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Cross-correlations between Renminbi and four major currencies in the Renminbi currency basket
Physica A 392 (2013) 1418–1428
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Cross-correlations between Renminbi and four major currencies in the Renminbi currency basket Gang-Jin Wang a , Chi Xie a,b,∗ a
College of Business Administration, Hunan University, Changsha 410082, China
b
Center of Finance and Investment Management, Hunan University, Changsha 410082, China
article
info
Article history: Received 20 June 2012 Received in revised form 9 August 2012 Available online 29 November 2012 Keywords: Econophysics Cross-correlations Detrended cross-correlation analysis Rolling windows Renminbi exchange rate Renminbi currency basket
abstract We investigate the cross-correlations between Renminbi (CNY) and four major currencies (USD, EUR, JPY, and KRW) in the Renminbi currency basket, i.e., the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Qualitatively, using a statistical test in analogy to the Ljung-Box test, we find that cross-correlations significantly exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Quantitatively, employing the detrended cross-correlation analysis (DCCA) method, we find that the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are weakly persistent. We use the DCCA crosscorrelation coefficient ρDCCA to quantify the level of cross-correlations and find the currency weight in the Renminbi currency basket is arranged in the order of USD > EUR > JPY > KRW. Using the method of rolling windows, which can capture the time-varying crosscorrelation scaling exponents, we find that: (i) CNY and USD are positively cross-correlated over time, but the cross-correlations of CNY–USD are anti-persistent during the US subprime crisis and the European debt crisis. (ii) The cross-correlation scaling exponents of CNY-EUR have the cyclical fluctuation with a nearly two-year cycle. (iii) CNY–JPY has longterm negative cross-correlations, during the European debt crisis, but CNY and KRW are positively cross-correlated. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The Renminbi (RMB), verbatim translated as ‘‘the people’s currency’’, is the currency of China. RMB is also known as the Chinese Yuan (CNY), and its currency symbol is denoted as ‘‘U ’’. Recently, there have been two major events in the RMB exchange rate: (i) On July 21, 2005, after a decade-long of pegging RMB to US dollar (USD), the People’s Bank of China (PBoC, i.e., the China’s central bank) announced that the RMB exchange rate would become ‘‘adjustable, based on market supply and demand with reference to exchange rate movements of currencies in a basket’’ [1,2]. Then, the governor of China’s central bank, Xiaochuan Zhou, revealed details of the RMB currency basket that ‘‘dominant amongst a raft of currencies are USD, the euro (EUR), the Japanese yen (JPY) and South Korea’s won (KRW)’’. (ii) Actually, during the US sub-prime crisis, RMB was re-pegged to USD. On June 19, 2010, PBoC announced that, it has ‘‘decided to proceed further with reform of the RMB exchange rate regime and to enhance the RMB exchange rate flexibility’’ [3]. Along with the rapid economic growth of China, the RMB exchange rate (or RMB currency basket) has spurred a large amount of attention in academic and industrial circles, even in the political arena [4]. Financial markets are considered as complex dynamic systems with a great number of interacting agents [5–7]. The foreign exchange markets, which represent the largest and most liquid financial market in the world, are extremely
∗ Corresponding author at: College of Business Administration, Hunan University, Changsha 410082, China. Tel.: +86 731 88823890; fax: +86 731 88823670. E-mail address: [email protected] (C. Xie). 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.11.035
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important [8]. One of the considerable characteristics of market dynamics is the presence of cross-correlations between financial variables [9]. Hence, in this study, we seek to investigate the cross-correlations between RMB and the major currencies in the RMB currency basket. In previous studies, many different approaches have been proposed to quantify the cross-correlations between financial entries, such as various clustering methods [8,10–12], the random matrix theory [13,14], and the cross-sample entropy [15,16]. A common assumption of the above-stated methods is that both of the analyzed time series are stationary. However, in the real-world, the financial time series are usually heterogeneous and nonstationary [17]. To overcome this limitation, many methods are developed to examine the auto-correlation and cross-correlation using the mono- and multifractal theory in various fields. For a noisy and nonstationary time series, Peng et al. [18] proposed the detrended fluctuation analysis (DFA) method to study the fractal structure of the DNA nucleotides sequence. After that, DFA was widely used to determine longrange dependence and auto-correlation of the nonstationary time series [19–26]. Vandewalle and Ausloos [27] devised an alternative approach of DFA, which is termed as the detrending moving average (DMA) algorithm, to investigate the longrange correlations of nonstationary time series. Based on DFA, Podobnik and Stanley [28] proposed the detrended crosscorrelation analysis (DCCA) to quantify power-law cross-correlations between simultaneously recorded nonstationary time series. Then, DCCA becomes a versatile and powerful method to investigate the cross-correlations between the financial variables [9,29–31]. For instance, Podobnik et al. [29] analyzed 14, 981 daily data of the Standard and Poor’s 500 Index in the period between 1950–2009, and found power-law cross-correlations between volume change and price change by means of DCCA. Based on DCCA, Siqueira et al. [9] examined 1, 908 daily observations of different stocks and commodities on the Brazilian market from August 10, 2000 to April 30, 2008, and found strong cross-correlations of the volatility time series in the Brazilian stock and commodity market. Lin et al. [30] analyzed the cross-correlations behavior in US and Chinese stock markets using DCCA. Their results showed that cross-correlations in Chinese stock markets (i.e., Shanghai stock market and Shenzhen stock market) are stronger than that of between US and Chinese stock markets. Recently, Zebende [32] proposed a new detrended cross-correlation coefficient (i.e., DCCA cross-correlation coefficient, ρDCCA ), which was defined in terms of DFA and DCCA, to quantify the level of cross-correlation between nonstationary time series. Vassoler and Zebende [33] applied the coefficient ρDCCA to analyze and quantify cross-correlations between air temperature and air relative humidity. Podobnik et al. [34] examined the statistical significance of the coefficient ρDCCA , and proposed an additional statistical test to quantify the cross-correlations between two power-law correlated time series. In this paper, we examine the cross-correlations between RMB (CNY) and four major currencies (i.e., USD, EUR, JPY, and KRW) in the RMB currency basket. We first make a preliminary analysis of the 5 currencies (CNY, USD, EUR, JPY, and KRW) from July 21, 2005 to May 25, 2012. Next, we qualitatively analyze the cross-correlations using the cross-correlation statistics proposed by Podobnik et al. [35]. Then, we employ DCCA and DCCA cross-correlation coefficient ρDCCA to study the presence of cross-correlations quantitatively. Finally, we use the method of rolling windows to capture the dynamics of the cross-correlations. The remainder of this paper is organized as follows. In the next section, we provide the methodologies of DCCA and DCCA cross-correlation coefficient ρDCCA . In Section 3, we present the data set and make a preliminary analysis. We show the main empirical results and some relevant discussions in Section 4. Finally, in Section 5 we draw some conclusions. 2. Methodology 2.1. Detrended cross-correlation analysis Detrended cross-correlation analysis (DCCA) is used to investigate the cross-correlations between two nonstationary time series, which can be described as follows [28,30]: Step 1. Consider two time series {x(t )} and {y(t )} of the same length N, where t = 1, 2, . . . , N. We can determine the profile as two new series, X (t ) =
t
(x(i) − ⟨x⟩) ,
Y (t ) =
i=1
t
(y(i) − ⟨y⟩) ,
t = 1, 2, . . . , N .
(1)
i =1
Step 2. Both of the profiles {X (t )} and {Y (t )} are divided into Ns = int(N /s) non-overlapping segments of equal length s. In this study, we set 10 ≤ s ≤ N /4. Step 3. For each non-overlapping segment v (1 ≤ v ≤ Ns ), the local trending functions of {Xv (t )} and {Yv (t )} are calculated as {X˜ v (t )} and {Y˜v (t )} by a least-square fit of the data, respectively. Step 4. The cross-correlation fluctuation variance for each segment is calculated as follows [17]: Fv2 (s) =
s 1 Xv (t ) − X˜ v (t ) Yv (t ) − Y˜v (t ) . s t =1
(2)
Then we average over all segments to obtain the cross-correlation fluctuation function:
FDCCA (s) =
Ns 1
Ns v=1
1/2 F v ( s) 2
.
