Physica A 391 (2012) 3503–3512

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Statistical properties of the yuan exchange rate indexDong-Hua Wang a,b,∗, Xiao-Wen Yu a, Yuan-Yuan Suo a

a School of Business, East China University of Science and Technology, Shanghai 200237, Chinab Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China

a r t i c l e i n f o

Article history:Received 24 October 2011Received in revised form 14 January 2012Available online 8 February 2012

Keywords:EconophysicsExchange rate indexProbability distributionLong-range correlationsMultifractal analysis

a b s t r a c t

We choice the yuan exchange rate index based on a basket of currencies as the effectiveexchange rate of the yuan and investigate the statistical properties of the yuan exchangerate index after China’s exchange rate system reform on the 21st July 2005. After dividingthe time series into two parts according to the change in the yuan exchange rate regimein July 2008, we compare the statistical properties of the yuan exchange rate index duringthese twoperiods.We find that the distribution of the two return series has the exponentialform.We also perform the detrendingmoving average analysis (DMA) and themultifractaldetrending moving average analysis (MFDMA). The two periods possess different degreesof long-range correlations, and the multifractal nature is also unveiled in these two timeseries. Significant difference is found in the scaling exponents τ(q) and singularity spectraf (α) of the two periods obtained from the MFDMA analysis. Besides, in order to detectthe sources of multifractality, shuffling and phase randomization procedures are appliedto destroy the long-range temporal correlation and fat-tailed distribution of the yuanexchange rate index respectively. We find that the fat-tailedness plays a critical role in thesources of multifractality in the first period, while the long memory is the major cause inthe second period. The results suggest that the change in China’s exchange rate regime inJuly 2008 gives rise to the different multifractal properties of the yuan exchange rate indexin these two periods, and thus has an effect on the effective exchange rate of the yuan afterthe exchange rate reform on the 21st July 2005.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

China is poised to set a new record in foreign trade in 2011 with its imports and exports adding up to 3.31 trillion USdollars till November. The European Union remained China’s top trading partner in the first 11 months and accounted for15.62% of China’s total foreign trade. The United States was ranked second with a 12.25% share of China’s imports andexports volume, followed by the Association of Southeast Asian Nations (ASEAN) with 9.94%, Japan with 9.43%, Hong Kongwith 7.69% and South Korean with 6.79% respectively. Along with the further reform and opening-up in China, the euro,Great British pound and Japanese yen also act as the invoicing and settlement currencies in China’s foreign trade besidesthe US dollar while the trade contacts between China and non-dollar countries becomemore andmore frequent. Because ofthe currency diversification in the foreign trade and investment, the yuan–dollar bilateral exchange rate can not reflect theactual level of the yuan exchange rate. For the purpose of investigating the actual level of the yuan exchange rate accuratelyand comprehensively, we shouldmove the focus from the yuan–dollar bilateral exchange rate to the effective exchange rateof the yuan.

∗ Correspondence to: 130Meilong Road, P.O. Box 114, School of Business, East China University of Science and Technology, Shanghai 200237, China. Tel.:+86 21 64253507.

E-mail address: dhwang@ecust.edu.cn (D.-H. Wang).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.01.054

3504 D.-H. Wang et al. / Physica A 391 (2012) 3503–3512

China has experienced several reforms in its exchange rate regime which has become a hot economic issue. On the 21stJuly 2005, the People’s Bank of China (PBC) announced a 2.1% appreciation of the yuan against the dollar and reformed theexchange rate regime by moving into a managed floating exchange rate regime based on market supply and demand withreference to a basket of currencies (see ‘‘Public announcement of the People’s Bank of China on reforming the RMB exchangerate regime’’, 21st July 2005, text issued by the People’s Bank of China). Since the modification, the yuan was pegged to abasket of currencies with a possible slow revaluation rather than solely pegged to dollar. In July 2008, China halted theyuan’s rise to cope with the global economic crisis. Since then, the yuan has been held at about 6.83 per dollar. On the 19thJune 2010, the PBC decided to proceed further with reform of the yuan exchange rate regime and to enhance its flexibility(see ‘‘Further reform the RMB exchange rate regime and enhance the RMB exchange rate flexibility’’, 19th June 2010, textissued by the People’s Bank of China). It means that the yuan has been pegged to a basket of currencies again. The exchangerate regime reformsmay affect the effective exchange rate of the yuan.Wewonder whether the change in China’s exchangerate regime in July 2008 has had an effect on the effective exchange rate of the yuan after the exchange rate reform on the21st July 2005.

