The Linkages of Carbon Spot-Futures: Evidence from EU-ETS in the Third Phase

: Based on the prices selected from European Energy Exchange (EEX) from 2013 to 2018, we investigate the inter-correlation of carbon spot and futures markets. Speciﬁcally, we adopt the widely used DCC-GARCH model and VAR-BEKK-GARCH model to conduct a comprehensive analysis on the carbon market, i.e., the dynamic correlation and volatility spillover between carbon spot and carbon futures. Moreover, we develop a hedge strategy based on the VAR-BEKK-GARCH model and calculate the hedging e ﬀ ectiveness (HE) value to evaluate the strategy performance. The empirical results show that (i) during our sample period, carbon spot and futures markets are highly correlated, (ii) carbon spot overﬂows to the futures market and vice versa, and (iii) the HE value is equal to 0.9370, indicating a good performance for the hedging strategy. Then, we provide further discussion on the relationship between carbon spot and futures markets by replacing our dataset with the data of phase II. The results do not change our conclusions on the dynamic correlation and volatility spillover. However, the HE value of phase III is higher than that of phase II, which indicates that the carbon futures market of phase III is not only an available market to hedge risk from the contemporaneous carbon spot market but also has a better hedge e ﬀ ectiveness than phase II.


Introduction
Currently, one of the most thought-provoking issues globally is global warming, given its potential damage to agricultural production, industrial manufacture and human sustainable development [1,2]. Thus, the Kyoto Protocol established three mechanisms for climate change mitigation. Among all of the three mechanisms, emission trading is a unique way for carbon reduction, as it reduces carbon emissions through a market-based mechanism, which makes carbon reduction a market activity [3]. Based on the Kyoto Protocol, the European Union established the famous European Union Emission Trading Scheme (EU-ETS) in 2005. To date, the EU-ETS has been operating successfully for three phases. The first phase was from 2005 to 2007, the subsequent phase was from 2008 to 2012, and the third phase began in 2013 and will end in late 2020. Furthermore, the fourth phase of the scheme will launch in 2021, lasting for 10 years until late 2030 [4]. Due to the vast trading volume and drastic price changes, the EU-ETS has become the largest carbon trading market and is viewed as a common financial market for analyzing financial characteristics and portfolio management [5].
The carbon product traded in the EU-ETS is named the European Union allowance (EUA). Although the EU-ETS is a professional market for carbon trading, spot and futures trading are already proceeding as a common financial market. Accordingly, EUA carbon spot and futures prices should be correlated. The existing literature shows that the spot price can be forecasted without bias according It is essential to note that existing studies mainly focused on the first two phases. Few studies paid attention to the operating third phase. Thus, there is a vacuum of current studies for the third phase regarding the linkages between carbon spot and futures. In this paper, we focus on the linkages during the third period to fill the gap.

Kyoto Protocol and EU-ETS
In December 1997, the Kyoto Protocol reached an agreement in Kyoto [2,29]. The Kyoto Protocol is an additional term of the United Nations Framework Convention on Climate Change (UNFCCC). The Kyoto Protocol is an international guideline of greenhouse gas emissions reduction for coping with the inevitable global warming after industrialization. However, the Kyoto Protocol is more binding compared with the UNFCCC. According to the protocol, the involved countries should control emissions of several greenhouse gases, e.g., carbon dioxide and sulfur hexafluoride [4]. As abatement technology varies across countries, not all countries can complete the tasks regulated by the protocol. Therefore, the protocol allows the related countries to conduct emission trading. Specifically, countries with excess emission allowances can sell their allowances to the countries with insufficient allowances [30]. According to mechanisms regulated by the Kyoto Protocol, i.e., emission trading, the European Union (EU) in 2005 made a far-reaching attempt to lead global emission reduction by establishing the well-known European Union Emission Trading Scheme (EU-ETS). Currently, the EU-ETS accounts for over seventy-five percent of carbon trading volume globally [6]. As the leader of global carbon reduction, the EU also approves the scheme to be linked with other countries' carbon trading markets [31].
One of the foundations of the EU-ETS is called "cap-and-trade." The total amount of greenhouse gases that can be emitted under the EU-ETS is called the "cap," while "trade" means that the emitters can trade their allowances. The cap reduces year by year in order to meet the requirements of carbon reduction [32]. Within the cap, participants can trade their allowances with others if necessary. After one company obtains the allowances, the company should hold enough allowances to cover its emissions during the year [33]. Otherwise, the company will be fined heavily by the government. If one company has additional allowances, it can store them for the futures or sell them to the companies without sufficient allowances for profits. Thus, emission trading accomplishes maximized emission reduction with minimized cost.

