Cross-correlation analysis between energy and carbon markets in China based on multifractal theory

Abstract:

Global climate change has become the greatest threat to mankind, endangering the ecological security of the earth and the long-term development of human society. Therefore, how to effectively reduce greenhouse gas emissions and curb the trend of global warming has become a common challenge facing all countries in the world. Fossil energy combustion is the fundamental cause of climate change. In this paper, the price return series of domestic energy and carbon markets are taken as the research objects. Firstly, the cross-correlation test is used to verify that there is an obvious cross-correlation between the return series of the energy and carbon markets. Based on this, the multifractal characteristics of energy market and carbon market cross-series are empirically studied. The empirical results show that there is an interactive correlation between energy and carbon markets and this relationship has multifractal characteristics; the interactive correlation between return series has multifractal characteristics. The long-range correlation between small fluctuations and large fluctuations and the fat tail distribution of return series are all reasons for the formation of multifractality. These conclusions will help to understand the non-linear dependence and potential dynamic mechanism between energy and carbon markets in China.

1. INTRODUCTION

Global climate change has become the greatest threat to mankind, seriously endangering the ecological security of the earth and the long-term development of human society. Therefore, how to reduce greenhouse gas emissions and curb the trend of global warming has become a new challenge facing all countries in the world.

In 2018, China’s greenhouse gas emissions accounted for 27% of the total global greenhouse gas emissions and have since become the world’s largest greenhouse gas emitter. At present, China’s total greenhouse gas emissions are about equal to the sum of the greenhouse gas emissions of EU countries and the USA, and the international pressure of emission reduction and energy demand are increasing dramatically. As a responsible big country, China actively promotes the construction of a community of human destiny. China should and can make its own contribution to the international community. On 29 October 2011, the General Office of the National Development and Reform Commission issued a circular on ‘Pilot Carbon Emission Trading’. Seven regions, Beijing, Tianjin, Shanghai, Chongqing, Guangdong, Hubei and Shenzhen, began the practice of carbon market pilot.

As a financial market, the carbon market is essentially the same as other financial markets, such as securities market and foreign exchange market. Economic development, system construction and supply–demand relationship will affect market development. At the same time, as a new financial market, China’s carbon market has obvious differences with the traditional financial market. It has not only common influence factors of the traditional financial market but also a series of unique factors such as fossil energy price, quota allocation system and extreme weather change. Carbon dioxide produced by fossil energy consumption is the core foothold of carbon emissions research, and the scale of carbon dioxide emissions is determined by factors such as the output scale of production sector, the proportion of various types of energy use and the actual utilization rate of energy in production sector. Relevant scholars have also confirmed that fossil energy burning accounts for more than 80% of the total emissions of greenhouse gases, and fossil energy is mainly coal, oil and natural gas [1], so these three types of fossil energy have become the focus of research. China’s reform in the field of energy is also deepening. The general goal of the modern energy system of ‘adhering to the green energy orientation and strengthening the clean energy industry’ has been put forward. The general direction of ‘actively promoting energy price reform’ has also been emphasized again. We should promote energy transformation, optimization and upgrading so as to fully and effectively reflect the market nature of energy prices. Also, the guidance of carbon quota trading price conforms to the law of market value.

Our contributions are threefold: first, ours is the first study to apply the rolling windows MF-DCCA method in the empirical analysis of multifractal cross-correlations between energy and carbon markets. Second, it reveals the multifractal strength of the cross-correlation between the two markets and the risk of the related markets. Third, our empirical results, to some extent, indicate that multifractality exists in the return series of energy and carbon markets.

