Volatility Spillovers and Causality of Carbon Emissions, Oil and Coal Spot and Futures for the EU and USA

Transcription

1 sustainability Article Volatility Spillovers Causality Carbon Emissions, Oil Coal Spot USA Chia-Lin Chang 1, Michael McAleer 2,3,4,5,6, * Guangdong Zuo 2 1 Department Applied Economics, Department Finance, National Chung Hsing University, Taichung 402, Taiwan; changchialin@ .nchu.edu.tw 2 Department Quantitative Finance, National Tsing Hua University, Hsinchu 300, Taiwan; guangdong.zuo@foxmail.com 3 Discipline Business Analytics, University Sydney Business School, Sydney, NSW 2006, Australia 4 Econometric Institute, Erasmus School Economics, Erasmus University Rotterdam, 3062 Rotterdam, The Nerls 5 Department Quantitative Economics, Complutense University Madrid, Madrid, Spain 6 Institute Advanced Sciences, Yokohama National University, Yokohama , Japan * Correspondence: michael.mcaleer@gmail.com Received: 28 July 2017; Accepted: 19 September 2017; Published: 2 October 2017 Abstract: Recent research shows that efts to limit climate change should focus on reducing emissions dioxide over or greenhouse gases or air pollutants. Many countries paying substantial attention to emissions to improve air quality public health. The largest source emissions from human activities in some countries in Europe elsewhere is from burning fossil fuels electricity, heat, transportation. The fuel emissions can influence each or. Owing to importance emissions ir connection to fossil fuels, possibility Granger causality in,, volatility emissions, crude oil coal have recently become very important research topics. For USA, available crude oil but re no emissions. For European Union (), re no coal or emissions, but re crude oil, coal emissions. For this reason, will be used to analyse Granger causality volatility spillovers in emissions, crude oil, coal. As estimators based on quasi-maximum likelihood estimators (QMLE) under incorrect assumption a normal distribution, we modify likelihood ratio (LR) test to a quasi-likelihood ratio test (QLR) to test multivariate conditional volatility Diagonal BEKK model, which estimates tests volatility spillovers, has valid regularity conditions asymptotic properties, against alternative Full BEKK model, which also estimates volatility spillovers, but has valid regularity conditions asymptotic properties only under null hyposis zero f-diagonal elements. Dynamic hedging strategies by using optimal hedge ratios suggested to analyse market fluctuations in volatility emissions, crude oil, coal. Keywords: emissions; fossil fuels; crude oil; coal; low targets; green energy; ; Granger causality; volatility spillovers; quasi likelihood ratio (QLR) test; diagonal BEKK; full BEKK; dynamic hedging JEL Classification: C58; L71; O13; P28; Q42 Sustainability 2017, 9, 1789; doi: /su

2 Sustainability 2017, 9, Sustainability 2017, 9, Sustainability 2017, 9, Introduction 1. Introduction Recent research shows that efts to limit climate change should focus on reducing emissions Recent research shows that efts to limit climate change should focus on reducing emissions dioxide dioxide over over or or greenhouse greenhouse gases or gases air pollutants. or air pollutants. Many countries Many countries paying substantially paying dioxide over or greenhouse gases or air pollutants. Many countries paying substantially greater attention greater to attention emissions to toemissions improveto airimprove quality air quality public health. public Carbon health. emissions Carbon substantially greater attention to emissions to improve air quality public health. Carbon emissions trading programs trading programs have been have established been established at international, international, regional, national, regional, national, sub-national levels subnational (see levels 1). (see 1). emissions trading programs have been established at international, regional, national, subnational levels (see 1). 1. Global mean With without dioxide mitigation. Source: [2] Rogelj Global Global mean mean temperatures. temperatures. With With without without dioxide dioxide mitigation. mitigation. Source: Source: [2] Rogelj [1] et Rogelj al. (2014). et al. (2014). et al. (2014). As can be seen from 1, in a scenario no dioxide mitigation, global temperatures As can be seen from 1, in a scenario no dioxide mitigation, global temperatures would be predicted to rise by over five degrees Celsius by 2100, but cutting emissions methane, would be predicted to rise by over five degrees Celsius by 2100, but cutting emissions methane, HFCs, black would reduce this rise to around one degree Celsius. The results suggest that HFCs, black would reduce this rise to around one degree Celsius. The results suggest that dioxide should certainly remain central to greenhouse gas emission cuts. dioxide should certainly remain central to greenhouse gas emission cuts. 2 shows that projects regions such as CDM (Clean Development Mechanism), 2 shows that projects regions such as CDM (Clean Development Mechanism), RGGI (Regional Greenhouse Gas Initiative), European Union (), countries like New Zeal, RGGI (Regional Greenhouse Gas Initiative), European Union (), countries like New Zeal, Australia, South Korea, State State Calinia in in USA, USA, Province Province Quebec Quebec inn Canada, inn Australia, South Korea, State Calinia in USA, Province Quebec inn Canada, have passed have passed implemented implemented programs programs to mitigate to mitigate emissions. emissions. Canada, have passed implemented programs to mitigate emissions. 2. Implementation programs to mitigate emissions. 2. Implementation programs to mitigate emissions.

3 Sustainability 2017, 9, The programs have operated in phases, with a pilot phase from 2005 to 2007 covering power sector certain heavy industries, a second phase from 2008 to 2012 exping coverage slightly, a third phase that adds a significant range industrial activities. The largest source emissions from human activities in some countries in Europe elsewhere is from burning fossil fuels electricity, heat, transportation. The price fuel influences emissions, but price emissions can also influence price fuel. Owing to importance emissions ir connection to fossil fuels, possibility [2] Granger (1980) causality in, volatility emissions, it is not surprising that crude oil coal have recently become a very important public policy issue, hence also a significant research topic. Energy markets have recently exped considerably due in large part to rapidly accelerating behaviour investors in financial markets. The synergy between financial energy markets is that financial aspect fossil fuels emissions need to be analysed more cfully by using advanced financial econometric methods. An important reference in field energy its consequences on financial markets empirical studies presented in [3] Ramos Veiga (2014). These macroeconomic variables include risk factors in oil industry, risk taking in airline industry,, volatility, shocks in oil industry, oil shock spillovers to stock market, equity, bond, volatility market risks. In a more microeconomic context, [4] Sawik, Faulin Pérez-Bernabeu (2017a) examine energy environment issues with respect to multi-criteria analysis multi-objective green logistics optimization. The optimality criteria presented in terms environmental costs, that is, minimization externality costs noise, pollution, fuel costs as compd with ir minimization. In a separate contribution, [5] Sawik, Faulin Pérez-Bernabeu (2017b) solve a multi-objective mulation problem by minimizing total distance, hence costs to a delivery company, amount CO 2 emissions. [6] Sawik, Faulin Pérez-Bernabeu (2017c) optimize a multi-criteria mulation green vehicle routing problems by mixed integer programming, specifically to decide best delivery route to minimize travel costs optimize transportation route a delivery company. The plan remainder paper is as follows. Section 2 discusses data emissions, oil that will be used in empirical analysis USA. Section 3 discusses methodological issues, including univariate multivariate conditional volatility models, Granger causality, volatility spillovers, optimal hedge ratios, causality in volatility, as well as an interesting novel adaptation likelihood ratio (LR) test to a quasi likelihood ratio (QLR) test Diagonal BEKK model against alternative a Full BEKK model. Section 4 examines alternative unit root tests that used to test stationarity in data. Granger Causality Spillovers in Returns Volatilities analysed in Section 5. Section 6 provides some concluding remarks. 2. Data The length sample period empirical analysis was dictated by availability data on, crude oil in United States America (USA). The emission trading market has longest trading period, but not. The USA is leader in developing a wide range financial derivatives, such as, financial, energy, commodities, but not emissions, where only available. Data emission, crude oil, coal available from 1 April 2008 to 20 May 2017, se will be analyzed in paper. Coal price in is available on a weekly basis. The emission crude oil have a high correlation with corresponding. The volume trades in market emissions is much smaller than in market, as shown in 3.

