Physica A 392 (2013) 1659–1670

Contents lists available at SciVerse ScienceDirect

Physica A

journal homepage: www.elsevier.com/locate/physa

Multifractal detrended cross-correlation analysis betweenthe Chinese stock market and surrounding stock marketsFeng Ma ∗, Yu Wei ∗∗, Dengshi HuangSchool of Economics & Management, Southwest Jiao Tong University, First Section of Northern Second Ring Road, Chengdu, Sichuan Province, China

a r t i c l e i n f o

Article history:Received 7 September 2012Received in revised form 10 November2012Available online 22 December 2012

Keywords:Stock marketsMultifractal detrended cross-correlationanalysis

Cross-correlationsRolling windows

a b s t r a c t

In this paper, we investigate the cross-correlations between the stock market in China andmarkets in Japan, South Korea and Hong Kong. We use not only the qualitative analysis ofthe cross-correlation test, but also the quantitative analysis of the MF-X-DFA. Our findingsconfirm the existence of cross-correlations between the stockmarket in China andmarketsin Japan, South Korea and Hong Kong, which have strongly multifractal features. We findthat the cross-correlations display the characteristic of multifractality in the short term.Moreover, the cross-correlations of small fluctuations are persistent and those of largefluctuations are anti-persistent in the short term, while the cross-correlations of all kindsof fluctuations are persistent in the long term. Furthermore, based on the multifractalspectrum, we also find that the multifractality of cross-correlation between stock marketsin China and Japan are stronger than those between China and South Korea, as well asbetween China and Hong Kong.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Since Hurst [1,2] derived the long memory of the hydrological time series from tide data, and Mandelbrot et al. [3] laidthe scrutinized mathematical foundation by introducing the Brownian movement and the Sectioning conception, researchof long memory has aroused widespread interests. In the last two decades, the methods of examining long memory arewidely applied to financial markets. For example, Peters [4] adopted the Rescaled Range analysis (R/S) in examining longmemory in historical stock markets. Peters [4] proved that the returns almost go with the property of high peak and fattail and found that financial markets in the US, the UK and the Japan have embodied the long-range auto-correlation whichimplied the market inefficiency. Using the R/S method, Cajueiro and Tabak [5] studied the long-range auto-correlations inemergingmarkets and found that themarkets becamemore andmore efficient over time. Barkoulas et al. [6] applied the R/Sin research of the interest rates and the stock market and obtain a similar conclusion. Lo [7] contended that R/S seemed tobe rather sensitive to the short term auto-correlation and the non-stationarity, which is likely to lead to a biased estimationof long memory parameters. In order to overcome the drawbacks, Peng et al. [8] put forward the DFA in studying the fractalstructure ofmolecular chains in deoxyribonucleic acid (DNA). Since then, DFA and itsmultifractal generalization,MF-DFA [9]have been widely used to detect the long-range auto-correlations in financial markets, including the stock markets [10,11],foreign exchange market [12,13], and gold market [14].

It has been found that financial time series are always cross-correlated [15–22]. To quantify power-law cross-correlationsin non-stationary time series, Podobnil and Stanley [23] extend the DFA into the DXA method. Subsequently, Zhou [24]proposed multifractal detrended cross-correlation analysis (MF-X-DFA) by combining MF-DFA and DXA approaches. Since

∗ Corresponding author. Tel.: +86 13882156028.∗∗ Corresponding author.

E-mail addresses:mafeng575@gmail.com (F. Ma), weiyusy@126.com (Y. Wei).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.12.010

1660 F. Ma et al. / Physica A 392 (2013) 1659–1670

then, DXA and MF-DXA method have been widely used in diverse fields including financial data [25,26], traffic flows[27–29], sunspot numbers and river flow fluctuations [30], and meteorological data [31]. Being of our interest, this methodis also applied to Chinese stock and commodity markets. For example, Wang et al. [32] studied the cross-correlationsbetween the A and B shares of the Chinese stockmarket. Cao et al. [33] discussed the cross-correlations between the Chinesestock market and the foreign exchange market of Chinese yuan. Li et al. [34] researched the cross-correlation properties ofagricultural futures markets between the China and US, and found that the cross-correlations are significantly multifractal.Yuan et al. [35] found that the cross-correlation between the Chinese stock price and trading volume is multifractal. As thedeep development of the methodology, Jiang and Zhou [36] introduced MF-X-DMA which is based on MF-DMA [37] andDMA [38]. Kristoufek proposed MF-HXA [39] based on the height–height correlation analysis of Barabasi and Vicsek [40]and Hedayatifar et al. [41].

This paper is focused upon the cross-correlations between the stock market in China and markets in Japan, South Koreaand Hong Kong by means of MF-X-DFA. Our contributions are fourfold. First, this paper is the first study which studiesmultifractal cross-correlations between the Chinese stock market and surrounding stock markets. Second, we not onlyqualitatively analyze the cross-correlations employing the statistics proposed by Podobnik et al. [42] and Podobnik et al. [43],but also quantitatively study the cross-correlations using the MF-X-DFA method. Third, we use the rolling windows toinvestigate the time-varying features ofmultifractal cross-correlations. Finally,we study themarket efficiency of the Chinesestock market using the MF-X-DFA method and the technique of rolling windows.

