Scaling analysis of stock markets Luping Bu and Pengjian Shang Citation: Chaos 24, 023107 (2014); doi: 10.1063/1.4871479 View online: http://dx.doi.org/10.1063/1.4871479 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Modified multidimensional scaling approach to analyze financial markets Chaos 24, 022102 (2014); 10.1063/1.4873523 Unraveling chaotic attractors by complex networks and measurements of stock market complexity Chaos 24, 013134 (2014); 10.1063/1.4868258 Multidimensional stock network analysis: An Escoufier's RV coefficient approach AIP Conf. Proc. 1557, 550 (2013); 10.1063/1.4823975 Hidden temporal order unveiled in stock market volatility variance AIP Advances 1, 022127 (2011); 10.1063/1.3598412 The values distribution in a competing shares financial market model AIP Conf. Proc. 519, 685 (2000); 10.1063/1.1291643
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CHAOS 24, 023107 (2014)
Scaling analysis of stock markets Luping Bu and Pengjian Shang School of Science, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
(Received 13 May 2013; accepted 4 April 2014; published online 16 April 2014) In this paper, we apply the detrended fluctuation analysis (DFA), local scaling detrended fluctuation analysis (LSDFA), and detrended cross-correlation analysis (DCCA) to investigate correlations of several stock markets. DFA method is for the detection of long-range correlations used in time series. LSDFA method is to show more local properties by using local scale exponents. DCCA method is a developed method to quantify the cross-correlation of two nonstationary time series. We report the results of auto-correlation and cross-correlation behaviors in three western countries and three Chinese stock markets in periods 2004–2006 (before the global financial crisis), 2007–2009 (during the global financial crisis), and 2010–2012 (after the global financial crisis) by using DFA, LSDFA, and DCCA method. The findings are that correlations of stocks are influenced by the economic systems of different countries and the financial crisis. The results indicate that there are stronger auto-correlations in Chinese stocks than western stocks in any period and stronger auto-correlations after the global financial crisis for every stock except Shen Cheng; The LSDFA shows more comprehensive and detailed features than traditional DFA method and the integration of China and the world in economy after the global financial crisis; When it turns to cross-correlations, it shows different properties for six stock markets, while for three Chinese stocks, it reaches the weakest cross-correlations during the global financial crisis. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4871479] V Nowadays, auto-correlations and cross-correlations have become increasing topics in many topics concerning time series analysis. And the interaction is highly nonlinear, unstable, and long-ranged for stock markets. Hence, the study of these properties is crucial for understanding the structure of stock markets. The main aim of our work is to describe the correlations in six stocks. Actually, first, we use two methods—detrended fluctuation analysis (DFA) and local scaling detrended fluctuation analysis (LSDFA)—to investigate the auto-correlations of six stocks in three periods: 2004–2006 (before the global financial crisis), 2007–2009 (during the global financial crisis), and 2010–2012 (after the global financial crisis). Then, we apply detrended cross-correlation analysis (DCCA) to investigate the cross-correlations between some stocks in three periods. The findings are that correlations of stocks are influenced by the economic systems of different countries and the financial crisis. We find that there are stronger auto-correlations in Chinese stocks than western stocks in any period and stronger auto-correlations after the global financial crisis for every stock except Shen Cheng; The LSDFA shows more comprehensive and detailed features than traditional DFA method and it shows the integration of China and the world in economy after the global financial crisis; When it turns to cross-correlations, it shows different properties for six stock markets, while for three Chinese stocks, it reaches the weakest crosscorrelations during the global financial crisis.
I. INTRODUCTION
Recently, auto-correlations and cross-correlations have become increasing topics in many fields concerning time 1054-1500/2014/24(2)/023107/7/$30.00
series analysis, ranging from finance1–3 to traffic,4–6 geophysics,7,8 and physiology.9,10 Nowadays, Many reports point out that all stock companies are correlated and interconnected. Stock markets have become an active research aspect because of theirs highly nonlinear, unstable, and long-ranged interaction. Hence, the study of auto-correlations and crosscorrelations of stocks is crucial for understanding the structure and property of stock markets. The DFA method invented by Peng has become a widely used method for the determination and detection of long-range correlations in time series.11–13 The a exponent of DFA is estimated as the overall slope of a variability function FðsÞ plotted versus the beat resolution s in a log–log scale. Although the DFA scaling coefficients have been used in a large number of studies, full validation of this approach has not been performed. In particular, the ranges for identification of a1 and a2 cannot be defined precisely and are usually determined by visual inspection of data. It is also unknown whether and how external factors alter the ranges for identification of the two coefficients. Moreover, describing auto-correlations among financial markets using a1 and a2 may be an inadequate oversimplification of a more complex phenomenon as these correlations might actually consist of a much lager number of scale. If this is the case, a whole spectrum of the local scaling exponent aðsÞ rather than just two coefficients would be required to describe the correlations accurately. Hence, previous studies on multiscale DFA14–18 inspired us to apply LSDFA to investigate the correlations. The proposed method-LSDFA is related to multiscale behaviors of time series, which share some common features with and ideas underlying the multifractal versions of