2. LITERATURE REVIEW

Obemdorfer (2009) studied the relationship between the share price and carbon price of major power companies in the European Union. It is believed that there is a significant positive correlation between them. At the same time, the impact of carbon price changes on the share price of power companies is asymmetric. Classification of specific time intervals can explain the whole impact process change [2]. Bunn and Fezzi (2007) used vector autoregression model to study the relationship among European Union carbon price, British electricity price and natural gas price. They believed that the change of natural gas price would have a significant impact on European Union carbon price and that the change of natural gas price and European Union carbon price would have a significant impact on British electricity price. For the definite effect [3], Benz and Truck (2009) studied on the basis of the GARCH model and elaborated on how to dynamically model the price of carbon dioxide emission quota [1]. Chevallier (2011) used the Bayesian self-incentive threshold autoregression model and Markov autoregression model to study the correlation between EU carbon price and macroeconomic variables, believing that there is a consistent change relationship between the two. At the same time, it also specifically analyzed the effect of Brent crude oil price on carbon price [4]. On this basis, Reboredo (2013) uses the time-varying Copula model to study the correlation between EU carbon emission futures price and Brent crude oil price, further confirming the stable and effective positive correlation between the two markets, but cannot verify whether the conduction effect between the two markets is effective [5]. Marvso et al. (2010) used the linear model to analyze the relationship between fossil energy price and carbon price, pointing out that controlling fossil energy cost can affect carbon price to a certain extent; after that, scholars began to concretely analyze the impact of energy price on carbon price [6]. For example, Hinterman (2010) used the econometric model to analyze European Union carbon price. The dependence between emission price and market basic index indicates that crude oil price is the key factor to control EUA price [7]. Keppler (2010) mainly studies the mechanism of price factors such as oil, natural gas and electricity affecting carbon price and finds that rising oil price will cause fluctuation of natural gas price, while natural gas causes fluctuation of natural gas price [8]. Rahman et al. (2012) also pointed out that the main cause of global warming could be attributed to the burning of fossil fuels such as coal, oil and natural gas [9]. Byun (2013) and Mehmet (2016) took the European Union carbon trading market and the international energy market as examples, from price fluctuations and markets [10, 11]. From the perspective of market risk, this paper analyses the market correlation between energy market and carbon trading market and finds that there is a time-varying relationship between them. MF-DFA is a new method proposed by Kantelhardt et al. [12] to verify the existence of long-term memory and multifractal features in non-linear time series. Compared with R/S analysis and modified R/S analysis, this method can avoid misjudgment of correlation and the results are more reliable. Wang et al. [13]studied the autocorrelation behavior of WTI crude oil price fluctuation from a multi-scale perspective by the MFDFA method. The results showed that the volatility of crude oil price exhibited long-term memory multi-fractal characteristics in a small time scale, but in a larger time scale, it exhibited single fractal characteristics. Chen et al. [13, 14] used the MFDFA and multifractal spectrum to analyze the multifractal characteristics and causes of the futures price return series of metals and agricultural products in China. The results showed that long-term memory was the main reason for the existence of multifractal characteristics in the market. Zhuang et al. [15, 16] analyzed the multifractality of Chinese stock market and its cross-correlation with WTI crude oil price. Matia et al. [17] used MFDFA method to test the daily prices of 29 commodities and 2449 stocks in the past 15 years. The results showed that both commodity prices and stock prices had multifractal characteristics and found that the multifractal spectrum of commodity prices is significantly wider than that of stock prices, which was mainly due to the strong correlation of commodity price fluctuations. Wei et al. used the MF-DCCA method to investigate the cross-correlations between crude oil spot and futures markets [18, 19].

In this paper, rolling windows MF-DCCA and other multi-fractal analysis methods are used to study the volatility and non-linear dynamic relationship between energy and carbon markets, revealing the intrinsic characteristics of market volatility, and using fractal characteristic statistics such as cross-correlation index and multi-fractal spectral width to measure market risk. This paper is organized as follows. Section 2 mainly focuses upon the description of MF-DFA, MF-DCCA and rolling window MF-DCCA. Section 3 describes the data of energy and carbon markets. Section 4 provides the empirical analysis. Section 5 is the conclusion of this paper.

3. METHODOLOGY

Zhou [21] combined the MF-DFA method with the DCCA method proposed by Podobnik [20], and proposes the MF-DCCA method, which is mainly used to study the correlation and multifractal characteristics of two non-stationary series, thus providing a new method for this study. We followed the methods of Zou et al. 2019 [23], and we suppose that there are two time series |${x}_i$|and |${y}_i$|⁠; we have |$i=1,2,3\dots, \dots L$|⁠, when L is the length of the two time series. We will introduce MF-DCCA method by the following steps:

The first step: determine the profile of time series

$$\begin{equation} X(i)=\sum \limits_{n=1}^i\left(x(n)-\overline{x}\right),Y(i)=\sum \limits_{n=1}^i\left(y(n)-\overline{y}\right),i=1,2,\dots, L \end{equation}$$

(1)

In which |$\overline{x}=\frac{1}{L}\sum\nolimits_{i=1}^Lx(i),\overline{y}=\frac{1}{L}\sum \nolimits_{i=1}^Ly(i)$|⁠.