This paper is organized as follows. Section 2 mainly focuses upon the description of MF-X-DFA. Section 3 describes thedata of four stock markets of interests. Section 4 provides the empirical analysis. Section 5 discusses the results based onrolling windows and Section 6 concludes.

2. Methodology

Assume that there are two series x(i) and y(i)(i = 1, 2, . . . ,N), let us introduce the MF-X-DFA method as follows.Step 1. Construct the profile

X(i) =

it=1

(x(t) − x), Y (i) =

it=1

(y(t) − y) (1)

where, x and y denote the average of the two whole time series x(i) and y(i).Step 2. The profiles X(i) and Y (i) are divided into Ns = [N/s] non-overlapping windows (or segments) of equal length s.Since the length N is not always a multiple of the considered time scale s. In order to not discard the Section of series, thesame procedure is repeated starting from the opposite end of each profile. Thus, 2Ns non-overlappingwindows are obtainedtogether.Step 3. The local trends Xν(i) and Y ν(i) for each segment ν(ν = 1, 2, 3, . . . , 2Ns) are evaluated by least squares fits of thedata, then the detrended covariance is determined by

F 2(s, v) =1s

ti=1

|X((v − 1)s + i) − Xv(i)| • |Y ((v − 1)s + i) − Y v(i)| (2)

for each segment ν, ν = 1, 2, . . . ,Ns and

F 2(s, v) =1s

ti=1

|X(N − (v − Ns)s + i) − Xv(i)| • |Y (N − (v − Ns)s + i) − Y v(i)| (3)

for each segment ν, ν = Ns + 1,Ns + 1, . . . , 2Ns. Then the trends Xν(i) and Yν(i) denote the fitting polynomial with orderm in each segment ν.Step 4. qth-order the fluctuation function as follows

Fq(s) =

1

2Ns

2Nsν=1

[F 2(s, ν)]q/2

1/q

. (4)

If q = 0, then

F0(s) = exp

1

4Ns

2Nsν=1

ln[F 2(s, ν)]

. (5)

When q = 2, MF-X-DFA is the standard of the DXA.

F. Ma et al. / Physica A 392 (2013) 1659–1670 1661

Step 5. Analyze the scaling behavior of the fluctuations by observing log–log plots Fq(s) verse s for each values of q. If thetwo series are long-range cross-correlated, Fq(s) will increase for values of s, we can obtain a power-law expression

Fq(s) ∼ sHxy(q). (6)

This can be presented as follows

log Fq(s) = Hxy(q) log(s) + log A (7)

the scaling exponentHxy(q) is known as the generalized cross-correlation exponent, describing the power-low relationship.Especially, if the two time series are equal, MF-X-DFA is theMF-DFA. when the scaling exponentHxy(q) is independent q, thecross-correlation between two series is monofractal. If the scaling exponent Hxy(q) is dependent on q, the cross-correlationbetween two series is multifractal. If the scaling exponent Hxy(q) > 0.5, the cross-correlations between the return fluctua-tions of the two series related to q are long-range persistent. If the scaling exponent Hxy(q) < 0.5, the cross-correlations be-tween the return fluctuations of the two series related to q are anti-persistent. IfHxy(q) = 0.5, there are no cross-correlationsbetween the two series. Furthermore, for positive q,Hxy(q) describes the scaling behavior of the segments with large fluc-tuations. On the contrary, for negative q,Hxy(q) describes the scaling behavior of the segments with small fluctuations.

According to the multifractal formalism, the Renyi exponent τxy(q) can be used to characterize the multifractal nature,

τxy(q) = qHxy(q) − 1. (8)

If the Renyi exponent τxy(q) is liner of the q, we can conclude that the two correlated series is monofractal, otherwise, it ismultifractal.

The multifractal spectra fxy(α) describes the singularity content of the time series one can obtain through the Legendretransform:

αxy(q) = Hxy(q) + qH ′

xy(q) (9)

fxy(α) = q(αxy − Hxy(q)) + 1. (10)

3. Data

Two stock markets exist in China, the Shanghai Stock Market and Shenzhen Stock Market. Relatively speaking, theShanghai Stock Market is more developed in the degree of openness, maturity, and the ability to receive global information.Consequently, we chose the Shanghai stock market as an example to study a Chinese stock market in this paper. Proceedingto the next step, we discuss the cross-correlation properties between stock markets in China, Japan and South Korea aswell as Hong Kong. The Japanese Stock Market is the second largest securities market in the world, occupying an extremelyimportant position in the world capital market, meanwhile, Japan is an economically powerful country not only amongChina’s neighboring countries, but also in the world. South Korea also has a developed security market as one of China’sneighboring countries, and it is one of the important financial centers in Asia. Hong Kong is an international financial center,and it is closely connected with the mainland’s financial market. This paper chose the Shanghai Stock Exchange Component(SSEC), Hengseng index (HIS), Nikkei 225 (N225), and Korea KOSPI index (KKI) as research objectives.