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C 2014 AIP Publishing LLC V
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DFA.19,20 They all require a multitude of scaling exponents for a full description of the scaling behavior of a time series. They are techniques for determining fractal scaling properties and detecting long-range correlations in noisy, nonstationary time series. However, there are also some differences between them. The multifractal versions of DFA describe the global multiscale correlation of time series under continuous scales and focus on the generalized scaling exponent HðqÞ and multifractal spectrum f ðaÞ. By contrast, LSDFA describes the local multiscale correlation of time series under discrete scales and focus on the spectrum of aðsÞ. In spite of successes noted in the literature, multifractal analysis has strict of noise in the signal. Multifractal analysis also requires basic assumptions such as nonstationarity and an acceptable level of noise in the signal. Multifractal analysis also requires initial assumptions about the presumed time scale of the problem investigated, which may lead to artifacts in some cases. For example, if a crossover falls within the scaling range, the results will be biased. If the scaling range includes only large scales, then information about the fractal properties at small scales may be missed, and these properties might be dramatically different and very interesting. Even if the MF-DFA and MF-DCCA are used separately for small and large scales, we would miss analysis of properties within the crossover range, and these properties might be very different from case to case. Hence, we propose LSDFA methods, which have fewer data prerequisites and initial assumptions, and thus yield a more concise analysis. In previous studies, there are many different methods used to quantify the cross-correlations of stock indices, for example, the applications of random matrix theory (RMT) to investigate the cross-correlations of stock price changes in literature.21,22 DCCA method is the extension of DFA using to quantify logrange cross-correlations between two non-stationary time series, which has been proposed recently in literature.23–25 In this paper, we study the correlation of a single stock market by using DFA and LSDFA. Lastly, we apply the DCCA to analyze different dynamics of cross-correlations between some stocks in three periods. The organization of this paper is as follows. Section II presents methods employed in this study. Section III describes the database used in our work briefly. Section IV is devoted to provide the detailed results of DFA, LSDFA, and DCCA of daily returns of three stocks in three periods. Section V presents the conclusions of this work. II. METHODS A. DFA method
DFA method has been used to investigate the long-range correlations in time series. The DFA procedure consists of five steps.26,27 Let us suppose that fxk g is a series of length N. Step 1: Determine the profile YðiÞ ¼
i X
ðxk hxiÞ;
i ¼ 1; …; N:
(1)
k¼1
Step 2: Cut the profile YðiÞ into Ns ½N=s nonoverlapping segments of equal length s. Since the record length N
need not be a multiple of the considered time scale s, a short part at the end of the profile will remain in most cases. In order not to disregard this part of the record, the same procedure is repeated starting from the other end of the record. Thus, 2Ns segments are obtained altogether. The subtraction of the mean hxi is not compulsory, since it would be eliminated by the later detrending in the third step anyway. Step 3: Calculate the local trend for each segment v by a least-square fit of the data. Then, we define the detrended time series for segment duration s, denoted by Ys ðiÞ, as the difference between the original time series and the fits Ys ðiÞ ¼ YðiÞ qv ðiÞ;
(2)
where qv ðiÞ is the fitting polynomial in vth the segment. Linear, cubic, or higher order polynomials can be used in the fitting procedure (DFA1, DFA2, and higher order DFA). Different fitting order will get different results. In order to get more precise results, we often use larger than 1-order to analyze the time series. Since the detrending of the time series is done by the subtraction of the polynomial fits from the profile, these methods differ in their capability of eliminating trends in the series. In n-th order DFA, trends of order n in the profile and of order n 1 in the original series are eliminated. We calculate, for each of the 2Ns segments, the variance Fs 2 ðvÞ ¼ hYs2 ðiÞi ¼
s 1X Y 2 ½ðv 1Þs þ i; s i¼1 s
v ¼ 1; 2; …; Ns ; (3)
Fs 2 ðvÞ ¼
s 1X Y 2 ½N ðv Ns Þs þ i; s i¼1 s
v ¼ Ns þ 1; Ns þ 2; …; 2Ns ;
(4)
of the detrended time series Ys ðiÞ by averaging over all data points i in the vth segment. Step 4: Average over all segments to obtain the fluctuation function "
#12 2Ns 1 X 2 F ðvÞ : FðsÞ ¼ 2Ns v¼1 s
(5)
It is apparent that FðsÞ will increase with increasing s. Of course, FðsÞ depends on the DFA order n. By construction, FðsÞ is only defined for s n þ 2. Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log-log plots FðsÞ versus s. If the series xk are long-range power-law correlated, FðsÞ increases, for large values of s, as a power-law FðsÞ sa :
(6)
The exponent a is called the scaling exponent or correlation exponent, represents the correlation properties of the signal. The value 0 < a < 0:5 indicates that larger and smaller values of the time series are more likely to alternate, while 0:5 < a < 1 indicates persistent long-range power-law
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correlations such that a large value in one variable is more likely to be followed by a large value in another variable and vice versa. a ¼ 0:5 denotes the absence of correlations. A special case of a ¼ 1 corresponds to 1=f noise. For a 1, correlations exist but cease to be of a power-law form; a ¼ 1:5 indicates Brown noise.7 B. LSDFA method