Data
In this paper, we collect our dataset for EUA carbon spot and futures prices from the European Energy Exchange (EEX) during the period from 1 January 2013 to 26 December 2018. As there are several futures contracts traded in the time period, we select the continuous monthly contracts composed of the contracts of the largest open interest in each month to represent the carbon futures price in phase III of the EU-ETS. The price trend of the EEX EUA spot and futures is depicted in the following Figure 1.
As described in Figure 1, the prices of EEX EUA spot and futures are generally increasing during the whole sample period, despite a decreasing period in the first half of 2016. The increasing may be mainly attributed to the emphasis on carbon reduction of the EU. Intuitively, the EUA spot price is less than €10 per ton in almost all of the sample period. Meanwhile, the EUA futures price is generally under €20 per ton before 2018. From Figure 1, it is obvious that the futures price is higher than the spot price as the yellow line representing the futures price is above the blue line representing the spot price, which may be mainly due to the more active transactions of the futures market than that of the spot market. Moreover, we can also find that the two prices change almost synchronously. For an in-depth comparative analysis of the two prices, we further calculate the correlation coefficient and the log returns of the prices by r t = log(p t /p t−1 ), and display the results in the following Table 1, Figures 2  and 3, respectively.
In this paper, we collect our dataset for EUA carbon spot and futures prices from the European Energy Exchange (EEX) during the period from 1 January 2013 to 26 December 2018. As there are several futures contracts traded in the time period, we select the continuous monthly contracts composed of the contracts of the largest open interest in each month to represent the carbon futures price in phase III of the EU-ETS. The price trend of the EEX EUA spot and futures is depicted in the following Figure 1.   As described in Figure 1, the prices of EEX EUA spot and futures are generally increasing during the whole sample period, despite a decreasing period in the first half of 2016. The increasing may be mainly attributed to the emphasis on carbon reduction of the EU. Intuitively, the EUA spot price is less than €10 per ton in almost all of the sample period. Meanwhile, the EUA futures price is generally under €20 per ton before 2018. From Figure 1, it is obvious that the futures price is higher than the spot price as the yellow line representing the futures price is above the blue line representing the spot price, which may be mainly due to the more active transactions of the futures market than that of the spot market. Moreover, we can also find that the two prices change almost synchronously. For an indepth comparative analysis of the two prices, we further calculate the correlation coefficient and the log returns of the prices by = log ( / ), and display the results in the following Table 1, Figure   2 and Figure 3, respectively.   The results in Table 1 show that the magnitude of the two correlation coefficients, i.e., the correlation of the two price series, and the correlation of the two return series, is close to one, which indicates that the price series and the return series of carbon spot and futures have a very close connection. Then, we can observe the two return series have almost the same tendency based on Sustainability 2020, 12, 2517 5 of 18 As described in Figure 1, the prices of EEX EUA spot and futures are generally increasing during the whole sample period, despite a decreasing period in the first half of 2016. The increasing may be mainly attributed to the emphasis on carbon reduction of the EU. Intuitively, the EUA spot price is less than €10 per ton in almost all of the sample period. Meanwhile, the EUA futures price is generally under €20 per ton before 2018. From Figure 1, it is obvious that the futures price is higher than the spot price as the yellow line representing the futures price is above the blue line representing the spot price, which may be mainly due to the more active transactions of the futures market than that of the spot market. Moreover, we can also find that the two prices change almost synchronously. For an indepth comparative analysis of the two prices, we further calculate the correlation coefficient and the log returns of the prices by = log ( / ), and display the results in the following Table 1, Figure   2 and Figure 3, respectively.   