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Multifractal detrended cross-correlation analysis of carbon and crude oil markets
Physica A 399 (2014) 113–125
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Physica A journal homepage: www.elsevier.com/locate/physa
Multifractal detrended cross-correlation analysis of carbon and crude oil markets Xiaoyang Zhuang ∗ , Yu Wei ∗∗ , Bangzheng Zhang School of Economics & Management, Southwest Jiaotong University, First Section of Northern Second Ring Road, Chengdu, China
highlights • We confirm the presence of cross-correlations between carbon and crude oil markets both quantitatively and qualitatively. • Based on the MF-DCCA approach, the nonlinear structure of the cross-correlations between carbon and crude oil markets is studied. • The MF-DFA method is used to investigate the multifractal behaviors of carbon markets and crude oil markets respectively.
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Article history: Received 8 June 2013 Received in revised form 20 November 2013 Available online 8 January 2014 Keywords: Multifractal detrended cross-correlation analysis Carbon market Crude oil markets
abstract The complex dynamics between carbon and crude oil markets have been an increasingly interesting area of research. In this paper, we try to take a fresh look at the cross-correlations between carbon and crude oil markets as well as their dynamic behavior employing multifractal detrended cross-correlation analysis. First, we find that the return series of carbon and crude oil markets are significantly cross-correlated. Second, we confirm the existence of multifractality for the return series of carbon and crude oil markets by the multifractal detrended fluctuation analysis. Third, based on the multifractal detrended cross-correlation analysis, we find the existence of power-law cross-correlations between carbon and crude oil markets. The cross-correlated behavior of small fluctuations is found to be more persistent than that of large fluctuations. At last, some relevant discussions and implications of the empirical results are presented. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In the last few decades, the dynamics of financial markets have been of great interest. As far as we know, the previous studies have confirmed that financial markets are extremely complex and exhibit dynamics and non-linear properties, such as multifractality and long-range correlation features [1–6]. For instance, Alvarez-Ramirez et al. [7] found the dynamics and multifractal effect of the oil market by analyzing the auto-correlations of international crude oil prices. Gu et al. [8] studied the multifractal structure of WTI and Brent crude oil markets by employing the multifractal detrended cross-correlation analysis (MF-DCCA) method. Ho et al. [9] conducted the multifractal analysis on the Taiwan Stock index and the Hang Seng index. Kyungsik et al. [10] confirmed the multifractal properties of foreign exchange markets. Ioan et al. [11] investigated the multifractal behavior of Central and Eastern European foreign exchange rates. In the previous works, various methods were developed to quantify the auto-correlation and cross-correlation behaviors of financial markets based on the monofractal and multifractal theory. Peng et al. [12] proposed the detrended fluctuation analysis (DFA) to explore the long-range auto-correlations of a non-stationary time series and widely used in financial
∗ ∗∗
Corresponding author. Corresponding author. Tel.: +86 13618001075. E-mail addresses: [email protected] (X. Zhuang), [email protected] (Y. Wei).
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time series analysis [13,14]. Then, DFA was extended into two important methods: one is the multifractal detrended fluctuation analysis (MF-DFA) proposed by Kantelhardt et al. [15], which is a powerful tool to investigate the multifractality of the financial time series [16–18]; the other is the detrended cross-correlation analysis (DCCA) proposed by Podobnik and Stanley [19], which can be used to quantify the cross-correlations between two non-stationary financial time series. To examine the multifractal characteristics of two cross-correlated non-stationary time series, Zhou [20] proposed the multifractal detrended cross-correlation analysis (MF-DCCA) based on MF-DFA and DCCA. After that, MF-DCCA was widely used to investigate the cross-correlations in financial markets [21–25]. For example, Wang et al. [25] investigated the crosscorrelations of West Texas Intermediate (WTI) crude oil spot and futures return series by means of MF-DCCA. On connecting the environment energy problem and the economic development, the mechanism of carbon market becomes a hot topic. In addition, with the rapid growth of the carbon market, carbon price fluctuations are increasingly important for market participants. The price of carbon is classically driven by the balance between supply and demand, and by other factors related to the market structure and institutional policies. On the supply side, the number of allowances distributed is determined by Member-State through National Allocation Plans. On the demand side, the use of CO2 allowances is a function of expected CO2 emissions. In turn the level of emissions depends on a large number of factors, such as unexpected fluctuations in energy demand and energy prices. The demand for allowances can be affected by economic growth and financial markets as well, but that latter impact needs to be further assessed in the academic community. In this paper, we try to take a fresh look at the cross-correlations between carbon and energy markets. Since fossil energy consumption has been considered as one of the main drivers of carbon prices, the price change of the energy market is a key to the international carbon market price formation mechanism. Mansanet-Bataller et al. [26] found that carbon prices are linked to the fossil fuel use. Alberola et al. [27] emphasized that the nature of the relationship between energy and carbon prices varies depending on the period under consideration. Convery and Redmond [28] found that an increase of fossil energy price will push the carbon price, and vice versa. Kanen [29] revealed that crude oil prices are the main thrust of the changes in natural gas prices; meanwhile fluctuations in natural gas prices will affect the tariff changes and ultimately impact the carbon price. Therefore, the analysis of cross-correlations between carbon and energy markets is essential and plays an important role in exploring the internal operation mechanism and forecasting the carbon price trend. In the paper, we choose crude oil markets as the representative of energy markets. Crude oil takes a big percentage in energy markets, and occupies a dominant position in the EU’s energy consumption. In addition, the previous studies have already confirmed that the presence of correlation and multifractality in crude oil markets and also implies inefficiency [30–32]. The organization of this paper is as follows. Section 2 introduces methods employed in this study. Section 3 briefly described the data used in our work. Section 4 provides the detailed empirical results. Section 5 presents some discussion. Then, the last section concludes. 2. Methodology There are two time series {x(i)} and {y(i)}, i = 1, 2, . . . , N, where N is the length of the series. The MF-DCCA method can be summarized as follows: First, calculate the profile X (i) =
i
(x(k) − x¯ ) ,
Y (i) =
i
(y(k) − y¯ ) ,
i = 1, 2, . . . , N ,
(1)
k=1
k=1
where x¯ and y¯ denote the averaging over the two whole time series x(i) and y(i). Second, divide the two profiles X (i) and Y (i) into Ns = int(N /s) non-overlapping segments of equal length s. Since the length N of the series is often not a multiple of the given time scale s, a short part at the end of each profile may remain. In order not to discard this part of the series, the same procedure is repeated starting from the opposite end of each profile. Thereby, 2Ns segments are obtained together. Then, estimate the local trends for each of the 2Ns segments by means of the mth order polynomial fit. Then the detrended covariance is determined by F 2 (s, λ) =
s 1 X(λ−1)s+j (j) − X˜ λ (j) Y(λ−1)s+j (j) − Y˜λ (j) s j =1
(2)
for each segment λ, λ = 1, 2, . . . , Ns and F 2 (s, λ) =
s 1 XN −(λ−Ns )s+j (j) − X˜ λ (j) YN −(λ−Ns )s+j (j) − Y˜λ (j) s j =1
(3)
for each segment λ, λ = Ns + 1, Ns + 2, . . . , 2Ns . Here, X˜ λ (j) and Y˜λ (j) denote the fitting polynomial with order m in segment λ. Then average over all segments to obtain the qth-order fluctuation function
Fq (s) =
2Ns 1
2Ns λ=1
q/2 F (s, λ) 2
1/q (4)
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for any q ̸= 0, and
F0 (s) = exp
2Ns 1
4Ns λ=1
ln F (s, λ)
2
(5)
for q = 0. Finally, analyze the scaling behavior of the fluctuations by observing log–log plots of Fq (s) versus s for each value of q. If the two series are long-range cross-correlated, Fq (s) will increase for large values of s, as a power-law Fq (s) ∝ sHxy (q) .