In this study, we extend DFA in order to estimate a local scaling exponent aðsÞ, and we name this method as LSDFA. The LSDFA procedure also consists of five steps. The first four steps are identical to the conventional DFA procedure. The fifth step is to determine the scaling behavior of the wave function by analyzing a local scaling exponent aðsk Þ defined as follows:16–18 aðsk Þ ¼
log½Fðskþ1 Þ log½Fðsk1 Þ ; log½skþ1 log½sk1
(7)
where sk is a subset of scale s. C. DCCA method
DCCA is a new method to quantify the crosscorrelations of two different non-stationary time series. This method is a generalization of the DFA method in which only one time series is analyzed. It was introduced by Podobnik and Stanley.24 The DCCA procedure also consists of five steps.28 The first three steps are similar to the DFA procedure. Consider two time series fxk g an fxk 0 g, where k ¼ 1; 2; …N and N is the length of the time series. Step 1: Determine the profile i X ðxk hxiÞ; YðiÞ ¼ i X ðx0 k hx0 iÞ;
III. DATA
The analyzed time series consists of six stock indices: three stocks indices of western countries, DAX (Deutscher Aktien Index) of Germany, FTSE100 of UK, and S&P500 of USA together with three Chinese stock indices, Heng Sheng of Hong Kong, Shang Zheng of Shang Hai, and Shen Cheng of Shen Zhen. The data are recorded every day of closing prices from January 2, 2004 to December 31, 2012 for a total of 2053 days. We exclude the asynchronous datum and reconnect the same parts of the original indices to obtain the same length because the different opening day of six stocks. Then, we split the 6 stock indices into three periods, which are parts of from 2004 to 2006 (Data1) and from 2007 to 2009 (Data2) and from 2010 to 2012 (Data3) obtained before, during, and after the global financial crisis, respectively. Denoting the stock market index as fxðtÞg, the logarithmic daily return is defined by gðtÞ ¼ logðxðtÞÞ logðxðt 1ÞÞ:
i ¼ 1; 2; …; N:
(8)
k¼1
Step 2: Cut the profile YðiÞ and Y 0 ðiÞ into 2Ns nonoverlapping segments of equal length s. Step 3: Calculate the local trend for each segment v by a least-square fit of the data. Then, we calculate the difference between the original time series and the fits. Step 4: Calculate the covariance of the residuals in each box s 1X 0 2 ðs; vÞ ¼ ðYk Y~k;v ÞðY 0 k Y~ k;v Þ; (9) fDCCA s k¼1 0 where Y~k;v and Y~ k;v are the fitting polynomials in segment v, respectively. Then average over all segments to obtain the fluctuation function ( )12 2Ns 1 X 2 f ðs; vÞ : (10) fDCCA ðsÞ ¼ 2Ns v¼1 DCCA
Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log-log plots FðsÞ versus s FDCCA ðsÞ sk :
(11)
(12)
The normalized daily return (NDR) is defined by RðtÞ ¼
k¼1
Y 0 ðiÞ ¼
The scaling exponent k represents the degree of the cross-correlation between the two time series xk and xk 0 . The value k ¼ 0:5 denotes the absence of cross-correlations. k > 0:5 indicates persistent long-range cross-correlations where a large value in one variable is likely to be followed by a large value in another variable, while k < 0:5 indicates anti-persistent cross-correlations where a large value in one variable is likely to be followed by a small value in another variable, and vice versa.
gðtÞ hgðtÞi ; r
(13)
where r is the standard deviation of the time series gðtÞ. In this paper, we use the NDRs of every stock indices in order to obtain a more meaningful result. The NDRs of every stock indices are showed in Fig. 1. IV. ANALYSIS AND RESULTS A. DFA of six stocks
The first thing we analyze is the correlation behaviors. This study investigates long-range power-law correlations in western and Chinese stock markets using NDRs by applying DFA. Here, we use log-log plots of FðsÞ versus s by firstorder fitting. It shows different slopes of NDRs of six stocks in every period. The scaling exponents of DFA method are listed in Table I. From Table I, we can see that the scaling exponents reach maximums in period 2010–2012 for six stocks except Shen Cheng indicating the strongest auto-correlations after the global financial crisis. For DAX the larger value appears in period 2007–2009, while for FTSE100, S&P500, Heng Sheng, and Shang Zheng it reaches larger values in period 2004–2006 indicating different properties of these five stocks during the global financial crisis. The scaling exponent reaches 0.4934 for S&P500 especially. For Shen Cheng, the
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FIG. 1. The normalized daily returns of closing prices of DAX, FTSE100, S&P500, Heng Sheng, Shang Zheng, and Shen Cheng in three periods.
scaling exponents reach the maximum in period 2004–2006 and a larger value in period 2010–2012 showing a stronger auto-correlation before the global financial crisis. As a whole, it shows stronger auto-correlations in Chinese stocks than western stocks in any period. However, it is difficult to evaluate details of fractal structures from Table I because small differences (the scaling exponents are approximately between 0.5 and 0.6) are shown between any two stocks, especially between Chinese stocks. In particular, this data representation does not allow us to easily evaluate the possible differences among conditions or among time series. As we will see in the following part, details on the fractal structure are much better quantified by the local scaling exponents aðsÞ. B. LSDFA of six stocks
In order to improve DFA method, we explain our application of LSDFA to investigate the local scaling exponents of NDRs of different stocks in this section. In this paper, we estimate FðsÞ on a subset sk ðk ¼ 1; :::; 21Þ spaced evenly on a log scale between s1 ¼ 4 and s21 ¼ 128 as the algorithm provided in Refs. 16–18. As Figs. 2–4 show, we make some comparative analyses of some stocks. The results are shown as follows. There are significant features of aðsÞ values of six stocks. The traditional DFA method fails to distinguish these characteristic. In Fig. 2, it shows disordered form in 2004–2006 and largest similarities between FTSE100 and S&P500 during TABLE I. Scaling exponents of DFA of six stocks.