The effective exchange rate of the yuan is aweighted average of the bilateral exchange rates of China and itsmajor tradingpartners, with the weight for each trading partner given by its share in trade. We choice the CIB-CNY Composite Index (CCI)which is a trade weighted index and compiled by the China Industrial Bank Research as the effective exchange rate. In orderto study the impact of the exchange rate policy modification in July 2008 on the effective exchange rate of the yuan after thereform on the 21st July 2005, we divide the time series (from 21st July 2005 to 18th June 2010) into two periods. Period D1starts from 21st July 2005 to 30th June 2008, and period D2 from 1st July 2008 to 18th June 2010.We compare the statisticalproperties of the CCI during Periods D1 and D2 to show this effect of the change in China’s exchange rate regime in July 2008on the effective exchange rate of the yuan.

The distribution is the commonly studied property of the exchange rate fluctuations in foreign exchange markets.Empirical analyses addressed that the distributions of fluctuations have diverse functional forms, such as the power laws[1–3], Gaussian function [4], and superimposed Gaussian function [5]. The scaling behavior in the fluctuations of exchangerates has been investigatedwith the detrended fluctuation analysis (DFA) [4,6–8]. For some foreign exchangemarkets, it hasbeen found that the exchange rate variations possess themultifractal nature by using themultifractal detrended fluctuationanalysis (MFDFA) [9], structure functions [10–13], and multifractal model of asset returns (MMAR) [14]. On the otherhand, a number of authors have explored exchange rate returns at various time horizons considering the seasonal factors[15–17]. Selçuk et al. also took into account those seasonal factors in the stock market returns [18]. For the high frequencydata set, there is strong seasonality due to the daily and weekly cycles of human activities. However, Fisher et al. providedempirical evidence that themultifractal properties in the exchange rate return series are fairly robust to seasonal adjustmentmethods [14].

Recently, more sophisticated methods in statistical and nonlinear physics have been applied to study the structure anddynamics in complex systems. Among them, a large number of methods have been successfully used to investigate themultifractal properties of the real time series fromdifferent financialmarkets. A popularmethod is the detrended fluctuationanalysis (DFA), which was originally invented to investigate the long-range dependence in coding and noncoding DNAnucleotides sequence [19]. The DFA algorithm was extended to multifractal detrended fluctuation analysis (MFDFA) toanalyze themultifractal time series [20]. These DFA andMFDFAmethodswere also generalized to analyze high-dimensionalfractals and multifractals [21].

The detrending moving average (DMA) algorithm is based on the moving average or mobile average technique [22].It was first proposed to estimate the Hurst exponent of self-affinity signals [23] and further developed to the detrendingmoving average by considering the second-order difference between the original signal and itsmoving average function [24].Carbone and co-workers extended the one-dimensional DMAmethod to higher dimensions to estimate the Hurst exponentsof higher-dimensional fractals [25,26]. Recently, the DMAmethod was extended to multifractal detrending moving average(MFDMA) and higher-dimensional versions to analyze multifractal time series [27].

The detrended cross-correlation analysis (DCCA) was proposed to investigate the long-range power-law crosscorrelations between two nonstationary time series [28–31]. The DCCA method was generalized to the multifractaldetrended cross-correlation analysis (MFDCCA), such as the MFDCCA based on the detrended fluctuation analysis(MFXDFA) [32] and the MFDCCA based on the detrending moving-average analysis (MFXDMA) [33], to analyze multifractalfeatures in the two nonstationary cross-correlated signals [34–37].

The remainder of the paper is organized as follows. Section 2 shows the data description. In Section 3, we investigatethe basic statistical properties of the yuan exchange rate index based on a basket of currencies. Sections 4 and 5 present theanalysis of the long-range correlations and multifractal properties of the yuan exchange rate index based on the DMA andMFDMA methods. Discussion and conclusions are provided in Section 6.