4 Sustainability 2017, 9, Sustainability 2017, 9, ,000 50,000 40,000 30,000 20,000 10, /10/ /10/ /10/ /10/ /10/2016 Carbon Volume Carbon Spot Volume Carbon Carbon volumes volumes European European Union Union () () December December May May Data crude oil available prior to However, data coal Data crude oil available prior to However, data coal emissions start from 17 July April 2008, respectively. Theree, data in emissions start from 17 July April 2008, respectively. Theree, data in empirical analysis European Union starts from latest date crude oil, empirical analysis European Union starts from latest date crude oil, emissions, namely 1 April emissions, Data namely, 1 April oil from 5 January 2016 to 20 May 2017 USA will also be Data analyzed, in paper, but oil data from 5 January 2016 to emissions 20 May 2017 not available USA will also be USA. analyzed Spot in paper, coal but data crude oil start prior to However, emissions data not available emissions USA. start Spot from 1 May coal Consequently, crude oil start prior price to data in However, empirical data analysis emissions USA starts start from from 1 May latest date Consequently, oil, price emissions, data in namely empirical 5 January analysis USA starts from latest The date transaction oil, markets emissions, units namely variables 5 January different. is Intercontinental The transaction Exchange markets allowance, units which variables is traded different. in ICE-ICE Europe is Intercontinental Commodities market Exchange is expressed allowance, in which Euros is traded per metric in ton. ICE-ICE coal Europe is ICE Commodities Rotterdam market Monthly Coal is expressed Contract, in Euros per is metric traded ton. in ICE-ICE coal is ICE Europe Rotterdam Commodities Monthly market. Coal oil Contract, is current is traded pipeline in export ICE-ICE quality Brent Europe blend, Commodities as supplied market. at Sullom Voe, oil is traded is in current ICE-ICE pipeline export Europe quality Commodities Brent blend, market, as supplied is expressed at Sullom in Voe, USDs is traded per bbl. in ICE-ICE Carbon Europe Commodities in market, USA given is expressed as United in USDs States per Carbon bbl. Dioxide RGGI Allowance, Carbon expressed in USDs in per USA allowance. given Coal as United States given Carbon as Dioxide Dow RGGI Jones Allowance, US Total Market expressed Coal Index, in USDs which per is allowance. expressed Coal in USD. Oil given as Dow given Jones as US Total West Market Texas Coal Intermediate Index, which Cushing is expressed Crude Oil, in which USD. is Oil expressed in USDs given per bbl. as All West Texas currency Intermediate units Cushing transmed Crude to USD Oil, which in is empirical expressed analysis. in USDs per bbl. All currency units transmed to USD The in endogenous empirical analysis. variables used in empirical analysis, where rate return The is obtained endogenous as variables first difference used in in empirical natural logarithm analysis relevant, where price data. rate The return mnemonics is obtained as, first coal difference, oil in denote, natural respectively, logarithm future relevant price data. The emission, mnemonics fr oil, coal in fr, oil European fr denote, respectively, Union. Similarly, future mnemonics emission, US, UScoal oil,usoil in European denote, respectively, Union. Similarly, mnemonics US emission, sr, UScoal sr, USoil oil in sr denote, USA. respectively, emission, oil in USA. The The variable variable sources sources definitions definitions given given in in Table Table 1, 1, with with respect respect to to USA, USA, as as well well as as ir ir transactions transactions markets, markets, descriptions descriptions data. data. For For USA, USA, available available crude crude oil oil but but re re no no or or emissions. emissions. For For,, re re no no coal coal or or emissions, emissions, but but re re crude crude oil, oil, emissions. emissions. For For this this reason, reason, will will be be used used to to analyse analyse Granger Granger causality causality volatility volatility spillovers spillovers in in emissions, emissions, crude crude oil, oil, coal. coal. This This will will be based be based on on Lagrange Lagrange multiplier multiplier test test univariate univariate causality causality in variance in variance (strictly, (strictly, causality causality in conditional in conditional volatility) volatility) [7] Hafner [7] Hafner Herwartz (2006), Herwartz (2006), more recently, more [8] recently, Chang [8] McAleer Chang (2017). McAleer An extension (2017). An to multivariate extension to multivariate tests causality tests in conditional causality in volatility conditional will volatility be a focus will be a paper. focus paper.

5 Sustainability 2017, 9, Table 1. Data Sources Definitions. Variable Name Definitions Transaction Market Description fr coal fr oil fr return coal return oil return ICE-ICE Europe Commodities ICE-ICE Europe Commodities ICE-ICE Europe Commodities US sr US return over counter UScoal sr US coal return over counter USoil sr US oil return over counter ICE A Contract R/MT ICE Rotterdam Monthly Coal Contract USD/MT Current pipeline export quality Brent blend as supplied at Sullom Voe USD/bbl United States Carbon Dioxide RGGI Allowance USD/Allowance Dow Jones US Total Market Coal Index USD West Texas Intermediate Cushing Crude Oil USD/bbl ICE is Intercontinental Exchange; A is allowance; MT is metric ton; RGGI (Regional Greenhouse Gas Initiative) is a CO 2 cap--trade emissions trading program that is comprised ten New Engl Mid-Atlantic States that will commence in 2009 aims to reduce emissions from power sector. RGGI will be first government mated CO 2 emissions trading program in USA. As estimators based on Quasi-Maximum Likelihood Estimators (QMLE) under incorrect assumption a normal likelihood function, we will modify likelihood ratio (LR) test to a novel quasi-likelihood ratio test (QLR). Definition QLR test statistic: QLR = 2 (quasi maximized log likelihood value under alternative hyposis quasi maximized log likelihood value under null hyposis). The QLR test statistic tests multivariate conditional volatility Diagonal BEKK model, which is used to estimate test spillovers, which has valid regularity conditions asymptotic properties, against alternative Full BEKK model, which is used to estimate spillovers, but has valid regularity conditions asymptotic properties only under null hyposis zero f-diagonal elements. Dynamic hedging strategies using optimal hedge ratios will be suggested to analyse market fluctuations in volatility emissions, crude oil, coal. The QLR statistic has an asymptotic chi-squd distribution under null hyposis, with degrees freedom (df ) equivalent to number f-diagonal terms in two m m matrices, weighting matrix, A, stability matrix, B, Full BEKK model, namely 2m(m 1). The descriptive statistics endogenous variables given in Table 2. The highest stard deviation over sample period is, followed by oil coal. Similarly, highest stard deviation US market is coal, followed by emission. Table 2. Descriptive Statistics 2 April May January May 2017 United States America (USA). Variable Mean Median Max Min SD Skewness Kurtosis Jarque-Bera fr ,434.2 coal fr ,155.8 oil fr US sr ,346.8 UScoal sr USoil sr The Jarque-Bera Lagrange multiplier statistic normality is based on testing empirical skewness kurtosis against ir normal counterparts.