(6)
The scaling exponent Hxy (q), known as the generalized cross-correlation exponent, describes the power-law relationship between two series, can be obtained by observing the slope of the log–log plots of Fq (s) versus s through the method of ordinary least squares (OLS). In particular, if {x(i)} is identical to {y(i)}, then MF-DCCA is equivalent to MF-DFA. Furthermore, the scaling exponent Hxy (q) has similar properties and interpretation to the generalized Hurst exponent H (q). If scaling exponent Hxy (q) is independent of q, the cross-correlations of two series are monofractal, otherwise, these are multifractal. If scaling exponent Hxy (q) > 0.5, the cross-correlations of the two time series related to q are long-range persistent, which implies that one price is likely to increase following an increase of the other price. If scaling exponent Hxy (q) < 0.5, the cross-correlations of the two time series related to q are anti-persistent, which implies that an increase of one price is likely to be followed by a decrease of the other price. If scaling exponent Hxy (q) = 0.5, there are no crosscorrelations or at most short-range cross-correlations between two time series. In order to measure the degree of multifractality [33], the financial risk measure ∆H is proposed as the following
∆H = Hmax (q) − Hmin (q).
(7)
Apparently, richer multifractality corresponds to higher variability of H (q). Therefore, the greater ∆H is, the stronger is the degree of multifractality, and then more inefficient is the market. According to Shadkhoo and Jafari [34], a similar relationship between classical Renyi exponent τxy (q) and q can be given by
τxy (q) = qHxy (q) − 1.
(8)
If the scaling exponent function τxy (q) is a linear function of q, the cross-correlations of the two time series are monofractal. Otherwise, these are multifractal. The following relationships can be obtained by a Legendre transform ′ α = Hxy (q) + qHxy (q) and fxy (α) = q(α − Hxy (q)) + 1.
(9)
Here, α is the Holder exponent or the singularity strength, which is used to characterize the singularities of the time series. The multifractal spectrum fxy (α) describes the singularity content of the time series. The strength of multifractality can be estimated by the width of singularity [35], which is given by
∆α = αmax − αmin .
(10)
3. Data In this study, two main carbon prices (Certification Emission Reduction (CER) and EU emission Allowance (EUA)) are selected. The daily closing price of each carbon price used in this study is available from Wind database. For CER, we take daily data from March 14, 2008 to December 15, 2012. For EUA, we take daily data from April 22, 2005 to December 15, 2012. As for crude oil markets, we chose daily spot prices of West Texas Intermediate (WTI) crude oil and Brent crude oil as representatives. After eliminating the non-matching missing data, we get the lengths of four pairs of series, CER & WTI, CER & Brent, EUA & WTI and EUA & Brent, as 1196, 1189, 1914 and 1919, respectively. We define the daily price return rt , the natural logarithmic difference of the daily closing price Pt , as rt = ln Pt − ln Pt −1 . Fig. 1 presents the curve of daily carbon prices of carbon and crude oil markets. In order to effectively investigate the cross-correlations between the two different kinds of markets, all the different units of the time series are converted to dollars per ton. Specifically, the returns of the daily price of the carbon market are adjusted to US dollars using the daily Euro/US dollar exchange rate. 4. Empirical results 4.1. Testing for the cross-correlations In order to investigate the nonlinear cross-correlations between carbon and crude oil markets, we apply a crosscorrelation statistic proposed by Podobnik et al.
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Fig. 1. Daily prices of carbon and crude oil markets. The blue and green lines represent the price series of carbon and crude oil markets, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
For two time series {x(i)} and {y(i)} , i = 1, 2, . . . , N, the test statistic is defined as Qcc (m) = N 2
m Xi2 , N −i i=1
(11)
where the cross-correlation function is N
xk yk−i
k=i+1
Xi =
N k=1
x2k
N
.
(12)
y2k
k=1
The cross-correlation statistic Qcc (m) is approximately χ 2 (m) distributed with m degrees of freedom. Generally, this statistic is employed for returns, not for original time series. The cross-correlation test agrees well with the χ 2 (m) distribution, if there were no cross-correlations between two return series. If the cross-correlation test exceeded the critical value of the χ 2 (m) distribution, the cross-correlations are significant at a special significance level. Fig. 2 shows the cross-correlation statistics Qcc (m) versus the degree of freedom m for the returns between the carbon and crude oil markets. The degrees of freedom vary from 1 to 1000. For comparison, the critical values for the χ 2 (m) distribution at the 5% level of significance are also provided in Fig. 2. Fig. 2 indicates that the cross-correlation statistics Qcc (m) of the four pairs are larger than the critical values of the χ 2 (m) distribution when m is larger than 2. Therefore, we can reject the null hypothesis of no cross-correlations, in other words, long-range cross-correlations exist between the carbon and crude oil markets.
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Fig. 2. Cross-correlation statistics Qcc (m) vs. m. The blue, black, green, purple and red lines represent the pairs of CER & WTI, CER & Brent, EUA & WTI, EUA & Brent and critical value, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 The value of ρDCCA for a given window size n. Size
16
32
64
128
256
CER & WTI CER & Brent EUA & WTI EUA & Brent
0.3526 0.4016 0.5872 0.5828
0.3545 0.4044 0.5906 0.5906
0.3674 0.4154 0.6019 0.6043
−0.0004
−0.2026 −0.1929
0.4196 0.6364 0.6390
0.0107 0.6728
In order to affirm our results above more carefully, we also apply another new method proposed by Podobnik et al. [36], 2 2 defined as the ratio between the detrended covariance function FDCCA and the detrended variance FDFA , the function reads as
ρDCCA =
2 FDCCA (n)
FDFA1 (n)FDFA2 (n)
.