2. Data description

The CIB-CNY Composite Index (CCI) considered in this work is compiled by the China Industrial Bank Research. It is basedon a basket of currencies including the Australian dollar (AUD), Argentine Peso (ARP), Canadian dollar (CAD), New Taiwandollar (TWD), euro (EUR), Hong Kong dollar (HKD), Indonesian Rupiah (IDR), Japanese yen (JPY), South Korean won (KRW),Malaysian ringgit (MYR), Russian rouble (RUB), Singapore dollar (SQD), Thai baht (THB), Great British pound (GBP), and USdollar (USD). By estimating the weights of the currencies in the basket and taking 3rd January 2005 as the base day (that

D.-H. Wang et al. / Physica A 391 (2012) 3503–3512 3505

100

105

110

115

120

125

130

135

t

P(t

)30/06/2008

a

0.015

0.01

0.005

0

0.005

0.01

0.015

t

r(t)

b30/06/2008

2006 2007 2008 2009 2010 2011 2006 2007 2008 2009 2010 2011

Fig. 1. The CIB-CNY Composite Index (a) and the daily log return (b) (21 July 2005–18 June 2010).

Table 1Basic statistics of the CIB-CNY Composite Index returns in the two periods.

D1 D2

Minimum −0.0070 −0.0135Maximum 0.0066 0.0125Mean 0.000114 0.000110Standard deviation 0.0020 0.0034Skewness −0.0760 −0.1515Kurtosis 3.2512 4.3889

is, on the 3rd January 2005, the index is equal to 100), the CIB-CNY Composite Index is calculated. The CIB-CNY CompositeIndex evolves with the fluctuations of the basket currencies. Fig. 1(a) shows the trajectory of the index from 21st July 2005to 18th June 2010.

In this work, we will analyze the return series of the daily CIB-CNY Composite Index since the reform in 2005. DenoteP(t) as the exchange rate index on trading day t , and then the daily log return r on day t can be defined as follows

r(t) = ln[P(t)] − ln[P(t − ∆t)], i = 1, 2, . . . ,N, (1)where the time interval ∆t = 1 day.

According to the change in the yuan exchange rate regime in July 2008, we divide the time series into two periods. PeriodD1 starts from 21st July 2005 to 30th June 2008, and period D2 from 1st July 2008 to 18th June 2010. So, the data set includes1270 data points. Fig. 1(b) shows the daily log return of the index from 21st July 2005 to 18th June 2010. A striking featureis observed that the amplitude of the envelope in D2 increases, which indicates that the CIB-CNY Composite Index in D2 ismore volatile.

The basic statistics of the CIB-CNY Composite Index variations in the two periods D1 and D2 are presented in Table 1. Wecan obtain that both of the variations in D1 and D2 possess peakedness and fat tails. The different standard deviations alsoindicate that the CIB-CNY Composite Index in D2 fluctuates more fiercely.

3. Probability distribution of CCI returns

In the past few years most econophysicists have focused their minds on the statistical properties by analyzing the pastdata through different physical conceptions and models, like scaling power and power law [9,12,38–42,3]. In the exchangerate market, such as Japan, the yen–dollar exchange rate shows a power law with anomalous scaling exponents whichare 0.92 (1 min) and 0.78 (10 min) [3]. Andersen et al. found that the distributions of deutschemark and yen returnsagainst the dollar are skewed to the right and leptokurtic, but that the distributions of logarithmic standard deviationsand correlations are approximately Gaussian [4]. The probability distribution function (PDF) of the daily fluctuation of theyuan–dollar exchange rate after 21st July 2005 is asymmetric, and also shows us that the negative part decays slower thanthe positive part [12]. In the stockmarket, the central parts of return ofNYSE stocks andNasdaq stocks have the above results,and further they give that the probability distribution follows the exponential distribution [39,40], while the tails of thoseensemble returns cannot bemodeled by the same exponential distribution as the center part [41]. The empirical probabilitydensity function of the ensemble returns which were traded in the Shanghai Stock Exchange (SHSE) and Shenzhen StockExchange (SZSE) in the period from February 1994 to September 2004 follows the exponential distribution, which showsus that the slopes are 76.1 ± 3.3 and 83.1 ± 3.5 respectively, when the ensemble returns are in the range [−0.06, 0] and[0, 0.06]. This can further tell us that the distribution is also asymmetric [42].