The second step: divide the profile time series |${x}_i$| and |${y}_i$| into |${L}_S=\mathit{\operatorname{int}}(L/s)$| non-overlapping segments of equal length |$s$|⁠. Since N is usually not an integer multiple of s, in order not to neglect the last series, the segmentation process is repeated from the tail of the series to obtain 2L_s sub-series.

The third step: The local trends |${X}^u(i)$| and |${Y}^u(i)$| for each segment u|$(u=1,2,3\dots, 2{L}_s)$| are evaluated by least squares fits of the time series, then fitting 2L_s sub-series with m-order polynomials, eliminating trend covariance

$$\begin{align} {F}^2\left(s,u\right)=&\ \frac{1}{s}\sum \limits_{i=1}^s\left|X\left(\left(u-1\right)s+i\right)-{\tilde{X}}_u(i)\right|\nonumber\\ &\cdot p \left|Y\left(\left(u-1\right)s+i\right)-{\tilde{Y}}_u(i)\right|, \end{align}$$

(2)

when |$u=1,2,3\dots, {L}_s$|⁠;

$$\begin{align} {F}^2\left(s,u\right)=&\ \frac{1}{s}\sum \limits_{i=1}^s\left|X\left(L-\left(u-{L}_s\right)s+i\right)-{\tilde{X}}_u(i)\right|\nonumber\\ &\cdot p \left|Y\left(L-\left(u-{L}_s\right)s+i\right)-{\tilde{Y}}_u(i)\right|, \end{align}$$

(3)

when |$u={L}_s+1,{L}_s+2,{L}_s+3\dots, 2{L}_s$|⁠. Therefore, the trends |${\tilde{X}}_u(i)$| and|${\tilde{Y}}_u(i)$| denote the fitting polynomial with order m in each part u.

The fourth step: calculate the qth-order detrended covariance function

$$\begin{equation} {F}_q(s)={\left\{\frac{1}{2{L}_s}\sum \limits_{u=1}^{2{L}_s}{\left[{F}^2\left(s,u\right)\right]}^{q/2}\right\}^{1/q}},q\ne 0 \end{equation}$$

(4)

$$\begin{equation} {F}_0(s)=\exp \left\{\frac{1}{4{L}_s}\sum \limits_{u=1}^{2{L}_s}\ln \left[{F}^2\left(s,u\right)\right]\right\},q=0. \end{equation}$$

(5)

The fifth step: for each fixed value of q, determine the scaling behavior of covariance fluctuations by observing log–log plots |${F}_q(s)$|⁠. If there is a long-range electricity law interaction between two series |${x}_i$| and |${y}_i$|⁠, then for a large value of s, we have

$$\begin{equation} {F}_q(s)\propto{s}^{H_{xy}(q)} \end{equation}$$

(6)

In which, |${H}_{xy}(q)$| is represented the cross-correlation exponent, describing the electricity-low relationship between two time series. Especially, when the two series |${x}_i$| and |${y}_i$| are identical, then MF-DCCA is simplified to MF-DFA. When q is equal to 2, MF-DCCA is the standard of the DFA. Besides, If the cross-correlation scaling exponent |${H}_{xy}(q)$| changes with q, the cross-correlation between the two sequences is multifractal. If the cross-correlation scaling exponent |${H}_{xy}(q)$| is not dependent on q, the cross-correlation between the two sequences is monofractal. If the cross-correlation scaling exponent |${H}_{xy}(q)>0.5$|⁠, the cross-correlations between the return fluctuations of the two series related to q are long-range persistent, which means that one rising price is likely to follow another rising price. If the cross-correlation scaling exponent |${H}_{xy}(q)<0.5$|⁠, the cross-correlations between the return fluctuations of the two series are anti-persistent, which means that one rising price is likely to follow another falling price. If the cross-correlation scaling exponent |${H}_{xy}(q)=0.5$|⁠, there is no cross-correlations between the return fluctuations of the two series related to q.

