund at time t is standard in such analysis. The price change from the previous close at a time step u is given by x(u) = ln(pt ) − ln(pt −1 ) = ln(pt /pt −1 )
(u = 1, 2, 3, . . . , N ).
(1)
The running deviation in price change from a mean x¯ over an interval u is y(u) =
u (x(u) − x¯ ) (u = 1, 2, 3, . . . , N ).
(2)
i
The length N of the time series is partitioned into n segments, each of length s, N = ns. A least squares method can be used to identify trends in running deviation over each segment k by a polynomial g (u). The average fluctuation Fk (s) in each subregion k is ks
1
[Fk (s)]2 =
[yk (u) − g (u)]2 .
s u=(k−1)s+1
(3)
The average moment of the fluctuation of order q over n segments of the time series is
F q ( s) =
n 1
n k=1
as q → 0
[Fk (s)]
1q q
ln F0 (s) =
(4) n 1
n k=1
Fk (s) ln Fk (s).
(5)
The power-law dependence of the q-th order moment of the fluctuation Fq (s) in interval s of the time series provides an estimate of the Hurst exponent h(q), i.e., Fq (s) ∝ sh(q) .
(6)
Thus, the Hurst exponent h(q) of the q-th order fluctuation moment can be evaluated from the slope of the ln Fq (s) versus ln s plot. The rate of change of the Hurst index with the order of the moment q defines the singularity exponent α , (using the Legendre transformation), i.e.,
α=
dh dq
(7)
3498
H. Wang et al. / Physica A 391 (2012) 3496–3502
Fig. 1. The return (x(t )) time series of GEM index and that of the computer and electronics sectors.
and the singularity spectrum f (α) f (α) = qα − α.
(8)
The width of fractal spectrum ∆α = αmax − αmin and the difference between maximum and minimum values of the singularity ∆f = f (αmin ) − f (αmax ) exponents, provide an estimate of the spread in changes in fractal patterns. The corresponding difference in the fractal spectrum provides an insight into the change in frequency. The detrended covariance fluctuation Fk (s) in two time series of y(u) and z (u) is calculated from the detrended covariance of each,
[Fk (s)] =
1
ks
s u=(k−1)s+1
[yk (u) − y˜ k (u)][zk (u) − z˜k (u)],
(9)
where y˜ k (u), z˜k (u) are least squares polynomials fit. The q-th order detrended covariance is,
Fyz (q, s) =
n 1
n k=1
Fyz (0, s) = exp
1/q q/ 2
Fk (s)
n 1
2n k=1
,
q ̸= 0
(10)
ln Fk (s) .
(11)
The Hurst index hyz for the detrended cross-correlation is then estimated from the power-law scaling relation, Fyz (q, s) ∝ shyz (q) .
(12)
3. Data analysis The data for the GEM index, computer sector and electronics sector, are selected for one year, i.e., from September 2010 through August 2011. The return in share price in time series (which is the change in price in logarithmic units), x(t ) of the GEM index, computer sector and electronics sector, is presented in Fig. 1. Data for some of the listed companies in electronics and computer sectors in GEM market are selected from 2010 to 2011 (see Fig. 2) Obviously, it is difficult to identify any trend due to noise in data in both Figs. 1 and 2. A multifractal analysis is therefore needed to see if there is any trend. We have analyzed fluctuation moments Fq (s) for a range of order q and evaluated the Hurst index h(q) and the singularity spectrum f (α). Fig. 3 shows the variation of the Hurst index h(q) with q for the GEM index and the computer and electronics sectors. Corresponding values of the singularity spectrum is presented in Fig. 4. Different values of the Hurst index h(q) for different order q clearly exhibits multifractality. Differences in the multifractal patterns among the GEM index and the computer and electronics sectors are more pronounced at higher order moments (q ≥ 6) (see Fig. 3). The multifractal patterns of the computer and electronics sectors are somewhat similar and differ (with lower h(q)) slightly from that of the GEM index. Further, the magnitude of h(q) is close to 0.5, which shows short-range correlations in the fluctuations in the share price. Fig. 5 shows the variation of the Hurst exponent with the order (q) of the fluctuation moment for three representative companies listed in the computer sector. The corresponding singularity spectrum of these companies is presented in Fig. 6.
H. Wang et al. / Physica A 391 (2012) 3496–3502
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Fig. 2. The return (x(t )) time series of selected IT companies.
Fig. 3. Variation of the Hurst index h(q) with q for the GEM index and the computer and electronics sectors.
Fig. 4. Variation of the singularity spectrum f (α) with the singularity exponent α for the GEM index and the computer and electronics sectors.