The sample time range is from 1 January 1997 to 31 December 2011. There is a large possibility that prices of stocksrise suddenly and drop sharply because Chinese stock markets are easily affected by policy. In 16 December 1996, a pricelimit measure was been carried out on the Chinese stock market, and the unsteady situation of the stock market had beencontrolled well since then. So data chosen after 1997 is strongly robust for the study. As a result, we choose trade datafrom the four areas in this article, the number of sample data is 3086. Stock returns are computed by the formula, Rt =

log(Pt)– log(Pt–1), where Pt is the closing price index at time t .The basic description of stockmarkets’ rate of return in the four areas is shown in Table 1. Themean of the four indexes is

close to zero, but their fluctuations aremore than zero. It is found from Jarque–Bera statistics TEST value that the assumptionthat all of the SSEC, N225, KKI and HIS’s distributions are Gaussian is rejected at the 1% significance level with skewnessdifferent fromzero andkurtosismore than3,which suggests that the four kinds of indexhave fat tail features.With respect tothe volatility, see in Table 1,we can find that theHongKong stockmarket ismore volatile comparedwith other stockmarkets,which means that the risk of the Hong Kong stock market is bigger. The Chinese stock market is less volatile compared withother stock markets, which can possibly explain that the Chinese stock market has a low degree of openness and the stocksare less interacted with international stocks compared with other stock markets.

4. Empirical results

4.1. Cross-correlation test

In order to quantify the cross-correlation between the Chinese stock market and Japanese, South Korean, Hong Kongstock market, we introduce a new cross-correlation statistic proposed by Podobnik et al. [42] in analogy to the Ljung-Box

1662 F. Ma et al. / Physica A 392 (2013) 1659–1670

Table 1Descriptive statistics for the returns of SSEI, N225, KKI and HSI.

Mean Max Min S.D Ske Kur J-B

SSEC 0.00013 0.09400 −0.09256 0.0167 −0.08894 7.05298 2115*N225 −0.00023 0.13234 −0.12111 0.01618 −0.26318 9.18918 4961*KKI 0.00016 0.11284 −0.12368 0.02013 −0.17850 6.4725 1566*HSI −0.00006 0.17247 −0.14734 0.01811 0.01851 12.0194 10460*

Note: * denote 1% significance levels; Symbols ‘‘Max’’, ‘‘Min’’, ‘‘S.D’’, ‘‘Ske’’, ‘‘Kur’’ denoteMaximum,Minimum, Stv. Dev, Skewness and Kurtosis respectively.

-3

-2

-1

0

1

2

3

4

log(m)

log(

Qcc

(m))

critical values

SSEC and KKISSEC and HSISSEC and N225

0 0.5 1 1.5 2 2.5 3

Fig. 1. The cross-correlation statistic between the Chinese and Japanese, South Korea, Hong Kong in return. The red line stands for the SSEC and KKI. Thegreen line stands for the SSEC and HIS. The black line stands for the SSEC and N225. The blue line stands for the critical values. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

test [44]. The cross-correlation statistic between two series {x(i)}, {y(i)}, which have the same lengthN , functions as follows

Ci =

Nk=i+1

xkyk−iN

k=1x2k

Nk=1

y2k

. (11)

Then the cross-correlation test statistic

Qcc(m) = N2mi=1

C2i

N − i(12)

which is approximately χ2(m) distributed with m degrees of freedom. If there are no cross-correlations between two timeseries, the cross-correlation test agrees well with the χ2(m) distribution. If the cross-correlations test exceeds the criticalvalue of the χ2(m) distribution, then the cross-correlations are significant at a special significance level. We describe thecritical value of the χ2(m) distribution at the 5% level of significance for the degrees of freedom varying from 1 to 1000.

Based on Eqs. (11)–(12), we can obtain the cross-correlation statistic (logarithmic form) Qcc(m) for the Chinese stockmarket and other three areas stock market respectively (Fig. 1). In Fig. 1, the cross-correlation statistic Qcc(m) for the threepairs (SSEC and N225, SSEC and KKI, SSEC and HSI) are always larger (or close to) than the critical values for the χ2(m)distribution at the 5% level of significance, which suggests there are existing long-range cross-correlations.

To affirm our results above more carefully in this paper, we also apply another new method proposed by Podobniket al. [43], defined as the ratio between the detrended covarivance covariance function F 2

DCCA and the detrend variance FDFA,the function as

ρDCCA =F 2DCCA(n)

FDFA1{n}FDFA2{n}. (13)

Then, the value of ρDCCA ranges between −1 ≤ ρDCCA ≤ 1. If ρDCCA is equal to zero, which means the two series have nocross-correlation, and it splits the level of cross-correlation between the positive and the negative case. We calculate thevalues of ρDCCA based on different values of window size n(n = 16, 32, 64, 128, 256) (see Table 2) that we can comparewiththe Table 1 (see Table 1 in Podobnik et al. [43]) and draw the conclusions that are consistent with the cross-correlation testas above.

F. Ma et al. / Physica A 392 (2013) 1659–1670 1663

Table 2The value of ρDCCA when for a given window size n.