(13)
The value of ρDCCA ranges between −1 ≤ ρDCCA ≤ 1. The two series have no cross-correlation if ρDCCA is equal to zero, otherwise they have cross-correlation. And, ρDCCA can identify the level of cross-correlation from the positive and the negative case. We calculate the value of ρDCCA based on different values of window size n (n = 16, 32, 64, 128, 256) (see Table 1) that we can draw the conclusions that consistent with the cross-correlation test as above. 4.2. Multifractal detrended cross-correlation analysis The cross-correlation test based on the statistics in Eq. (11) can only test the existence of cross-correlations between carbon and crude oil markets qualitatively. In order to affirm our results obtained above, we also apply the MF-DCCA method proposed by Podobnik and Stanley to test the presence of cross-correlation quantitatively. Fig. 3 shows the log–log plots of fluctuation function Fq (s) versus time scale s between the carbon and crude oil markets. As shown in Fig. 3, for different values of q, all the curves exhibit linearity during a span, indicating that there exists a powerlaw cross-correlation in each pair of daily return series. The power-law relationship indicates that a large price change in the crude oil markets is more likely to be followed by a large price change in carbon market, and vice versa. In order to explore the different changes of the cross-correlation exponent with varying values of q, we display the scaling cross-correlation exponents Hxy (q) with q varying from −5 to 5 in Fig. 4 (blue star lines). We can see that the scaling exponents Hxy (q) decrease with the increase of q nonlinearly, implying that multifractality exists in the cross-correlations between carbon and crude oil markets. The scaling exponents Hxy (q) for q < 0 are larger than those for q > 0, indicating that the cross-correlated behavior of small fluctuations is more persistent than that of large fluctuations. For q = 2, the scaling exponent Hxy (q) is just similar to the Hurst exponent calculated from the method of MF-DFA. The cross-correlation exponents Hxy (2) for CER & WTI and CER & Brent both are slightly larger than 0.5, implying that their cross-correlations are weakly persistent. The cross-correlation exponents Hxy (2) of EUA & WTI and EUA & Brent are smaller than 0.5, indicating that their cross-correlations are anti-persistent.
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Fig. 3. Log–log plots of F (s) vs. s between carbon and crude oil markets. The lines from bottom to top are corresponding to the plots with q varying from −5 to 5.
Fig. 4 also shows that, for different values of q, the scaling exponents are different. In other words, for different values of q, the cross-correlated behaviors of the four return pairs are different. We can find that for EUA & WTI, the scaling exponents decrease from about 0.7 to 0.2, implying that the cross-correlated behavior shows strong multifractality. Moreover, the
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Fig. 4. The nonlinear relationship of H (q) vs. q between carbon and crude oil markets. The red circle and purple diamond curves, estimated from MFDFA, and the blue star curves, estimated from MF-DCCA, represent the scaling exponents of carbon and crude oil markets and the cross-correlated ones, respectively. The black lines are the average of the scaling exponents in the two markets. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
scaling exponents for q < 0 are larger than 0.5, indicating the cross-correlated behavior of small fluctuations is persistent. The scaling exponents for q > 1 are smaller than 0.5, demonstrating that the cross-correlated behaviors of large fluctuations are anti-persistent. In other words, the small fluctuations of WTI price will lead to similar fluctuations in the EUA market, while the large fluctuations of WTI price will result in small fluctuations in the EUA market. The other three return pairs between carbon and crude oil markets have similar cross-correlation behaviors. 4.3. Multifractal detrended fluctuation analysis In order to make a comparison, we also estimate the generalized Hurst exponent H (q) of each separate series by means of the MF-DFA method. The generalized Hurst exponents H (q) describe the persistence of auto-correlation in each series. The empirical results are shown with the red circle and purple diamond lines in Fig. 4. We can find that H (q) is not a constant for each of the underlying series, which means they are multifractal. Further observation shows that the multifractal behaviors of small fluctuations are persistent for q < 0, while the multifractal behaviors of large fluctuations are anti-persistent for q > 2. Based on Eq. (8), the Renyi exponent τ (q) is estimated (see Fig. 5). The exponent τ (q) is nonlinearly dependent on q, which is another piece of evidence that multifractality exists in the relationship of carbon and crude oil markets. To describe multifractality better, we further investigate the multifractal strength by means of Eq. (9) (see Fig. 6). It is widely known that the width of the multifractal spectrum is zero only if the system under study is monofractal. In Fig. 6, the multifractal spectra of all the series are not a point, which reveals that multifractality exists not only in carbon and crude oil markets separately but also in the cross-correlated markets. Fig. 6 shows that there are negative dimensions for some pair of the series. More discussions about the negative fractal dimension can be found in [37,38]. In particular, Mandelbrot [38] considers that positive dimensions f (α) are shown to define a ‘‘typical’’ distribution of the measure, whereas the negative f (α) determines the sampling variability. In this case, f (α) related to some large positive values of α is negative, implying more noise (left part of the multifractal spectrum). Based on MF-DCCA and MF-DFA, the scaling behaviors of large fluctuations are indicated by the H (q) related to positive values of q. Thus, we can conclude that the large fluctuations bring higher variability to the markets.
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Fig. 5. The relationship of τ (q) and q between carbon and crude oil markets. The red circle and purple diamond curves, estimated from MF-DFA, and the blue star curves, estimated from MF-DCCA, represent the relationship between carbon and crude oil markets and the cross-correlated ones, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2 Multifractal degree ∆H and multifractal spectra widths ∆α .