The same value of the Hurst exponent h(q) at q = 2 shows similarity in the overall patterns in fluctuations in their share prices. The differences in h(q) of these companies increases with higher order (Fig. 5). For example, the correlation among the higher order fluctuations in share price of the Leshi Internet Information and Technology Corporation remains relatively long-range in time (with h(q) > 0.6) while that of the Beijing Ultrapower Software Company becomes short-range (with h(q) < 0.5) at higher q (Fig. 5). The spread of singularity spectrum (Fig. 6) does not seem to follow the trend in Hurst’s measures of the multifractality. For example, the long-range and short correlations lead to a relatively narrow width of the spectrum (lower risk) for the Leshi Internet Information and Technology Corporation and East Money Information Company.
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Fig. 5. Variation of the Hurst index h(q) with q for selected corporations in the computer sector.
Fig. 6. Variation of the singularity spectrum f (α) with the singularity exponent α for selected corporations in the computer sector.
Fig. 7. Variation of the Hurst index h(q) with q for selected corporations in the electronics sector.
Variations of the Hurst index h(q) with the order of the fluctuation moment q and that of the singularity spectrum with α are presented in Figs. 7 and 8, respectively. Dependence of h(q) on q of the Chengdu Galaxy Magnet Company is similar to that of the Zhuhai Orbital Control Engineering at q ≤ 0. The differences in the magnitude of h(q) of the two companies increases at higher order moments, i.e., at q ≥ 4 where the range of correlation in fluctuation in share price of the Chengdu Galaxy Magnet seems to remain long range while that of the Zhuhai Orbital Control Engineering seems to become short range. The corresponding range of correlation in the share price fluctuation of the EVE Energy Company remains short range at higher order (q) (see Fig. 7). The range of correlation in fluctuations of the lower order moments (0 ≤ q ≤ 3) of all of the companies remains larger than that of the higher order fluctuations.
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Fig. 8. Variation of the singularity spectrum f (α) with the singularity exponent α for selected corporations in the electronics sector.
Fig. 9. Variation of the scaling exponent hxy (q) for the cross-correlated share price between computer and electronics sectors with the order q of the fluctuation moment along with corresponding Hurst exponent hx (q) and hy (q) in these sectors.
The singularity spectrum of share price fluctuation of the Chengdu Galaxy Magnet is similar to that of the Zhuhai Orbital Control Engineering with their spread much narrower than that of the EVE Energy Company. Consequently, the investment in EVE Energy Company appears to be comparatively more risky. In order to study the cross-correlation in the share price between computer and electronics sectors, we have examined the Hurst exponent (hxy (q)) as a function of the fluctuation moment of order q using Eq. (12) with the detrended crosscorrelation analysis (DCCA) described above [24–28]. Fig. 9 shows the variation of hxy (q) with q. Corresponding Hurst exponents hx (q) and hy (q) for the share price in computer and electronics sectors respectively are also included for the comparison. At low moment (q = 0, 1, 2), the magnitude of hxy (q) is somewhat higher than the Hurst exponent in individual sectors but still less than 0.5. The value of the cross-correlated Hurst exponent hxy (q) decays slightly faster than hx (q) in the computer sector and slower than hy (q) in the electronics sector which show some correlated bias towards the computer sector. 4. Summary and conclusions Patterns of the fluctuation moments of the share prices of an index (GEM, computer and electronics sectors) and that of the individual companies for about a year exhibit multifractality. Our analysis consists of evaluating the Hurst exponent h(q) and its scaling with the order of the share price fluctuation moment q and the associated singularity spectrum f (α). We find that the trends in price return of each index and that of the individual company are described by a specific Hurst index h(q) which shows variations, though generally small but detectable, with the order q of the fluctuation moments at higher q ≥ 0. The spread in the singularity spectrum could be a measure of relative risk. Among the share price fluctuations of the indices, we find that the electronics sector is relatively more risky than the computer sector while the GEM appears to have the lowest risk. In the computer sector, trade and investment in Leshi Internet Information and Technology Corporation appears to have lower risk than that in Beijing Ultrapower Software Company and East Money Information Company. In the electronics sector, the risk of investment in Chengdu Galaxy Magnet and Zhuhai Orbital Control Engineering are similar and
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appears to be lower than that in EVE Energy Company. Thus, the multifractal analysis of the trends in share prices in time series could be a useful tool to identify the relative risk of investments in both indices as well as in individual companies. The detrended cross-correlation analysis is consistent with the multi-fractal characteristics and shows strong correlation in share price between computer and electronics sectors. Finally, the results of our cross-correlation analysis are consistent with recent finding by Wang et al. [32] based on the analysis of 48 world stocks using a time lag random matrix theory. Acknowledgments We thank Diana Lovejoy for careful reading and corrections. We thank referee for pointing out the important development on cross-correlations in multiple time series based on random matrix theory. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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