In order to obtainmore information of the exchange rate index, we also display the empirical probability density functionf (r) of the returns r . We find that the main part of function f (r) has the following form

f (r) ∼ exp(−kDi±|r|), i = 1, 2. (2)

3506 D.-H. Wang et al. / Physica A 391 (2012) 3503–3512

The probability distribution function f (r) is presented in Fig. 2. Using the list-squares fitting method, we obtain theexponents kD1− = 703.20 ± 35.16, kD2− = 444.05 ± 22.20, in the range −0.01 < r < 0 for the negative returns, andkD1+ = 813.95 ± 40.70, kD2+ = 438.18 ± 21.91, in the range 0 < r < 0.01 for the positive ones. Judging from Fig. 2, wealso find that the fluctuations of CIB-CNY Composite Index in D2 are more volatile.

The similar results have been found in early works. Lee et al. and Canning et al. reported that distribution of annualgrowth rates for countries of a given GDP decays with ‘‘fatter’’ tails than for a Gaussian [43,44]. For US firms with similarsales, the distribution of growth rates has an exponential form [45]. Podobnik et al. studied annual logarithmic growth ratesof various economic variables and found that the distributions can be approximated by exponential (Laplace) distributionsin the central parts and power-law distributions in the tails [46].

4. Long-range correlations

4.1. The DMA algorithm

Many techniques have been proposed to investigate the presence of long-range correlations in time series. Among them,the most frequently used methods are the rescaled range analysis (R/S) [47–49], the detrended fluctuation analysis (DFA)[19,50–53], and very recently the detrending moving average (DMA) analysis [23,24,54]. In this paper, we apply the DMAalgorithm to detect the long-range correlations in the yuan exchange rate index.

For the sake of clarity, wewill briefly review theDMAalgorithm. TheDMAalgorithmhas been first introduced to estimatetheHurst exponent of self-affine signals [23] and further developed to the detrendingmoving average (DMA) algorithm [24].

First, at time t , the cumulative sum is calculated as follows

y(t) =

ti=1

r(i), t = 1, 2, . . . ,N. (3)

The moving average function is determined as follows Ref. [55],

y(t) =1n

⌈(n−1)(1−θ)⌉k=−⌊(n−1)θ⌋

y(t − k), (4)

where the position parameter θ varies from 0 to 1, n is the length of each sliding window, ⌊x⌋ is the largest integer notgreater than x, and ⌈x⌉ is the smallest integer not less than x. Varying θ , the moving average functiony(t) contains differentinformation. θ = 0 refers to the backward moving average which depends only on the past points of the time series [56,27], θ = 0.5 refers to the centered moving average which is obtained by half-past and half-future data points [56,27], andθ = 1 refers to the forward moving average which depends only on the future data points [27]. We use θ = 1 in this work.

The residual series is obtained by subtracting the trendy(i) from y(i),ϵ(i) = y(i) −y(i), (5)

where n − ⌊(n − 1)θ⌋ 6 i 6 N − ⌊(n − 1)θ⌋.Next, divide the residual series intoNn parts of equal size n, whereNn corresponds to the integer part of (N/n−1). Define

each part as ϵυ , so that ϵυ(i) = ϵ(l + i), for 1 6 i 6 n, where l = (υ − 1)n. Calculate the root-mean-square function Fυ(n)with the size n

[fυ(n)]2 =1n

ni=1

[ϵυ(i)]2 . (6)

The overall detrended fluctuation function is estimated as follows

[F(n)]2 =1Nn

Nni=1

[fυ(n)]2 . (7)

The power-law relation between the overall fluctuation function F(n) and n,

F(n) ∼ nH , (8)where H is the Hurst exponent.

The Hurst exponent takes values from 0 to 1 (0 ≤ H ≤ 1). If H = 0.5, the time series is uncorrelated. If 0 < H < 0.5,the time series is anti-persistent with negative autocorrelation. If 0.5 < H < 1, the time series is persistent with positiveautocorrelation.

4.2. Results

Fig. 3 plots theDMA fluctuation function F(n) of the return series of the CIB-CNYComposite Index. Nice power-law scalingrelations are observed.We obtain that DMA scaling exponents ofD1 andD2 areH1 = 0.598±0.013 andH2 = 0.626±0.012.