This paper improves the interval segmentation method of MF-DCCA by using rolling window technology, which is embodied in Step 3. In Step 3, rolling window MF-DCCA uses rolling window technology for continuous overlapping interval segmentation, resulting in a sharp increase in sub-intervals (from L_s or 2L_s to L-s + 1). The function is expressed as

$$\begin{equation} {F}_q(s)={\left\{\frac{1}{L-s+1}\sum \limits_{u=1}^{L-s+1}{\left[{F}^2\left(s,u\right)\right]}^{q/2}\right\}^{1/q}},q\ne 0 \end{equation}$$

(7)

$$\begin{equation} {F}_0(s)=\exp \left\{\frac{1}{2\left(L-s+1\right)}\sum \limits_{u=1}^{L-s+1}\ln \left[{F}^2\left(s,u\right)\right]\right\},q=0 \end{equation}$$

(8)

ΔH can be used to measure the degree of multifractality between two series, which is expressed as

$$\begin{equation} \varDelta H={H}_{\mathrm{max}}(q)-{H}_{\mathrm{min}}(q). \end{equation}$$

(9)

Figure 1

Daily prices of energy and carbon markets.

Obviously, the higher variability of H(q) is, the richer multifractality will be. Therefore, the larger ΔH is, the stronger the degree of multifractality will be, and then the market is becoming more inefficient.

It is pointed out that there is the following relationship between the qth-order cross-correlation exponent |${H}_{xy}(q)$| and the multifractal quality exponent (Renyi exponent)|${\tau}_{xy}(q)$| [18].

$$\begin{equation} {\tau}_{xy}(q)={qH}_{xy}(q)-1 \end{equation}$$

(10)

If the scaling exponent formula |${\tau}_{xy}(q)$| is linear to q, the cross-correlations of the series are considered to be monofractal. If the scaling exponent formula |${\tau}_{xy}(q)$| is not linear to q, the cross-correlations of the two time series are considered to be multifractal. Through Legendre transformation, the following relations can be obtained:

$$\begin{equation} \alpha ={H}_{xy}(q)+q{H}_{xy}^{\prime }(q) \end{equation}$$

(11)

$$\begin{equation} {f}_{xy}\left(\alpha \right)=q\left(\alpha -{H}_{xy}(q)\right)+1 \end{equation}$$

(12)

In the method MF-DFA, |$\varDelta h={h}_{\mathrm{max}}(q)-{h}_{\mathrm{min}}(q)$| can be used to describe the degree of multifractality. The greater |$\varDelta h$| is, the stronger the multifractality will be, which indicates a large fluctuation in the market. The multifractal spectrum |${f}_{xy}(\alpha )$| describes the singularity content of the time series. In addition, the width of multifractal spectrum reflects the size of market fluctuation. The larger the |$\varDelta \alpha$| is, the more uneven the distribution of time series is and the stronger the multifractal character is. |$\varDelta \alpha$| is expressed as

$$\begin{equation} \varDelta \alpha ={\alpha}_{\mathrm{max}}-{\alpha}_{\mathrm{min}} \end{equation}$$

(13)

4. INDICATORS AND DATA SOURCES

This paper uses the rolling window MF-DCCA method to study the multifractal characteristics of the relationship between energy and carbon markets in China. The daily settlement price of energy and carbon markets used in our research is available from Wind database. In June 2013, Shenzhen Carbon Emission Trading Market started, with a relatively high degree of marketization, but there was only one transaction data in the first two months. Therefore, this paper selected the closing price of the transaction from August 2013 to July 2019 as the sample data. This paper chooses Energy Sector Index (index code: 882001) to represent the income of China’s energy market, which covers 78 energy listed companies. Wind information compiled energy index is a very effective comprehensive index, which can better reflect the development of China’s energy market. Therefore, this paper chooses the closing price of energy index from August 2013 to July 2019 as the sample data. After preliminary processing of the data, 1342 sample data (From Wind database) were used in the study. Figure 1 shows the fluctuation of energy price and carbon price during this period.