6 Sustainability 2017, 9, The have different degrees skewness. The oil in US markets, coal in USA skewed to left, indicating that se series have longer left tails (extreme losses) than right tails (extreme gains). However, or all skewed to right, especially emission return in USA, which value skewness is high, indicating that se series have more extreme gains than extreme losses. These stylized facts should be interest to participants in commodity markets. All price distributions have kurtosis that is significantly higher than three, implying that higher probabilities extreme market movements in eir direction (gains or losses) occur in se markets, with greater frequency in practice than would be expected under normal distribution. In market, highest kurtosis is coal, followed by oil. For US market, highest kurtosis is, followed by coal. The Jarque-Bera Lagrange multiplier statistic is based on testing empirical skewness kurtosis against ir normal counterparts, confirms non-normal distributions all series. 3. Methodology Although financial energy almost certainly stationary, empirical analysis will commence with tests unit roots based on ADF, DF-GLS, KPSS. This will be followed by an analysis estimation univariate GARCH multivariate diagonal BEKK models (see [9] Baba et al. (1985) [10] Engle Kroner (1995)), from which conditional covariances will be used testing co-volatility spillovers, that is, Granger causality in conditional volatility. Despite empirical applications a wide range conditional volatility models in numerous papers in empirical finance, re oretical problems associated with virtually all m. The CCC ([11] Bollerslev (1990)), VARMA-GARCH ([12] Ling McAleer (2003), its asymmetric counterpart, VARMA-AGARCH [13] McAleer et al. (2009)), models have static conditional covariances correlations, which means that accommodating volatility spillovers is not possible. Apart from diagonal version, multivariate Full BEKK model conditional covariances has been shown to have no regularity conditions, hence no statistical properties (see [14] McAleer et al. (2008) [15] Chang McAleer (2017b), discussion below, furr details). Theree, spillovers can be considered only special case Diagonal BEKK. The multivariate DCC model (purported) conditional correlations has been shown to have no regularity conditions, hence no statistical properties (see [16] Hafner McAleer (2014) [17] McAleer (2017) furr details). The analysis univariate multivariate conditional volatility models below is a summary what has been presented in literature (see, example [18] Caporin McAleer (2012) [19] Chang et al. (2015), especially [20] Chang et al. (2017)), although a comprehensive discussion Full Diagonal BEKK models is not available in any published source. In particular, application quasi likelihood ratio (QLR) test Diagonal BEKK model as null hyposis against alternative hyposis a Full BEKK model does not seem to have been considered in literature. The first step in estimating multivariate models is to obtain stardized residuals from conditional mean shocks. For this reason, most widely used univariate conditional volatility model, namely GARCH, will be presented briefly, followed by two most widely estimated multivariate conditional covariance models, namely Diagonal Full BEKK models Univariate Conditional Volatility Consider conditional mean financial, as follows: y t = E(y t I t 1 ) + ε t (1) where financial, y t = logp t, represent log-difference in financial commodity or agricultural, P t, I t 1 is inmation set at time t 1, ε t is a conditionally heteroskedastic

7 Sustainability 2017, 9, error term, or shock. In order to derive conditional volatility specifications, it is necessary to specify stochastic processes underlying shocks, ε t. The most popular univariate conditional volatility model, GARCH model, is discussed below. Now consider rom coefficient AR (1) process underlying return shocks, ε t : ε t = φ t ε t 1 + η t (2) where φ t iid(0, α), α 0, η t iid(0, ω), ω 0, η t = ε t / h t is stardized residual, with h t defined below. [21] Tsay (1987) derived ARCH (1) model [22] Engle (1982) [23] Bollerslev (1986) from Equation (2) as: h t E(ε 2 t I t 1 ) = ω + αε 2 t 1 (3) where h t represents conditional volatility, I t 1 is inmation set available at time t 1. A lagged dependent variable, h t 1, is typically added to Equation (3) to improve sample fit: h t E(ε 2 t I t 1 ) = ω + αε 2 t 1 + βh t 1 (4) From specification Equation (2), it is clear that both ω α should be positive, as y unconditional variances two different stochastic processes. Given non-normality shocks, Quasi-Maximum Likelihood Estimators (QMLE) parameters have been shown to be consistent asymptotically normal in several papers. For example [12] Ling McAleer (2003) showed that QMLE a generalized ARCH(p,q) (or GARCH(p,q)) is consistent if second moment is finite. A sufficient condition QMLE GARCH(1,1) in Equation (4) to be consistent asymptotically normal is α + β < 1. In general, pros asymptotic properties follow from fact that GARCH can be derived from a rom coefficient autoregressive process. Ref. [13] McAleer et al. (2008) give a general pro asymptotic normality multivariate models that based on proving that regularity conditions satisfy conditions given in [24] Jeanau (1998) consistency, conditions given in Theorem in [25] Amemiya (1985) asymptotic normality Multivariate Conditional Volatility The multivariate extension univariate ARCH GARCH models is given in [9] Baba et al. (1985) [10] Engle Kroner (1995) ( caveats regarding Full BEKK, see [15] Chang McAleer (2017b)). In order to establish volatility spillovers in a multivariate framework, it is useful to define multivariate extension relationship between shocks stardized residuals, that is, η t = ε t / h t. The multivariate extension Equation (1), namely y t = E(y t I t 1 ) + ε t, can remain unchanged by assuming that three components now m 1 vectors, where m is number financial assets. The multivariate definition relationship between ε t η t is given as: ε t = D 1/2 t η t (5) where D t = diag(h 1t, h 2t,..., h mt ) is a diagonal matrix comprising univariate conditional volatilities. Define conditional covariance matrix ε t as Q t. As m 1 vector, η t, is assumed to be iid all m elements, conditional correlation matrix ε t, which is equivalent to conditional correlation matrix η t, is given by Γ t. Theree, conditional expectation (5) is defined as: Q t = D 1/2 t Γ t D 1/2 t (6) Equivalently, conditional correlation matrix, Γ t, can be defined as: Γ t =D 1/2 t Q t D 1/2 t. (7)