D.-H. Wang et al. / Physica A 391 (2012) 3503–3512 3507

Fig. 2. (Color online) Plot of the empirical probability density function f (r) in the main range for both the negative and positive returns (for two parts ofdata). The diagonal crosses are plotted for the first part of data (D1), and squares for the second part (D2). The solid lines are the least squares fits to thedata in the ranges [−0.01, 0] and [0, 0.01], respectively. The exponents are kD1− = 703.20 ± 35.16, kD2− = 444.05 ± 22.20 for the negative returns, andkD1+ = 813.95 ± 40.70, kD2+ = 438.18 ± 21.91 for the positive ones.

Fig. 3. (Color online) Detection of long-range correlations in the return time series of the CIB-CNY Composite Index using the detrending moving averageanalysis.

We shuffle the return time series of the CIB-CNY Composite Index and perform the DMA analysis. The results are alsoillustrated in Fig. 3. We find that, for the shuffled data, the DMA scaling exponents of D1 and D2 are H ′

1 = 0.500 ± 0.009and H ′

2 = 0.494 ± 0.009. According to Fig. 3, it is obvious that the CCI return series of D1 and D2 exhibit varying degrees oflong-range correlations, while the shuffled series behave similarly to the uncorrelated ones.

In previous works, the currency exchange rates in Chile, India, Korea, Australia, Canada, the United States and so on showpositive autocorrelation. The exponents of these exchange rates are larger than 0.5 [57,58]. On the contrary, Tsonis et al.found the dollar–yen exchange rate is anti-persistent at time scales shorter than 10min and exhibits pure Brownianmotionfor time scales longer than 10 min [59].

4.3. Distribution of the Hurst exponents of shuffled data

In order to further study the long-range correlations in D1 and D2, we will compare the Hurst exponents of originaland shuffled data. We shuffle the return series and get the Hurst exponent of shuffled data using the DMA method. Afterrepeating the above 10,000 times, we obtain 10,000 Hurst exponents of shuffled data. We present the basic statistics of theHurst exponents of shuffled data in Table 2 and plot the empirical probability density function of them in Fig. 4.

Compared with D1, the mean of the Hurst exponents of shuffled data in D2 approximates to 0.5 and the differencesbetween the Hurst exponents of original and shuffled data in D2 are greater. It provides the evidence of higher degree oflong-range correlations in D2. The similar Hurst exponents of original and shuffled data in D1 illustrate that the degree oflong-range correlations in D1 is weaker than that in D2. The dash areas in Fig. 4 are the probabilities of the Hurst exponentsof shuffled data which are greater than the Hurst exponents of original data (that is, H1 = 0.598 and H2 = 0.626). We canget P(x > H1) = 0.193 from the Fig. 4(a) and P(x > H2) = 0.083 from the Fig. 4(b), which also indicates that the Hurstexponents of original and shuffled data in D2 are more significantly different and the period D2 possesses a higher degree oflong-range correlations than D1.

3508 D.-H. Wang et al. / Physica A 391 (2012) 3503–3512

Table 2Basic statistics of the Hurst exponents of shuffled data.

D1 D2

Minimum 0.3133 0.1989Maximum 0.7677 0.7740Mean 0.5440 0.5141Standard deviation 0.0631 0.0823Skewness −0.1031 −0.1523Kurtosis 2.9846 2.8749

ba

Fig. 4. (Color online) Empirical probability density function of the Hurst exponents of shuffled data in D1 (a) and D2 (b). The black hollow circles representtheHurst exponents of shuffled data. The continuous red solid line is a normal distribution density function fit, and the red dashed line is theHurst exponentof original data in D1 (a) or D2 (b).

5. Multifractal analysis

In this section, we investigate whether the return series of the yuan exchange rate index possesses multifractal nature.

5.1. The MFDMA algorithm

In the recent days, there have beenmany studies showing that financial markets exhibit multifractal nature [10,11,60,61,20,62–70]. The DMA algorithm was extended to multifractal detrending moving average (MFDMA) to analyze multifractaltime series [27,71]. TheMFDMA technique has been applied to study the intertrade durations of a stock and its warrant [72],the US foreign exchange rates [73], and the cross correlations of stock index returns and volatilities [33]. For MFDMA, theqth-order fluctuation function is generalized to the following form [27],

Fq(n) =

1Nn

Nnυ=1

F qυ(n)

1q

, (9)

where q can take any real values except zero. When q = 0, according to L’Hôspital’s rule, there is

ln[F0(n)] =1Nn

Nnυ=1

ln[Fυ(n)]. (10)