a of DFA
DAX
FTSE100
S&P500
Heng Sheng
Shang Zheng
Shen Cheng
a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
0.5367 0.5507 0.5566
0.5494 0.5095 0.5508
0.5171 0.4934 0.5172
0.5722 0.5084 0.5813
0.5745 0.5505 0.5761
0.5800 0.5567 0.5705
and after the global financial crisis. While for DAX, it shows consistency with other two stocks during 2007–2009, which demonstrates the closer correlations during the crisis for capitalist countries. It is in line with literature.29,30 In Fig. 3, there is small significant difference between Shang Zheng and Shen Cheng at every scale in any period. The fluctuation of Heng Sheng is strongest in period 2007–2009. Heng Sheng is more identical to other two stocks in period 2010–2012. The results show that the similarities of mainland China and the relation between mainland China and Hong Kong is more identical. In Fig. 4, it shows significant similarities between S&P500 and Heng Sheng in every period, while there are smaller difference between S&P500 and three Chinese stocks after the crisis, which shows the integration of China and the world in economy after the global financial crisis. C. DCCA method
In this section, we explain our application of DCCA to investigate the cross-correlations between NDRs of six stocks. For simplicity, we select the returns and sðiÞði ¼ 1; :::; 21Þ spaced evenly on a log scale between sð1Þ ¼ 4 and sð21Þ ¼ 128 and log-log plots of FðsÞ versus s by first-order fitting to analyse and investigate. Tables II–IV show the scaling exponents of DCCA between every two stocks. The results are listed as follows. For DAX and S&P500 as well as FTSE100 and S&P500, it shows the stronger and strongest cross-correlations during and after the global financial crisis, respectively. For DAX and FTSE100, it indicates the stronger and strongest crosscorrelations after and during the global financial crisis, separately (Table II). It shows different properties for three Chinese stock markets. Heng Sheng and Shang Zheng shows the stronger and strongest cross-correlations after and before the global financial crisis, separately; Heng Sheng and Shen Cheng reach the stronger and strongest cross-correlations before and after the global financial crisis, respectively;
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FIG. 2. The local scaling exponents of DAX, FTSE100, and S&P500 indices in three periods.
FIG. 3. The local scaling exponents of Heng Sheng, Shang Zheng, and Shen Cheng indices in three periods.
FIG. 4. The local scaling exponents of S&P500, Heng Sheng, Shang Zheng, and Shen Cheng indices in three periods.
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TABLE II. The scaling exponents of DCCA of every two western stocks indices.
a of DCCA
DAX and FTSE100
DAX and S&P500
FTSE100 and S&P500
1.0149 1.0656 1.0495
0.9714 1.0251 1.1807
0.9037 1.0356 1.0859
a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
TABLE III. The scaling exponents of DCCA of every two Chinese stocks indices.
a of DCCA a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
Heng Sheng and Shang Zheng
Heng Sheng and Shen Cheng
Shang Zheng and Shen Cheng
1.0494 0.6452 0.9592
0.9153 0.7407 0.9787
0.8330 0.8850 0.8007
TABLE IV. The scaling exponents of DCCA of S&P500 and every Chinese stock indices.
a of DCCA a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
S&P500 and Heng Sheng
S&P500 and Shang Zheng
S&P500 and Shen Cheng
0.8264 0.9136 0.9492
0.9723 0.9385 0.9114
0.8968 1.0313 0.9537
Shang Zheng and Shen Cheng show the stronger and strongest cross-correlations before and during the global financial crisis, separately (Table III). S&P500 and Heng Sheng show the stronger and strongest cross-correlations during and after the global financial crisis, separately; S&P500 and Shang Zheng reach the stronger and strongest cross-correlations during and before the global financial crisis, respectively; S&P500 and Shen Cheng show the stronger and strongest crosscorrelations after and during the global financial crisis, separately (Table IV). For three western stocks, it shows the larger values than what between other two stocks in any periods; For three Chinese stocks, it reaches the smallest values during the global financial crisis. However, an interesting thing is that there is no significant relation between LSDFA and DCCA method, it is possible because the two methods are different construction and lead to different properties of correlations. The financial crisis effects the correlations of stocks, which causes the complexity of stocks, it is mainly because the depressed situation and decrease of rebound of stock markets and the control ability of government.
V. CONCLUSIONS
In this paper, we investigate the correlation behaviors of stock indices. First, we analyze the traditional DFA to find differences between six stocks in three periods. Furthermore, we apply LSDFA to improve the effectiveness of DFA. Finally, we study scale exponents of DCCA of stock series.