Size 16 32 64 128 256

N225 and SSEC 0.417276 0.340212 0.325824 0.315191 0.457323KKI and SSEC 0.360618 0.380055 0.359532 0.377693 0.517081HSI and SSEC 0.388704 0.426381 0.465153 0.4095 0.646076

Table 3Cross-correlation exponents for three pairs with q varying from −10 to 10.

q N225 KKI HSIs∗ = 126 s∗ = 178 s∗ = 151s < s∗ s > s∗ s < s∗ s > s∗ s < s∗ s > s∗

−10 0.546761 0.615715 0.56709 0.498851 0.529327 0.523902−9 0.542505 0.607921 0.562635 0.502564 0.528417 0.518479−8 0.538471 0.599701 0.558119 0.50758 0.52788 0.512671−7 0.53491 0.591244 0.553652 0.514251 0.527841 0.506648−6 0.532045 0.582843 0.549358 0.522964 0.528398 0.500788−5 0.529982 0.574893 0.545363 0.534091 0.529547 0.495833−4 0.528644 0.567867 0.541765 0.547886 0.51888 0.493048−3 0.527798 0.562255 0.538607 0.564342 0.524132 0.49418−2 0.527132 0.558465 0.535847 0.58302 0.524132 0.500882−1 0.526322 0.556709 0.533344 0.602961 0.536352 0.5135440 0.525075 0.556914 0.530848 0.622784 0.529686 0.5303471 0.523143 0.561493 0.528047 0.641027 0.52548 0.5478482 0.520337 0.561493 0.524626 0.656583 0.519957 0.5630693 0.516538 0.564642 0.520345 0.668947 0.513536 0.5748654 0.511719 0.567621 0.5151 0.678161 0.506614 0.583515 0.505969 0.570089 0.508946 0.684603 0.499506 0.5897596 0.499489 0.571896 0.502072 0.688775 0.492438 0.594327 0.492556 0.573029 0.494756 0.691175 0.485563 0.5977098 0.485456 0.573557 0.487293 0.668947 0.47898 0.600279 0.478433 0.573583 0.479944 0.656583 0.472748 0.6022310 0.471666 0.573214 0.472904 0.641027 0.466901 0.603737

4.2. MF-X-DFA

Based on the Eqs. (12)–(13), we can only test for the presence of cross-correlation qualitatively, so we use to the MF-X-DFA method to estimate a quantitative cross-correlation exponent.

Fig. 2 shows the log–log plots of log Fq(s) versus log(s) for the Chinese stockmarket and Japanese, South Korean, andHongKong stock markets, respectively as q = −10, −9, . . . , 10. As can be seen from Fig. 2, for different q, all the curves exhibitlinearity during a span, which can tell us that power-lower cross-correlation exists between the Chinese stock market andJapanese (and South Korean and Hong Kong) stock markets. Through observing the linear trend of the curves, we can findthat the curves undergo a fundamental change in a point, which defines the ‘‘crossover’’, s∗. A ‘‘crossover’’ is introducedto distinguish features of the long term (s > s∗) and the short term (s < s∗). The short term behavior of the financialmarket is easily influenced by external market factors such as major events while the long term behavior is determined byinternal factors, and with time evolving, the short term shocks gradually decay in terms of the effects of long term supplyand demand mechanism in the market.

According to the log–log plots Fig. 2, we can find the ‘‘crossover’’ of SSEC andN225 located at approximately log(s∗) = 2.1(126 days, about 6.3 months), the ‘‘crossover’’ of SSEC and KKI at about log(s∗) = 2.25 (178 days, about 8.9 months), andthe ‘‘crossover’’ of SSEC and HIS at about log(s∗) = 2.18 (151 days, about 7.55 months). Furthermore, we also find the‘‘crossover’’ of the three pairs located at approximately log(s∗) = 2.2 nearby.

In order to explore the different changes of cross-correlation exponent in the long term and short term, we calculate thescaling cross-correlation exponent with step 1 to step 5 again. The results can be seen in Table 3.

When q = 2, the bivariate Hurst exponent Hxy(2) has similar properties and interpretation as a univariate Hurst ex-ponent [39]. When q is equal to 2, the cross-correlation exponent Hxy(2) of the SSEC and N225 is 0.520337 for s < s∗,implying that the cross-correlation of Chinese and Japanese stock markets are weakly persistent in the short term. For thecross-correlation exponent Hxy(2) of the rest of SSEC and KKI, SSEC and HIS are 0.524626 and 0.519957 respectively. In allthese cases, s < s∗, displaying similar cross-correlation behavior to that of Chinese and Japanese stock markets. For s > s∗,the cross-correlation exponent of SSEC and N225 is 0.561493, which is close to 0.5 and shows a more strongly persistentcross-correlation in the long term than in the short term. The same results for the SSEC and HIS. While the cross-correlationexponent Hxy(2) of SSEC and KKI is 0.656583, a value bigger than other pairs, we can conclude that the long term cross-correlation behavior of China and South Korea is more strong than in Japanese and Hong Kong stock markets. Furthermore,we also obtain that the Chinese and South Korean stock market has a stronger cross-correlation behavior in the long termthan the short term.

1664 F. Ma et al. / Physica A 392 (2013) 1659–1670

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-2.6

-2.4

-2.2

-2

-1.8

-1.6

-1.4

-1.2

logF

q(s)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8-2.5

-2

-1.5

-1

-2.5

-2

-1.5

-1

logF

q(s)

logF

q(s)

A SSEC and N225

logs

B SSEC and KKI

logs

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8logs

C SSEC and HSI

Fig. 2. Log–log plot of log Fq(s) versus log(s) for three pairs. A stands for SSEC and N225. B stands for SSEC and KKI. C stands for SSEC and HIS.