(3)
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Step 5. By observing the log–log plots FDCCA (s) vs. s, we can determine the scaling behavior of the fluctuation function. If the original series {x(t )} and {y(t )} are power-law cross-correlated, then FDCCA (s) ∝ sα ,
(4)
where the cross-correlation scaling exponent α can be obtained by the slope of log–log plot of FDCCA (s) vs. s via ordinary least squares (OLS) [9]. In general, there are three cases of α : (i) If α < 0.5, the cross-correlations between the two time series are anti-persistent (negative). This implies that if there is an increase of one price, then it is likely to be followed by a decrease of the other price [36], and vice versa. (ii) If α > 0.5, the cross-correlations between the two time series are persistent (positive). This means that if one price has been an increase or decrease, then it is likely to be followed by an increase or decrease of the other price, respectively [36]. (iii) If α = 0.5, there are no cross-correlations between the two time series, and the change of one price cannot affect the behavior of the other price [30,36]. Especially, if the time series {x(t )} is identical to {y(t )}, DCCA is equivalent to DFA, i.e., the detrended function FDCCA (s) reduces to FDFA (s):
FDFA (s) =
Ns 1
Ns v=1
1/2 Fv (s)
.
2
(5)
At this point, Fv2 (s) = 1/s t =1 (Xv (t ) − X˜ v (t ))2 . If the original series {x(t )} is power-law auto-correlated, the autocorrelation detrended function FDFA (s) vs. s follows a power law, FDFA (s) ∝ sα .
s
2.2. DCCA cross-correlation coefficient ρDCCA In order to quantify the level of cross-correlation, Zebende [32,33] proposed the DCCA cross-correlation coefficient ρDCCA , 2 defined as the ratio between the detrended covariance function FDCCA (s) of Eq. (3) and two detrended variance functions FDFA (s) of Eq. (5), i.e.,
ρDCCA =
2 FDCCA
FDFA{x(t )} FDFA{y(t )}
,
(6)
where ρDCCA is a dimensionless coefficient that ranges between −1 ≤ ρDCCA ≤ 1. If two time series are completely crosscorrelated (anti cross-correlated) then ρDCCA = 1(−1), and if there are no cross-correlations between the two time series then ρDCCA = 0. 3. Data and preliminary analysis We choose the daily foreign exchange (FX) rates of the five currencies (CNY, USD, EUR, JPY, and KRW) from July 21, 2005 to May 25, 2012. Keskin et al. [8] argued that ‘‘one of the problems in FX research is that currencies are priced against each other so no independent numeraire exists. . . Gold was considered, but rejected due to its high volatility’’. But Jang et al. [11] pointed out: ‘‘. . . Special Drawing Right (SDR), which is a potential claim on the freely usable currencies of International Monetary Fund (IMF) members’’. Therefore, we choose SDR as the numeraire. The daily FX rates data are provided by the Pacific Exchange Rate Service (http://fx.sauder.ubc.ca/data.html). Let Pi (t ) denote the price of FX rate of currency i on day t. The daily return of currency i, ri (t ), is calculated by the logdifference of the price Pi (t ), i.e., ri (t ) = ln(Pi (t )) − ln(Pi (t − 1)).
(7)
Fig. 1 provides the graphical representation of returns of the five currencies (i.e., CNY, USD, EUR, JPY, and KRW). One can find that a common large fluctuation zone of the five returns in Fig. 1: before, during and after October 2008, which may be the worst days of the US sub-prime crisis. We present the descriptive statistics of the five returns in Table 1. The mean values of the five returns are very close to zero, and quite small by comparison with the standard deviations. The standard deviation of the return series of EUR is the largest among the five returns. The high volatility of EUR may be affected by the European debt crisis. The Jarque–Bera statistics reject the null hypothesis of the Gaussian distribution at the 1% significance level. This phenomenon is also accompanied by non-zero skewness and kurtosis larger than three, which indicates that the five returns are fat-tailed. 4. Results and discussion 4.1. Cross-correlation test In order to quantify the cross-correlations between CNY and four major currencies in the RMB currency basket (henceforth, we denote the four pairs of cross-correlations as CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW, respectively),
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0.05 CNY 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
0.05 USD 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
0.05 Returns
EUR 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
0.05 JPY 0 −0.05 July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
−0.05 KRW 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08 Time (date)
May 06, 10
Dec. 09, 11
Fig. 1. Returns of the five currencies (i.e., CNY, USD, EUR, JPY, and KRW). Table 1 Descriptive statistics of returns of the five currencies.
−5
Mean (×10 ) Maximum Minimum S.D. (×10−3 ) Skewness Kurtosis Jarque–Bera (×103 ) Observations
CNY
USD
EUR
JPY
KRW
−5.1841
−1.0250
0.0107 −0.0049 1.4140 0.3921 5.9026 0.6460* 1715
0.0114 −0.0053 1.4063 0.4483 6.6064 0.9868* 1715
1.9112 0.0079 −0.0088 1.6509 −0.0038 6.0016 0.6438* 1715
7.2530 0.0138 −0.0214 0.0027 −0.4818 8.6907 2.3805* 1715
4.4365 0.0429 −0.0478 0.0039 −0.1727 33.3240 65.7178* 1715
Notes: S.D. stands for ‘‘Standard deviation’’. The Jarque–Bera statistics test for the null hypothesis of normality in the sample returns distribution. * Indicates rejection of the null hypothesis at the 1% significance level.
we employ a new cross-correlation test proposed by Podobnik et al. [35], which is in analogy to the Ljung–Box test [37] and widely used in financial markets [36,38–43]. For two time series, {x(t )|t = 1, 2, . . . , N } and {y(t )|t = 1, 2, . . . , N }, the cross-correlation statistic is defined by Qcc (m) = N 2
m C 2 (t ) , N −t t =1
(8)
where the cross-correlation function C (t ) is defined as N
x(k)y(k − t )
k=t +1
C (t ) =
N k=1
x2
(k)
N k=1
. y2
(k)
(9)
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105 4
10
Qcc(m)
103
Test statistics (CNY-USD) Test statistics (CNY-EUR) Test statistics (CNY-JPY) Test statistics (CNY-KRW) Critical values
102 101 m=42
100 10-1 100
102
101
103
m Fig. 2. Log–log plots of test statistics Qcc (m) vs. degrees of freedom m.