The results in Table 1 show that the magnitude of the two correlation coefficients, i.e., the correlation of the two price series, and the correlation of the two return series, is close to one, which indicates that the price series and the return series of carbon spot and futures have a very close connection. Then, we can observe the two return series have almost the same tendency based on Figure 2 and Figure 3. For example, the two series were maximized at almost the same time during 2013. Furthermore, the returns fluctuate around zero. The return series changed seriously at the The results in Table 1 show that the magnitude of the two correlation coefficients, i.e., the correlation of the two price series, and the correlation of the two return series, is close to one, which indicates Sustainability 2020, 12, 2517 6 of 18 that the price series and the return series of carbon spot and futures have a very close connection. Then, we can observe the two return series have almost the same tendency based on Figures 2 and 3. For example, the two series were maximized at almost the same time during 2013. Furthermore, the returns fluctuate around zero. The return series changed seriously at the beginning of 2013, which may have been caused by the inconclusive policy orientation as the global carbon reduction environment changed in phase III. Moreover, some basic descriptive statistics of the two return series are shown in the following Table 2. As summarized in the above Table 2, the maximum value of the EEX carbon spot and futures returns are 0.0915 and 0.1019, respectively. Meanwhile, the minimum values of the two returns are −0.1939 and −0.1834. The average returns of the two series are all approximately 0.0004. The magnitude of standard deviations is close to zero and the mean values are even smaller. Therefore, it is high for the volatilities of the two return series of carbon spot and futures. The kurtosis of carbon spot and futures returns exceeds three, and the skewness of these returns is not equal to zero. Such characteristics indicate that carbon spot and futures returns series have the characteristics of most financial time series [2,5]-namely, the series are not normally distributed and have heavy tails [5].
In most of the cases, the univariate GARCH model is apparently successful in estimating and forecasting the volatility of the financial time series data by existing literature [34][35][36][37][38]. Besides, according to previous researches [35,38,39], GARCH (1, 1) is selected as the appropriate model for the volatility estimating. We assume a generalized error distribution and calculate the conditional variances of carbon spot and futures and show the results in Figures 4 and 5.
Sustainability 2020, 12, 2517 6 of 18 carbon reduction environment changed in phase III. Moreover, some basic descriptive statistics of the two return series are shown in the following Table 2. As summarized in the above Table 2, the maximum value of the EEX carbon spot and futures returns are 0.0915 and 0.1019, respectively. Meanwhile, the minimum values of the two returns are −0.1939 and −0.1834. The average returns of the two series are all approximately 0.0004. The magnitude of standard deviations is close to zero and the mean values are even smaller. Therefore, it is high for the volatilities of the two return series of carbon spot and futures. The kurtosis of carbon spot and futures returns exceeds three, and the skewness of these returns is not equal to zero. Such characteristics indicate that carbon spot and futures returns series have the characteristics of most financial time series [2,5]-namely, the series are not normally distributed and have heavy tails [5].
In most of the cases, the univariate GARCH model is apparently successful in estimating and forecasting the volatility of the financial time series data by existing literature [34][35][36][37][38]. Besides, according to previous researches [35,38,39], GARCH (1, 1) is selected as the appropriate model for the volatility estimating. We assume a generalized error distribution and calculate the conditional variances of carbon spot and futures and show the results in Figure 4 and Figure 5.   Intuitively, carbon spot and futures returns were volatile in the first half of 2013, the conditional variances reached the maximum in the sample period, and the entrance of Phase III may explain the phenomenon as significant changes occurred in the cap-and-trade system and the prohibition of banking of residual allowances [40]. Besides, the results in Figures 4 and 5 consist with the finding of Kim [40] and Zhang [2], whose research sample period covers Phase III. In addition, it can also be noticed that compared with the variances in each calendar year, the variation at the beginning of each year is much more volatile, which may be ascribed to the "yearly complaint event." From the above two figures, we may also find that the two conditional volatility series have almost the same tendency especially when it comes to extreme value. Intuitively, carbon spot and futures returns were volatile in the first half of 2013, the conditional variances reached the maximum in the sample period, and the entrance of Phase III may explain the phenomenon as significant changes occurred in the cap-and-trade system and the prohibition of banking of residual allowances [40]. Besides, the results in Figure 4 and 5 consist with the finding of Kim [40] and Zhang [2], whose research sample period covers Phase III. In addition, it can also be noticed that compared with the variances in each calendar year, the variation at the beginning of each year is much more volatile, which may be ascribed to the "yearly complaint event." From the above two figures, we may also find that the two conditional volatility series have almost the same tendency especially when it comes to extreme value.