The finding is that correlations of stock dynamics are influenced by the economic systems of different countries and the crises. There are stronger auto-correlations in Chinese stocks than western stocks in any period. The LSDFA shows more comprehensive and detailed features and the integration of China and the world in economy after the global financial crisis, which is not found from traditional DFA method. For three Chinese stocks, it reaches the weakest crosscorrelations during the global financial crisis. There are already many methods to detect the scaling behaviors and predict the stock markets. Preis, Schneider, and Stanley have reported scaling exponents in time domain on various time scales.31,32 It is a good way to distinguish the financial crisis using various scales and has a very good practicability. Recent work demonstrates that there is predictive power for weekly stock market returns in data generated by internet users online.33 Learning more prediction functions from these methods, we may improve the methods used in our paper to predict the stock indices and financial crisis. We can also apply the methods to other regions to distinguish or recognize different types of time series. We believe that such studies are relevant for a better understanding of the stock market mechanisms. ACKNOWLEDGMENTS
Financial support by China National Science (61071142, 61371130), Beijing National Science (4122059), and the National High Technology Research Development Program of China (863 Program) (2011AA110306) is gratefully acknowledged. 1
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CHAOS 24, 023107 (2014)
Scaling analysis of stock markets Luping Bu and Pengjian Shang School of Science, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
(Received 13 May 2013; accepted 4 April 2014; published online 16 April 2014) In this paper, we apply the detrended fluctuation analysis (DFA), local scaling detrended fluctuation analysis (LSDFA), and detrended cross-correlation analysis (DCCA) to investigate correlations of several stock markets. DFA method is for the detection of long-range correlations used in time series. LSDFA method is to show more local properties by using local scale exponents. DCCA method is a developed method to quantify the cross-correlation of two nonstationary time series. We report the results of auto-correlation and cross-correlation behaviors in three western countries and three Chinese stock markets in periods 2004–2006 (before the global financial crisis), 2007–2009 (during the global financial crisis), and 2010–2012 (after the global financial crisis) by using DFA, LSDFA, and DCCA method. The findings are that correlations of stocks are influenced by the economic systems of different countries and the financial crisis. The results indicate that there are stronger auto-correlations in Chinese stocks than western stocks in any period and stronger auto-correlations after the global financial crisis for every stock except Shen Cheng; The LSDFA shows more comprehensive and detailed features than traditional DFA method and the integration of China and the world in economy after the global financial crisis; When it turns to cross-correlations, it shows different properties for six stock markets, while for three Chinese stocks, it reaches the weakest cross-correlations during the global financial crisis. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4871479] V Nowadays, auto-correlations and cross-correlations have become increasing topics in many topics concerning time series analysis. And the interaction is highly nonlinear, unstable, and long-ranged for stock markets. Hence, the study of these properties is crucial for understanding the structure of stock markets. The main aim of our work is to describe the correlations in six stocks. Actually, first, we use two methods—detrended fluctuation analysis (DFA) and local scaling detrended fluctuation analysis (LSDFA)—to investigate the auto-correlations of six stocks in three periods: 2004–2006 (before the global financial crisis), 2007–2009 (during the global financial crisis), and 2010–2012 (after the global financial crisis). Then, we apply detrended cross-correlation analysis (DCCA) to investigate the cross-correlations between some stocks in three periods. The findings are that correlations of stocks are influenced by the economic systems of different countries and the financial crisis. We find that there are stronger auto-correlations in Chinese stocks than western stocks in any period and stronger auto-correlations after the global financial crisis for every stock except Shen Cheng; The LSDFA shows more comprehensive and detailed features than traditional DFA method and it shows the integration of China and the world in economy after the global financial crisis; When it turns to cross-correlations, it shows different properties for six stock markets, while for three Chinese stocks, it reaches the weakest crosscorrelations during the global financial crisis.
I. INTRODUCTION
Recently, auto-correlations and cross-correlations have become increasing topics in many fields concerning time 1054-1500/2014/24(2)/023107/7/$30.00
series analysis, ranging from finance1–3 to traffic,4–6 geophysics,7,8 and physiology.9,10 Nowadays, Many reports point out that all stock companies are correlated and interconnected. Stock markets have become an active research aspect because of theirs highly nonlinear, unstable, and long-ranged interaction. Hence, the study of auto-correlations and crosscorrelations of stocks is crucial for understanding the structure and property of stock markets. The DFA method invented by Peng has become a widely used method for the determination and detection of long-range correlations in time series.11–13 The a exponent of DFA is estimated as the overall slope of a variability function FðsÞ plotted versus the beat resolution s in a log–log scale. Although the DFA scaling coefficients have been used in a large number of studies, full validation of this approach has not been performed. In particular, the ranges for identification of a1 and a2 cannot be defined precisely and are usually determined by visual inspection of data. It is also unknown whether and how external factors alter the ranges for identification of the two coefficients. Moreover, describing auto-correlations among financial markets using a1 and a2 may be an inadequate oversimplification of a more complex phenomenon as these correlations might actually consist of a much lager number of scale. If this is the case, a whole spectrum of the local scaling exponent aðsÞ rather than just two coefficients would be required to describe the correlations accurately. Hence, previous studies on multiscale DFA14–18 inspired us to apply LSDFA to investigate the correlations. The proposed method-LSDFA is related to multiscale behaviors of time series, which share some common features with and ideas underlying the multifractal versions of