8 Sustainability 2017, 9, Equation (6) is useful if a model Γ t is available purposes estimating Q t, whereas (7) is useful if a model Q t is available purposes estimating Γ t. Equation (6) is convenient a discussion volatility spillover effects, while both Equations (6) (7) instructive a discussion asymptotic properties. As elements D t consistent asymptotically normal, consistency Q t in (6) depends on consistent estimation Γ t, whereas consistency Γ t in (7) depends on consistent estimation Q t. As both Q t Γ t products matrices, with inverses in (7), neir QMLE Q t nor Γ t will be asymptotically normal based on definitions given in Equations (6) (7) Diagonal BEKK The Diagonal BEKK model can be derived from a vector rom coefficient autoregressive process order one, which is multivariate extension univariate process given in Equation (2): ε t = Φ t ε t 1 + η t (8) where ε t η t m 1 vectors, Φ t is an m m matrix rom coefficients, Φ t iid(0, A), A is positive definite, η t iid(0, C), C is an m m matrix. Vectorization a full matrix A to vec A can have dimension as high as m 2 m 2, whereas vectorization a symmetric matrix A to vech A can have a smaller dimension m(m + 1)/2 m(m + 1)/2. In a case where A is a diagonal matrix, with a ii > 0 all i = 1,..., m bjj < 1 all j = 1,..., m, so that A has dimension m m, [13] McAleer et al. (2008) showed that multivariate extension GARCH(1,1) from Equation (8) is given as Diagonal BEKK model, namely: Q t = CC + Aε t 1 ε t 1 A + BQ t 1 B (9) where A B both diagonal matrices, though last term in Equation (9) need not come from an underlying stochastic process. The diagonality positive definite matrix A is essential matrix multiplication as ε t 1 ε t 1 is an m m matrix; orwise, Equation (9) could not be derived from vector rom coefficient autoregressive process in Equation (8) Full, Triangular Hadamard BEKK The full BEKK model in [9] Baba et al. (1985) [10] Engle Kroner (1995), who do not derive model from an underlying stochastic process, is presented as: Q t = CC + Aε t 1 ε t 1 A + BQ t 1 B (10) except that A (possibly) B in Equation (10) now both full matrices, rar than diagonal matrices that were derived in Equation (9) by using stochastic process in Equation (8). The full BEKK model can be replaced by triangular or Hadamard (element-by-element multiplication) BEKK models, with similar problems identification (lack ) existence. A fundamental technical problem is that full, triangular, Hadamard BEKK models cannot be derived from any known underlying stochastic processes, which means that re no regularity conditions (except by assumption) checking internal consistency alternative models, consequently no valid asymptotic properties QMLE associated parameters (except by assumption). Moreover, as number parameters in a full BEKK model can be as much as 3m(m + 1)/2, curse dimensionality will be likely to arise, which means that convergence estimation algorithm can become problematic less reliable when re is a large number parameters to be estimated.

9 Sustainability 2017, 9, As a matter empirical fact, estimation full BEKK can be problematic even when m is as low as five financial assets. Such computational difficulties do not arise Diagonal BEKK model. Convergence estimation algorithm is more likely when number commodities is less than four, though this is neverless problematic in terms interpretation. Theree, in empirical analysis, in order to investigate volatility spillover effects, solution is to use Diagonal BEKK model estimation. A quasi likelihood ratio (QLR) test is developed to test multivariate conditional volatility Diagonal BEKK model in Equation (9) (where A B both diagonal matrices), which has valid regularity conditions asymptotic properties, against alternative Full BEKK model in Equation (10) (where A B in now both full matrices), which has valid regularity conditions asymptotic properties only under null hyposis zero f-diagonal elements. The quasi likelihood ratio (QLR) test null Diagonal BEKK model against alternative Full BEKK model does not yet seem to have been presented in literature Granger Causality, Volatility Spillovers, Optimal Hedge Ratios [13] McAleer et al. (2008) showed that QMLE parameters Diagonal BEKK model were consistent asymptotically normal, so that stard statistical inference on testing hyposes is valid. Moreover, as Q t in (9) can be estimated consistently, Γ t in Equation (7) can also be estimated consistently. The Diagonal BEKK model is given as Equation (9), where matrices A B given as: a 11 0 A = a mm, B = b b mm (11) The Diagonal BEKK model permits a test Co-volatility Spillover effects, which is effect a shock in commodity j at t 1 on subsequent co-volatility between j anor commodity at t. Given Diagonal BEKK model, as expressed in Equations (9) (10), subsequent co-volatility must only be between commodities j i at time t. [19] Chang et al. (2015) define Full Partial Volatility Covolatility Spillovers in context Diagonal Full BEKK models. Volatility spillovers defined as delayed effect a shock in one asset on subsequent volatility or covolatility in anor asset. Theree, a model relating Q t to shocks is essential, this will be addressed in following sub-section. Spillovers can be defined in terms full volatility spillovers full covolatility spillovers, as well as partial covolatility spillovers, as follows, i, j, k = 1,..., m: (1) Full volatility spillovers: (2) Full covolatility spillovers: Q iit / ε kt 1, k = i; (12) Q ijt / ε kt 1, i = j, k = i, j; (13) (3) Partial covolatility spillovers: Q ijt / ε kt 1, i = j, k = eir i or j. (14) Full volatility spillovers occur when shock from financial asset k affects volatility a different financial asset i. Full covolatility spillovers occur when shock from financial asset k affects covolatility between two different financial assets, i j. Partial covolatility spillovers occur when shock from financial asset k affects covolatility between two financial assets, i j, one which can be asset k.