∆α
∆H CER & WTI CER & Brent EUA & WTI EUA & Brent
Carbon
Oil
Cross
Carbon
Oil
Cross
1.2188 1.1129 1.4411 1.1323
1.1737 0.9464 1.1891 1.0116
0.5046 0.4319 0.9658 0.5947
1.0022 0.9009 0.9538 1.0205
0.8352 0.7343 0.8049 0.7896
0.3701 0.3494 0.8248 0.5225
The width of the multifractal spectrum can be regarded as an estimate of multifractal strength, the numerical results of the widths are listed in Table 2. The widths of cross-correlation multifractal spectra for analyzed return series between carbon and crude oil markets are significantly nonzero. The multifractality degree of cross-correlations is smaller than those of auto-correlations, which may be due to the fact that the portfolio containing carbon and crude oil assets are affected in opposite directions by some external market factors, which decrease the volatility. That is to say, the portfolio has much less associated risk. Besides, the degrees of multifractality ∆H are calculated according to Eq. (7) (see Table 2). From Table 2, we can find that the multifractality degrees of cross-correlations are mostly smaller than those of auto-correlations. Moreover, the degrees of multifractality of the underlying series of the carbon market are larger than those of crude oil markets, implying that the
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Fig. 6. Multifractal spectrum for carbon and crude oil markets. The red circle and purple diamond curves, estimated from MF-DFA, and the blue star curves, estimated from MF-DCCA, represent the scaling exponents of carbon and crude oil markets and the cross-correlated ones, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
volatilities of carbon markets are more violent than crude oil markets and that carbon markets are more inefficient than crude oil markets. 4.4. The source of multifractality There are two main sources of multifractality [39,40]. The first is the long-range correlations of large and small fluctuations. The other is the fat-tail distributions of the return series. To verify the contribution of long-range correlations, we can compare the degrees of multifractality of the original series to that of the shuffled series. The shuffling procedures are as follows. First, generate pairs (p, q) of random integer numbers with p, q ≤ N, where N is the length of the time series to be shuffled. Then swap entries p and q. At last, repeat the above two steps 20N times to ensure that the original series are fully shuffled. The popular method of verifying the contribution of fat-tail distribution is comparing the degrees of multifractality between the original and the surrogated series which can be created by Fourier phase randomization [41]. In this paper, we create the surrogated series by the method employed in Ref. [40]. For a given distribution, we generate a series of random numbers {ri , i = 1, 2, . . . , N }, which are rearranged to get the rearranged series {yi , i = 1, 2, . . . , N }, which has the same rank ordering as the original series {xi , i = 1, 2, . . . , N } [42]. In other words, {yi } should rank n if and only if {xi } ranks n [42]. Fig. 7 provides the variation of H (q) of the original series, shuffled series and surrogated series in carbon and crude oil markets. Apparently, the variations of H (q) for the original series, shuffled series and surrogated series can be distinguished
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Fig. 7. Scaling exponents H (q) of the original, shuffled and surrogated series in carbon and crude oil markets.
clearly, which means both long-range correlations and fat-tail distributions play important roles in the contributions of multifractality.
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5. Discussion 5.1. The rolling sample test The method of rolling windows began to be widely used to investigate many topics regarding the financial market after the influential work by Cajueiro and Tabak in [43], for example market efficiency [44]. It can provide additional insights into the evaluation of time series dynamics. However, the selection of window size should be cautious for different research purposes. Wang et al. [45] used 250 data points as the rolling windows, while Cao et al. [46] used the length of several years. In this paper, we take the same measure of Wang et al. [45], choosing 250 days as the length of rolling windows to investigate the dynamics of short term cross-correlations (see Fig. 8). The results are very similar, implying that the conclusion of this research is robust. 5.2. A binomial measure from p-model According to the assertion of Podobnik and Stanley [19], the cross-correlation exponent is equal to the average of the individual Hurst exponents for two fractionally autoregressive integrated moving average (ARFIMA) processes sharing the same random noise, when q = 2. Zhou [20] found that for two time series constructed by a binomial measure from the proposed model (p-model), there exists the following relationship: Hxy (q) = [Hxx (q) + Hyy (q)]/2.
(14)
Hence, we also calculate the average scaling exponents of carbon and crude oil markets (black lines in Fig. 4). We find that the average scaling exponents (black lines in Fig. 4) are larger than the cross-correlation exponents for q < 0 and smaller for q > 0. 5.3. Implications In this paper, we focus on the cross-correlations between carbon and crude oil markets. Firstly, the evidence from the cross-correlation statistic test shows that the existence of cross-correlations is significant at the 5% level. This result indicates that oil price volatility has significant influence on the carbon price. So, the price change of crude oil can affect the volatility of the carbon market. That is helpful in dodging the carbon market risks. Secondly, the findings based on the MFDCCA method reveal that the cross-correlations of the return series between carbon and crude oil markets are multifractal, indicating the nonlinear structure in the cross-correlations between the two markets. Thus, conventional linear models (e.g., correlation coefficient analysis) are not suitable for measuring the cross-correlations between carbon and crude oil markets. The nonlinear models should be more appealing in the empirical analysis of carbon and oil markets. Thirdly, the cross-correlated behavior of small fluctuations (for q < 0) differs from that of large fluctuations (for q > 0), implying that the MF-DCCA method can be modified and proposed as a statistic measurement for detecting the cross-correlations among different fluctuations in these markets. According to efficient market hypothesis, the financial markets can react quickly to new information and thus the market price is unpredictable. Based on the analysis of the multifractality, we can confirm that both the carbon market and the crude oil market are existing multifractal. And from the multifractality degree, we can infer that the carbon market is less efficient than the crude oil market. In other words, the carbon market is much more vulnerable to the external shock. So, the volatility of carbon price is much fiercer, and the risks of carbon market are much bigger. For investors, the carbon market is more risky. As an emerging market, the mechanism of carbon market is imperfection. There are many defects in the carbon market, such as not enough free market transaction and excessive government intervention. So, how to improve the transactions, minimize the interference of non-market factors and prompt to form a stable internal price formation mechanism are to be studied next. Besides, since the carbon markets have a short history, we have to admit that there may be a finite-size effect of the enlargement of multifractality for the limited size of the sample [47]. This also may need further study in the future. 6. Conclusion In this paper, we investigate the cross-correlations between carbon and crude oil markets mainly by means of multifractal detrended cross-correlation analysis (MF-DCCA). The conclusions are summarized as follows. First of all, we find that the cross-correlations between carbon and crude oil markets are significant based on the analysis of the statistic Qcc (m) as well as the MF-DCCA method. Secondly, empirical evidence from MF-DCCA shows that each pair of series between the two markets is cross-correlated. The cross-correlations of small fluctuations differed from that of large fluctuations. The cross-correlation exponents are smaller than the average exponents when q < 0 and larger when q > 0. Thirdly, the cross-correlated behavior between the return series of carbon and crude oil markets are multifractal, implying that conventional linear models, say
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Fig. 8. Scaling exponents H (2) between carbon and crude oil markets with a rolling window. The window length is fixed at 250 days.
vector auto-regression models, are generally unsuitable for describing the relationship between the carbon and crude oil markets. When using the nonlinear model to analyze the cross-correlations between carbon and crude oil markets, one should also consider the structural changes caused by the shock of events. At last, based on the analysis of multifractality degree, we can infer that carbon and crude oil markets are not efficient at present and the volatility of the carbon price is more violent than that of crude oil price.