If the cross-correlation exponent Hxy(q) varies with different values of q, the research two times series is multifractal;otherwise, it ismonofractal. For s < s∗, with q varying from−10 to 10, the cross-correlation exponentHxy(q) of the SSEC andN225 decreases from 0.5468 to 0.4716, implying that the Chinese and Japanese stock markets display the characteristic ofmultifractality in the short term. When q < 0, the cross-correlation exponent Hxy(q) is larger than 0.5, which indicates thatthe cross-correlation behavior of small fluctuations is persistent in the short term. However, for q > 6, the cross-correlationexponentHxy(q) is smaller than 0.5, implying that the cross-correlation behavior of large fluctuations is anti-persistent in theshort term. The stock markets in China, South Korea and Hong Kong respectively have the same cross-correlation behaviorsin the short term as in China and Japan.

F. Ma et al. / Physica A 392 (2013) 1659–1670 1665

Fig. 3. The relationship τxy(q) and q of the Chinese and Japanese, South Korean and Hong Kong stock markets respectively. A1, B1, C1 stand for therelationship between τxy(q) and q in the short term, A2, B2, C2 stand for the relationship between τxy(q) and q in the long term. Red line and green linefrom theMF-DFA, blue line from theMF-DCCA. (For interpretation of the references to colour in this figure legend, the reader is referred to the web versionof this article.)

However, for s > s∗, the cross-correlation exponent Hxy(q) is more weakly dependent on the values of q in the long termthan short. For all q, we can find that the cross-correlation exponent Hxy(q) is always larger (or close to) 0.5, which showsthat three pairs are persistent in the long term. Based on Eq. (8), we can obtain the Renyi exponent τxy(q) and different valuesof q in the short term and in the long term, see Fig. 3.

1666 F. Ma et al. / Physica A 392 (2013) 1659–1670

0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1SSEC and N225SSEC and KKISSEC and HSI

Fig. 4. The multifractal spectrum fxy(α) and α. The blue line stands for SSEC and N225. The black line stands for SSEC and KKI. The red line stands for SSECand HIS. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

From Fig. 3, we can find the Renyi exponent τxy(q) is a nonlinear curve of A1, B1 and C1 dependent on q. We can prove theChinese stock market and other three stock markets showed a multifractal cross-correlation in the short term. We also findthat Renyi exponent τxy(q) is nearly linear in the long term, which demonstrates that the Chinese stock market and otherthree stock markets show a monofractal cross-correlation in the long term.

In this paper, we use Eqs. (9)–(10) to calculate the multifractal spectrum of the China and Japan, South Korea and HongKong, respectively (Fig. 4). If a multifractal spectrum appears as a point, it is monofractal. Fig. 4 shows that the multifractalspectrum of the stock markets in China and Japan, South Korea and Hong Kong respectively are not a point, so we can provethat between three pairs stockmarkets exhibit amultifractal characteristic.Moreover, thewidth of themultifractal spectrumcan be used to estimate the strength of multifractality. From Fig. 4, we can obtain that the strength of multifractality of thestockmarket in China and Japan is stronger than the strength of multifractality in China and South Korea, as well as in Chinaand Hong Kong.

However, finite size effects in time series may also cause a visible difference between H(xy)(q_{max}) and H(xy)(q_{min}), amplified additionally by memory effects in series. These effects however do not describe the ‘‘real’’ multi-fractality and are a result of multifractal bias that should be subtracted before any further conclusions are drawn [45,46].Refs. [45,46]’s approach is based on the Fourier filtering method (FFM) and used the multifractal detrended fluctuationanalysis (MF-DFA). In this paper, we take the simulated approach that procedures follow.

First, we generate two independent time series with the standard normal distribution. The length of each series isequivalent to those employed in the empirical analysis (sample size is 3086). We denote the simulating series as {li} and{mi}(i = 1, 2, . . . , 3086). Theoretically, these two series are monofractal and have no cross-correlation (H = 0.5).

Second, we performmultifractal detrended cross-correlation analysis on these two series.We calculate theH(xy)(q(10))and H(xy)(q(−10)) and get the Delta H(xy).

Third, we repeat the first two steps 100 times to get 100 values of the Delta H(xy). Then, we use the t-statistic to judgethe significance of Delta H(xy) obtained from the empirical analysis. We find that the Delta H(xy) (SSEC and N225, KKI,HSI respectively) is significantly different from the values based on the simulating method. In a word, the multifractalitiesamong these stock markets are significant.

5. Discussion

5.1. Rolling windows

The method of rolling windows began to be widely used to investigate many topics regarding financial markets afterthe influential work in Cajueiro and Tabak [5], such as market efficiency [47,48] and risk management [49,50]. For differentresearch purposes, the length of rolling windows is not fixed. Cajueiro and Tabak [5] and Tabak and Cajueiro [51] used thelength of several years to analyze the evolution of long term correlation, whereasWang at al. [32] only used 250 data pointsas the rolling windows. Grech and Mazur [49] argued that if the length of the rolling window is too large, the calculatedexponent may lose its locality. Grech and Mazur [49] also argued that the local exponent at a given time t depends on thetime-window length. Furthermore, when rather short time series are used, the universal multifractal hypothesis might bemisleading [49]. Thus, for different purposes, the selection of the length of rollingwindow should be careful. In this paper, wetake the same measure of Wang et al. [32], choosing 250 days as the length of rolling windows to investigate the dynamicsof short term cross-correlations (see in Fig. 5).