10 Sustainability 2017, 9, When m = 2, only spillovers (1) (3) possible as full covolatility spillovers depend on existence a third financial asset. This leads to definition a Co-volatility Spillover Effect as: H ij,t ε j,t 1 = a ii a jj ε i,t 1, i = j As a ii > 0 all i, a test co-volatility spillover effect is given as a test null hyposis: H 0 : a ii a jj = 0 which is a test significance estimate a ii a jj in following co-volatility spillover effect, as ε i,t 1 = 0: H ij,t ε j,t 1 = a ii a jj ε i,t 1, i = j. If H 0 is rejected against alternative hyposis, H 1 : a ii a jj = 0, re is a spillover from shock commodity j at t 1 to co-volatility between commodities i j at t that depends only on shock commodity i at t 1. It should be emphasized that shock commodity j at t 1 does not affect co-volatility spillover commodity j on co-volatility between commodities i j at t. Moreover, spillovers can do vary each observation t 1, so that empirical results average co-volatility spillovers will be presented, based on average return shocks over sample period. Granger (1980) [2] causality is based on following vector AR (VAR(m,n)) models: x(t) = a 0 + a 1 x(t 1) + + a m x(t m) + b 1 y(t 1) + + b n y(t n) + u(t), (15) y(t) = c 0 + c 1 y(t 1) + + c n y(t n) + d 1 x(t 1) + + d m x(t m) +v(t) (16) The null hyposis Granger non-causality y(t 1) on x(t) is based on testing: H 0 : b i = 0 all i = 1, n in Equation (12), while null hyposis Granger non-causality x(t) on y(t 1) is based on testing: H 0 : d i = 0 all i = 1, m in Equation (13). In empirical analysis, m = n = 1 as data used. For multivariate conditional mean equation: y it = E(y it I t 1 ) + ε it,i = 1, 2,, m (17) bivariate rom coefficient autoregressive process ε it is given as: ε it = φ it ε it 1 +φ jt ε jt 1 +η it, i = j (18) where φ it iid(0, α i ), α i 0, φ jt iid(0,α j ), α j 0, η it iid(0,ω i ), ω i 0, η it = ε it / h it is stardized residual, h it is conditional volatility obtained by setting φ jt = 0 in bivariate Equation (15): ε it = φ it ε it 1 + η it ( ) E ε 2 it I t 1 h it = ω i + α i ε 2 it 1 Adding anor commodity, as in bivariate Equation (15), gives:

11 Sustainability 2017, 9, ( E while adding first-order lags h it h jt gives: ε it = φ it ε it 1 + φ jt ε jt 1 + η it, i = j ε 2 it ) I t 1 h it = ω i + α i ε 2 it 1 + α jε 2 jt 1 h it = ω i + α i ε 2 it 1 + α jε 2 jt 1 + β ih it 1 + β j h jt 1 where α i 0, α j 0, β i ( 1, 1), β j ( 1, 1) The null hyposis non-causality in volatility is given as a test : H 0 : α j = β j = 0 Based on empirical results, dynamic hedging strategies using optimal hedge ratios will be suggested to analyse market fluctuations in volatility emissions, crude oil, coal. Using hedge ratio: R H,t = R S,t γ t R F,t its variance, namely: var(r H,t Ω t 1 ) = var(r S,t Ω t 1 ) 2γ t cov(r S,t, R F,t Ω t 1 ) + γ 2 t var(r F,t Ω t 1 ) optimal hedge ratio is given as: γ t Ω t 1 = cov(r S,t, R F,t Ω t 1 )/var(r F,t Ω t 1 ) An extension recent research on realized matrix-exponential stochastic volatility with asymmetry, long memory, spillovers, in [26] Asai, Chang McAleer (2017), to multivariate conditional volatility models, especially use matrix-exponential transmation to ensure a positive definite covariance matrix, will enable a significant extension univariate Granger causality tests to be extended to multivariate Granger causality tests. This would be a novel extension paper. 4. Unit Root Tests In order to evaluate characteristics data, we investigate wher shocks to a series temporary or permanent in nature. We will use ADF test ([27] Dickey Fuller, 1979; [28] Dickey Fuller, 1982; [29] Said Dickey, 1984), DF-GLS test ([30] Elliott et al., 1996), KPSS test ([31] Kwiatkowski et al., 1992) to test unit roots in individual series. The ADF DF-GLS tests designed to test null hyposis a unit root, while KPSS test is used null hyposis stationarity. In Table 3, based on ADF test results, large negative values in all cases indicate a rejection null hyposis unit roots at 1% level. Based on KPSS test, small positive values in all cases do not reject null hyposis stationary at 1% level. For DF-GLS test, emissions coal in, emissions in USA, reject null hyposis unit roots at 1% level. However, results coal oil do not reject null hyposis. It should be noted that, USA, a relatively small sample size 357 observations is used.

12 Sustainability 2017, 9, Table 3. Unit Root Tests 2 April May January May 2017 USA. Variables ADF DF-GLS KPSS fr * 3.09 * 0.05 * coal fr * * 0.12 * oil fr * * US sr * * UScoal sr * * USoil sr * * * Denotes null hyposis a unit root is rejected at 1%. 5. Granger Causality Spillovers in Returns Volatilities Table 4 reports results [2] Granger (1980) causality spillover tests in, with one lag being used throughout empirical analysis. There is no evidence bidirectional Granger causality between coal. However, oil in has a causal effect on emissions in. For USA, emissions has a causal effect on coal, as well as on oil. Table 4. Granger Causality Test Returns 2 April May January May 2017 USA. Variables Lags Outcome A Does Not Cause B Null Hyposis B Does Not Cause A A B F-Test p-value F-Test p-value fr coal fr 1 fr coal fr fr oil fr 1 fr oil fr US sr UScoal sr 1 US sr UScoal sr US sr USoil sr 1 US sr USoil sr Estimates DBEKK Full BEKK models Carbon, Coal, Oil given in Table 5. The estimates weighting coefficients, A(1,1), similar two models, but estimates weighting coefficients A(2,2) A(3,3) different two models. Similar comments apply to estimates matrix stability coefficients, B(1,1), B(2,2), B(3,43), respectively. Table 5. DBEKK Full BEKK Carbon, Coal, Oil 2 April May DBEKK C A B CARBON fr *** ** *** *** *** (0.055) (0.010) (0.024) (0.025) (0.009) COAL fr *** *** *** (0.010) (0.075) (0.007) (0.001) OIL fr *** *** (0.077) (0.013) (0.003) Full BEKK C A B CARBON fr *** * *** 0.014*** *** (0.055) (0.038) (0.072) (0.023) (0.004) (0.006) (0.009) (0.007) (0.010) COAL fr *** *** *** *** *** (0.068) (0.103) (0.029) (0.011) (0.017) (0.036)) (0.015) (0.023) *** ** *** 0189 *** *** *** OIL fr (0.101) (0.026) (0.013) (0.010) (0.024) (0.011) (0.015) 1. A = a 11 a 12 a 13 a 21 a 22 a 23, B = b 11 b 12 b 13 b 21 b 22 b 23, C = c 11 c 12 c 13 c 21 c 22 c Stard errors in a 31 a 32 a 33 b 31 b 32 b 33 c 31 c 32 c 33 pnses, *** denotes significant at 1%, ** denotes significant at 5%, * denotes significant at 10%.

