Emmanuel Hache
email: emmanuel.hache@ifpen.fr.

Paris Anthony

Market Efficiency and Optimal Hedging Strategy for the US Ethanol Market

Keywords: JEL Classification: Q41, Q42, G15, C41 Ethanol prices, Futures markets, Markov-switching regime models, Hedge ratio

The aim of this paper is to study the ethanol price dynamics in the US market and find the optimal hedging strategy. To this end, we first attempt to identify the long-term relationship between ethanol spot prices and the prices of futures contracts on the Chicago Board of Trade (CBOT). Then, we model the short-term dynamics between these two prices using a Markov-switching vector error correction model (Ms-VECM). Finally, accounting for the variance dynamics using a Gjr-MGarch error structure, we compute a time-varying hedge ratio and determine the optimal hedging strategy in the US ethanol market.

Introduction

Ethanol is derived from various agricultural products (cassava, corn, hemp, sugar beet or sugarcane) and has been increasingly added to gasoline blends for several reasons: (i) it helps to reduce greenhouse gases emissions (GHG) in the transportation sector, (ii) if produced with agricultural feedstock, ethanol can be seen as a renewable energy, and (iii) from a technical point of view, the use of ethanol helps to boost octane numbers and leads to an improvement in thermal engine efficiency. All these factors have contributed to the development of ethanol's use worldwide. Such recent evolution calls for a detailed investigation of the ethanol market to fully understand its dynamics. Specifically, the aim of this paper is to study the ethanol price dynamics and determine the optimal hedging strategy on the US market.

Ethanol policy is a story that has many chapters in the past 40 years in the US. Ethanol inclusion in US gasoline blends began in 1908 when the Model-T Ford could be customized to run on gasoline or alcohol. It was not until the late Seventies, however, that the meaningful inclusion of ethanol came about. The first government involvement for ethanol was the Energy Tax Act of 1978 (a tax exemption for adding ethanol to the gasoline blend) in the wake of geopolitical concerns in the oil market with the 2 nd world oil shock. The Surface Transportation Assistance Act of 1982 and the Tax Reform Act of 1984 gave an impetus for ethanol inclusion despite a decrease of the tax exemption during the 1992-2000 period with the Omnibus Budget Recollection Act. The Renewable Fuel Standard (RFS) program, created by the Energy Policy Act of 2005 and extended by the Energy Independence and Security Act of 2007, has led to a new expansion of the US ethanol market. Ethanol production and consumption have since been multiplied by four between 2005 and 2016, increasing approximately from 300 to 1,200 million gallons.

Since 2009 the US has become a net exporter in the ethanol market. According to the US Census Bureau, the Department of Commerce, and the Department of Agriculture, the US exported 836 million gallons of ethanol in 2015 (5.7% of total US ethanol production) and imported 93 million gallons of fuel ethanol (less than 1% of US ethanol consumption). Canada (30% of US exports), Brazil (14%), Philippines (9%), China (8%), and India (6%) are the top destinations of US ethanol in 2015. Brazil also remains the main supplier for the US with 73% of the imported ethanol volume in 2015. This export-import structure within the ethanol market with Brazil can be easily explained by the RFS and California Low Carbon Fuel Standard (LCFS) targets put in place for the reduction of GHG emissions that impose more stringent requirements. As mentioned by the Energy Information Administration, 1 life cycle analysis (LCA) studies demonstrate that ethanol from sugarcane has 1 https://www.eia.gov/todayinenergy/detail.php?id=25312 1 a better scoring in terms of GHG emissions than products based on corn feedstock. It contributes to the substitution of corn-ethanol production from the countryside with imported sugarcane-ethanol from Brazil. The ethanol market structure is already driven by (i) the inclusion policy of different countries, (ii) energy prices and especially the evolution of the crude oil price, and (iii) the regulatory framework. But recent changes prove that the production process (ethanol is derived from different agricultural products) could also impact the international market structure and ethanol price dynamics. Ethanol prices registered several ups and downs since 2008, with prices ranging from $1.47 per gallon to more than $4 per gallon following the volatility observed in energy and agricultural prices.