From Figure 1, we can see that energy prices experienced a major adjustment around June 2015, and then prices fell, while carbon prices fluctuated slightly, showing a downward trend in the overall volatility.

We define the daily price return |${r}_t$|⁠, the natural logarithmic difference of the daily settlement price |${P}_t$|⁠, as |${r}_t=\ln ({P}_t)-\ln ({P}_{t-1})$|⁠. In order to have an intuitive impression of the sample time series, the statistics of daily price returns of energy and carbon markets are first analyzed separately. The descriptive statistics summarize the general behavior of all four time series which is given in Table 1.

Descriptive statistics for the price returns of energy and carbon markets.

Table 1

Descriptive statistics for the price returns of energy and carbon markets.

	REIP	RSZA
Mean	−5.46E-05	−0.0008
Median	0.00042	0
Maximum	0.081	2.29131
Minimum	−0.129773	−2.257008
Std. Dev.	0.019077	0.261472
Skewness	−0.993387	0.107645
Kurtosis	9.394246	36.1174
Jarque-Bera	2505.081	61284.16
Probability	0	0

Figure 2

Price returns of energy and carbon markets.

It can be seen from Table 2 that we can see that the standard deviation of carbon price series is greater than that of energy price series, which indicates that the dispersion degree of carbon price series deviating from the mean value is greater than that of energy price series. Jarque–Bera statistics show that all series reject the hypothesis of normal distribution under 1% significance. The skewness of energy price and carbon price series is −0.993387 and 0.107645, respectively, and the kurtosis is 9.394246 and 36.1174, respectively. Therefore, all series show the characteristics of spikes and thick tails.

5. EMPIRICAL RESULTS

5.1. Cross-correlation test

This paper used cross-correlation statistic |${Q}_{cc}(m)$| to study the non-linear cross-correlations between returns and trading volume of European carbon futures markets. Here, for two series |${x}_i$| and |${y}_i$|⁠, we have |$i=1,2,3\dots, \dots L$|⁠; the test statistic and the cross-correlation function are expressed as the following two functions:

$$\begin{equation} {Q}_{cc}(m)={N}^2\sum \limits_{i=1}^m\frac{{X_i}^2}{N-i}, \end{equation}$$

(14)

$$\begin{equation} {X}_i=\frac{\sum \limits_{k=i+1}^N{x}_k{y}_{k-i}}{\sqrt{\sum \limits_{k=1}^N{x_k}^2\sum \limits_{k=1}^N{y}_k^2}} \end{equation}$$

(15)

If the cross-correlation statistic |${Q}_{cc}(m)$| approximately obeys the chi-square distribution with degree of freedom m, there are no cross-correlations between two time return series; otherwise, the cross-correlations are significant at a special significance level.

Figure 3 shows the log–log plot of cross-correlation statistic |${Q}_{cc}(m)$|⁠, in which the degree of freedom m is taken from 1 to 1000 as the control. The critical value of chi-square distribution at 95% confidence level is also given in Figure 2. Obviously, the cross-correlation statistic |${Q}_{cc}(m)$| deviates from the critical value of chi-square distribution with degree of freedom m. Therefore, the original hypothesis of no cross-correlation is rejected, which indicates that there is an interactive correlation between energy and carbon markets.

Figure 3

Log–log plot of test statistics |${Q}_{cc}(m)$|⁠.

To make the results more accurate, in this paper, we have also used another statistical test method proposed by Podobnik et al. to quantify the level of cross-correlations between two time series, which is defined as the ratio between function of detrended covariance |${F}_{\mathrm{DCCA}}^2$| and the detrended variance |${F}_{\mathrm{DFA}}^2$|⁠. The difference between the original walk and the local trend is ‘detrended walk’. According to the ‘detrended walk’, we calculate the covariance of the residuals in each box:

$$\begin{equation} {f}_{\mathrm{DCCA}}^2\left(n,i\right)\equiv 1/\left(n-1\right)\sum \limits_{k=i}^{i+n}\left({R}_k-{\tilde{R}}_{k,i}\right)\left({R}_k^{\prime }-{{\tilde{R}}^{\prime}}_{k,i}\right) \end{equation}$$