13 Sustainability 2017, 9, Given differences in two three weighting coefficients in A in Table 5, it is not particularly surprising that quasi likelihood ratio (QLR) test in Table 6 null hyposis, DBEKK, against alternative hyposis, Full BEKK, leads to rejection null hyposis that f-diagonal elements A B zero. The calculated chi-squd statistic with 12 degrees freedom, at 34.32, is greater than critical value at 1% level. Theree, DBEKK is rejected, but Full BEKK is not appropriate as it is valid only under null hyposis zero f-diagonal coefficients weighting matrix A stability matrix B. In short, Diagonal BEKK model is rejected, but full BEKK model is not an appropriate replacement. Table 6. Quasi Likelihood Ratio (QLR) Test DBEKK Full BEKK 2 April May Quasi Log-likelihood value DBEKK 14, Quasi Log-likelihood value Full BEKK 14, QLR test statistic Critical value at 1% with 12 df Estimates DBEKK Full BEKK models US Carbon, Coal, Oil Spot given in Table 7. The estimates three weighting coefficients, A(1,1), A(2,2), A(3,3), reasonably similar two models, as estimates stability coefficients B(1,1) B(2,2), though estimates B(3,3) different two models. In view similarities in estimates three weighting coefficients in A in Table 7, quasi likelihood ratio (QLR) test in Table 8 null hyposis, DBEKK, against alternative hyposis, Full BEKK, leads to non-rejection null hyposis that f-diagonal elements A B zero, as compd with outcome in Table 6. The calculated chi-squd statistic with 12 degrees freedom, at 22.18, is less than critical value at 1% level. Theree, DBEKK is not rejected against Full BEKK, which is valid only under null hyposis zero f-diagonal coefficients weighting matrix A stability matrix B. In short, Diagonal BEKK model is empirically supported by data. Table 7. DBEKK Full BEKK US Carbon, Coal, Oil Spot 6 January May DBEKK C A B CARBON sr *** *** *** (0.105) (0.294) (0.332) (0.073) (0.038) COAL sr ** *** *** (0.314) (0.154) (0.034) (0.008) OIL sr *** *** (1.029) (0.0035) (0.010) Full BEKK C A B CARBON sr *** *** *** *** (0.092) (0.606) (0.178) (0.054) (0.089) (0.064) (0.025) (0.112) (0.063) COAL sr *** *** *** (0.528) (0.715) (0.033) (0.058) (0.041) (0.046) (0.056) (0.044) *** *** ** OIL sr (0.721) (0.049) (0.092) (0.060) (0.080) (0.074) (0.082) 1. A = a 11 a 12 a 13 a 21 a 22 a 23, B = b 11 b 12 b 13 b 21 b 22 b 23, C = c 11 c 12 c 13 c 21 c 22 c Stard errors in a 31 a 32 a 33 b 31 b 32 b 33 c 31 c 32 c 33 pnses, *** denotes significant at 1%, ** denotes significant at 5%.

14 Sustainability 2017, 9, Table 8. QLR Test DBEKK Full BEKK US Spot 6 January May Quasi Log-likelihood value DBEKK Quasi Log-likelihood value Full BEKK QLR test statistic Critical value at 1% with 12 df In light discussion based on Equations (14), partial co-volatility spillovers with DBEKK presented in Table 9. Based on estimates weighting matrix A, six eight partial co-volatility spillovers negative, which means that a shock in one emission, or oil will have a one-period delayed negative impact on conditional correlation between itself one or two commodities. Two eight partial co-volatility spillovers positive, so an opposite effect will be observed. Table 9. Partial Co-volatility Spillovers with DBEKK USA 2 April May January May 2017 USA. Market USA ( ) Hij,t ε k,t 1 Average Co-Volatility Spillovers j = k = coal fr, i = fr = j = k = fr, i = coal fr = j = k = oil fr, i = fr = j = k = fr, i = oil fr = j = k = coal sr, i = sr = j = k = sr, i = coal sr = j = k = oil sr, i = sr = j = k = sr, i = oil sr = Co-volatility Spillovers: H ij,t ε k,t 1 = a ii a jj ε i,t 1. Given discussion based on Equations (12) (13), full co-volatility spillovers with DBEKK presented in Table 10. Based on estimates weighting matrix A, two six full co-volatility spillovers negative, which means that a shock in one emission, or oil will have a one-period delayed negative impact on conditional correlation between two or commodities. Two six full co-volatility spillovers positive, so an opposite effect will be observed, while two six full co-volatility spillovers zero, in which case re will be no spillovers. Table 10. Full Co-volatility Spillovers with Full BEKK USA 2 April May January May 2017 USA. Market USA ( ) Hij,t ε k,t 1 j = coal fr, i = fr k = oil fr j = oil fr, i = fr k = coal fr, 0 j = coal fr, i = oil fr k = fr j = coal sr, i = sr k = oil sr j = oil sr, i = sr k = coal sr j = coal sr, i = oil sr k = sr 0 Co-Volatility Spillovers Co-volatility Spillovers: H ij,t ε k,t 1 spillover 0 is to three decimal places. = a ii a jk ε i,t 1 + a ij a jk ε j,t 1 + a ik a ji ε i,t 1 + a ik a jj ε j,t 1 + 2a ik a jk ε k,t 1. A co-volatility The results full co-volatility spillovers in Table 10 not as clear or as helpful as in case partial co-volatility spillovers in Table 9, as estimates f-diagonal elements in weighting matrix A not especially large.

15 j = coal, Co-volatility Spillovers: i = oil,, =a a ε, k = +a a ε, 0 +a a ε, +a a ε, + 2a a ε,. A co-volatility spillover 0 is to three decimal places. The results full co-volatility spillovers in Table 10 not as clear or as helpful as in case partial co-volatility spillovers in Table 9, as estimates f-diagonal elements in weighting matrix A not especially large. The The unconditional unconditional conditional conditional volatility volatility,, oil oil shown in 4a f, while unconditional conditional volatility, oil shown in 4a f, while unconditional conditional volatility, oil USA USA shown shownin in 5a f. 5a f.the Theconditional conditionalvolatility volatilityestimates estimates ecasts ecasts unconditional unconditional volatilities. volatilities. Both Both figures figures show show that that re re is is aasignificant significantdifference differencebetween between conditional unconditional volatilities. As one purposes paper is to use conditional conditional unconditional volatilities. As one purposes paper is to use conditional volatilities volatilities to toecast ecastoptimal optimal hedge hedge ratios ratios various various,,any anydifferences differences between unconditional conditional volatilities is based on unconditional volatilities between unconditional conditional volatilities is based on unconditional volatilitiesbeing being unpredictable unpredictableas ascompd compdto to conditional conditionalvolatilities. volatilities. Sustainability 2017, 9, 1789 Sustainability 2017, 9, (e) (f) Unconditional Unconditional (a,c,e) (a,c,e) Conditional Conditional (b,d,f) (b,d,f) Volatility Volatility Carbon, Carbon, Coal, Coal, Oil Oil 2 April May April May 2017.

16 (e) (f) Sustainability 2017, 4. Unconditional 9, 1789 (a,c,e) Conditional (b,d,f) Volatility Carbon, Coal, Oil April May (e) (f) Unconditional (a,c,e) Conditional (b,d,f) (b,d,f) Volatility Carbon, Coal, Coal, Oil OilSpot Spot USA USA6 6January January May May2017. The conditional co-volatility correlations, oil The conditional co-volatility correlations, oil shown in 6a f, while conditional co-volatility correlations, oil shown in 6a f, while conditional co-volatility correlations, oil USA shown in 7a f. Both figures show that re substantial USA shown in 7a f. Both figures show that re substantial differences in correlations conditional co-volatility across two markets time periods, oil.