Podobnik et al. [35] indicated that, the cross-correlation statistic Qcc (m) is approximately χ 2 (m) distributed with m degrees of freedom. It can be used to ‘‘test the null hypothesis of none of the first m cross-correlation coefficients is different from zero’’ [35]. Fig. 2 shows the log–log plots of cross-correlation statistics Qcc (m) vs. degrees of freedom m for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW, where the degrees of freedom vary from 100 to 103 . To make a comparison, as illustrated in Fig. 2, we also present the critical values for the χ 2 (m) distribution at the 5% level of significance. In Fig. 2, when m ≥ 3, the cross-correlation test statistics of CNY–KRW are always larger than the critical values. It seems that for m < 42, Qcc (m) of CNY–USD, CNY–EUR, and CNY–JPY are all smaller than the critical values but close to them. Nevertheless, when m ≥ 42, all the statistics Qcc (m) have the trend of being near or equal to the critical values with m increasing, even larger than the critical values. Podobnik et al. [35] proposed that ‘‘if for a broad range of values of m the values of the test of Eq. (8) between the two time series are larger than the critical values of the χ 2 (m) distribution, the cross-correlations are considered significant’’. Hence, we can reject the null hypothesis of no cross-correlations. That is to say, cross-correlations significantly exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. 4.2. Cross-correlation analysis Podobnik et al. [35] suggested that the cross-correlation test of Eq. (8) can only be used to qualitatively demonstrate the existence of nonlinear cross-correlation; while DCCA can present a quantitative cross-correlation test by estimating the cross-correlation scaling exponent. Thus, in this subsection, we first use the DCCA to investigate the cross-correlations quantitatively for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. We show the log–log plots of FDCCA (s) vs. time scale s for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW in Fig. 3. The cross-correlation scaling exponents (i.e., the slopes of regression lines) are also presented in Fig. 3. One can find that all of the cross-correlation scaling exponents are greater than 0.5 but very close to 0.5, which implies that cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are weakly persistent (positive). Interestingly, it is CNY–EUR not CNY–USD, has the largest scaling exponent. Actually, when the four cross-correlation scaling exponents are very close, we can only use the scaling exponent to test the cross-correlation is either positive or negative, and cannot employ the scaling exponent to quantify the level of cross-correlation. Therefore, we then use the DCCA cross-correlation coefficient ρDCCA to quantify the level of cross-correlations. Fig. 4 plots the DCCA cross-correlation coefficient ρDCCA vs. time scale s for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. From Fig. 4, we can find that CNY–USD has the largest values of ρDCCA among the four pairs of currencies. As a comparison, we illustrate the descriptive statistics of the four ρDCCA series in Table 2. For the DCCA cross-correlation coefficient ρDCCA , Podobnik et al. [34] proposed an additional statistical test to quantify the cross-correlations between two power-law correlated time series. According to Table 1 in Ref. [34], the critical value for ρDCCA for each time scale s is not greater than 0.4 when T is 1000 (or 2000) [43]. From Table 2, we can find that the minimum of the four ρDCCA series is 0.5212. Hence, it is another piece of evidence that cross-correlations exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. In Table 2, the mean values of the ρDCCA series for four pairs of currencies are arranged in the order of CNY–USD > CNY–EUR > CNY–JPY > CNY–KRW. This finding implies that the currency weight in the RMB currency basket is arranged in the order of USD > EUR > JPY > KRW. On the contrary, the values of standard deviation of the ρDCCA series are arranged in the order of CNY–USD < CNY–EUR < CNY–JPY < CNY–KRW, which suggests that the stability of the currency weight in the RMB currency basket is arranged in the order of USD > EUR > JPY > KRW. The Jarque–Bera statistics of ρDCCA series of CNY–USD, CNY–JPY, and CNY–EUR reject the null hypothesis of the Gaussian distribution at the 10%, 5%, and 1% significance level, respectively, also as evidenced by non-zero skewness and kurtosis larger than three. These results indicate that the
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CBY-USD
10-2
1423
CNY-EUR
FDCCA(S)
FDCCA(S)
10-2
10-3 101
10-3
102
101
102
S
S CNY-KRW
CNY-JPY
10-2
FDCCA(S)
FDCCA(S)
10-2
10-3 101
10-3
102
101
102
S
S
Fig. 3. Log–log plots of cross-correlation fluctuation function FDCCA (s) vs. time scale s. Table 2 Descriptive statistics of four ρDCCA series.
Mean Maximum Minimum Standard deviation Skewness Kurtosis Jarque–Bera Observations
CNY–USD
CNY–EUR
CNY–JPY
CNY–KRW
0.9653 0.9777 0.9418 0.0069 −0.4872 4.2164 5.0607* 50
0.8742 0.9261 0.8431 0.0174 1.1134 3.9952 12.3936*** 50
0.6411 0.6944 0.6038 0.0200 0.8606 3.7082 7.2170** 50
0.6136 0.7745 0.5212 0.0618 0.4569 2.7380 1.8826 50
Notes: * Denotes 10% significance level. ** Denotes 5% significance level. *** Denotes 1% significance level.
three ρDCCA series are fat-tailed. However, the ρDCCA series of CNY–KRW is a normal distribution because its Jarque–Bera statistic cannot reject the null hypothesis of the Gaussian distribution and its shewness and kurtosis are close to zero and three, respectively. 4.3. Rolling windows analysis To capture the dynamics of cross-correlations, we employ the method of rolling windows to investigate the evolution of cross-correlations over time. The method of rolling windows is used to examine the temporal evolution of the Hurst (scaling) exponent α at different scales, which is also called as the local Hurst (scaling) exponent [22,25,35,44], or a rolling test [36,45]. The detailed introduction of the rolling windows method can be seen in Ref. [45]. In previous works, many scholars used the method of rolling windows to analyze financial market dynamics, and discussed the selection of window size [22,36,39]. Grech and Mazur [22] pointed out the local scaling exponent at a given time t depends on window size. About the choice of window size, Liu and Wan [36] suggested that: on the one hand, one
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CNY-USD
CNY-EUR
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 0
0 101
101
102
102 S
S CNY-JPY
CNY-KRW
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0 101
0 101
102
102 S
S
Fig. 4. The DCCA cross-correlation coefficient ρDCCA vs. time scale s.
CNY−USD 0.7 0.65
June 26, 09−June 24, 10
0.6 0.55 0.5 0.45 Dec. 03, 08−Dec. 01, 09
0.4
Sept. 29, 10−Sept. 27, 11
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 5. Time-varying cross-correlation scaling exponents for CNY–USD.
should select large window size (e.g., in Ref. [23], the window size is fixed to be four years) to investigate the general trend of long-term mark dynamics (e.g., market efficiency); on the other hand, one should choose small window size (e.g., Refs. [36,38,46] set the window size to be one year) to analyze the influences of exogenous events (e.g., seasonal factors, financial crisis and economic cycling) on market short-term dynamics. Similar to Refs. [36,38,46], in this study, we set the window size to be 250 business days (roughly equal to one trading year). The step is a single day. The graphical representations of time-varying cross-correlation scaling exponents for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are shown in Figs. 5–8, respectively. The time in x-axis stands for the date of the beginning and the last day in each window. In Fig. 5, we can find that most of the cross-correlation scaling exponents of CNY–USD are larger than 0.5, which indicates that CNY and USD are positively cross-correlated. However, one can see two periods in Fig. 5, whose cross-correlation scaling exponents are smaller than 0.5, one is from December 2008 to June 2010, and the other is after September 2010. In other words, the cross-correlations of CNY–USD are anti-persistent (negative) during the US sub-prime crisis and the European debt crisis.
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CNY−EUR 0.7 Original data 10th fitting curve
July 09, 09−July 07, 10
Jan. 04, 06−Jan. 04, 07
0.65
Dec. 04, 07−Dec. 01, 08
0.6 0.55 0.5 0.45 0.4 Apr. 09, 07−Apr. 04, 08
Dec. 12, 08−Dec. 04, 09
Sept. 20, 10−Sept. 16, 11
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 6. Time-varying cross-correlation scaling exponents for CNY–EUR.
CNY−JPY 0.7 0.65 Apr. 24, 06−Apr. 24, 07
Nov. 05, 07−Oct. 31, 08
0.6 May 04, 10−May 02, 11
0.55 0.5 0.45 June 22, 09−June 18, 10
0.4 Mar. 06, 07−Mar. 03, 08
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 7. Time-varying cross-correlation scaling exponents for CNY–JPY.
CNY−KRW 0.7 0.65
Oct. 21, 10−Oct. 19, 11
Mar. 09, 07−Mar. 06, 08
0.6 0.55 0.5 0.45 0.4 Apr. 26, 06−Apr. 26, 07
Feb. 16, 09−Feb. 12, 10
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 8. Time-varying cross-correlation scaling exponents for CNY–KRW.