(16)

The detrended covariance function is as

$$\begin{equation} {F}_{\mathrm{DCCA}}(n)\equiv \sqrt{\frac{1}{N-n}\sum \limits_{i=1}^{N-n}{f}_{\mathrm{DCCA}}^2\left(n,i\right)} \end{equation}$$

(17)

$$\begin{equation} {F}_{\mathrm{DFA}}(n)\equiv \sqrt{\frac{1}{N-n}\sum \limits_{i=1}^{N-n}{f}_{\mathrm{DFA}}^2\left(n,i\right)} \end{equation}$$

(18)

The formula is as follows:

$$\begin{equation} {\rho}_{\mathrm{DCCA}}=\frac{F_{\mathrm{DCCA}(n)}^2}{F_{\mathrm{DFA}1(n)}{F}_{\mathrm{DFA}2(n)}} \end{equation}$$

(19)

In the above formula, the value of |${\rho}_{\mathrm{DCCA}}$| ranges between |$-1\le{\rho}_{\mathrm{DCCA}}\le 1$|⁠. If |${\rho}_{\mathrm{DCCA}}=0$|⁠, there is no cross-correlation between two time series, If |${\rho}_{\mathrm{DCCA}}=1$|⁠, it means there has a perfect cross-correlation between two time series and if |${\rho}_{\mathrm{DCCA}}=-1$|⁠, which indicates a perfect anti cross-correlation existing in the time series. In this paper, we have calculated the DCCA cross-correlation coefficients, and Figure 4 shows |${\rho}_{\mathrm{DCCA}}$| for all the bivariate time series for different scale size n. From Figure 4, we can see that energy and carbon prices have long-range cross-correlation. Because of the finite size of time series, even if there is no cross-correlation, |${\rho}_{\mathrm{DCCA}}$| is not equal to 0. This cross-correlation coefficient test is used to show the existence of cross-correlation. Therefore, in order to find out whether the cross-correlation is long-range or anti-correlation, the DCCA method and its variants are needed to applied to our study.

Figure 4

The DCCA cross-correlation coefficients of energy and carbon markets.

Figure 5

Log–log plots of |${F}_{xy}\Big(q,s\Big)$| with rolling windows vs. time scale.

5.2. Empirical analysis of MF-DCCA method

Cross-correlation statistics Q_cc(m) can only qualitatively test the existence of cross-correlations between energy and carbon markets. Based on the MF-DCCA method mentioned above, this section will test whether there is a non-linear dependence and multifractal feature between energy and carbon markets in China. In order to affirm our results obtained above, the MF-DCCA method proposed by Podobnik and Stanley (2008) is quantitatively applied to test the presence of cross-correlation [21].

According to the rolling windows multifractal detrended cross-correlation (MF-DCCA) method, Figure 5 gives the log–log plots of the fluctuation function |${F}_{xy}\Big(q,s\Big)$| versus time scale between energy and carbon markets. As we can see from Figure 5, for different q values (q = −10, −8,... ,8,10), all curves are approximately linear under large scales, which further shows that there is power-law cross-correlation in the two return series.

Table 3 shows that the variation of the interaction correlation index H_xy(q) with q from −10 to 10. It can be seen that H_xy(q) decreases non-linearly with the increase in q value, which indicates that the interactive correlation between the two markets has multifractal characteristics. As can be seen from Table 3, H_xy(q) is lesser than 0.5 for all q values, indicating that the interaction between energy and carbon markets has a negative sustainability in the long run.