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Acknowledgments The authors are grateful for the financial support from the National Natural Science Foundation of China (Nos. 71071131, 71090402 and 71371157), the Fundamental Research Funds for the Central Universities (Nos. SWJTU11ZT30 and SWJTU11CX137), and the doctoral program of higher education fund special research project (20120184110020). References [1] D.O. Cajueiro, B.M. Tabak, Testing for long-range dependence in world stock markets, Chaos Solitons Fractals 37 (2008) 918–927. [2] G. Oh, C. Eom, S. Havlin, W.-S. Jung, F. Wang, H.E. Stanley, S. Kim, A multifractal analysis of Asian Foreign exchange markets, Eur. Phys. J. B 85 (6) (2012) 1–6. [3] K. Matia, Y. Ashkenazy, H.E. Stanley, Multifractal properties of price fluctuations of stocks and commodities, Europhys. Lett. 61 (3) (2003) 422–428. [4] S. Kumar, N. Deo, Multifractal properties of the Indian financial market, Physica A 388 (2009) 1593–1602. [5] S.P. Chen, L.Y. 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Physica A journal homepage: www.elsevier.com/locate/physa
Multifractal detrended cross-correlation analysis of carbon and crude oil markets Xiaoyang Zhuang ∗ , Yu Wei ∗∗ , Bangzheng Zhang School of Economics & Management, Southwest Jiaotong University, First Section of Northern Second Ring Road, Chengdu, China
highlights • We confirm the presence of cross-correlations between carbon and crude oil markets both quantitatively and qualitatively. • Based on the MF-DCCA approach, the nonlinear structure of the cross-correlations between carbon and crude oil markets is studied. • The MF-DFA method is used to investigate the multifractal behaviors of carbon markets and crude oil markets respectively.
article
info
Article history: Received 8 June 2013 Received in revised form 20 November 2013 Available online 8 January 2014 Keywords: Multifractal detrended cross-correlation analysis Carbon market Crude oil markets
abstract The complex dynamics between carbon and crude oil markets have been an increasingly interesting area of research. In this paper, we try to take a fresh look at the cross-correlations between carbon and crude oil markets as well as their dynamic behavior employing multifractal detrended cross-correlation analysis. First, we find that the return series of carbon and crude oil markets are significantly cross-correlated. Second, we confirm the existence of multifractality for the return series of carbon and crude oil markets by the multifractal detrended fluctuation analysis. Third, based on the multifractal detrended cross-correlation analysis, we find the existence of power-law cross-correlations between carbon and crude oil markets. The cross-correlated behavior of small fluctuations is found to be more persistent than that of large fluctuations. At last, some relevant discussions and implications of the empirical results are presented. © 2014 Elsevier B.V. All rights reserved.
1. Introduction In the last few decades, the dynamics of financial markets have been of great interest. As far as we know, the previous studies have confirmed that financial markets are extremely complex and exhibit dynamics and non-linear properties, such as multifractality and long-range correlation features [1–6]. For instance, Alvarez-Ramirez et al. [7] found the dynamics and multifractal effect of the oil market by analyzing the auto-correlations of international crude oil prices. Gu et al. [8] studied the multifractal structure of WTI and Brent crude oil markets by employing the multifractal detrended cross-correlation analysis (MF-DCCA) method. Ho et al. [9] conducted the multifractal analysis on the Taiwan Stock index and the Hang Seng index. Kyungsik et al. [10] confirmed the multifractal properties of foreign exchange markets. Ioan et al. [11] investigated the multifractal behavior of Central and Eastern European foreign exchange rates. In the previous works, various methods were developed to quantify the auto-correlation and cross-correlation behaviors of financial markets based on the monofractal and multifractal theory. Peng et al. [12] proposed the detrended fluctuation analysis (DFA) to explore the long-range auto-correlations of a non-stationary time series and widely used in financial
∗ ∗∗
Corresponding author. Corresponding author. Tel.: +86 13618001075. E-mail addresses: [email protected] (X. Zhuang), [email protected] (Y. Wei).
0378-4371/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physa.2013.12.048
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time series analysis [13,14]. Then, DFA was extended into two important methods: one is the multifractal detrended fluctuation analysis (MF-DFA) proposed by Kantelhardt et al. [15], which is a powerful tool to investigate the multifractality of the financial time series [16–18]; the other is the detrended cross-correlation analysis (DCCA) proposed by Podobnik and Stanley [19], which can be used to quantify the cross-correlations between two non-stationary financial time series. To examine the multifractal characteristics of two cross-correlated non-stationary time series, Zhou [20] proposed the multifractal detrended cross-correlation analysis (MF-DCCA) based on MF-DFA and DCCA. After that, MF-DCCA was widely used to investigate the cross-correlations in financial markets [21–25]. For example, Wang et al. [25] investigated the crosscorrelations of West Texas Intermediate (WTI) crude oil spot and futures return series by means of MF-DCCA. On connecting the environment energy problem and the economic development, the mechanism of carbon market becomes a hot topic. In addition, with the rapid growth of the carbon market, carbon price fluctuations are increasingly important for market participants. The price of carbon is classically driven by the balance between supply and demand, and by other factors related to the market structure and institutional policies. On the supply side, the number of allowances distributed is determined by Member-State through National Allocation Plans. On the demand side, the use of CO2 allowances is a function of expected CO2 emissions. In turn the level of emissions depends on a large number of factors, such as unexpected fluctuations in energy demand and energy prices. The demand for allowances can be affected by economic growth and financial markets as well, but that latter impact needs to be further assessed in the academic community. In this paper, we try to take a fresh look at the cross-correlations between carbon and energy markets. Since fossil energy consumption has been considered as one of the main drivers of carbon prices, the price change of the energy market is a key to the international carbon market price formation mechanism. Mansanet-Bataller et al. [26] found that carbon prices are linked to the fossil fuel use. Alberola et al. [27] emphasized that the nature of the relationship between energy and carbon prices varies depending on the period under consideration. Convery and Redmond [28] found that an increase of fossil energy price will push the carbon price, and vice versa. Kanen [29] revealed that crude oil prices are the main thrust of the changes in natural gas prices; meanwhile fluctuations in natural gas prices will affect the tariff changes and ultimately impact the carbon price. Therefore, the analysis of cross-correlations between carbon and energy markets is essential and plays an important role in exploring the internal operation mechanism and forecasting the carbon price trend. In the paper, we choose crude oil markets as the representative of energy markets. Crude oil takes a big percentage in energy markets, and occupies a dominant position in the EU’s energy consumption. In addition, the previous studies have already confirmed that the presence of correlation and multifractality in crude oil markets and also implies inefficiency [30–32]. The organization of this paper is as follows. Section 2 introduces methods employed in this study. Section 3 briefly described the data used in our work. Section 4 provides the detailed empirical results. Section 5 presents some discussion. Then, the last section concludes. 2. Methodology There are two time series {x(i)} and {y(i)}, i = 1, 2, . . . , N, where N is the length of the series. The MF-DCCA method can be summarized as follows: First, calculate the profile X (i) =
i
(x(k) − x¯ ) ,
Y (i) =
i
(y(k) − y¯ ) ,
i = 1, 2, . . . , N ,
(1)
k=1
k=1
where x¯ and y¯ denote the averaging over the two whole time series x(i) and y(i). Second, divide the two profiles X (i) and Y (i) into Ns = int(N /s) non-overlapping segments of equal length s. Since the length N of the series is often not a multiple of the given time scale s, a short part at the end of each profile may remain. In order not to discard this part of the series, the same procedure is repeated starting from the opposite end of each profile. Thereby, 2Ns segments are obtained together. Then, estimate the local trends for each of the 2Ns segments by means of the mth order polynomial fit. Then the detrended covariance is determined by F 2 (s, λ) =
s 1 X(λ−1)s+j (j) − X˜ λ (j) Y(λ−1)s+j (j) − Y˜λ (j) s j =1
(2)
for each segment λ, λ = 1, 2, . . . , Ns and F 2 (s, λ) =
s 1 XN −(λ−Ns )s+j (j) − X˜ λ (j) YN −(λ−Ns )s+j (j) − Y˜λ (j) s j =1
(3)
for each segment λ, λ = Ns + 1, Ns + 2, . . . , 2Ns . Here, X˜ λ (j) and Y˜λ (j) denote the fitting polynomial with order m in segment λ. Then average over all segments to obtain the qth-order fluctuation function
Fq (s) =
2Ns 1
2Ns λ=1
q/2 F (s, λ) 2
1/q (4)
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for any q ̸= 0, and
F0 (s) = exp
2Ns 1
4Ns λ=1
ln F (s, λ)
2
(5)
for q = 0. Finally, analyze the scaling behavior of the fluctuations by observing log–log plots of Fq (s) versus s for each value of q. If the two series are long-range cross-correlated, Fq (s) will increase for large values of s, as a power-law Fq (s) ∝ sHxy (q) .