Due to the increase of ethanol production and consumption in the US in the first part of the last decade, futures contracts on corn-based ethanol were launched on March 2005 on the Chicago Board of Trade (CBOT). 2 Derivatives markets allow commercial players to reduce their price risk exposure with various hedging strategies and different tools (futures contracts, options, etc.). These tools protect against adverse price movements in order to reduce the risk of loss in the business. In this context, the optimal hedging strategy is to minimize the variance of the hedge portfolio containing spot and futures contracts. A variety of questions has been asked regarding the derivatives strategy, which is related to traders' behavior, speculation, price volatility, etc.

The motivations of this paper are threefold. Firstly, considering the role the ethanol market could play in the transportation sector for its own energy transition, we study the long-term relationship between ethanol spot prices and the prices of futures contracts on the CBOT; allowing us to investigate the weak form of the efficient market hypothesis. 3 To the best of our knowledge, this is the first article focusing on ethanol in this research field. Secondly, we have a methodological motivation and contribution. Indeed, we compute a wide range of time-varying hedge ratios 4 with different econometric models to look for the optimal hedging strategy for ethanol commercial players. We consider adjustments to long-term equilibrium and regime shifts governed by a Markov chain (as [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF]) and shortrun dynamics between spot and futures price changes (as in Salvador and Arago 2 CME Group is the world's leading and most diverse derivatives marketplace, made up of four markets, CME, CBOT, NYMEX, and COMEX. Each market offers a wide range of global benchmarks across major asset classes.

3 An efficient market is characterized by prices that reflect all available information. The weak form of market efficiency considers only historical price or return series in the information set [START_REF] Fama | Efficient capital markets: A review of theory and empirical work[END_REF]. 4 One of them is the hedge ratio which is initially defined as the estimated coefficient between spot and futures price changes based on Ordinary Least Squares (OLS) estimation [START_REF] Ederington | The hedging performance of the new futures markets[END_REF] i.e., the ratio of the unconditional spot and futures price changes covariance over the unconditional variance of the futures price changes. It provides the number of futures contracts to buy or sell for one unit of the underlying asset (in the case of this article: ethanol) to minimize the variance of the hedged portfolio returns.

(2014)). In addition, we extend the work of [START_REF] Salvador | Measuring hedging effectiveness of index futures contracts: Do dynamic models outperform static models? a regime-switching approach[END_REF] by allowing short-run dynamics between prices to be state-dependent on price volatility. [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF] proposes the Markov-switching model while [START_REF] Krolzig | Statistical analysis of cointegrated var process with markovian regime shifts[END_REF] extends this specification to the vector autoregressive model. By including structural breaks in the variance equation, we take into account the high volatility persistence [START_REF] Lamoureux | Persistence in variance, structural change, and the garch model[END_REF]. With structural breaks in the short-run dynamics, we allow for time-varying behavior in the adjustment to the equilibrium and the short-run dynamic processes. We then include an informational link between mean and volatility processes across each market state [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF]. Finally, relying on the Gjr framework [START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF], we introduce asymmetric behavior to the variance process (see also [START_REF] Brooks | The effect of asymmetries on optimal hedge ratios[END_REF]) to take into account different responses to new information according to the past shocks sign. Therefore, we estimate a Markov-switching vector error correction model with a Gjr-MGarch error structure (Ms-VECM-Gjr-MGarch). To overcome Johansen (1988)'s approach drawbacks,5 we use Nielsen (2010)'s nonparametric cointegration approach to analyze its ability to improve hedging strategy. As Nielsen (2010)'s nonparametric cointegration procedure does not require model specification, we assume non-linear dynamics in short-run and variance equations. Thirdly, we check the performance of a cross-hedging strategy6 with the gasoline futures market. Indeed, [START_REF] Franken | Cash ethanol cross-hedging opportunities[END_REF] highlight its efficiency while [START_REF] Dahlgran | Inventory and transformation hedging effectiveness in corn crushing[END_REF] concludes there is a lower performance with this market compared to the ethanol futures market.