Scaling exponents H_xy(q) for return series with q varying from −10 to 10

Table 3

Scaling exponents H_xy(q) for return series with q varying from −10 to 10

q	REIP/RSZA	REIP/REIP	RSZA/RSZA
−10	0.4987	0.7826	0.4591
−9	0.4914	0.7750	0.4508
−8	0.4831	0.7662	0.4408
−7	0.4736	0.7557	0.4288
−6	0.4628	0.7434	0.4141
−5	0.4505	0.7287	0.3958
−4	0.4367	0.7110	0.3728
−3	0.4216	0.6899	0.3438
−2	0.4051	0.6641	0.3081
−1	0.3866	0.6322	0.2654
0	0.3654	0.5916	0.2145
1	0.3399	0.5403	0.1486
2	0.3100	0.4810	0.0672
3	0.2778	0.4233	−0.0066
4	0.2473	0.3751	−0.0601
5	0.2212	0.3379	−0.0977
6	0.1998	0.3096	−0.1249
7	0.1824	0.2879	−0.1454
8	0.1683	0.2710	−0.1613
9	0.1567	0.2575	−0.1741
10	0.1471	0.2465	−0.1846
|$\Delta \mathrm{H}(q)$|	0.3516	0.5362	0.6437

Figure 6

The relationship of H_xy(q) with rolling windows vs. q between energy and carbon markets.

For q = 2, the cross-correlation index H_xy(q) has a similar interpretation to the classical Hurst index. The cross-correlation of yield series is slightly greater than 0.5, indicating that the cross-correlation of the series is weak and persistent, and that the cross-correlation of H_xy(q) is significantly higher than 0.5, indicating that the cross-correlation of extreme small fluctuations is strong and persistent. When H_xy(q) is significantly less than 0.5, it shows that the cross-correlation of extreme large fluctuations is strong and anti-persistent. As can be seen from Figure 6, H_xy(q) is less than 0.5, which indicates that the cross-correlation of large fluctuations between the two markets is anti-persistent.

Figure 7

The relationship of |$\tau (q)$|~|$q$| with rolling windows between energy and carbon markets.

As can be seen from Table 3 and Figure 6, the scaling exponent H_xy(q) is strongly dependent on q. Therefore, the relationship between energy and carbon markets shows non-linear dependence and multifractality. When q changes from −10 to +10, the scaling exponents H_xy(q) of two markets decreases from 0.987 to 0.1471. The fact that H_xy(q) is significantly not a constant also shows that the relationship of energy and carbon markets has obvious multifractality. As can be seen from Table 3, the Hxy of REIP/RSZA is the smallest, indicating that the portfolio of energy market and carbon market bears the smallest financial risk.

5.3. Analysis of multifractal detrended fluctuation

From formula (8), in this paper we calculate|$\tau (q)$| (Renyi exponent). As is shown in Figure 7, the exponent |$\tau (q)$| we calculated from Equation (8) is non-linearly dependent on |$q$|⁠, which can also indicate that multifractality does exist in the relationship of energy and carbon markets. Figure 7 shows the relationship between quality exponent |$\tau (q)$| and |$q$| of the series in energy and carbon markets. From Figure 7, we can see that they are all non-linear increasing, and the two lines of REIP/RSZA and RSZA/RSZA are similar, which shows that the non-linear characteristic of the interaction correlation between the two series is similar, and the multi-fractal strength of the interaction correlation is similar.

In order to study the multifractality between price and volume in European carbon futures markets better, we have to get the multifractal strength by using the formula (9). As we all know, only if the system under study is monofractal can the width of the multifractal spectrum be zero. Figure 8 is a multifractal spectrum of the return series. From Figure 8, we can see that the width of multifractal spectrum is greater than 0, which indicates that there is multifractal in the cross-correlation between the two sequences. When the value of α is large, the multifractal spectral function f(α) appears negative, which indicates that the interaction correlation between markets has greater multifractality.

Figure 8

Multifractal spectrum with rolling windows for energy and carbon markets.

Table 4 shows the calculation results of the multifractal spectral width alpha of the return series of energy and carbon prices. From Table 4, we can see that the multifractal spectrum width of the cross-correlation sequence is greater than 0, and the multifractal spectrum width of the cross-correlation sequence is obviously smaller than that of the auto-correlation sequence. The results show that the multifractal intensity of the cross-correlation sequence is weaker than that of the auto-correlation series.

Multifractal spectra widths Δα.