DCC-GARCH model
We select the DCC-GARCH model to identify the dynamic correlation between carbon spot and futures. The model has been developed by Engle [41], which is a natural extension of the constant correlation coefficient (CCC)-GARCH [42] model. There are two advantages of the DCC-GARCH model. On the one hand, the model detects possible changes in conditional correlations over time [43,44]; on the other hand, the model accounts the heteroscedasticity as the correlation coefficients of standardized residuals are estimated [45]. The variances are adjusted depending on the variance equation, and thus, the DCC estimator is unbiased. Hence, the DCC-GARCH model has become an excellent measure for correlation [46]. There are two steps of the DCC estimation process; the first step is to estimate a univariate GARCH model, and the second step refers to the estimation of timevarying conditional correlations. The model is defined in the following equations (1), (2), (3), (4) and (5).
where X and represents the past observations and the conditional returns, respectively. H is the time-varying conditional variances, ε is the vector for the standardized residuals, R is a symmetric conditional correlation matrix, D is a diagonal matrix containing standard deviations of the returns estimated by the univariate GARCH model, and h , (i = 1, ⋯ , N) represents the diagonal

DCC-GARCH Model
We select the DCC-GARCH model to identify the dynamic correlation between carbon spot and futures. The model has been developed by Engle [41], which is a natural extension of the constant correlation coefficient (CCC)-GARCH [42] model. There are two advantages of the DCC-GARCH model. On the one hand, the model detects possible changes in conditional correlations over time [43,44]; on the other hand, the model accounts the heteroscedasticity as the correlation coefficients of standardized residuals are estimated [45]. The variances are adjusted depending on the variance equation, and thus, the DCC estimator is unbiased. Hence, the DCC-GARCH model has become an excellent measure for correlation [46]. There are two steps of the DCC estimation process; the first step is to estimate a univariate GARCH model, and the second step refers to the estimation of time-varying conditional correlations. The model is defined in the following Equations (1)-(5).
where X t and µ t represents the past observations and the conditional returns, respectively. H t is the time-varying conditional variances, ε t is the vector for the standardized residuals, R t is a symmetric conditional correlation matrix, D t is a diagonal matrix containing standard deviations of the returns estimated by the univariate GARCH model, and h ii,t (i = 1, . . . , N) represents the diagonal elements of H t . In the above equation, δ i,t is calculated as δ i,t = ε i,t / h i,t , where ε i,t is the ith element in ε t and h i,t is the ith element of the vector (h ii,1 , . . . ,h ii,t ). Q represents the unconditional correlation matrixes. Besides, the model requires both α and β in Equation (5) are non-negative, and α + β < 1. Q * t is a diagonal matrix with the square root of the ith diagonal element of Q t on its corresponding location. For more details, you may refer to Celık [47].