The rest of the paper is organized as follows. Section 2 briefly reviews the literature on storable commodity market efficiency and hedging-ratio estimation. In section 3 we present data and the Markov-switching vector error correction model (Ms-VECM-Gjr-MGarch). Section 4 presents empirical results on the efficient market hypothesis and the optimal hedging strategy. The main conclusions are summarized in the final section.

A brief overview of literature

Following the works of [START_REF] Kaldor | Speculation and economic stability[END_REF], [START_REF] Working | Theory of the inverse carrying charge in futures markets[END_REF], [START_REF] Brennan | The supply of storage[END_REF] and [START_REF] Telser | Futures trading and the storage of cotton and wheat[END_REF], spot and futures prices of a storable commodity should be equal. The difference between these prices is explained by the cost of storage and the interest rate as,

F T t = S t exp[(r t + s)(T -t)] (1)
and with a log-transformation,

f T t = s t + (r t + s)(T -t) (2)
Here, F T t (resp. f T t ) is the price (resp. log-price) of futures contract at the time t for a maturity T . S t (resp. s t ) is the spot price (resp. log-price) at the same date. r t and s refer, respectively, to the risk-free interest rate and the cost of carry, this latter is supposed to be constant. According to the aforementioned works, the difference between spot and futures prices is instantaneously compensated by arbitrageurs.

This hypothesis has been relaxed by [START_REF] Garbade | Price movements and price discovery in futures and cash markets[END_REF]. They mention that arbitrageurs operate in the markets if the spread between these prices is large enough to enlarge their profits according to the transaction and information costs. Therefore, the unit relationship between spot and futures prices is only valid in the long term.The spot and futures markets are thus efficient if prices are cointegrated as in [START_REF] Chowdhury | Futures market efficiency: Evidence from cointegration tests[END_REF] or [START_REF] Lai | A cointegration test for market efficiency[END_REF]. In addition, [START_REF] Garbade | Price movements and price discovery in futures and cash markets[END_REF] show that futures markets integrate new information faster than in the underlying spot market, leading to a causality from futures to spot prices. It helps the price discovery process registered in commodities markets which leads to informational efficiency for physical and financial markets. [START_REF] Figuerola-Ferretti | Modelling and measuring price discovery in commodity markets[END_REF] extent this model by integrating the convenience yield, i.e., the premium attributed by agents for physically holding the commodity instead of holding a futures contract. It depends on various market characteristics in the spot market (weather conditions, geopolitical unrest, transaction costs, etc.).7 With a constant free-risk interest rate, one-period futures contract and the approximation of the convenience yield, y t , used by these authors, as

y t = γ 1 s t -γ 2 f t (3) equation (2) becomes f t = 1 -γ 1 1 -γ 2 s t + r + s 1 -γ 2 (4)
Their theoretical framework allows a long-term relationship, i.e., a cointegrating relationship, with a non-unit coefficient between spot and futures prices. In addition, they mention that the coefficient value depends on the spot market condition. The parameter is greater (resp. smaller) than unity if the spot market is in contango (resp. backwardation).