F. Ma et al. / Physica A 392 (2013) 1659–1670 1667

500 1000 1500 2000 2500 30000.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

point

Hxy

(2)

A SSEC and N225

500 1000 1500 2000 2500 3000

500 1000 1500 2000 2500 3000

B SSEC and KKI

point

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Hxy

(2)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Hxy

(2)

point

C SSEC and HSI

Fig. 5. The cross-correlation exponent Hxy(q) with the lengths of rolling windows at 250 days. A stands for SSEC and N225. B stands for SSEC and KKI. Cstands for SSEC and HIS.

5.2. A binomial measure from P-model

Zhou [24] observed that for two time series constructed by a binomial measure from the p-model, the followingrelationship exists:

Hxy(q) = [Hxx(q) + Hyy(q)]/2. (14)

1668 F. Ma et al. / Physica A 392 (2013) 1659–1670

-10 -8 -6 -4 -2 0 2 4 6 8 100.4

0.45

0.5

0.55

0.6

0.65

q

A SSEC and N225

Hxx(q)-SSEC

Hyy(q)-N225

Hxy(q)

[Hxx(q)+Hyy(q)]/2

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.35

0.4

0.45

0.5

0.55

0.6

Hxx(q)-SSEC

Hyy(q)-KKI

Hxy(q)

[Hxx(q)+Hyy(q)]/2

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.35

0.4

0.45

0.5

0.55

0.6

q

H(q

)H

(q)

H(q

)

C SSEC and HSI

Hxx(q)-SSEC

Hyy(q)-HSI

Hxy(q)

(Hxx(q)+Hyy(q))/2

B SSEC and KKI

q

Fig. 6. The relationship of Hxx(q),Hyy(q),Hxy(q), [Hxx(q) + Hyy(q)]/2 and different q.

The generalized Hurst exponents Hxx(q) and Hyy(q) of the Chinese stock market and Japanese stock market (South Koreanand Hong Kong stock markets) calculated by MF-DFA. Fig. 6 shows that the stock markets in China, Japan, South Korea andHong Kong are all exhibit the multifractal features. If the four stock markets do not have this characteristic, Hxx(q) andHyy(q) are a constant, not varying from the different values of q. For each q, the right side is not equal to the left side forEqs. (13). When q < 0, we can find that the scaling exponents Hxy(q) are always less than the average scaling exponents[Hxx(q) + Hyy(q)]/2. But for q > 0, Hxy(q) is always bigger than the average scaling exponents [Hxx(q) + Hyy(q)]/2.

F. Ma et al. / Physica A 392 (2013) 1659–1670 1669

5.3. Implications

In this paper, we have investigated the cross-correlations between the Chinese stock market and its surrounding stockmarkets, which can reveal some important modeling implications for the Chinese stock market and surrounding stockmarkets as well. First, the cross-correlations between the return series of the Chinese stock market and surrounding stockmarkets are multifractal and nonlinear, which implies that conventional linear models, such as the linear regression modeland vector auto-regression model cannot be used to describe the dynamics of cross-correlations between the Chinese stockmarket and its surrounding stock markets. Second, by introducing the concept of ‘‘crossover’’, we find that the short termcross-correlation behavior is different from long term situations, which means modeling the cross-correlations betweenstock markets should consider different terms. Third, cross-correlations embody time-varying features. Taking this intoconsideration the traditional models with a constant coefficient cannot capture the nature of the relationship between theChinese stock market and its surrounding stock markets, which provides an empirical basis for building nonlinear models.

As an emerging market, the Chinese stock market becomes more and more efficient over time through some importantreforms, especially the shareholder structure in listed companies. As shown in Fig. 5, the scaling exponent is close to 0.5,slowly after the shareholder structure in listed companies. However, the Chinese stock market is still not mature as it iseasily affected by market external factors, such as regulated policies imposed by the government, financial crises and so on.In this paper, we can see that the cross-correlations exponents between the Chinese stock market and surrounding stockmarkets seem to be smaller than 0.5, which is mainly caused by the global crisis (see Fig. 5). To improve the efficiency of theChinese stockmarket andweaken the cross-correlations between the Chinese stockmarket and surrounding stockmarkets,the government needs to do a great deal of work in the future.

6. Conclusion

In this paper, we used multifractal detrended cross-correlation analysis to investigate the cross-correlation propertiesbetween the stockmarkets in China, Japan, South Korea and Hong Kong respectively. Some conclusions can be found below:

First, we used not only qualitative analysis of the cross-correlation test, but also the quantitative analysis of the MF-X-DFA, confirming that the stock markets in China, Japan, South Korea and Hong Kong respectively are cross-correlated tofurther study their multifractal features.