VAR-BEKK-GARCH Model
According to existing research, the DCC-GARCH model can only capture the inter-relationship between different financial markets. However, the direction of the relationship estimated by the model, i.e., volatility spillovers from one market to other markets, cannot be inferred from the DCC-GARCH model [47]. For an in-depth understanding of the relationship between the carbon spot and futures market, we adopt a widely used bivariate BEKK-GARCH model [48][49][50][51][52] to test the direction of the relationship. The BEKK-GARCH model allows the conditional variances and co-variances to be interacted. Besides, the BEKK-GARCH model does not need to estimate so much parameters. Furthermore, the co-variance matrix is positive defined in the calculation process. Moreover, the VAR model is a linear predictability model, which allows the return on assets to be predicted by past values of predictors. According to previous studies [53][54][55], the VAR model is widely used in estimating and predicting asset returns as it can effectively reduce the transaction cost. For the perfect fitness of the VAR model on asset return, we update the mean equation of the BEKK-GARCH model by employing the VAR model. In this paper, we select the VAR-BEKK-GARCH model to analyze the volatility spillovers between carbon spot and futures markets. Specifically, the VAR-BEKK model can be summarized as follows in Equations (6)-(8).
where R t represents the return matrix, c is the conditional return, and ε t refers to the residual. As for H t , it is the conditional-variance-covariance matrix. And the matrix B is the parameter matrix with conditional variances, representing the relationships of the variances between different assets. The matrix A captures the ARCH effect, and the matrix D reflects the GARCH effect. For more details, you may refer to Yu [56].

Dynamic Correlation of the EEX EUA Carbon Spot and Futures
A prerequisite of GARCH modeling is that the series need to be stable. Thus, before establishing the GARCH models for analyzing the correlation between EUA carbon spot and futures, it is essential to test the stationarity of the two return series. We exhibit the stationarity results by the augmented Dickey-Fuller (ADF) test in the following Table 3. As seen in Table 3, the results for the return series of carbon spot and futures prices reject the null hypothesis of the ADF test as the values of the t-statistic are equal to 0.0000, which means that the two series are both stationary. Thus, this approach can be further used for GARCH modeling. According to existing researches [57][58][59], DCC-GARCH (1, 1) can efficiently capture the dynamic correlation between assets. Thus, in this paper, we chose the model to summarize the dynamic correlation between carbon spot and futures and the related result is shown in Figure 6. As seen in Table 3, the results for the return series of carbon spot and futures prices reject the null hypothesis of the ADF test as the values of the t-statistic are equal to 0.0000, which means that the two series are both stationary. Thus, this approach can be further used for GARCH modeling. According to existing researches [57][58][59], DCC-GARCH (1, 1) can efficiently capture the dynamic correlation between assets. Thus, in this paper, we chose the model to summarize the dynamic correlation between carbon spot and futures and the related result is shown in Figure 6. We conduct a T-test to further identify whether the mean value of the correlation series is equal to a constant, i.e., 0. The results in Table 4 indicate that the two markets are indeed correlated because the null hypothesis is rejected as the p-value of the t-test is equal to zero. Table 4. T-test for the dynamic correlation coefficient.

Null hypothesis t-statistic p-value Conclusion
The mean value of the correlation series is equal to 0 436.9691 0.0000 Refuse As presented in Figure 6, the EUA carbon spot and EUA futures are highly correlated. The correlation coefficients are over 0.9 for almost all of the sample period. Evidently, extreme values appeared in the second half of 2015 and 2016 and the beginning of 2018. The abnormalities may be attributed to the following reasons. First, the leaders of the G7 countries showed support on carbon reduction with a clear attitude for the first time in June 2015; they pledged to realize a 40-70% carbon reduction in 2050 compared with 2010. The announcement may account for the volatility of futures prices compared with the ordinary moments, as seen in Figure 3 in the time period of June 2015. Meanwhile, the spot price did not change considerably, as shown in Figure 2, which may be the reason for the reduced correlation at that time. Second, in December 2016, the European Parliament Committee on Environment, Public Health and Food Safety reached an agreement on a reform scheme for the EU-ETS after 2020. As shown in Figures 2 and 3, the agreement caused severe changes in carbon spot and futures prices, given that the EU approved that the emitters could save allowances for the next stage, which may account for the abnormal value in the second half of 2016. Third, at the end of 2017, the EU reached an agreement on the regulatory framework of the EU-ETS in Phase IV, and it may be one of the reasons for the drastic decrease in the correlation.

Volatility Spillover between the EEX EUA Carbon Spot and Futures
In this section, we first show the lag length selection criteria of the VAR model in the following Table 5. We conduct a T-test to further identify whether the mean value of the correlation series is equal to a constant, i.e., 0. The results in Table 4 indicate that the two markets are indeed correlated because the null hypothesis is rejected as the p-value of the t-test is equal to zero. Table 4. T-test for the dynamic correlation coefficient.