(6)
The scaling exponent Hxy (q), known as the generalized cross-correlation exponent, describes the power-law relationship between two series, can be obtained by observing the slope of the log–log plots of Fq (s) versus s through the method of ordinary least squares (OLS). In particular, if {x(i)} is identical to {y(i)}, then MF-DCCA is equivalent to MF-DFA. Furthermore, the scaling exponent Hxy (q) has similar properties and interpretation to the generalized Hurst exponent H (q). If scaling exponent Hxy (q) is independent of q, the cross-correlations of two series are monofractal, otherwise, these are multifractal. If scaling exponent Hxy (q) > 0.5, the cross-correlations of the two time series related to q are long-range persistent, which implies that one price is likely to increase following an increase of the other price. If scaling exponent Hxy (q) < 0.5, the cross-correlations of the two time series related to q are anti-persistent, which implies that an increase of one price is likely to be followed by a decrease of the other price. If scaling exponent Hxy (q) = 0.5, there are no crosscorrelations or at most short-range cross-correlations between two time series. In order to measure the degree of multifractality [33], the financial risk measure ∆H is proposed as the following
∆H = Hmax (q) − Hmin (q).
(7)
Apparently, richer multifractality corresponds to higher variability of H (q). Therefore, the greater ∆H is, the stronger is the degree of multifractality, and then more inefficient is the market. According to Shadkhoo and Jafari [34], a similar relationship between classical Renyi exponent τxy (q) and q can be given by
τxy (q) = qHxy (q) − 1.
(8)
If the scaling exponent function τxy (q) is a linear function of q, the cross-correlations of the two time series are monofractal. Otherwise, these are multifractal. The following relationships can be obtained by a Legendre transform ′ α = Hxy (q) + qHxy (q) and fxy (α) = q(α − Hxy (q)) + 1.
(9)
Here, α is the Holder exponent or the singularity strength, which is used to characterize the singularities of the time series. The multifractal spectrum fxy (α) describes the singularity content of the time series. The strength of multifractality can be estimated by the width of singularity [35], which is given by
∆α = αmax − αmin .
(10)
3. Data In this study, two main carbon prices (Certification Emission Reduction (CER) and EU emission Allowance (EUA)) are selected. The daily closing price of each carbon price used in this study is available from Wind database. For CER, we take daily data from March 14, 2008 to December 15, 2012. For EUA, we take daily data from April 22, 2005 to December 15, 2012. As for crude oil markets, we chose daily spot prices of West Texas Intermediate (WTI) crude oil and Brent crude oil as representatives. After eliminating the non-matching missing data, we get the lengths of four pairs of series, CER & WTI, CER & Brent, EUA & WTI and EUA & Brent, as 1196, 1189, 1914 and 1919, respectively. We define the daily price return rt , the natural logarithmic difference of the daily closing price Pt , as rt = ln Pt − ln Pt −1 . Fig. 1 presents the curve of daily carbon prices of carbon and crude oil markets. In order to effectively investigate the cross-correlations between the two different kinds of markets, all the different units of the time series are converted to dollars per ton. Specifically, the returns of the daily price of the carbon market are adjusted to US dollars using the daily Euro/US dollar exchange rate. 4. Empirical results 4.1. Testing for the cross-correlations In order to investigate the nonlinear cross-correlations between carbon and crude oil markets, we apply a crosscorrelation statistic proposed by Podobnik et al.
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Fig. 1. Daily prices of carbon and crude oil markets. The blue and green lines represent the price series of carbon and crude oil markets, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
For two time series {x(i)} and {y(i)} , i = 1, 2, . . . , N, the test statistic is defined as Qcc (m) = N 2
m Xi2 , N −i i=1
(11)
where the cross-correlation function is N
xk yk−i
k=i+1
Xi =
N k=1
x2k
N
.
(12)
y2k
k=1
The cross-correlation statistic Qcc (m) is approximately χ 2 (m) distributed with m degrees of freedom. Generally, this statistic is employed for returns, not for original time series. The cross-correlation test agrees well with the χ 2 (m) distribution, if there were no cross-correlations between two return series. If the cross-correlation test exceeded the critical value of the χ 2 (m) distribution, the cross-correlations are significant at a special significance level. Fig. 2 shows the cross-correlation statistics Qcc (m) versus the degree of freedom m for the returns between the carbon and crude oil markets. The degrees of freedom vary from 1 to 1000. For comparison, the critical values for the χ 2 (m) distribution at the 5% level of significance are also provided in Fig. 2. Fig. 2 indicates that the cross-correlation statistics Qcc (m) of the four pairs are larger than the critical values of the χ 2 (m) distribution when m is larger than 2. Therefore, we can reject the null hypothesis of no cross-correlations, in other words, long-range cross-correlations exist between the carbon and crude oil markets.