Table 4

Multifractal spectra widths Δα.

	|$\varDelta \alpha$|			|$\varDelta H$|
	Energy	Carbon	Cross	Energy	Carbon	Cross
REIP/RSZA	0.700177	0.811131	0.501324	0.536151	0.643691	0.351609

5.4. The causes of multifractality

Through the above analysis, we know that the return series of energy and carbon markets has obvious multifractal characteristics. Generally speaking, the multifractal characteristics are mainly caused by two reasons: one is the fat tail probability distribution of the return series itself; the other is the long-range correlation between the large and small fluctuations of the return rate. The most important index to measure the strength of multifractal is the width of multifractal spectrum. Figure 9 is the generalized Hurst exponent is obtained by using rolling window MF-DCCA method to analyze the multifractal characteristics of the original, the shuffled and the surrogated series of energy market and carbon market. Figure 9 includes two sub-graphs, one above and the other below. Each sub-graph contains three curves representing the original, the shuffled and the surrogated series, respectively. According to the meanings of the three series, we know that the shuffled series eliminates the influence of long-range correlation on the multifractal characteristics of the return series, and the surrogated series eliminates the influence of fat tail distribution on the multifractal characteristics of the return series. This means that by comparing the multifractal spectral widths of three series, we can roughly estimate the contribution of the persistent correlation and the fat tail distribution to the multifractal characteristics. From Figure 9, we can see that the original series, the shuffled and the surrogated series can be clearly distinguished from each other, which shows that the long-range correlation and fat tail distribution are the reasons for the multifractal nature of the interactive correlation.

5.5. DISCUSSION

We have investigated the cross-correlations between energy and carbon markets in China. The results show that the return series were strongly cross-correlated in the short-term. Moreover, short-term cross-correlations displayed high multifractality indicating that the cross-correlations were easily to be affected by large and small fluctuations. With time varying, the short-term cross-correlations between series changed largely indicating that the optimal portfolio varied over time. As for investors, they should not hold the portfolio with fixed weight of each asset. As an emerging market, the Chinese carbon market becomes more and more efficient over time through some important reforms. However, the Chinese carbon market is still not mature as it is easily affected by market external factors, such as energy market, regulated policies imposed by the government and so on.

6. CONCLUSION

Firstly, the cross-correlation test is used to verify that there is an obvious cross-correlation between the return series of the energy and carbon markets. Based on this, the multi-fractal characteristics of energy market and carbon market cross-series are empirically studied. The main conclusions of this paper are as follows:

(1) Using the rolling windows MF-DCCA method to study the cross-series of energy and carbon markets under the overall trend, we find that there is a non-linear dependence between energy and carbon markets. The cross-series of energy and carbon markets has significant multifractal characteristics. For different levels of volatility, the cross-series of energy and carbon markets show long-term memory, while the long-term memory of small fluctuations is significantly stronger.

Figure 9

Scaling exponents H(q) of the original, shuffled and surrogated series in energy and carbon markets.

(2) By randomly arranging and phase adjusting the original series of energy and carbon markets, it is found that the two methods weaken the multifractal characteristics of the cross-sequence, especially the phase adjustment. This shows that the long-term memory and the thick-tailed probability distribution are the main reasons for the occurrence of the cross-series between energy market and carbon market. The main factor of multifractality is the probability distribution with thick tail, which plays a more important role.

This study reveals that although energy prices and carbon prices are negatively correlated for a long time under normal circumstances, the relationship is non-linear and dynamic, with complex multifractal characteristics. The relationship will show different trends in different periods with the changes of national political and economic situations, and the intensity of correlation will also change with the changes of time. These results will help to fully understand the complexity of price fluctuations in energy and carbon markets, provide valuable reference for effective measurement and prevention of market risks and investment decisions.

In summary, the purpose of this study is to find out the beneficial implications for the development of China’s carbon market and energy market by studying the relationship between the carbon market and the energy market: 1. The government and relevant regulatory agencies must continuously improve the transparency and timely The announcement will help increase and improve the effectiveness of macroeconomic regulation. 2. Accelerate the upgrading of the energy industry and vigorously promote economized production. 3. Adjust the structure of the energy industry in a timely manner and increase the production proportion of high value-added export products. Fourth, large-scale introduction of high-tech, increase the intensity of independent research and development, increase the technical knowledge and talent reserves, and fully launch green development.

Acknowledgements

This research was supported by the National Social Science Foundation of China (NSSFC) under Grant No. 19GBL183. The authors sincerely thank the anonymous referees as well as the editors.