Null Hypothesis t-Statistic p-Value Conclusion
The mean value of the correlation series is equal to 0 436.9691 0.0000 Refuse As presented in Figure 6, the EUA carbon spot and EUA futures are highly correlated. The correlation coefficients are over 0.9 for almost all of the sample period. Evidently, extreme values appeared in the second half of 2015 and 2016 and the beginning of 2018. The abnormalities may be attributed to the following reasons. First, the leaders of the G7 countries showed support on carbon reduction with a clear attitude for the first time in June 2015; they pledged to realize a 40-70% carbon reduction in 2050 compared with 2010. The announcement may account for the volatility of futures prices compared with the ordinary moments, as seen in Figure 3 in the time period of June 2015. Meanwhile, the spot price did not change considerably, as shown in Figure 2, which may be the reason for the reduced correlation at that time. Second, in December 2016, the European Parliament Committee on Environment, Public Health and Food Safety reached an agreement on a reform scheme for the EU-ETS after 2020. As shown in Figures 2 and 3, the agreement caused severe changes in carbon spot and futures prices, given that the EU approved that the emitters could save allowances for the next stage, which may account for the abnormal value in the second half of 2016. Third, at the end of 2017, the EU reached an agreement on the regulatory framework of the EU-ETS in Phase IV, and it may be one of the reasons for the drastic decrease in the correlation.

Volatility Spillover between the EEX EUA Carbon Spot and Futures
In this section, we first show the lag length selection criteria of the VAR model in the following Table 5. According to the information criterion, the lag length selected in the VAR model is 8. As for the variance equation, we choose the widely used (1, 1) as the lag length for the ARCH and GARCH term [48][49][50][51][52]. The estimation results of the VAR (8)-BEKK (1, 1)-GARCH model is exhibited in the following Table 6. In Table 6, A (i, j), B (i, j) and C (i, j) are the elements of the matrix A, B, and C described in Equation (9) in Secton 4.3. As for the volatility spillover, the parameters of A (i, j) and B (i, j) are of interest, as the parameters reflect the ARCH effect and GARCH effect, respectively. According to the parameters, we can obtain the following conclusions. First, the current conditional variance is positively affected by its previous ARCH term as the parameters of A (1, 1) and A (2,2) are statistically significant at the 1% level. Second, the current conditional variance is closely related to the GARCH term, i.e., the conditional variance in the last period, as B (1, 1) and B (2, 2) are still significant. Third, the parameters of A (1, 1), A (2, 2), B (1, 1), and B (2, 2) are all significant, which means that carbon spot and futures all show volatility clustering.
Investigating the off-diagonal elements of matrix A and B, which capture cross-market effects, namely shock and volatility spillovers, respectively, we have several interesting discoveries. First, the ARCH term has a negative impact on other's conditional variances since A (1, 2) and A (2, 1) are statistically negative, while the GARCH term has a positive effect as B (1, 2) and B (2, 1) are statistically significant. Second, there exists a bi-directional volatility spillover between carbon spot and futures as the parameters of A (1, 2), B (1, 2), A (2, 1) and B (2, 1) are all significant at the level of 1%. It means that the information embedded in price innovations of carbon spot as well as carbon futures indeed transmit to volatility. The reason of the volatility spillover is that the micro-foundation, market investors, and information source are almost the same for carbon spot and futures. These common factors will lead to volatility linkages by altering expectations, and thereby affecting asset demand, or cross-market hedging [60,61]. Moreover, carbon futures exhibits a higher degree of volatility spillover to carbon spot than reverse spillover mechanism since the values of A (1, 2) and B (1, 2) over that of A (2, 1) and B (2,1). It infers that carbon futures have the function of price discovery to some extent. The existence of volatility spillover helps to construct the optimal hedging ratio, which benefits the carbon investors in the futures. We elaborate this point in the next section.
To further verify the validity of the volatility spillover exhibited in Table 6, we conduct the Wald test shown in Table 7. The results of the Wald test support the above conclusions that the volatility of carbon futures can impact on that of carbon spot, vice versa.