Second, we use the log–log plots of Fq(s) versus time scaling s to judge the ‘‘crossover’’. An interesting phenomenon isfoundwhere the ‘‘crossover’’ is surrounding the point of log(s∗) = 2.2 for the stockmarkets in China, Japan, South Korea andHong Kong respectively, which can possibly explain that the stock markets in Japan, South Korea and Hong Kong could beinfluenced by each other. We find that the cross-correlations display the characteristic of multifractality in the short term.Moreover, the cross-correlations of small fluctuations are persistent, and those of large fluctuations are anti-persistent inthe short term, while the cross-correlations of all kinds of fluctuations are persistent in the long term. Furthermore, basedon the multifractal spectrum, we also find that the multifractality of cross-correlation between stock markets in China andJapan are stronger than those between China and South Korea, as well as between China and Hong Kong.

Finally, when q < 0, the cross-correlation exponents for Chinese, Japanese, South Korean and Hong Kong stock marketsrespectively are smaller than their average exponents, but larger than the average exponents when q > 0.

Acknowledgments

We sincerely thank the main Editor H.E. Stanley and the three anonymous reviewers for their helpful comments andsuggestions. The author is grateful for the financial support from the National Natural Science Foundation of China (Nos.70771097, 71071131 and 71090402), the Program for New Century Excellent Talents in University (No. NCET-08-0826), theProgram for Changjiang Scholars and Innovative Research Teams in Universities (No. PCSIRT0860) and the FundamentalResearch Funds for the Central Universities (Nos. SWJTU11ZT30 and SWJTU11CX137), the doctoral program of highereducation fund special research project (20120184110020).

References

[1] H.E. Hurst, Long term storage capacity of reservoirs, Transactions of the American Society of Civil Engineers 116 (1951) 770–808.[2] H.E. Hurst, A suggested statistical model of some time series which occur in nature, Nature 180 (1957) 494.[3] B.B. Mandelbrot, J.W. Van Ness, Fractional Brownian motions, fractional Brownian noises and applications, SIAM Review 10 (1968) 422–437.[4] E.E. Peter, Fractal Market Analysis Applying Chaos Theory to Investment and Economics, John Wiley & Sons, New York, 1994.[5] D.O. Cajueiro, B.M. Tabak, Evidence of long range dependence in Asian equity markets: the role of liquidity and market restrictions, Physica A 342

(2004) 656–664.[6] J.T. Barkoulas, C.F. Baum, N. Travlos, Long memory in the Greek stock market, Applied Financial Economics 10 (2000) 177–184.[7] A.W. Lo, Long term memory in stock market price, Econometrica 59 (1991) 1279–1313.[8] C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simon, H.E. Stanley, A.L. Coldberger, Mosaic organization of DNA nucleotide, Physical Review E 49 (1994)

1685–1689.[9] J.W. Kantelhardt, S.A. Zschiegner, E. Koscienlny-Bunde, S. Havlin, Multifractal detrended fluctuation analysis of nonstationary time series, Physica A

316 (2002) 87–114.[10] M.R. Eldridge, C. Bernbarde, I. Mulvey, Evidence of Chaos in the S&P 500 cash index, Advances in Futures and Options Research 6 (1993) 179–192.[11] M.T. Greene, B.D. Fieltz, Long term dependence in common stock returns, Journal of Financial Economics 4 (1997) 249–339.

1670 F. Ma et al. / Physica A 392 (2013) 1659–1670

[12] F.G. Schmitt, L. Ma, T. Angounou, Multifractal analysis of the dollar–yuan and euro–yuan exchange rates before and after the reform of the peg,Quantitative Finance 11 (2011) 505–513.

[13] D.H. Wang, X.W. Yu, Y.Y. Suo, Statistical properties of the yuan exchange rate index, Physica A 391 (2012) 3503–3512.[14] Y.D. Wang, Y. Wei, C.F. Wu, Analysis of the efficiency andmultifractality of gold markets based onmultifractal detrended fluctuation analysis, Physica

A 390 (2011) 297–308.[15] B. LeBaron, W.B. Arthur, R. Palmer, Time series properties of an artificial stock market, Journal of Economic Dynamics & Control 23 (1999) 1487–1516.[16] S. Arianos, A. Carbone, Cross-correlation of long-range correlated series, Journal of Statistical Mechanics: Theory and Experiment (2009) P03037.[17] AkihikoUtsugi, KazusumiIno,Masaki Oshikawa, Randommatrix theory analysis of cross-correlations in financialmarkets, Physical ReviewE 70 (2004)

026110.[18] B. Podobnik, D. Horvatic, A.L. Ng, H.E. Stanley, P.C. Ivanov, Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes,

Physica A 387 (2008) 3954–3959.[19] Boris Podobnik, DavorHorvatic, Alexander Petersen, H. Eugene Stanley, Cross-correlations between volume change and price change, Proceedings of

the National Academy of Sciences of the USA 106 (2009) 22079–22084.[20] B. Podobnik, D. Wang, D. Horvatic, I. Grosse, H.E. Stanley, Time-lag cross-correlations in collective phenomena, Europhysics Letters 90 (2010) 68001.[21] Duan Wang, Boris Podobnik, DavorHorvatic, H. Eugene Stanley, Quantifying and modeling long-range cross correlations in multiple time series with

applications to world stock indices, Physical Review E 83 (2011) 046121.[22] B. Podobnik, D.F. Fu, H.E. Stanley, P. ChIvanov, Power-law autocorrelated stochastic processes with long-range cross-correlations, European Physical