17 Sustainability 2017, 9, differences in correlations conditional co-volatility across two markets time periods , oil. Sustainability 2017,2017, 9, 1789 Sustainability 9, 1789 differences in correlations conditional co-volatility across two markets time periods, oil. (e)(e) (f)(f) Conditional Co-volatility (a,c,e) Correlations (b,d,f) Carbon, Coal, 6. Conditional Co-volatility (a,c,e) Correlations Correlations (b,d,f) OilOil 6.6.Conditional Co-volatility (a,c,e) (b,d,f) Carbon, Carbon,Coal, Coal, Oil 2 April May April May April May Cont.

18 Sustainability 2017, 9, Sustainability 2017, 9, Sustainability 2017, 9, (e) (f) (e) (f) 7. Conditional Co-volatility (a,c,e) Correlations (b,d,f) Carbon, Coal, Oil Spot USA 7. Conditional Co-volatility (a,c,e) Correlations (b,d,f) Carbon, Coal, Oil Spot USA 6 January May Conditional Co-volatility (a,c,e) Correlations (b,d,f) Carbon, Coal, Oil Spot USA 6 January May January May The optimal hedge ratios, oil, optimal hedge The optimal hedge ratios, oil, optimal hedge ratios, oil oil USA, given in s 8a f hedge 9a f, The optimal hedge ratios,, optimal respectively. The hedge ratios covariances in between two assets changes ratios, oilshow how USA, in 8a f s 9a f, ratios, oil USA, given ingiven s 8a f 9a f, respectively. relativeratios to variance covariances hedging show thatchanges reassets isrelative substantial The hedge ratios show how instrument. infigures between two changes The respectively. hedge show how in covariances Both between two assets to variation variance optimal hedge that relative toin ratios, hedging instrument. Both figures show that reemissions, is substantial variance hedging instrument. Both so figures show that re is substantial variation in optimal oil should be contemporaneously emissions, simultaneously portfolio that variation in hedge ratios, that a emissions, hedge ratios, so thatoptimal considered so in oil should belinks considered, volatilitiescontemporaneously emissions to use fossil fuels. oil, should be considered simultaneously in a portfolio that links contemporaneously simultaneously in a portfolio that links,, volatilities,, volatilities emissions to use fossil fuels. emissions to use fossil fuels. 8. Cont.

19 Sustainability 2017, 9, 1789 Sustainability Sustainability2017, 2017,9,9, (e) (e) (f) (f) 8. Optimal Hedge Ratios Carbon (a,b), Coal (c,e), Oil (d,f) 2 April Optimal Ratios Carbon Carbon(a,b), (a,b), Coal (c,e), Oil (d,f) April OptimalHedge Hedge Ratios Coal (c,e), Oil (d,f) 2 April 19 May May May (e) (f) 9. Optimal Hedge Ratios Carbon (a,b), Coal (c,e), Oil (d,f) Spot USA 6 January (e) (f) May Optimal Hedge Ratios Carbon (a,b), Coal (c,e), Oil (d,f) Spot USA 6 January 9. Optimal Hedge Ratios Carbon (a,b), Coal (c,e), Oil (d,f) Spot USA 6 January May a d show optimal hedge ratios Finally, May both coal oil USA. In all cases, optimal hedge ratios vary substantially, Finally, show optimal ratios which suggests that10a d it would be sensible to usehedge both markets to hedge emission Finally, 10a d show optimal hedge ratios hedge both coal oil USA. all cases, optimal ratios vary substantially, in against both coal oil pricein in USA. both coalsuggests oil that it sensible USA. In all cases, to optimal hedge ratios vary substantially, which would be to use both markets hedge emission which suggests that both it would use both markets to hedge emission in against coal be sensible oil to price in USA. in against both coal oil price in USA.

20 Sustainability 2017, 9, Sustainability 2017, 9, Optimal Optimal Hedge Hedge Ratios Ratios Carbon Carbon (a,b) (a,b),, Coal Coal Oil Oil Spot Spot USA USA 2 April April May May Concluding Remarks 6. Concluding Remarks The paper discussed recent research that showed efts to limit climate change have been The paper discussed recent research that showed efts to limit climate change have been focusing on reduction dioxide emissions over or greenhouse gases or air pollutants. focusing on reduction dioxide emissions over or greenhouse gases or air pollutants. Many countries have paid great attention to emissions in order to improve air quality Many countries have paid great attention to emissions in order to improve air quality public health. The largest source emissions from human activities in many countries in public health. The largest source emissions from human activities in many countries in Europe around world has been from burning fossil fuels. The both fuel Europe around world has been from burning fossil fuels. The both fuel emissions can do have simultaneous contemporaneous effects on each or. emissions can do have simultaneous contemporaneous effects on each or. Owing to importance emissions ir interconnection to, financial Owing to importance emissions ir interconnection to, financial, associated volatilities fossil fuels, possibility Granger causality in, associated volatilities fossil fuels, possibility Granger causality in,, volatility emissions, it is not surprising that crude oil,, volatility emissions, it is not surprising that crude oil ir interactions with emission, volatility, have recently become very ir interactions with emission, volatility, have recently become very important public policy an associated research topic. important public policy an associated research topic. For USA, available crude oil but re no For USA, available crude oil but re no or emissions. For, re no coal or or emissions. For, re no coal emissions, but re crude oil, emissions. For this or emissions, but re crude oil, emissions. For reason, were used to analyse Granger causality volatility spillovers in this reason, were used to analyse Granger causality volatility spillovers in emissions, crude oil, coal. emissions, crude oil, coal. quasi likelihood ratio (QLR) test was developed to test multivariate conditional volatility A quasi likelihood ratio (QLR) test was developed to test multivariate conditional volatility Diagonal BEKK model, which has valid regularity conditions asymptotic properties, against Diagonal BEKK model, which has valid regularity conditions asymptotic properties, against alternative Full BEKK model, which has valid regularity conditions asymptotic properties only alternative Full BEKK model, which has valid regularity conditions asymptotic properties under null hyposis zero f-diagonal elements. In short, Full BEKK has no desirable only under null hyposis zero f-diagonal elements. In short, Full BEKK has no desirable mamatical or statistical properties, except eir under null hyposis zero f-diagonal mamatical or statistical properties, except eir under null hyposis zero f-diagonal elements weighting matrix, or simply by assumption. elements weighting matrix, or simply by assumption. In empirical analysis, DBEKK was rejected against Full BEKK model In empirical analysis, DBEKK was rejected against Full BEKK model,, but DBEKK was not rejected against Full BEKK US. Theree, furr work but DBEKK was not rejected against Full BEKK US. Theree, furr work would would seem to be required DBEKK in case, whereas DBEKK is seem to be required DBEKK in case, whereas DBEKK is empirically empirically supported by data US. supported by data US. Dynamic hedging strategies using optimal hedge ratios were suggested to analyse market fluctuations in volatility emissions, crude oil, coal