In Fig. 6, an interesting finding is that the cross-correlation scaling exponents of CNY–EUR have the cyclical fluctuation. The cycle length is near to two years. We also present a 10th fitting curve of the time-varying scaling exponents in Fig. 6, which can better fit the original data. The fitting curve may be used to forecast the trend of cross-correlations of CNY–EUR. Another interesting finding is that after July 2007, as shown in Figs. 5 and 6, the cross-correlation scaling exponents of CNY–USD and CNY–EUR have the same trend. From Fig. 7, we can find that most of cross-correlation scaling exponents of CNY–JPY are smaller than 0.5, which implies that CNY and JPY are negatively cross-correlated. Especially, during the European debt crisis (i.e., after May 2010 in Fig. 7),
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CNY
USD
10-2
FDFA(S)
FDFA(S)
10-2
10-3 101
10-3 101
102
102 S
S EUR 10
KRW
JPY
-2
10-2
10-3101
102
FDFA(S)
FDFA(S)
FDFA(S)
10-2
101
S
102
101
S
102 S
Fig. 9. Log–log plots of auto-correlation fluctuation function FDFA (s) vs. time scale s.
CNY–JPY has long-term negative cross-correlations. On the contrary, during the European debt crisis (i.e., after October 2010 in Fig. 8), CNY and KRW are positively cross-correlated. 4.4. Discussion Podobnik and Stanley [28,35] and Zhou [47] demonstrated, both numerically and analytically, that the cross-correlation scaling exponent is approximately equal to the average of auto-correlation scaling exponents for two fractionally autoregressive integrated moving average (ARFIMA) processes sharing the same i.i.d. Gaussian process. Namely, the following relationship exists among αij , αi , and αj 1 :
αij ≈ (αi + αj )/2,
(10)
where αij is the cross-correlation scaling exponent between currency i and j estimated by DCCA, αi and αj are the autocorrelation scaling exponents of currency i and j estimated by DFA, respectively. (αi + αj )/2 is denoted as the average scaling exponent. In order to evaluate the relationship between a bivariate cross-correlation scaling exponent and the average scaling exponent, we first estimate the auto-correlation scaling exponents of the five currencies by DFA and show the graphical representations in Fig. 9. We can find that the auto-correlation scaling exponents of CNY, USD, EUR, and JPY are greater than 0.5, i.e., the four currencies are positively auto-correlated. However, KRW is negatively auto-correlated because its auto-correlation scaling exponent is 0.4245, smaller than 0.5. Based on the auto-correlation scaling exponents of the five currencies, we then calculate the average scaling exponents (αi + αj )/2 and provide the results in Table 3. For a comparison, we also present the cross-correlation scaling exponents αij in Table 3, which also can be seen in Fig. 3. From Table 3, one can see that the cross-correlation scaling exponents αij for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are larger than the average scaling exponents (αi + αj )/2 of the individual auto-correlation scaling exponents αi and αj . This finding suggests that cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are stronger than the
1 Actually, in Ref. [47] by Zhou, Eq. (10) should be α = (α + α )/2. ij i j
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Table 3 Scaling exponents αij estimated by DCCA and average scaling exponents (αi + αj )/2 estimated by DFA.
αij (αi + αj )/2
CNY–USD
CNY–EUR
CNY–JPY
CNY–KRW
0.5405 0.5399
0.5458 0.5405
0.5301 0.5254
0.5206 0.4804
auto-correlations of the individual currency (i.e., CNY, USD, EUR, JPY, and KRW), respectively. For CNY–USD, CNY–EUR, and CNY–JPY, αij are larger than (αi +αj )/2, but they are approximately equal and confirm the conclusion of Eq. (10). However, for CNY–KRW, the cross-correlation αij is 0.5206 larger than 0.5, but the average scaling exponent (αi + αj )/2 is 0.4804 smaller than 0.5, which violates the conclusion of Eq. (10). Therefore, it is interesting and important to investigate the relationship among αij , αi , and αj for future work. Many previous works confirmed that ‘‘multifractals have been a ‘stylized fact’ in financial markets’’, such as in Refs. [38–43,46,47]. Various methods have been developed to characterize the properties of multifractals. For a nonstationary time series, based on DFA, Kantelhardt et al. [48] proposed the multifractal DFA (MF-DFA) approach to study the multifractality. Gu and Zhou [49] extended DFA and MF-DFA to investigate high-dimensional fractals and multifractals. Gu and Zhou [50] also extended DMA to multifractal DMA (MF-DMA) to study multifractal time series and multifractal surfaces. Then, both MF-DFA and MF-DMA were combined with DCCA to examine the cross-correlations between two nonstationary time series, which are termed as multifractal DCCA (MF-DCCA) [47] and multifractal DMA cross-correlation analysis (MF-XDMA) [51], respectively. Recently, MF-DCCA has become a widely technical tool to analyze cross-correlations in financial markets [17,38–43]. For example, based on MF-DCCA, Wang et al. [38] studied the cross-correlations between Chinese A-share and B-share markets. Their results indicated that strong multifractality existed in the short-term and weak multifractality existed in the long-term. Wang et al. [39] also examined the cross-correlations of WTI crude oil spot and futures markets using MF-DCCA. Their findings showed that the cross-correlations are strongly multifractal for small time scales but nearly monofractal for large time scales. Cao et al. [43] investigated the cross-correlations between the Chinese foreign exchange market and stock market via MF-DCCA and found that multifractality existed. Inspired by Ref. [43], in our future work we plan to investigate the cross-correlations between RMB and four major currencies in the RMB currency basket using MF-DCCA or MF-XDMA. 5. Conclusions In summary, we investigate the cross-correlations behavior between RMB and four major currencies (USD, EUR, JPY, and KRW) in the RMB currency basket. In other words, we examine the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Employing a statistical test proposed by Podobnik et al. [35], which can qualitatively test the existence of cross-correlations, we find that the cross-correlations significantly exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Using DCCA, which can quantitatively examine the presence of cross-correlations, we find that the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are weakly persistent. We also use the DCCA cross-correlation coefficient ρDCCA to quantify the level of cross-correlations and find the currency weight in the RMB currency basket is arranged in the order of USD > EUR > JPY > KRW. In addition, employing the method of rolling windows, which can capture the dynamics of cross-correlations, we find some interesting results: (i) Most of the cross-correlation scaling exponents of CNY–USD are larger than 0.5, which indicates that CNY and USD are positively cross-correlated. However, the cross-correlations of CNY–USD are anti-persistent during the US sub-prime crisis and the European debt crisis. (ii) The cross-correlation scaling exponents of CNY–EUR have the cyclical fluctuation with a nearly two-year cycle. (iii) Most of the cross-correlation scaling exponents of CNY–JPY are smaller than 0.5, which implies that CNY and JPY are negatively cross-correlated. Especially, during the European debt crisis, CNY–JPY has long-term negative cross-correlations. On the contrary, CNY and KRW are positively cross-correlated during the European debt crisis. Finally, we also make some investigations on the relationship between a bivariate cross-correlation scaling exponent and the average scaling exponent, and we discuss some relevant researches of multifractal cross-correlation analysis for future work. Acknowledgments We would like to thank the Editor and anonymous reviewers for their constructive comments. We thank Dr. Feng Han and Chang-Qing Luo for helpful discussions. This work was supported by the National Social Science Foundation of China (Grant No. 07AJL005), the National Soft Science Research Program of China (Grant No. 2010GXS5B141), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0916), and the Science Fund for Innovative Groups of Natural Science Foundation of Hunan Province of China (Grant No. 09JJ7002). G.-J. Wang expresses thanks for the partial support from the Scholarship Award for Excellent Doctoral Student granted by the Ministry of Education of China.