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DFA.19,20 They all require a multitude of scaling exponents for a full description of the scaling behavior of a time series. They are techniques for determining fractal scaling properties and detecting long-range correlations in noisy, nonstationary time series. However, there are also some differences between them. The multifractal versions of DFA describe the global multiscale correlation of time series under continuous scales and focus on the generalized scaling exponent HðqÞ and multifractal spectrum f ðaÞ. By contrast, LSDFA describes the local multiscale correlation of time series under discrete scales and focus on the spectrum of aðsÞ. In spite of successes noted in the literature, multifractal analysis has strict of noise in the signal. Multifractal analysis also requires basic assumptions such as nonstationarity and an acceptable level of noise in the signal. Multifractal analysis also requires initial assumptions about the presumed time scale of the problem investigated, which may lead to artifacts in some cases. For example, if a crossover falls within the scaling range, the results will be biased. If the scaling range includes only large scales, then information about the fractal properties at small scales may be missed, and these properties might be dramatically different and very interesting. Even if the MF-DFA and MF-DCCA are used separately for small and large scales, we would miss analysis of properties within the crossover range, and these properties might be very different from case to case. Hence, we propose LSDFA methods, which have fewer data prerequisites and initial assumptions, and thus yield a more concise analysis. In previous studies, there are many different methods used to quantify the cross-correlations of stock indices, for example, the applications of random matrix theory (RMT) to investigate the cross-correlations of stock price changes in literature.21,22 DCCA method is the extension of DFA using to quantify logrange cross-correlations between two non-stationary time series, which has been proposed recently in literature.23–25 In this paper, we study the correlation of a single stock market by using DFA and LSDFA. Lastly, we apply the DCCA to analyze different dynamics of cross-correlations between some stocks in three periods. The organization of this paper is as follows. Section II presents methods employed in this study. Section III describes the database used in our work briefly. Section IV is devoted to provide the detailed results of DFA, LSDFA, and DCCA of daily returns of three stocks in three periods. Section V presents the conclusions of this work. II. METHODS A. DFA method
DFA method has been used to investigate the long-range correlations in time series. The DFA procedure consists of five steps.26,27 Let us suppose that fxk g is a series of length N. Step 1: Determine the profile YðiÞ ¼
i X
ðxk hxiÞ;
i ¼ 1; …; N:
(1)
k¼1
Step 2: Cut the profile YðiÞ into Ns ½N=s nonoverlapping segments of equal length s. Since the record length N
need not be a multiple of the considered time scale s, a short part at the end of the profile will remain in most cases. In order not to disregard this part of the record, the same procedure is repeated starting from the other end of the record. Thus, 2Ns segments are obtained altogether. The subtraction of the mean hxi is not compulsory, since it would be eliminated by the later detrending in the third step anyway. Step 3: Calculate the local trend for each segment v by a least-square fit of the data. Then, we define the detrended time series for segment duration s, denoted by Ys ðiÞ, as the difference between the original time series and the fits Ys ðiÞ ¼ YðiÞ qv ðiÞ;
(2)
where qv ðiÞ is the fitting polynomial in vth the segment. Linear, cubic, or higher order polynomials can be used in the fitting procedure (DFA1, DFA2, and higher order DFA). Different fitting order will get different results. In order to get more precise results, we often use larger than 1-order to analyze the time series. Since the detrending of the time series is done by the subtraction of the polynomial fits from the profile, these methods differ in their capability of eliminating trends in the series. In n-th order DFA, trends of order n in the profile and of order n 1 in the original series are eliminated. We calculate, for each of the 2Ns segments, the variance Fs 2 ðvÞ ¼ hYs2 ðiÞi ¼
s 1X Y 2 ½ðv 1Þs þ i; s i¼1 s
v ¼ 1; 2; …; Ns ; (3)
Fs 2 ðvÞ ¼
s 1X Y 2 ½N ðv Ns Þs þ i; s i¼1 s
v ¼ Ns þ 1; Ns þ 2; …; 2Ns ;
(4)
of the detrended time series Ys ðiÞ by averaging over all data points i in the vth segment. Step 4: Average over all segments to obtain the fluctuation function "
#12 2Ns 1 X 2 F ðvÞ : FðsÞ ¼ 2Ns v¼1 s
(5)
It is apparent that FðsÞ will increase with increasing s. Of course, FðsÞ depends on the DFA order n. By construction, FðsÞ is only defined for s n þ 2. Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log-log plots FðsÞ versus s. If the series xk are long-range power-law correlated, FðsÞ increases, for large values of s, as a power-law FðsÞ sa :
(6)
The exponent a is called the scaling exponent or correlation exponent, represents the correlation properties of the signal. The value 0 < a < 0:5 indicates that larger and smaller values of the time series are more likely to alternate, while 0:5 < a < 1 indicates persistent long-range power-law
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correlations such that a large value in one variable is more likely to be followed by a large value in another variable and vice versa. a ¼ 0:5 denotes the absence of correlations. A special case of a ¼ 1 corresponds to 1=f noise. For a 1, correlations exist but cease to be of a power-law form; a ¼ 1:5 indicates Brown noise.7 B. LSDFA method