Literature about the estimation of an optimal hedge ratio has been developed since the seminal work of [START_REF] Ederington | The hedging performance of the new futures markets[END_REF] who proposes using the estimated coefficient between changes in spot and futures prices with an ordinary least square estimator (OLS). However, this hedge ratio is unsatisfactory for many markets [START_REF] Cecchetti | Estimation of the optimal futures hedge[END_REF][START_REF] Myers | Generalized optimal hedge ratio estimation[END_REF]. [START_REF] Baillie | Bivariate garch estimation of the optimal commodity futures hedge[END_REF] and [START_REF] Kroner | Time-varying distributions and dynamic hedging with foreign currency futures[END_REF] state that the hedge ratio should be time-varying based on the time-varying distribution of many asset prices. They propose computing this dynamic optimal hedge ratio (δ t ) for each period by taking into account all past information (Ω t-1 ) such as

δ t |Ω t-1 = σ t-1 (ΔF t-1 , ΔS t-1 ) σ 2 t-1 (ΔF t-1 ) (5)
Many studies estimate those conditional covariance (σ t-1 ) and variance (σ 2 t-1 ) with the multivariate Garch model proposed by [START_REF] Engle | Multivariate simultaneous generalized arch[END_REF] as, for instance, [START_REF] Kroner | Time-varying distributions and dynamic hedging with foreign currency futures[END_REF], [START_REF] Garcia | Simultaneously determined, timevarying hedge ratios in the soybean complex[END_REF] or [START_REF] Kavussanos | Hedging in the futures market[END_REF] and conclude there has been an improvement of the hedging strategy with the dynamic hedge ratio compared to the constant formulation. The improvement degree depends on the market and the futures maturity studied [START_REF] Tse | Some recent developments in futures hedging[END_REF].

The estimation of the dynamic hedge ratio should integrate the possible existence of a cointegrating relationship between spot and futures prices. [START_REF] Kroner | Time-varying distributions and dynamic hedging with foreign currency futures[END_REF], [START_REF] Ghosh | Cointegration and error correction models: Intertemporal causality between index and futures prices[END_REF], [START_REF] Chou | Hedging with the nikkei index futures: The conventional model versus the error correction model[END_REF] or [START_REF] Lien | The effect of the cointegration relationship on futures hedging: A note[END_REF] highlighted an underestimated hedge ratio if this characteristic is not accounted for. In addition, [START_REF] Brooks | The effect of asymmetries on optimal hedge ratios[END_REF] show the improvements of the hedge ratio effectiveness with the integration of the asymmetric volatility response against positive and negative shocks, i.e., the leverage effect. Furthermore, the conditional mean [START_REF] Sarno | The cost of carry model and regime shifts in stock index futures markets: An empirical investigation[END_REF] and variance [START_REF] Lamoureux | Persistence in variance, structural change, and the garch model[END_REF] estimations can be biased if regime shifts exist. Thus, the hedge ratio effectiveness can be improved by integrating regime shifts in the estimation. Lee and Yoder (2007a,b) include regime shifts in the variance process and show an improvement -but not always significant -of the hedge ratio effectiveness. [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF] extent this model by integrating regime shifts in variance and conditional mean processes and highlight a significant effectiveness improvement for most of the markets studied. Finally, [START_REF] Salvador | Measuring hedging effectiveness of index futures contracts: Do dynamic models outperform static models? a regime-switching approach[END_REF] propose incorporating (i) the regime shifts, the cointegrating relationship and the leverage effect in the same model in order to estimate an optimal dynamic hedge ratio, as well as (ii) the short-run dynamics between spot and futures price changes.

The literature concerning hedging strategies on energy markets is well developed with, for instance, [START_REF] Yang | Asymmetric effect of basis on dynamic futures hedging: Empirical evidence from commodity markets[END_REF] for heating and crude oil markets, [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF] on crude oil, unleaded gasoline and heating oil markets, [START_REF] Hanly | Managing energy price risk using futures contracts: A comparative analysis[END_REF] with WTI and Brent crude oils, natural gas, unleaded gasoline, heating oil and gasoil. However, the literature on hedging strategies on ethanol market is very scarce. [START_REF] Franken | Cash ethanol cross-hedging opportunities[END_REF] highlight the cross-hedging efficiency between ethanol spot price and unleaded gasoline futures markets. However, while they correct the estimation for autocorrelation and heteroscedasticity, they do not incorpo-rate the error correction term, regime switching and time-varying variance process. Finally, [START_REF] Dahlgran | Inventory and transformation hedging effectiveness in corn crushing[END_REF] compares direct hedging for ethanol commercial agents with cross-hedging strategy with unleaded and Reformulated Gasoline Blendstock for Oxygen Blending (RBOB gasoline) futures markets. He demonstrates that the direct hedging strategy outperforms cross-hedging for a four-week, and more, hedge horizon.