21 Sustainability 2017, 9, Dynamic hedging strategies using optimal hedge ratios were suggested to analyse market fluctuations in volatility emissions, crude oil, coal. It was suggested that emissions, oil should be considered contemporaneously simultaneously in a portfolio that links,, volatilities emissions to use fossil fuels. It would also be sensible to use in both markets to hedge emission price in against both coal oil price in USA. Acknowledgments: The authors most grateful helpful comments suggestions two referees. For financial support, first author wishes to thank National Science Council, Ministry Science Technology (MOST), Taiwan, second author is most grateful to Australian Research Council National Science Council, Ministry Science Technology (MOST), Taiwan. Author Contributions: Chia-Lin Chang Michael McAleer conceived designed ideas. McAleer wrote first draft. Guangdong Zuo collected data estimated models. Chang McAleer analysed data discussed empirical results. McAleer revised paper. Conflicts Interest: The authors decl no conflict interest. References 1. Rogelj, J.; Meinshausen, M.; Sedláček, J.; Knutti, R. Implications potentially lower climate sensitivity on climate projections policy. Environ. Res. Lett. 2014, 9, [CrossRef] 2. Granger, C.W.J. Testing causality: A personal viewpoint. J. Econ. Dyn. Control 1980, 2, [CrossRef] 3. Ramos, S.; Veiga, H. (Eds.) The Interrelationship between Financial Energy Markets. In Lecture Notes in Energy; Springer: Berlin, Germany, 2014; Volume Sawik, B.; Faulin, J.; Pérez-Bernabeu, E. A multicriteria analysis green VRP: A case discussion distribution problem a Spanish retailer. Transp. Res. Proced. 2017, 22, [CrossRef] 5. Sawik, B.; Faulin, J.; Pérez-Bernabeu, E. Multi-objective traveling salesman transportation problem with environmental aspects. In Applications Management Science; Lawrence, K.D., Kleinman, G., Eds.; Emerald: Bingley, UK, 2017; Volume 18, pp Sawik, B.; Faulin, J.; Pérez-Bernabeu, E. Selected multi-criteria green vehicle routing problems. In Applications Management Science; Lawrence, K.D., Kleinman, G., Eds.; Emerald: Bingley, UK, 2017; Volume 18, pp Hafner, C.M.; Herwartz, H. A Lagrange multiplier test causality in variance. Econ. Lett. 2006, 93, [CrossRef] 8. Chang, C.-L.; McAleer, M. A simple test causality in volatility. Econometrics 2017, 5, 15. [CrossRef] 9. Baba, Y.; Engle, R.F.; Kraft, D.; Kroner, K.F. Multivariate Simultaneous Generalized ARCH; Unpublished manuscript; Department Economics, University Calinia: San Diego, CA, USA, Engle, R.F.; Kroner, K.F. Multivariate simultaneous generalized ARCH. Econom. Theory 1995, 11, [CrossRef] 11. Bollerslev, T. Modelling coherence in short-run nominal exchange rate: A multivariate generalized ARCH approach. Rev. Econ. Stat. 1990, 72, [CrossRef] 12. Ling, S.; McAleer, M. Asymptotic ory a vector ARMA-GARCH model. Econom. Theory 2003, 19, [CrossRef] 13. McAleer, M.S.; Hoti, F. Chan Structure asymptotic ory multivariate asymmetric conditional volatility. Econom. Rev. 2009, 28, [CrossRef] 14. McAleer, M.F.; Chan, S.; Hoti, O. Lieberman Generalized autoregressive conditional correlation. Econom. Theory 2008, 24, [CrossRef] 15. Chang, C.-L.; McAleer, M. The Fiction Full BEKK; Tinbergen Institute Discussion Paper ; Tinbergen Institute: Amsterdam, The Nerls, Hafner, C.; McAleer, M. A One Line Derivation DCC: Application a Vector Rom Coefficient Moving Average Process; Tinbergen Institute Discussion Paper ; Tinbergen Institute: Amsterdam, The Nerl, McAleer, M. Stationarity Invertibility a Dynamic Correlation Matrix; Tinbergen Institute Discussion Paper ; Tinbergen Institute: Amsterdam, The Nerl, Caporin, M.; McAleer, M. Do we really need both BEKK DCC? A tale two multivariate GARCH models. J. Econ. Surv. 2012, 26, [CrossRef]

22 Sustainability 2017, 9, Chang, C.-L.; Li, Y.-Y.; McAleer, M. Volatility Spillovers between Energy Agricultural Markets: A Critical Appraisal Theory Practice; Tinbergen Institute Discussion Paper /III; Tinbergen Institute: Amsterdam, The Nerl, Chang, C.-L.; McAleer, M.; Wang, Y.-A. Modelling volatility spillovers bio-ethanol, sugarcane corn. Renew. Sustain. Energy Rev. 2017, 81, [CrossRef] 21. Tsay, R.S. Conditional heteroscedastic time series models. J. Am. Stat. Assoc. 1987, 82, [CrossRef] 22. Engle, R.F. Autoregressive conditional heteroskedasticity with estimates variance United Kingdom inflation. Econometrica 1982, 50, [CrossRef] 23. Bollerslev, T. Generalized autoregressive conditional heteroscedasticity. J. Econom. 1986, 31, [CrossRef] 24. Jeanau, T. Strong consistency estimators multivariate ARCH models. Econom. Theory 1998, 14, [CrossRef] 25. Amemiya, T. Advanced Econometrics; Harvard University Press: Cambridge, MA, USA, Asai, M.; Chang, C.-L.; McAleer, M. Realized stochastic volatility with general asymmetry long memory. J. Econom. 2017, 199, [CrossRef] 27. Dickey, D.A.; Fuller, W.A. Distribution estimators autoregressive time series with a unit root. J. Am. Stat. Assoc. 1979, 74, Dickey, D.A.; Fuller, W.A. Likelihood ratio statistics autoregressive time series with a unit root. Econometrica 1981, 49, [CrossRef] 29. Said, S.E.; Dickey, D.A. Testing unit roots in autoregressive-moving average models unknown order. Biometrika 1984, 71, [CrossRef] 30. Elliott, G.; Ronberg, T.J.; Stock, J.H. Efficient tests an autoregressive unit root. Econometrica 1996, 64, [CrossRef] 31. Kwiatkowski, D.; Phillips, P.C.B.; Schmidt, P.; Shin, Y. Testing null hyposis stationarity against alternative a unit root: How sure we that economic time series have a unit root? J. Econom. 1992, 54, [CrossRef] 2017 by authors. Licensee MDPI, Basel, Switzerl. This article is an open access article distributed under terms conditions Creative Commons Attribution (CC BY) license (