In this study, we extend DFA in order to estimate a local scaling exponent aðsÞ, and we name this method as LSDFA. The LSDFA procedure also consists of five steps. The first four steps are identical to the conventional DFA procedure. The fifth step is to determine the scaling behavior of the wave function by analyzing a local scaling exponent aðsk Þ defined as follows:16–18 aðsk Þ ¼
log½Fðskþ1 Þ log½Fðsk1 Þ ; log½skþ1 log½sk1
(7)
where sk is a subset of scale s. C. DCCA method
DCCA is a new method to quantify the crosscorrelations of two different non-stationary time series. This method is a generalization of the DFA method in which only one time series is analyzed. It was introduced by Podobnik and Stanley.24 The DCCA procedure also consists of five steps.28 The first three steps are similar to the DFA procedure. Consider two time series fxk g an fxk 0 g, where k ¼ 1; 2; …N and N is the length of the time series. Step 1: Determine the profile i X ðxk hxiÞ; YðiÞ ¼ i X ðx0 k hx0 iÞ;
III. DATA
The analyzed time series consists of six stock indices: three stocks indices of western countries, DAX (Deutscher Aktien Index) of Germany, FTSE100 of UK, and S&P500 of USA together with three Chinese stock indices, Heng Sheng of Hong Kong, Shang Zheng of Shang Hai, and Shen Cheng of Shen Zhen. The data are recorded every day of closing prices from January 2, 2004 to December 31, 2012 for a total of 2053 days. We exclude the asynchronous datum and reconnect the same parts of the original indices to obtain the same length because the different opening day of six stocks. Then, we split the 6 stock indices into three periods, which are parts of from 2004 to 2006 (Data1) and from 2007 to 2009 (Data2) and from 2010 to 2012 (Data3) obtained before, during, and after the global financial crisis, respectively. Denoting the stock market index as fxðtÞg, the logarithmic daily return is defined by gðtÞ ¼ logðxðtÞÞ logðxðt 1ÞÞ:
i ¼ 1; 2; …; N:
(8)
k¼1
Step 2: Cut the profile YðiÞ and Y 0 ðiÞ into 2Ns nonoverlapping segments of equal length s. Step 3: Calculate the local trend for each segment v by a least-square fit of the data. Then, we calculate the difference between the original time series and the fits. Step 4: Calculate the covariance of the residuals in each box s 1X 0 2 ðs; vÞ ¼ ðYk Y~k;v ÞðY 0 k Y~ k;v Þ; (9) fDCCA s k¼1 0 where Y~k;v and Y~ k;v are the fitting polynomials in segment v, respectively. Then average over all segments to obtain the fluctuation function ( )12 2Ns 1 X 2 f ðs; vÞ : (10) fDCCA ðsÞ ¼ 2Ns v¼1 DCCA
Step 5: Determine the scaling behavior of the fluctuation functions by analyzing log-log plots FðsÞ versus s FDCCA ðsÞ sk :
(11)
(12)
The normalized daily return (NDR) is defined by RðtÞ ¼
k¼1
Y 0 ðiÞ ¼
The scaling exponent k represents the degree of the cross-correlation between the two time series xk and xk 0 . The value k ¼ 0:5 denotes the absence of cross-correlations. k > 0:5 indicates persistent long-range cross-correlations where a large value in one variable is likely to be followed by a large value in another variable, while k < 0:5 indicates anti-persistent cross-correlations where a large value in one variable is likely to be followed by a small value in another variable, and vice versa.
gðtÞ hgðtÞi ; r
(13)
where r is the standard deviation of the time series gðtÞ. In this paper, we use the NDRs of every stock indices in order to obtain a more meaningful result. The NDRs of every stock indices are showed in Fig. 1. IV. ANALYSIS AND RESULTS A. DFA of six stocks
The first thing we analyze is the correlation behaviors. This study investigates long-range power-law correlations in western and Chinese stock markets using NDRs by applying DFA. Here, we use log-log plots of FðsÞ versus s by firstorder fitting. It shows different slopes of NDRs of six stocks in every period. The scaling exponents of DFA method are listed in Table I. From Table I, we can see that the scaling exponents reach maximums in period 2010–2012 for six stocks except Shen Cheng indicating the strongest auto-correlations after the global financial crisis. For DAX the larger value appears in period 2007–2009, while for FTSE100, S&P500, Heng Sheng, and Shang Zheng it reaches larger values in period 2004–2006 indicating different properties of these five stocks during the global financial crisis. The scaling exponent reaches 0.4934 for S&P500 especially. For Shen Cheng, the
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FIG. 1. The normalized daily returns of closing prices of DAX, FTSE100, S&P500, Heng Sheng, Shang Zheng, and Shen Cheng in three periods.
scaling exponents reach the maximum in period 2004–2006 and a larger value in period 2010–2012 showing a stronger auto-correlation before the global financial crisis. As a whole, it shows stronger auto-correlations in Chinese stocks than western stocks in any period. However, it is difficult to evaluate details of fractal structures from Table I because small differences (the scaling exponents are approximately between 0.5 and 0.6) are shown between any two stocks, especially between Chinese stocks. In particular, this data representation does not allow us to easily evaluate the possible differences among conditions or among time series. As we will see in the following part, details on the fractal structure are much better quantified by the local scaling exponents aðsÞ. B. LSDFA of six stocks
In order to improve DFA method, we explain our application of LSDFA to investigate the local scaling exponents of NDRs of different stocks in this section. In this paper, we estimate FðsÞ on a subset sk ðk ¼ 1; :::; 21Þ spaced evenly on a log scale between s1 ¼ 4 and s21 ¼ 128 as the algorithm provided in Refs. 16–18. As Figs. 2–4 show, we make some comparative analyses of some stocks. The results are shown as follows. There are significant features of aðsÞ values of six stocks. The traditional DFA method fails to distinguish these characteristic. In Fig. 2, it shows disordered form in 2004–2006 and largest similarities between FTSE100 and S&P500 during TABLE I. Scaling exponents of DFA of six stocks.