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Fig. 2. Cross-correlation statistics Qcc (m) vs. m. The blue, black, green, purple and red lines represent the pairs of CER & WTI, CER & Brent, EUA & WTI, EUA & Brent and critical value, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 The value of ρDCCA for a given window size n. Size
16
32
64
128
256
CER & WTI CER & Brent EUA & WTI EUA & Brent
0.3526 0.4016 0.5872 0.5828
0.3545 0.4044 0.5906 0.5906
0.3674 0.4154 0.6019 0.6043
−0.0004
−0.2026 −0.1929
0.4196 0.6364 0.6390
0.0107 0.6728
In order to affirm our results above more carefully, we also apply another new method proposed by Podobnik et al. [36], 2 2 defined as the ratio between the detrended covariance function FDCCA and the detrended variance FDFA , the function reads as
ρDCCA =
2 FDCCA (n)
FDFA1 (n)FDFA2 (n)
.
(13)
The value of ρDCCA ranges between −1 ≤ ρDCCA ≤ 1. The two series have no cross-correlation if ρDCCA is equal to zero, otherwise they have cross-correlation. And, ρDCCA can identify the level of cross-correlation from the positive and the negative case. We calculate the value of ρDCCA based on different values of window size n (n = 16, 32, 64, 128, 256) (see Table 1) that we can draw the conclusions that consistent with the cross-correlation test as above. 4.2. Multifractal detrended cross-correlation analysis The cross-correlation test based on the statistics in Eq. (11) can only test the existence of cross-correlations between carbon and crude oil markets qualitatively. In order to affirm our results obtained above, we also apply the MF-DCCA method proposed by Podobnik and Stanley to test the presence of cross-correlation quantitatively. Fig. 3 shows the log–log plots of fluctuation function Fq (s) versus time scale s between the carbon and crude oil markets. As shown in Fig. 3, for different values of q, all the curves exhibit linearity during a span, indicating that there exists a powerlaw cross-correlation in each pair of daily return series. The power-law relationship indicates that a large price change in the crude oil markets is more likely to be followed by a large price change in carbon market, and vice versa. In order to explore the different changes of the cross-correlation exponent with varying values of q, we display the scaling cross-correlation exponents Hxy (q) with q varying from −5 to 5 in Fig. 4 (blue star lines). We can see that the scaling exponents Hxy (q) decrease with the increase of q nonlinearly, implying that multifractality exists in the cross-correlations between carbon and crude oil markets. The scaling exponents Hxy (q) for q < 0 are larger than those for q > 0, indicating that the cross-correlated behavior of small fluctuations is more persistent than that of large fluctuations. For q = 2, the scaling exponent Hxy (q) is just similar to the Hurst exponent calculated from the method of MF-DFA. The cross-correlation exponents Hxy (2) for CER & WTI and CER & Brent both are slightly larger than 0.5, implying that their cross-correlations are weakly persistent. The cross-correlation exponents Hxy (2) of EUA & WTI and EUA & Brent are smaller than 0.5, indicating that their cross-correlations are anti-persistent.
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Fig. 3. Log–log plots of F (s) vs. s between carbon and crude oil markets. The lines from bottom to top are corresponding to the plots with q varying from −5 to 5.
Fig. 4 also shows that, for different values of q, the scaling exponents are different. In other words, for different values of q, the cross-correlated behaviors of the four return pairs are different. We can find that for EUA & WTI, the scaling exponents decrease from about 0.7 to 0.2, implying that the cross-correlated behavior shows strong multifractality. Moreover, the
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Fig. 4. The nonlinear relationship of H (q) vs. q between carbon and crude oil markets. The red circle and purple diamond curves, estimated from MFDFA, and the blue star curves, estimated from MF-DCCA, represent the scaling exponents of carbon and crude oil markets and the cross-correlated ones, respectively. The black lines are the average of the scaling exponents in the two markets. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
scaling exponents for q < 0 are larger than 0.5, indicating the cross-correlated behavior of small fluctuations is persistent. The scaling exponents for q > 1 are smaller than 0.5, demonstrating that the cross-correlated behaviors of large fluctuations are anti-persistent. In other words, the small fluctuations of WTI price will lead to similar fluctuations in the EUA market, while the large fluctuations of WTI price will result in small fluctuations in the EUA market. The other three return pairs between carbon and crude oil markets have similar cross-correlation behaviors. 4.3. Multifractal detrended fluctuation analysis In order to make a comparison, we also estimate the generalized Hurst exponent H (q) of each separate series by means of the MF-DFA method. The generalized Hurst exponents H (q) describe the persistence of auto-correlation in each series. The empirical results are shown with the red circle and purple diamond lines in Fig. 4. We can find that H (q) is not a constant for each of the underlying series, which means they are multifractal. Further observation shows that the multifractal behaviors of small fluctuations are persistent for q < 0, while the multifractal behaviors of large fluctuations are anti-persistent for q > 2. Based on Eq. (8), the Renyi exponent τ (q) is estimated (see Fig. 5). The exponent τ (q) is nonlinearly dependent on q, which is another piece of evidence that multifractality exists in the relationship of carbon and crude oil markets. To describe multifractality better, we further investigate the multifractal strength by means of Eq. (9) (see Fig. 6). It is widely known that the width of the multifractal spectrum is zero only if the system under study is monofractal. In Fig. 6, the multifractal spectra of all the series are not a point, which reveals that multifractality exists not only in carbon and crude oil markets separately but also in the cross-correlated markets. Fig. 6 shows that there are negative dimensions for some pair of the series. More discussions about the negative fractal dimension can be found in [37,38]. In particular, Mandelbrot [38] considers that positive dimensions f (α) are shown to define a ‘‘typical’’ distribution of the measure, whereas the negative f (α) determines the sampling variability. In this case, f (α) related to some large positive values of α is negative, implying more noise (left part of the multifractal spectrum). Based on MF-DCCA and MF-DFA, the scaling behaviors of large fluctuations are indicated by the H (q) related to positive values of q. Thus, we can conclude that the large fluctuations bring higher variability to the markets.
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Fig. 5. The relationship of τ (q) and q between carbon and crude oil markets. The red circle and purple diamond curves, estimated from MF-DFA, and the blue star curves, estimated from MF-DCCA, represent the relationship between carbon and crude oil markets and the cross-correlated ones, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 2 Multifractal degree ∆H and multifractal spectra widths ∆α .