Hedge Ratios and Hedging Effectiveness
In this section, we examine the implications of the aforementioned results of the VAR-BEKK-GARCH model for hedging strategy and portfolio management. Based on the modern portfolio theory, Johnson [62] proposed the well-known minimum variance hedge ratio model for minimizing the risk associated with a certain portfolio. Assume asset j can be used as a hedge for another asset i. The return of a hedged portfolio can be written as follows in Equation (9).
where R H,t represents the return of the hedged portfolio, γ f,t is the hedge ratio, R s,t and R f,t represent the return of spot and futures return respectively. According to Johnson [62], Baillie [63], and Kroner [64], the optimal holding weight of the carbon spot and futures can be determined by the following equations in (10), (11) and (12).
In the above expressions, w sf,t represents the weight of carbon futures in one unit portfolio including carbon spot and futures. h sf t represents the co-variance between carbon spot and futures, which is estimated by the multivariate GARCH model. h ss t and h ff t denote the variance of carbon spot and futures, respectively. γ * f,t is the optimal hedge ratio. In order to evaluate the effectiveness of the constructed hedging strategy, we further calculate a hedging effectiveness (HE) value, which is adopted by Ku [65]. The calculation of HE value is shown as follows. HE = var unhedged − var hedged var unhedged (13) where the variances of the hedge portfolio, namely var hedged , are obtained from the variance of the rate of return, R H,t . The variance of the unhedged portfolio is the variance of spot returns (see, for example, Ripple and Moosa [66]. The closer the HE tends to one, the higher hedging effectiveness and better risk reduction. Some basic results of the above-mentioned equations are shown in Table 8. As can be seen from Table 8, the optimal hedge ratio for carbon futures is 0.9468. This means that a unit of long position in carbon spot can be hedged by shorting 0.9468 of carbon futures in average. Moreover, the weight for carbon futures in the portfolio including carbon spot and futures is 0.4208, while the weight for carbon spot is 0.5292. Furthermore, the HE value is 0.9370, which indicates that around 94% of price return variance of the carbon spot can be effectiveness hedged by taking a short position in carbon futures into the portfolio