Journal B 56 (2007) 47–52.[23] B. Podobnik, H.E. Stanley, Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series, Physical Review Letters

100 (2008) 084102.[24] W.X. Zhou, Multifractal detrended cross-correlation analysis for two nonstationary signals, Physical Review E 77 (2008) 066211.[25] L.-Y. He, S.-P. Chen, Multifractal detrended cross-correlation analysis of agricultural futures markets, Chaos, Solitons and Fractals 44 (2011) 355–361.[26] L.-Y. He, S.-P. Chen, Nonlinear bivariate dependency of price–volume relationships in agricultural commodity futures markets: a perspective from

multifractal detrended cross-correlation analysis, Physica A 390 (2011) 297–308.[27] N. Xu, P. Shang, S. Kamae, Modeling traffic flow correlation using DFA and DCCA, Nonlinear Dynamics 61 (2010) 207–216.[28] Xiaojun Zhao, Pengjian Shang, Aijing Lin, Gang Chen, Multifractal Fourier detrended cross-correlation analysis oftraffic signals, Physica A 390 (2011)

3670–3678.[29] G.F. Zebende, P.A. da Silva, A. Machado Filho, Study of cross-correlation in a self-affine time series of taxi accidents, Physica A 390 (2011) 1677–1683.[30] S. Hajian, M. SadeghMovahed, Multifractal detrended cross-correlation analysis of sunspot numbers and river flow fluctuations, Physica A 389 (2010)

4942–4957.[31] DavorHorvatic, H. Eugene Stanley, Boris Podobnik, Detrended cross-Correlation analysis for non-stationary time series with periodic trends,

Europhysics Letters 94 (2011) 18007.[32] Y.D. Wang, Y. Wei, C.F. Wu, Cross-correlations between Chinese A-share and B-share markets, Physica A 389 (2010) 5468–5478.[33] G.X. Cao, L.B. Xu, J. Cao, Multifractal detrended cross-correlations between the Chinese exchange market and stock market, Physica A 391 (2012)

4855–4866.[34] Z.H. Li, X.S. Lu, Cross-correlations between agricultural commodity futures markets in the US and China, Physica A 391 (2012) 3930–3941.[35] Y. Yuan, X. Zhuang, Z. Liu, Price–volume multifractal analysis and its application in Chinese stock markets, Physica A 391 (2012) 3484–3495.[36] Z.Q. Jiang, W.X. Zhou, Multifractal detrending moving average cross-correlation analysis, Physical Review E 84 (1) (2011) 016106.[37] G.F. Gu, W.X. Zhou, Detrending moving average algorithm for multifractals, Physical Review E 82 (2010) 011136.[38] E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Second-order moving average and scaling of stochastic time series, European Physical Journal B 27

(2002) 197–200.[39] L. Kristoufek, Multifractal height cross-correlation analysis: a newmethod for analyzing long-range cross-correlations, Europhysics Letters 95 (2011)

68001.[40] A.-L. Barabasi, T. Vicsek, Multifractality of self-affine fractals, Physica Review A 44 (1991) 2730–2733.[41] L. Hedayatifar, M. Vahabi, G.R. Jafari, Coupling detrended fluctuation analysis for analyzing coupled nonstationary signals, Physical Review E 84 (2011)

021138.[42] B. Podobnik, I. Grosse, D. Horvatic, S. Ilic, P.C. Ivanov, H.E. Stanley, Quantifying cross-correlations using local and global detrended approaches, The

European Physical Journal B 17 (2009) 243–250.[43] B. Podobnik, Z.Q. Jiang, W.X. Zhou, H.E. Stanley, Statistical tests for power-law cross-correlated processes, Physical Review E 84 (2011) 066118.[44] G.M. Ljung, G.E.P. Box, On a measure of a lack of fit in time series models, Biometrika 65 (1978) 297–303.[45] D. Grech, G. Pamula, Multifractal background noise of monofractal signals, Actaphysica Polonica A121 (2012) B-34.[46] D. Grech, G. Pamula, How much multifractality is included in monofractal signals? arXiv:1108.1951v2 [physics.data-an].[47] Y.D.Wang, L.L. Liu, R.B. Gu, Analysis of efficiency for Shenzhen stockmarket based onmultifractaldetrended fluctuation analysis, International Review

of Financial Analysis 18 (2009) 271–276.[48] J. Alvarez-Ramirez, J. Alvarez, E. Rodriguez, Short termpredictability of crude oilmarkets: a detrended fluctuation analysis approach, Energy Economic

30 (2008) 2645–2656.[49] D. Grech, Z. Mazur, Can one make any crash prediction in finance using the local Hurst exponent idea? Physica A 336 (2004) 133–145.[50] D. Grech, G. Pamuła, The local Hurst exponent of the financial time series in the vicinity of crashes on the Polish stock exchange market, Physica A

387 (2008) 4299–4308.[51] D.O. Cajueiro, B.M. Tabak, Long-range dependence and multifractality in the term structure of LIBOR interest rates, Physica A 373 (2007) 603–614.