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Cross-correlations between Renminbi and four major currencies in the Renminbi currency basket Gang-Jin Wang a , Chi Xie a,b,∗ a
College of Business Administration, Hunan University, Changsha 410082, China
b
Center of Finance and Investment Management, Hunan University, Changsha 410082, China
article
info
Article history: Received 20 June 2012 Received in revised form 9 August 2012 Available online 29 November 2012 Keywords: Econophysics Cross-correlations Detrended cross-correlation analysis Rolling windows Renminbi exchange rate Renminbi currency basket
abstract We investigate the cross-correlations between Renminbi (CNY) and four major currencies (USD, EUR, JPY, and KRW) in the Renminbi currency basket, i.e., the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Qualitatively, using a statistical test in analogy to the Ljung-Box test, we find that cross-correlations significantly exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Quantitatively, employing the detrended cross-correlation analysis (DCCA) method, we find that the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are weakly persistent. We use the DCCA crosscorrelation coefficient ρDCCA to quantify the level of cross-correlations and find the currency weight in the Renminbi currency basket is arranged in the order of USD > EUR > JPY > KRW. Using the method of rolling windows, which can capture the time-varying crosscorrelation scaling exponents, we find that: (i) CNY and USD are positively cross-correlated over time, but the cross-correlations of CNY–USD are anti-persistent during the US subprime crisis and the European debt crisis. (ii) The cross-correlation scaling exponents of CNY-EUR have the cyclical fluctuation with a nearly two-year cycle. (iii) CNY–JPY has longterm negative cross-correlations, during the European debt crisis, but CNY and KRW are positively cross-correlated. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The Renminbi (RMB), verbatim translated as ‘‘the people’s currency’’, is the currency of China. RMB is also known as the Chinese Yuan (CNY), and its currency symbol is denoted as ‘‘U ’’. Recently, there have been two major events in the RMB exchange rate: (i) On July 21, 2005, after a decade-long of pegging RMB to US dollar (USD), the People’s Bank of China (PBoC, i.e., the China’s central bank) announced that the RMB exchange rate would become ‘‘adjustable, based on market supply and demand with reference to exchange rate movements of currencies in a basket’’ [1,2]. Then, the governor of China’s central bank, Xiaochuan Zhou, revealed details of the RMB currency basket that ‘‘dominant amongst a raft of currencies are USD, the euro (EUR), the Japanese yen (JPY) and South Korea’s won (KRW)’’. (ii) Actually, during the US sub-prime crisis, RMB was re-pegged to USD. On June 19, 2010, PBoC announced that, it has ‘‘decided to proceed further with reform of the RMB exchange rate regime and to enhance the RMB exchange rate flexibility’’ [3]. Along with the rapid economic growth of China, the RMB exchange rate (or RMB currency basket) has spurred a large amount of attention in academic and industrial circles, even in the political arena [4]. Financial markets are considered as complex dynamic systems with a great number of interacting agents [5–7]. The foreign exchange markets, which represent the largest and most liquid financial market in the world, are extremely
∗ Corresponding author at: College of Business Administration, Hunan University, Changsha 410082, China. Tel.: +86 731 88823890; fax: +86 731 88823670. E-mail address: [email protected] (C. Xie). 0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.11.035
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important [8]. One of the considerable characteristics of market dynamics is the presence of cross-correlations between financial variables [9]. Hence, in this study, we seek to investigate the cross-correlations between RMB and the major currencies in the RMB currency basket. In previous studies, many different approaches have been proposed to quantify the cross-correlations between financial entries, such as various clustering methods [8,10–12], the random matrix theory [13,14], and the cross-sample entropy [15,16]. A common assumption of the above-stated methods is that both of the analyzed time series are stationary. However, in the real-world, the financial time series are usually heterogeneous and nonstationary [17]. To overcome this limitation, many methods are developed to examine the auto-correlation and cross-correlation using the mono- and multifractal theory in various fields. For a noisy and nonstationary time series, Peng et al. [18] proposed the detrended fluctuation analysis (DFA) method to study the fractal structure of the DNA nucleotides sequence. After that, DFA was widely used to determine longrange dependence and auto-correlation of the nonstationary time series [19–26]. Vandewalle and Ausloos [27] devised an alternative approach of DFA, which is termed as the detrending moving average (DMA) algorithm, to investigate the longrange correlations of nonstationary time series. Based on DFA, Podobnik and Stanley [28] proposed the detrended crosscorrelation analysis (DCCA) to quantify power-law cross-correlations between simultaneously recorded nonstationary time series. Then, DCCA becomes a versatile and powerful method to investigate the cross-correlations between the financial variables [9,29–31]. For instance, Podobnik et al. [29] analyzed 14, 981 daily data of the Standard and Poor’s 500 Index in the period between 1950–2009, and found power-law cross-correlations between volume change and price change by means of DCCA. Based on DCCA, Siqueira et al. [9] examined 1, 908 daily observations of different stocks and commodities on the Brazilian market from August 10, 2000 to April 30, 2008, and found strong cross-correlations of the volatility time series in the Brazilian stock and commodity market. Lin et al. [30] analyzed the cross-correlations behavior in US and Chinese stock markets using DCCA. Their results showed that cross-correlations in Chinese stock markets (i.e., Shanghai stock market and Shenzhen stock market) are stronger than that of between US and Chinese stock markets. Recently, Zebende [32] proposed a new detrended cross-correlation coefficient (i.e., DCCA cross-correlation coefficient, ρDCCA ), which was defined in terms of DFA and DCCA, to quantify the level of cross-correlation between nonstationary time series. Vassoler and Zebende [33] applied the coefficient ρDCCA to analyze and quantify cross-correlations between air temperature and air relative humidity. Podobnik et al. [34] examined the statistical significance of the coefficient ρDCCA , and proposed an additional statistical test to quantify the cross-correlations between two power-law correlated time series. In this paper, we examine the cross-correlations between RMB (CNY) and four major currencies (i.e., USD, EUR, JPY, and KRW) in the RMB currency basket. We first make a preliminary analysis of the 5 currencies (CNY, USD, EUR, JPY, and KRW) from July 21, 2005 to May 25, 2012. Next, we qualitatively analyze the cross-correlations using the cross-correlation statistics proposed by Podobnik et al. [35]. Then, we employ DCCA and DCCA cross-correlation coefficient ρDCCA to study the presence of cross-correlations quantitatively. Finally, we use the method of rolling windows to capture the dynamics of the cross-correlations. The remainder of this paper is organized as follows. In the next section, we provide the methodologies of DCCA and DCCA cross-correlation coefficient ρDCCA . In Section 3, we present the data set and make a preliminary analysis. We show the main empirical results and some relevant discussions in Section 4. Finally, in Section 5 we draw some conclusions. 2. Methodology 2.1. Detrended cross-correlation analysis Detrended cross-correlation analysis (DCCA) is used to investigate the cross-correlations between two nonstationary time series, which can be described as follows [28,30]: Step 1. Consider two time series {x(t )} and {y(t )} of the same length N, where t = 1, 2, . . . , N. We can determine the profile as two new series, X (t ) =
t
(x(i) − ⟨x⟩) ,
Y (t ) =
i=1
t
(y(i) − ⟨y⟩) ,
t = 1, 2, . . . , N .
(1)
i =1
Step 2. Both of the profiles {X (t )} and {Y (t )} are divided into Ns = int(N /s) non-overlapping segments of equal length s. In this study, we set 10 ≤ s ≤ N /4. Step 3. For each non-overlapping segment v (1 ≤ v ≤ Ns ), the local trending functions of {Xv (t )} and {Yv (t )} are calculated as {X˜ v (t )} and {Y˜v (t )} by a least-square fit of the data, respectively. Step 4. The cross-correlation fluctuation variance for each segment is calculated as follows [17]: Fv2 (s) =
s 1 Xv (t ) − X˜ v (t ) Yv (t ) − Y˜v (t ) . s t =1
(2)
Then we average over all segments to obtain the cross-correlation fluctuation function:
FDCCA (s) =
Ns 1
Ns v=1
1/2 F v ( s) 2
.