a of DFA
DAX
FTSE100
S&P500
Heng Sheng
Shang Zheng
Shen Cheng
a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
0.5367 0.5507 0.5566
0.5494 0.5095 0.5508
0.5171 0.4934 0.5172
0.5722 0.5084 0.5813
0.5745 0.5505 0.5761
0.5800 0.5567 0.5705
and after the global financial crisis. While for DAX, it shows consistency with other two stocks during 2007–2009, which demonstrates the closer correlations during the crisis for capitalist countries. It is in line with literature.29,30 In Fig. 3, there is small significant difference between Shang Zheng and Shen Cheng at every scale in any period. The fluctuation of Heng Sheng is strongest in period 2007–2009. Heng Sheng is more identical to other two stocks in period 2010–2012. The results show that the similarities of mainland China and the relation between mainland China and Hong Kong is more identical. In Fig. 4, it shows significant similarities between S&P500 and Heng Sheng in every period, while there are smaller difference between S&P500 and three Chinese stocks after the crisis, which shows the integration of China and the world in economy after the global financial crisis. C. DCCA method
In this section, we explain our application of DCCA to investigate the cross-correlations between NDRs of six stocks. For simplicity, we select the returns and sðiÞði ¼ 1; :::; 21Þ spaced evenly on a log scale between sð1Þ ¼ 4 and sð21Þ ¼ 128 and log-log plots of FðsÞ versus s by first-order fitting to analyse and investigate. Tables II–IV show the scaling exponents of DCCA between every two stocks. The results are listed as follows. For DAX and S&P500 as well as FTSE100 and S&P500, it shows the stronger and strongest cross-correlations during and after the global financial crisis, respectively. For DAX and FTSE100, it indicates the stronger and strongest crosscorrelations after and during the global financial crisis, separately (Table II). It shows different properties for three Chinese stock markets. Heng Sheng and Shang Zheng shows the stronger and strongest cross-correlations after and before the global financial crisis, separately; Heng Sheng and Shen Cheng reach the stronger and strongest cross-correlations before and after the global financial crisis, respectively;
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FIG. 2. The local scaling exponents of DAX, FTSE100, and S&P500 indices in three periods.
FIG. 3. The local scaling exponents of Heng Sheng, Shang Zheng, and Shen Cheng indices in three periods.
FIG. 4. The local scaling exponents of S&P500, Heng Sheng, Shang Zheng, and Shen Cheng indices in three periods.
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TABLE II. The scaling exponents of DCCA of every two western stocks indices.
a of DCCA
DAX and FTSE100
DAX and S&P500
FTSE100 and S&P500
1.0149 1.0656 1.0495
0.9714 1.0251 1.1807
0.9037 1.0356 1.0859
a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
TABLE III. The scaling exponents of DCCA of every two Chinese stocks indices.
a of DCCA a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
Heng Sheng and Shang Zheng
Heng Sheng and Shen Cheng
Shang Zheng and Shen Cheng
1.0494 0.6452 0.9592
0.9153 0.7407 0.9787
0.8330 0.8850 0.8007
TABLE IV. The scaling exponents of DCCA of S&P500 and every Chinese stock indices.
a of DCCA a1 (2004–2006) a2 (2007–2009) a3 (2010–2012)
S&P500 and Heng Sheng
S&P500 and Shang Zheng
S&P500 and Shen Cheng
0.8264 0.9136 0.9492
0.9723 0.9385 0.9114
0.8968 1.0313 0.9537
Shang Zheng and Shen Cheng show the stronger and strongest cross-correlations before and during the global financial crisis, separately (Table III). S&P500 and Heng Sheng show the stronger and strongest cross-correlations during and after the global financial crisis, separately; S&P500 and Shang Zheng reach the stronger and strongest cross-correlations during and before the global financial crisis, respectively; S&P500 and Shen Cheng show the stronger and strongest crosscorrelations after and during the global financial crisis, separately (Table IV). For three western stocks, it shows the larger values than what between other two stocks in any periods; For three Chinese stocks, it reaches the smallest values during the global financial crisis. However, an interesting thing is that there is no significant relation between LSDFA and DCCA method, it is possible because the two methods are different construction and lead to different properties of correlations. The financial crisis effects the correlations of stocks, which causes the complexity of stocks, it is mainly because the depressed situation and decrease of rebound of stock markets and the control ability of government.
V. CONCLUSIONS
In this paper, we investigate the correlation behaviors of stock indices. First, we analyze the traditional DFA to find differences between six stocks in three periods. Furthermore, we apply LSDFA to improve the effectiveness of DFA. Finally, we study scale exponents of DCCA of stock series.
The finding is that correlations of stock dynamics are influenced by the economic systems of different countries and the crises. There are stronger auto-correlations in Chinese stocks than western stocks in any period. The LSDFA shows more comprehensive and detailed features and the integration of China and the world in economy after the global financial crisis, which is not found from traditional DFA method. For three Chinese stocks, it reaches the weakest crosscorrelations during the global financial crisis. There are already many methods to detect the scaling behaviors and predict the stock markets. Preis, Schneider, and Stanley have reported scaling exponents in time domain on various time scales.31,32 It is a good way to distinguish the financial crisis using various scales and has a very good practicability. Recent work demonstrates that there is predictive power for weekly stock market returns in data generated by internet users online.33 Learning more prediction functions from these methods, we may improve the methods used in our paper to predict the stock indices and financial crisis. We can also apply the methods to other regions to distinguish or recognize different types of time series. We believe that such studies are relevant for a better understanding of the stock market mechanisms. ACKNOWLEDGMENTS
Financial support by China National Science (61071142, 61371130), Beijing National Science (4122059), and the National High Technology Research Development Program of China (863 Program) (2011AA110306) is gratefully acknowledged. 1
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