If the time series r(t) possesses scaling properties, there is a power-law relation between Fq(n) and n,

Fq(n) ∼ nh(q), (11)

where h(q), the generalized Hurst exponent, is called theMFDMA exponent. The h(q) exponent is related to themultifractalscaling exponent τ(q) by

τ(q) = qh(q) − 1. (12)

If the multifractal exponent τ(q) is a nonlinear function of q, the time series has multifractal nature. The singularitystrength function α(q) which characterizes the singularities of a time series and the multifractal spectrum f (α) whichdescribes the singularity content of a time series can be obtained easily via the Legendre transform

α(q) = h(q) + qh′(q)f (q) = q[α − h(q)] + 1. (13)

D.-H. Wang et al. / Physica A 391 (2012) 3503–3512 3509

a b

c d

Fig. 5. (Color online) Multifractal analysis of the CIB-CNY Composite Index return series obtained from the MFDMA method in D1 and D2 . We show thefluctuation functions Fq(n) (a) of the raw return series, the generalized Hurst exponents h(q) (b), the scaling exponents τ(q) (c) and themultifractal spectraf (α) (d) of the raw, shuffled and surrogated series.

5.2. Results

We now apply the MFDMAmethod to analyze the multifractal properties of the CIB-CNY Composite Index return series.The results are illustrated in Fig. 5. It is found that both periods exhibit multifractal nature.

The fluctuation functions Fq(n) with different scales n are calculated for q = −4, q = 0, and q = 4. Fig. 5(a) showsthe fluctuation functions for the two periods D1 and D2. We can find that the fluctuation function and the scale n have apower-law relation. Fig. 5(b) shows that the generalized Hurst exponents h(q) decrease with q.

The scaling exponents τ(q) of multifractal time series are nonlinear, while τ(q) of monofractal ones exhibit linearity.Moreover, the nonlinearity of the τ(q) curve indicates the strength of multifractality. For the visualization of the scalingproperties, the corresponding τ(q) curves of different time series D1 and D2 are shown in Fig. 5(c). We observe thatboth curves are nonlinear, which confirms the existence of multifractality in the CIB-CNY Composite Index return series.Furthermore, the nonlinear properties of the curves D1 and D2 are not the same, which implies the different degrees ofmultifractal strength in D1 and D2.

In Fig. 5(d) we present the multifractal spectra f (α) with respect to the singularity strength α for D1 and D2. Definethe width of the multifractal singularity strength function as ∆α, ∆α = αmax − αmin, which also indicates the degree ofmultifractality. If the value of∆α gets close to zero, the time series is nearly monofractal while if∆α is large, the time serieshas multifractal nature. We obtain from Fig. 5(d) that the ∆α values of D1 and D2 are 0.3476 and 0.4962 respectively. Theresults also provide the evidence of the different multifractality degrees in D1 and D2.

All of the above properties show the multifractal nature in the return series of the CIB-CNY Composite Index. However,the periods D1 and D2 exhibit the different degrees of multifractality. According to the τ(q) curves and the values of ∆α, theperiod D2 has stronger multifractality, which may be caused by the change in the yuan exchange rate regime in July 2008between the two periods.

5.3. Origin of the multifractality

In Section 5.2, we find that the CIB-CNY Composite Index return series possessmultifractality.Wewill further investigatethe origin of the multifractality. The sources of multifractal nature in financial time series are the fat tails in the distribution

3510 D.-H. Wang et al. / Physica A 391 (2012) 3503–3512

Table 3Multifractality degrees of the original, shuffled and surrogated series.

∆h ∆α

Original Shuffled Surrogated Original Shuffled Surrogated

D1 0.2032 0.1744 0.0726 0.3476 0.3320 0.1500D2 0.2991 0.0408 0.2776 0.4962 0.0999 0.4565

and/or the long-range temporal correlation [20]. There are two popular methods applied to quantify the contributions ofthese two sources, shuffling and phase randomization. All correlations are destroyed by the shuffling procedure for thedata are put into random order, while the distributions remain unchanged [62]. The phase randomization procedure caneliminate the fat-tailed distribution, but the linear properties are preserved [74,75]. Furthermore, we employ twomeasures,∆h (∆h = h(q)max − h(q)min) and ∆α, to quantify the degree of multifractality. If the multifractality is caused by the long-range temporal correlation, the degree of multifractality becomes significantly weaker after the shuffling procedure butonly decreases slightly after the phase randomization procedure. On the contrary, if the source of the multifractality is thefat-tailed distribution, the multifractality degree of the surrogated series created by the phase randomization procedureweakens more remarkably than that of the shuffled series.