(3)
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Step 5. By observing the log–log plots FDCCA (s) vs. s, we can determine the scaling behavior of the fluctuation function. If the original series {x(t )} and {y(t )} are power-law cross-correlated, then FDCCA (s) ∝ sα ,
(4)
where the cross-correlation scaling exponent α can be obtained by the slope of log–log plot of FDCCA (s) vs. s via ordinary least squares (OLS) [9]. In general, there are three cases of α : (i) If α < 0.5, the cross-correlations between the two time series are anti-persistent (negative). This implies that if there is an increase of one price, then it is likely to be followed by a decrease of the other price [36], and vice versa. (ii) If α > 0.5, the cross-correlations between the two time series are persistent (positive). This means that if one price has been an increase or decrease, then it is likely to be followed by an increase or decrease of the other price, respectively [36]. (iii) If α = 0.5, there are no cross-correlations between the two time series, and the change of one price cannot affect the behavior of the other price [30,36]. Especially, if the time series {x(t )} is identical to {y(t )}, DCCA is equivalent to DFA, i.e., the detrended function FDCCA (s) reduces to FDFA (s):
FDFA (s) =
Ns 1
Ns v=1
1/2 Fv (s)
.
2
(5)
At this point, Fv2 (s) = 1/s t =1 (Xv (t ) − X˜ v (t ))2 . If the original series {x(t )} is power-law auto-correlated, the autocorrelation detrended function FDFA (s) vs. s follows a power law, FDFA (s) ∝ sα .
s
2.2. DCCA cross-correlation coefficient ρDCCA In order to quantify the level of cross-correlation, Zebende [32,33] proposed the DCCA cross-correlation coefficient ρDCCA , 2 defined as the ratio between the detrended covariance function FDCCA (s) of Eq. (3) and two detrended variance functions FDFA (s) of Eq. (5), i.e.,
ρDCCA =
2 FDCCA
FDFA{x(t )} FDFA{y(t )}
,
(6)
where ρDCCA is a dimensionless coefficient that ranges between −1 ≤ ρDCCA ≤ 1. If two time series are completely crosscorrelated (anti cross-correlated) then ρDCCA = 1(−1), and if there are no cross-correlations between the two time series then ρDCCA = 0. 3. Data and preliminary analysis We choose the daily foreign exchange (FX) rates of the five currencies (CNY, USD, EUR, JPY, and KRW) from July 21, 2005 to May 25, 2012. Keskin et al. [8] argued that ‘‘one of the problems in FX research is that currencies are priced against each other so no independent numeraire exists. . . Gold was considered, but rejected due to its high volatility’’. But Jang et al. [11] pointed out: ‘‘. . . Special Drawing Right (SDR), which is a potential claim on the freely usable currencies of International Monetary Fund (IMF) members’’. Therefore, we choose SDR as the numeraire. The daily FX rates data are provided by the Pacific Exchange Rate Service (http://fx.sauder.ubc.ca/data.html). Let Pi (t ) denote the price of FX rate of currency i on day t. The daily return of currency i, ri (t ), is calculated by the logdifference of the price Pi (t ), i.e., ri (t ) = ln(Pi (t )) − ln(Pi (t − 1)).
(7)
Fig. 1 provides the graphical representation of returns of the five currencies (i.e., CNY, USD, EUR, JPY, and KRW). One can find that a common large fluctuation zone of the five returns in Fig. 1: before, during and after October 2008, which may be the worst days of the US sub-prime crisis. We present the descriptive statistics of the five returns in Table 1. The mean values of the five returns are very close to zero, and quite small by comparison with the standard deviations. The standard deviation of the return series of EUR is the largest among the five returns. The high volatility of EUR may be affected by the European debt crisis. The Jarque–Bera statistics reject the null hypothesis of the Gaussian distribution at the 1% significance level. This phenomenon is also accompanied by non-zero skewness and kurtosis larger than three, which indicates that the five returns are fat-tailed. 4. Results and discussion 4.1. Cross-correlation test In order to quantify the cross-correlations between CNY and four major currencies in the RMB currency basket (henceforth, we denote the four pairs of cross-correlations as CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW, respectively),
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0.05 CNY 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
0.05 USD 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
0.05 Returns
EUR 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
0.05 JPY 0 −0.05 July 22, 05
Feb. 28, 07
Oct. 01, 08
May 06, 10
Dec. 09, 11
−0.05 KRW 0 −0.05
July 22, 05
Feb. 28, 07
Oct. 01, 08 Time (date)
May 06, 10
Dec. 09, 11
Fig. 1. Returns of the five currencies (i.e., CNY, USD, EUR, JPY, and KRW). Table 1 Descriptive statistics of returns of the five currencies.
−5
Mean (×10 ) Maximum Minimum S.D. (×10−3 ) Skewness Kurtosis Jarque–Bera (×103 ) Observations
CNY
USD
EUR
JPY
KRW
−5.1841
−1.0250
0.0107 −0.0049 1.4140 0.3921 5.9026 0.6460* 1715
0.0114 −0.0053 1.4063 0.4483 6.6064 0.9868* 1715
1.9112 0.0079 −0.0088 1.6509 −0.0038 6.0016 0.6438* 1715
7.2530 0.0138 −0.0214 0.0027 −0.4818 8.6907 2.3805* 1715
4.4365 0.0429 −0.0478 0.0039 −0.1727 33.3240 65.7178* 1715
Notes: S.D. stands for ‘‘Standard deviation’’. The Jarque–Bera statistics test for the null hypothesis of normality in the sample returns distribution. * Indicates rejection of the null hypothesis at the 1% significance level.
we employ a new cross-correlation test proposed by Podobnik et al. [35], which is in analogy to the Ljung–Box test [37] and widely used in financial markets [36,38–43]. For two time series, {x(t )|t = 1, 2, . . . , N } and {y(t )|t = 1, 2, . . . , N }, the cross-correlation statistic is defined by Qcc (m) = N 2
m C 2 (t ) , N −t t =1
(8)
where the cross-correlation function C (t ) is defined as N
x(k)y(k − t )
k=t +1
C (t ) =
N k=1
x2
(k)
N k=1
. y2
(k)
(9)
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105 4
10
Qcc(m)
103
Test statistics (CNY-USD) Test statistics (CNY-EUR) Test statistics (CNY-JPY) Test statistics (CNY-KRW) Critical values
102 101 m=42
100 10-1 100
102
101
103
m Fig. 2. Log–log plots of test statistics Qcc (m) vs. degrees of freedom m.