Further Discussion
In this section, we provide further discussion on the correlation between carbon spot and carbon futures markets by replacing our dataset with the trading data in phase II of the EU-ETS. Due to the data availability, the data scope for the prices of carbon spot and futures are from 27 October 2009 to 31 December 2012. Some basic characteristics of the new data in phase II is shown in Table 9. Then, we employ the same methods as in Section 5 to further discuss the correlation between carbon spot and futures, and the detailed results are shown as follows.
The ADF test results of the two return series and the estimation results of the DCC-GARCH model are exhibited in the following Table 10 and Figure 7. Specifically, the results of the ADF test show that the returns of carbon spot and futures are still stationary and thus can be used for subsequent GARCH modeling. The results of dynamic correlation indicate that there exists an obvious dynamic linkage between carbon spot and futures. Similar to our research for phase III of the EU-ETS, we still conduct a T-test to further certify whether the two markets are correlated, and the results shows the correlation between carbon spot and futures to be dynamic.  We further show the lag length selection process of the VAR model in Table 11. As shown in Table 11, the lag length selected for the VAR model in phase II of the EU ETS is 4. The estimation results for VAR (4)-BEKK (1, 1)-GARCH model is shown in Table 12. The results exhibit that there exists a volatility spillover from carbon futures to carbon spot in phase II of the EU ETS as the parameters of A (1, 2) and B (1,2) are significant at the level of 1%. The reverse volatility spillover from carbon spot to carbon futures is relatively weaker as the parameters of B (2, 1) is not significant. Moreover, the results of the Wald test shown in Table 13 further support this conclusion for the rejection of the null hypothesis of volatility spillover from carbon spot to carbon futures. Thus, the volatility spillover effect in phase II of the EU ETS is relative weaker than that in phase III. This further demonstrates the importance of researching the phase III of the EU ETS as the more volatility spillover between carbon spot and futures.  We further show the lag length selection process of the VAR model in Table 11. As shown in Table 11, the lag length selected for the VAR model in phase II of the EU ETS is 4. The estimation results for VAR (4)-BEKK (1, 1)-GARCH model is shown in Table 12. The results exhibit that there exists a volatility spillover from carbon futures to carbon spot in phase II of the EU ETS as the parameters of A (1, 2) and B (1, 2) are significant at the level of 1%. The reverse volatility spillover from carbon spot to carbon futures is relatively weaker as the parameters of B (2, 1) is not significant. Moreover, the results of the Wald test shown in Table 13 further support this conclusion for the rejection of the null hypothesis of volatility spillover from carbon spot to carbon futures. Thus, the volatility spillover effect in phase II of the EU ETS is relative weaker than that in phase III. This further demonstrates the importance of researching the phase III of the EU ETS as the more volatility spillover between carbon spot and futures.
Similar with the analysis in the above section, we further provide the hedge ratio and HE value to evaluate the effectiveness of the hedge strategy in phase II of the EU ETS. And the results are shown in the following Table 14. Here, we can conclude that the hedging effectiveness of phase II is lower than that of phase III for the HE value is obviously under 0.9370-namely, the HE value in phase III. In general, the correlation between carbon spot and futures becomes closer in phase III of the EU ETS when compared with the phase II, as the mean value of dynamic correlation shown in Figure 6 is relatively higher than the correlation in Figure 7. Moreover, the volatility effect is more intense in phase III of the EU ETS than that in phase III for the parameters of A (2, 1) is significant at a level of 10% and that of B (2, 1) is not significant. As to the hedging strategy based on the VAR-BEKK-GARCH model, it is obvious that the hedging strategy performs better in phase III than phase II as the HE value is relatively higher in phase III. Thus, we can deduce the conclusions that the carbon futures market in phase III of the EU ETS is a more applicable market to hedge risk from carbon spot market when compared with the market in phase II.

Conclusions
In this paper, we collect our data set of carbon spot and futures market in phase III of the EU-ETS to analyze the dynamic correlation and volatility spillover of the two markets. The novelty of this paper is that it extends the time scale of existing research to phase III of the EU-ETS and makes a comprehensive analysis for the carbon spot-futures linkages. Our empirical results of the DCC-GARCH model show that carbon spot and carbon futures are highly correlated. Meanwhile, the empirical results of the VAR-BEKK-GARCH model indicate that there exist significantly bi-directional volatility spillovers between the two carbon markets. Besides, we construct a hedge strategy based on the results of VAR-BEKK-GARCH model, and the results denote that the carbon futures market is a good market for investors who want to hedge risk from carbon spot market. Moreover, we make a further exploration on the correlation of carbon spot and futures by replacing our data set with the trading data in phase II of the EU-ETS. The results do not change our conclusions on dynamic correlation and volatility spillover. But another interesting issue is found that carbon futures market in phase III of the EU ETS can better hedge the risk from carbon spot market when compared with phase II. Our empirical results further help to better understand the carbon spot-futures linkages in the EU-ETS. Besides, our results benefit those investors who invest in both carbon spot and futures market to effectively hedge risk.
There are indeed alternative methods to estimate dynamic correlation relationship between carbon spot and futures or even more assets. Such as, Orthogonal GARCH (O-GARCH) model [2,5], Copula-DCC-GARCH model [67], ADCC-EGARCH model [68], et al. In this paper, we only select the DCC-GARCH model as the basic estimating method to explore the relationship between carbon spot and futures. And these alternative methods will be carefully employed in our futures research. Moreover, in Section 5.3, we construct a portfolio composed of carbon assets without any other assets. Thus, we try to include more assets, i.e., energy assets, financial assets and commodity assets, into the portfolios in the futures research, which may be benefit the investors globally.
Author Contributions: H.C. collected the data and wrote the original manuscript; Z.L. provided the methodologies used in this paper; Y.Z. implemented the empirical process in this paper; Y.W. provided the idea and polished the paper. All authors have read and agreed to the published version of the manuscript.