The generalized Hurst exponents h(q), the scaling exponents τ(q) and the multifractal spectra f (α) of the shuffledand surrogated series are also provided in Fig. 5. We find that the generalized Hurst exponents, scaling exponents andmultifractal spectra of the shuffled series in D1 change a little while those of the surrogated series are significantly differentcompared with the raw series. However, the h(q), τ(q) and f (α) of the shuffled series in D2 change markedly while thoseof the surrogated series are similar with the original series. Table 3 shows the multifractality degrees of the original,shuffled and surrogated series of the CIB-CNY Composite Index returns. We obtain that ∆h varies from 0.2032 (original)to 0.1744 (shuffled) and 0.0726 (surrogated), ∆α from 0.3476 (original) to 0.3320 (shuffled) and 0.1500 (surrogated) in D1.It indicates that the multifractality degree of the surrogated series in D1 decreases much more significantly than that of theshuffled series. For D2,∆h changes from 0.2991 (original) to 0.0408 (shuffled) and 0.2776 (surrogated), and∆α from 0.4962(original) to 0.0999 (shuffled) and 0.4565 (surrogated), which means the multifractality degree of the shuffled series in D2reduces much more markedly than that of the surrogated series. These results imply that the fat-tailedness plays a moresignificant role in the sources of the observed multifractality in D1 than that in D2, and the long memory is the major causeof multifractality in D2.

In earlier works which have studied the origin of multifractal nature in financial returns, different results are presented.Long memory plays an important role in the returns of the Iranian rial–US dollar exchange rate [9] and Shanghai StockExchange Composite (SSEC) Index [27], while the fat-tailedness in the distribution has a major impact on the daily returnsof the Dow Jones Industrial Average [76].

6. Conclusion

In this paper, we firstly study the statistical properties of the logarithmic returns of the CIB-CNY Composite Index, whichacts as the effective exchange rate of the yuan. In order to reveal the possible effect of the exchange rate policy modificationin July 2008 on the effective exchange rate of the yuan after the exchange rate regime reformon the 21st July 2005,we dividethe CIB-CNY Composite Index return series into two portions for comparison. We have observed that the logarithmic returnseries of D1 and D2 can be well modeled by the exponential distribution. The detrending moving average analysis showsthat the two return series possess positive autocorrelation, but the degrees of long-range correlations in the two periods aredifferent. With the MFDMA algorithm, the returns series of D1 and D2 exhibit the multifractal nature. We also employ theshuffling and phase randomization procedures to obtain the origins of the multifractality. The results provide the evidencethat the periods D1 and D2 possess the different degrees of multifractality and the fat-tailedness plays a major role in thesources of multifractality in D1, while the long memory is the significant cause of multifractality in D2. It indicates that thechange in the yuan exchange rate regime in July 2008 caused the different multifractal properties in D1 and D2. Thus, we canconclude that the exchange rate regime modification in July 2008 has an effect on the effective exchange rate of the yuanafter the reform on the 21st July 2005. The recent work [12] has centered on the statistical properties of the dollar–yuanand euro–yuan exchange rates before and after the change of the peg on the 21st July 2005. It concluded that the changeof the peg had exerted an influence on the dollar–yuan exchange rate, while the euro–yuan exchange rate was not affectedby the change. In this work, we focused on the CIB-CNY Composite Index based on a basket of currencies and used the timeseries after China’s exchange rate system reform on 21 July 2005.Moreover, this paper presented the result that the CIB-CNYComposite Index exhibit different multifractal properties when the regime changes.

Acknowledgments

We thank Gao-Feng Gu, Zhi-Qiang Jiang, Fei Ren, and Wei-Xing Zhou for helpful discussions. This research is supportedby the National Science Foundation of China (Grant No. 71171083) and the Humanities and Social Sciences Fund sponsoredby the Ministry of Education of the Peoples Republic of China (Grant No. 09YJC630075).

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