Podobnik et al. [35] indicated that, the cross-correlation statistic Qcc (m) is approximately χ 2 (m) distributed with m degrees of freedom. It can be used to ‘‘test the null hypothesis of none of the first m cross-correlation coefficients is different from zero’’ [35]. Fig. 2 shows the log–log plots of cross-correlation statistics Qcc (m) vs. degrees of freedom m for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW, where the degrees of freedom vary from 100 to 103 . To make a comparison, as illustrated in Fig. 2, we also present the critical values for the χ 2 (m) distribution at the 5% level of significance. In Fig. 2, when m ≥ 3, the cross-correlation test statistics of CNY–KRW are always larger than the critical values. It seems that for m < 42, Qcc (m) of CNY–USD, CNY–EUR, and CNY–JPY are all smaller than the critical values but close to them. Nevertheless, when m ≥ 42, all the statistics Qcc (m) have the trend of being near or equal to the critical values with m increasing, even larger than the critical values. Podobnik et al. [35] proposed that ‘‘if for a broad range of values of m the values of the test of Eq. (8) between the two time series are larger than the critical values of the χ 2 (m) distribution, the cross-correlations are considered significant’’. Hence, we can reject the null hypothesis of no cross-correlations. That is to say, cross-correlations significantly exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. 4.2. Cross-correlation analysis Podobnik et al. [35] suggested that the cross-correlation test of Eq. (8) can only be used to qualitatively demonstrate the existence of nonlinear cross-correlation; while DCCA can present a quantitative cross-correlation test by estimating the cross-correlation scaling exponent. Thus, in this subsection, we first use the DCCA to investigate the cross-correlations quantitatively for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. We show the log–log plots of FDCCA (s) vs. time scale s for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW in Fig. 3. The cross-correlation scaling exponents (i.e., the slopes of regression lines) are also presented in Fig. 3. One can find that all of the cross-correlation scaling exponents are greater than 0.5 but very close to 0.5, which implies that cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are weakly persistent (positive). Interestingly, it is CNY–EUR not CNY–USD, has the largest scaling exponent. Actually, when the four cross-correlation scaling exponents are very close, we can only use the scaling exponent to test the cross-correlation is either positive or negative, and cannot employ the scaling exponent to quantify the level of cross-correlation. Therefore, we then use the DCCA cross-correlation coefficient ρDCCA to quantify the level of cross-correlations. Fig. 4 plots the DCCA cross-correlation coefficient ρDCCA vs. time scale s for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. From Fig. 4, we can find that CNY–USD has the largest values of ρDCCA among the four pairs of currencies. As a comparison, we illustrate the descriptive statistics of the four ρDCCA series in Table 2. For the DCCA cross-correlation coefficient ρDCCA , Podobnik et al. [34] proposed an additional statistical test to quantify the cross-correlations between two power-law correlated time series. According to Table 1 in Ref. [34], the critical value for ρDCCA for each time scale s is not greater than 0.4 when T is 1000 (or 2000) [43]. From Table 2, we can find that the minimum of the four ρDCCA series is 0.5212. Hence, it is another piece of evidence that cross-correlations exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. In Table 2, the mean values of the ρDCCA series for four pairs of currencies are arranged in the order of CNY–USD > CNY–EUR > CNY–JPY > CNY–KRW. This finding implies that the currency weight in the RMB currency basket is arranged in the order of USD > EUR > JPY > KRW. On the contrary, the values of standard deviation of the ρDCCA series are arranged in the order of CNY–USD < CNY–EUR < CNY–JPY < CNY–KRW, which suggests that the stability of the currency weight in the RMB currency basket is arranged in the order of USD > EUR > JPY > KRW. The Jarque–Bera statistics of ρDCCA series of CNY–USD, CNY–JPY, and CNY–EUR reject the null hypothesis of the Gaussian distribution at the 10%, 5%, and 1% significance level, respectively, also as evidenced by non-zero skewness and kurtosis larger than three. These results indicate that the
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CBY-USD
10-2
1423
CNY-EUR
FDCCA(S)
FDCCA(S)
10-2
10-3 101
10-3
102
101
102
S
S CNY-KRW
CNY-JPY
10-2
FDCCA(S)
FDCCA(S)
10-2
10-3 101
10-3
102
101
102
S
S
Fig. 3. Log–log plots of cross-correlation fluctuation function FDCCA (s) vs. time scale s. Table 2 Descriptive statistics of four ρDCCA series.
Mean Maximum Minimum Standard deviation Skewness Kurtosis Jarque–Bera Observations
CNY–USD
CNY–EUR
CNY–JPY
CNY–KRW
0.9653 0.9777 0.9418 0.0069 −0.4872 4.2164 5.0607* 50
0.8742 0.9261 0.8431 0.0174 1.1134 3.9952 12.3936*** 50
0.6411 0.6944 0.6038 0.0200 0.8606 3.7082 7.2170** 50
0.6136 0.7745 0.5212 0.0618 0.4569 2.7380 1.8826 50
Notes: * Denotes 10% significance level. ** Denotes 5% significance level. *** Denotes 1% significance level.
three ρDCCA series are fat-tailed. However, the ρDCCA series of CNY–KRW is a normal distribution because its Jarque–Bera statistic cannot reject the null hypothesis of the Gaussian distribution and its shewness and kurtosis are close to zero and three, respectively. 4.3. Rolling windows analysis To capture the dynamics of cross-correlations, we employ the method of rolling windows to investigate the evolution of cross-correlations over time. The method of rolling windows is used to examine the temporal evolution of the Hurst (scaling) exponent α at different scales, which is also called as the local Hurst (scaling) exponent [22,25,35,44], or a rolling test [36,45]. The detailed introduction of the rolling windows method can be seen in Ref. [45]. In previous works, many scholars used the method of rolling windows to analyze financial market dynamics, and discussed the selection of window size [22,36,39]. Grech and Mazur [22] pointed out the local scaling exponent at a given time t depends on window size. About the choice of window size, Liu and Wan [36] suggested that: on the one hand, one
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CNY-USD
CNY-EUR
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2 0
0 101
101
102
102 S
S CNY-JPY
CNY-KRW
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0 101
0 101
102
102 S
S
Fig. 4. The DCCA cross-correlation coefficient ρDCCA vs. time scale s.
CNY−USD 0.7 0.65
June 26, 09−June 24, 10
0.6 0.55 0.5 0.45 Dec. 03, 08−Dec. 01, 09
0.4
Sept. 29, 10−Sept. 27, 11
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 5. Time-varying cross-correlation scaling exponents for CNY–USD.
should select large window size (e.g., in Ref. [23], the window size is fixed to be four years) to investigate the general trend of long-term mark dynamics (e.g., market efficiency); on the other hand, one should choose small window size (e.g., Refs. [36,38,46] set the window size to be one year) to analyze the influences of exogenous events (e.g., seasonal factors, financial crisis and economic cycling) on market short-term dynamics. Similar to Refs. [36,38,46], in this study, we set the window size to be 250 business days (roughly equal to one trading year). The step is a single day. The graphical representations of time-varying cross-correlation scaling exponents for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are shown in Figs. 5–8, respectively. The time in x-axis stands for the date of the beginning and the last day in each window. In Fig. 5, we can find that most of the cross-correlation scaling exponents of CNY–USD are larger than 0.5, which indicates that CNY and USD are positively cross-correlated. However, one can see two periods in Fig. 5, whose cross-correlation scaling exponents are smaller than 0.5, one is from December 2008 to June 2010, and the other is after September 2010. In other words, the cross-correlations of CNY–USD are anti-persistent (negative) during the US sub-prime crisis and the European debt crisis.
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CNY−EUR 0.7 Original data 10th fitting curve
July 09, 09−July 07, 10
Jan. 04, 06−Jan. 04, 07
0.65
Dec. 04, 07−Dec. 01, 08
0.6 0.55 0.5 0.45 0.4 Apr. 09, 07−Apr. 04, 08
Dec. 12, 08−Dec. 04, 09
Sept. 20, 10−Sept. 16, 11
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 6. Time-varying cross-correlation scaling exponents for CNY–EUR.
CNY−JPY 0.7 0.65 Apr. 24, 06−Apr. 24, 07
Nov. 05, 07−Oct. 31, 08
0.6 May 04, 10−May 02, 11
0.55 0.5 0.45 June 22, 09−June 18, 10
0.4 Mar. 06, 07−Mar. 03, 08
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 7. Time-varying cross-correlation scaling exponents for CNY–JPY.
CNY−KRW 0.7 0.65
Oct. 21, 10−Oct. 19, 11
Mar. 09, 07−Mar. 06, 08
0.6 0.55 0.5 0.45 0.4 Apr. 26, 06−Apr. 26, 07
Feb. 16, 09−Feb. 12, 10
0.35 July 22, 05−July 21, 06
July 23, 07−July 18, 08
July 20, 09−July 16, 10 May 30, 11−May 25, 12
Time (date) Fig. 8. Time-varying cross-correlation scaling exponents for CNY–KRW.