∆α
∆H CER & WTI CER & Brent EUA & WTI EUA & Brent
Carbon
Oil
Cross
Carbon
Oil
Cross
1.2188 1.1129 1.4411 1.1323
1.1737 0.9464 1.1891 1.0116
0.5046 0.4319 0.9658 0.5947
1.0022 0.9009 0.9538 1.0205
0.8352 0.7343 0.8049 0.7896
0.3701 0.3494 0.8248 0.5225
The width of the multifractal spectrum can be regarded as an estimate of multifractal strength, the numerical results of the widths are listed in Table 2. The widths of cross-correlation multifractal spectra for analyzed return series between carbon and crude oil markets are significantly nonzero. The multifractality degree of cross-correlations is smaller than those of auto-correlations, which may be due to the fact that the portfolio containing carbon and crude oil assets are affected in opposite directions by some external market factors, which decrease the volatility. That is to say, the portfolio has much less associated risk. Besides, the degrees of multifractality ∆H are calculated according to Eq. (7) (see Table 2). From Table 2, we can find that the multifractality degrees of cross-correlations are mostly smaller than those of auto-correlations. Moreover, the degrees of multifractality of the underlying series of the carbon market are larger than those of crude oil markets, implying that the
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Fig. 6. Multifractal spectrum for carbon and crude oil markets. The red circle and purple diamond curves, estimated from MF-DFA, and the blue star curves, estimated from MF-DCCA, represent the scaling exponents of carbon and crude oil markets and the cross-correlated ones, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
volatilities of carbon markets are more violent than crude oil markets and that carbon markets are more inefficient than crude oil markets. 4.4. The source of multifractality There are two main sources of multifractality [39,40]. The first is the long-range correlations of large and small fluctuations. The other is the fat-tail distributions of the return series. To verify the contribution of long-range correlations, we can compare the degrees of multifractality of the original series to that of the shuffled series. The shuffling procedures are as follows. First, generate pairs (p, q) of random integer numbers with p, q ≤ N, where N is the length of the time series to be shuffled. Then swap entries p and q. At last, repeat the above two steps 20N times to ensure that the original series are fully shuffled. The popular method of verifying the contribution of fat-tail distribution is comparing the degrees of multifractality between the original and the surrogated series which can be created by Fourier phase randomization [41]. In this paper, we create the surrogated series by the method employed in Ref. [40]. For a given distribution, we generate a series of random numbers {ri , i = 1, 2, . . . , N }, which are rearranged to get the rearranged series {yi , i = 1, 2, . . . , N }, which has the same rank ordering as the original series {xi , i = 1, 2, . . . , N } [42]. In other words, {yi } should rank n if and only if {xi } ranks n [42]. Fig. 7 provides the variation of H (q) of the original series, shuffled series and surrogated series in carbon and crude oil markets. Apparently, the variations of H (q) for the original series, shuffled series and surrogated series can be distinguished
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Fig. 7. Scaling exponents H (q) of the original, shuffled and surrogated series in carbon and crude oil markets.
clearly, which means both long-range correlations and fat-tail distributions play important roles in the contributions of multifractality.
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5. Discussion 5.1. The rolling sample test The method of rolling windows began to be widely used to investigate many topics regarding the financial market after the influential work by Cajueiro and Tabak in [43], for example market efficiency [44]. It can provide additional insights into the evaluation of time series dynamics. However, the selection of window size should be cautious for different research purposes. Wang et al. [45] used 250 data points as the rolling windows, while Cao et al. [46] used the length of several years. In this paper, we take the same measure of Wang et al. [45], choosing 250 days as the length of rolling windows to investigate the dynamics of short term cross-correlations (see Fig. 8). The results are very similar, implying that the conclusion of this research is robust. 5.2. A binomial measure from p-model According to the assertion of Podobnik and Stanley [19], the cross-correlation exponent is equal to the average of the individual Hurst exponents for two fractionally autoregressive integrated moving average (ARFIMA) processes sharing the same random noise, when q = 2. Zhou [20] found that for two time series constructed by a binomial measure from the proposed model (p-model), there exists the following relationship: Hxy (q) = [Hxx (q) + Hyy (q)]/2.
(14)
Hence, we also calculate the average scaling exponents of carbon and crude oil markets (black lines in Fig. 4). We find that the average scaling exponents (black lines in Fig. 4) are larger than the cross-correlation exponents for q < 0 and smaller for q > 0. 5.3. Implications In this paper, we focus on the cross-correlations between carbon and crude oil markets. Firstly, the evidence from the cross-correlation statistic test shows that the existence of cross-correlations is significant at the 5% level. This result indicates that oil price volatility has significant influence on the carbon price. So, the price change of crude oil can affect the volatility of the carbon market. That is helpful in dodging the carbon market risks. Secondly, the findings based on the MFDCCA method reveal that the cross-correlations of the return series between carbon and crude oil markets are multifractal, indicating the nonlinear structure in the cross-correlations between the two markets. Thus, conventional linear models (e.g., correlation coefficient analysis) are not suitable for measuring the cross-correlations between carbon and crude oil markets. The nonlinear models should be more appealing in the empirical analysis of carbon and oil markets. Thirdly, the cross-correlated behavior of small fluctuations (for q < 0) differs from that of large fluctuations (for q > 0), implying that the MF-DCCA method can be modified and proposed as a statistic measurement for detecting the cross-correlations among different fluctuations in these markets. According to efficient market hypothesis, the financial markets can react quickly to new information and thus the market price is unpredictable. Based on the analysis of the multifractality, we can confirm that both the carbon market and the crude oil market are existing multifractal. And from the multifractality degree, we can infer that the carbon market is less efficient than the crude oil market. In other words, the carbon market is much more vulnerable to the external shock. So, the volatility of carbon price is much fiercer, and the risks of carbon market are much bigger. For investors, the carbon market is more risky. As an emerging market, the mechanism of carbon market is imperfection. There are many defects in the carbon market, such as not enough free market transaction and excessive government intervention. So, how to improve the transactions, minimize the interference of non-market factors and prompt to form a stable internal price formation mechanism are to be studied next. Besides, since the carbon markets have a short history, we have to admit that there may be a finite-size effect of the enlargement of multifractality for the limited size of the sample [47]. This also may need further study in the future. 6. Conclusion In this paper, we investigate the cross-correlations between carbon and crude oil markets mainly by means of multifractal detrended cross-correlation analysis (MF-DCCA). The conclusions are summarized as follows. First of all, we find that the cross-correlations between carbon and crude oil markets are significant based on the analysis of the statistic Qcc (m) as well as the MF-DCCA method. Secondly, empirical evidence from MF-DCCA shows that each pair of series between the two markets is cross-correlated. The cross-correlations of small fluctuations differed from that of large fluctuations. The cross-correlation exponents are smaller than the average exponents when q < 0 and larger when q > 0. Thirdly, the cross-correlated behavior between the return series of carbon and crude oil markets are multifractal, implying that conventional linear models, say
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Fig. 8. Scaling exponents H (2) between carbon and crude oil markets with a rolling window. The window length is fixed at 250 days.
vector auto-regression models, are generally unsuitable for describing the relationship between the carbon and crude oil markets. When using the nonlinear model to analyze the cross-correlations between carbon and crude oil markets, one should also consider the structural changes caused by the shock of events. At last, based on the analysis of multifractality degree, we can infer that carbon and crude oil markets are not efficient at present and the volatility of the carbon price is more violent than that of crude oil price.
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