In Fig. 6, an interesting finding is that the cross-correlation scaling exponents of CNY–EUR have the cyclical fluctuation. The cycle length is near to two years. We also present a 10th fitting curve of the time-varying scaling exponents in Fig. 6, which can better fit the original data. The fitting curve may be used to forecast the trend of cross-correlations of CNY–EUR. Another interesting finding is that after July 2007, as shown in Figs. 5 and 6, the cross-correlation scaling exponents of CNY–USD and CNY–EUR have the same trend. From Fig. 7, we can find that most of cross-correlation scaling exponents of CNY–JPY are smaller than 0.5, which implies that CNY and JPY are negatively cross-correlated. Especially, during the European debt crisis (i.e., after May 2010 in Fig. 7),
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CNY
USD
10-2
FDFA(S)
FDFA(S)
10-2
10-3 101
10-3 101
102
102 S
S EUR 10
KRW
JPY
-2
10-2
10-3101
102
FDFA(S)
FDFA(S)
FDFA(S)
10-2
101
S
102
101
S
102 S
Fig. 9. Log–log plots of auto-correlation fluctuation function FDFA (s) vs. time scale s.
CNY–JPY has long-term negative cross-correlations. On the contrary, during the European debt crisis (i.e., after October 2010 in Fig. 8), CNY and KRW are positively cross-correlated. 4.4. Discussion Podobnik and Stanley [28,35] and Zhou [47] demonstrated, both numerically and analytically, that the cross-correlation scaling exponent is approximately equal to the average of auto-correlation scaling exponents for two fractionally autoregressive integrated moving average (ARFIMA) processes sharing the same i.i.d. Gaussian process. Namely, the following relationship exists among αij , αi , and αj 1 :
αij ≈ (αi + αj )/2,
(10)
where αij is the cross-correlation scaling exponent between currency i and j estimated by DCCA, αi and αj are the autocorrelation scaling exponents of currency i and j estimated by DFA, respectively. (αi + αj )/2 is denoted as the average scaling exponent. In order to evaluate the relationship between a bivariate cross-correlation scaling exponent and the average scaling exponent, we first estimate the auto-correlation scaling exponents of the five currencies by DFA and show the graphical representations in Fig. 9. We can find that the auto-correlation scaling exponents of CNY, USD, EUR, and JPY are greater than 0.5, i.e., the four currencies are positively auto-correlated. However, KRW is negatively auto-correlated because its auto-correlation scaling exponent is 0.4245, smaller than 0.5. Based on the auto-correlation scaling exponents of the five currencies, we then calculate the average scaling exponents (αi + αj )/2 and provide the results in Table 3. For a comparison, we also present the cross-correlation scaling exponents αij in Table 3, which also can be seen in Fig. 3. From Table 3, one can see that the cross-correlation scaling exponents αij for CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are larger than the average scaling exponents (αi + αj )/2 of the individual auto-correlation scaling exponents αi and αj . This finding suggests that cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are stronger than the
1 Actually, in Ref. [47] by Zhou, Eq. (10) should be α = (α + α )/2. ij i j
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Table 3 Scaling exponents αij estimated by DCCA and average scaling exponents (αi + αj )/2 estimated by DFA.
αij (αi + αj )/2
CNY–USD
CNY–EUR
CNY–JPY
CNY–KRW
0.5405 0.5399
0.5458 0.5405
0.5301 0.5254
0.5206 0.4804
auto-correlations of the individual currency (i.e., CNY, USD, EUR, JPY, and KRW), respectively. For CNY–USD, CNY–EUR, and CNY–JPY, αij are larger than (αi +αj )/2, but they are approximately equal and confirm the conclusion of Eq. (10). However, for CNY–KRW, the cross-correlation αij is 0.5206 larger than 0.5, but the average scaling exponent (αi + αj )/2 is 0.4804 smaller than 0.5, which violates the conclusion of Eq. (10). Therefore, it is interesting and important to investigate the relationship among αij , αi , and αj for future work. Many previous works confirmed that ‘‘multifractals have been a ‘stylized fact’ in financial markets’’, such as in Refs. [38–43,46,47]. Various methods have been developed to characterize the properties of multifractals. For a nonstationary time series, based on DFA, Kantelhardt et al. [48] proposed the multifractal DFA (MF-DFA) approach to study the multifractality. Gu and Zhou [49] extended DFA and MF-DFA to investigate high-dimensional fractals and multifractals. Gu and Zhou [50] also extended DMA to multifractal DMA (MF-DMA) to study multifractal time series and multifractal surfaces. Then, both MF-DFA and MF-DMA were combined with DCCA to examine the cross-correlations between two nonstationary time series, which are termed as multifractal DCCA (MF-DCCA) [47] and multifractal DMA cross-correlation analysis (MF-XDMA) [51], respectively. Recently, MF-DCCA has become a widely technical tool to analyze cross-correlations in financial markets [17,38–43]. For example, based on MF-DCCA, Wang et al. [38] studied the cross-correlations between Chinese A-share and B-share markets. Their results indicated that strong multifractality existed in the short-term and weak multifractality existed in the long-term. Wang et al. [39] also examined the cross-correlations of WTI crude oil spot and futures markets using MF-DCCA. Their findings showed that the cross-correlations are strongly multifractal for small time scales but nearly monofractal for large time scales. Cao et al. [43] investigated the cross-correlations between the Chinese foreign exchange market and stock market via MF-DCCA and found that multifractality existed. Inspired by Ref. [43], in our future work we plan to investigate the cross-correlations between RMB and four major currencies in the RMB currency basket using MF-DCCA or MF-XDMA. 5. Conclusions In summary, we investigate the cross-correlations behavior between RMB and four major currencies (USD, EUR, JPY, and KRW) in the RMB currency basket. In other words, we examine the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Employing a statistical test proposed by Podobnik et al. [35], which can qualitatively test the existence of cross-correlations, we find that the cross-correlations significantly exist in CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW. Using DCCA, which can quantitatively examine the presence of cross-correlations, we find that the cross-correlations of CNY–USD, CNY–EUR, CNY–JPY, and CNY–KRW are weakly persistent. We also use the DCCA cross-correlation coefficient ρDCCA to quantify the level of cross-correlations and find the currency weight in the RMB currency basket is arranged in the order of USD > EUR > JPY > KRW. In addition, employing the method of rolling windows, which can capture the dynamics of cross-correlations, we find some interesting results: (i) Most of the cross-correlation scaling exponents of CNY–USD are larger than 0.5, which indicates that CNY and USD are positively cross-correlated. However, the cross-correlations of CNY–USD are anti-persistent during the US sub-prime crisis and the European debt crisis. (ii) The cross-correlation scaling exponents of CNY–EUR have the cyclical fluctuation with a nearly two-year cycle. (iii) Most of the cross-correlation scaling exponents of CNY–JPY are smaller than 0.5, which implies that CNY and JPY are negatively cross-correlated. Especially, during the European debt crisis, CNY–JPY has long-term negative cross-correlations. On the contrary, CNY and KRW are positively cross-correlated during the European debt crisis. Finally, we also make some investigations on the relationship between a bivariate cross-correlation scaling exponent and the average scaling exponent, and we discuss some relevant researches of multifractal cross-correlation analysis for future work. Acknowledgments We would like to thank the Editor and anonymous reviewers for their constructive comments. We thank Dr. Feng Han and Chang-Qing Luo for helpful discussions. This work was supported by the National Social Science Foundation of China (Grant No. 07AJL005), the National Soft Science Research Program of China (Grant No. 2010GXS5B141), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT0916), and the Science Fund for Innovative Groups of Natural Science Foundation of Hunan Province of China (Grant No. 09JJ7002). G.-J. Wang expresses thanks for the partial support from the Scholarship Award for Excellent Doctoral Student granted by the Ministry of Education of China.
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