Data and methodology

As stressed above, our article deals with the relationship between the spot prices and the futures prices of ethanol. As transaction volumes have risen, in particular for the shortest maturities, we focus on the relationship between the spot prices and the prices for the two-month futures contracts. The data studied are relative to the ethanol in the North American market: the spot price for ethanol (Argus Ethanol USGC barge/rail fob Houston), the futures prices of ethanol on the CBOT, and the transaction volumes and open interest for the same market, (weekly market business reports of the Commodity Futures Trading Commission [CFTC]). Apart from the spot price of ethanol, these pieces of information are all in the public domain. The data cover the period from July 2008 to December 2016, corresponding to 468 weekly observations. The prices are expressed in US dollars per gallon and are logtransformed.

Table 1 presents some descriptive statistics and tests results. Unit root tests confirm the stationarity of spot and futures prices series in their first-difference. 8 In addition, the [START_REF] Ljung | On a measure of lack of fit in time series models[END_REF] and ARCH tests confirm the presence of autocorrelation in most cases and heteroscedasticity, respectively. These characteristics justify the choice of a specification with autoregressive terms and heteroscedastic errors.

We apply the Johansen (1988)'s test to check the existence of a long-term relationship with unit cointegrating vectors and to estimate the conditional mean with a Markov switching vector error correction model (Ms-VECM) within a bivariate framework. The inclusion of a multivariate generalized autoregressive conditional heteroscedasticity (MGarch) error structure allows us to compute the dynamic hedge ratio. By including a long-term equilibrium, we eliminate the bias in the hedge ratio estimation mentioned by [START_REF] Kroner | Time-varying distributions and dynamic hedging with foreign currency futures[END_REF] and [START_REF] Ghosh | Cointegration and error correction models: Intertemporal causality between index and futures prices[END_REF]. In addition, the nonlinear specification avoids estimation bias due to the existence of multiple regimes in the mean [START_REF] Sarno | The cost of carry model and regime shifts in stock index futures markets: An empirical investigation[END_REF] and variance [START_REF] Lamoureux | Persistence in variance, structural change, and the garch model[END_REF] equations. Furthermore, the dynamic hedge ratio computed with this specification outperforms OLS hedge ratio in many energy markets [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF]. Finally, we take into account the leverage effect within the Gjr framework. It should be emphasized that the [START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF] cointegration test requires assumptions regarding the short-run dynamics that must follow a linear process. Using Johansen (1988)'s procedure with a non-linear short-run specification may lead to bias in both cointegration test results and long-term estimations, generating in turn a bias on the short-run and conditional variance estimations. To overcome these major drawbacks, we rely on Nielsen (2010)'s nonparametric variance ratio testing approach as this methodology does not require assumptions in the short-run specification. 9 The nonparametric variance ratio trace statistic is defined by

Λ n,r (d 1 ) = T 2d 1 n-r j=1 λ j (6)
where λ j , j = 1, ..., n, are the eigenvalues, listed by increasing order, of the observed (n × T ) time series matrix, r is the cointegration rank tested and d 1 is a summation parameter fixed to 0.1. 10 The eigenvalues of the price series matrix are given by the solutions of

|λB T -A T | = 0 (7) with A T = T t=1 Z t Z t B T = T t=1 Zt Z t ( 8
)
where Zt is the fractional difference of Z t truncated by d 1 . Z t is our time series matrix after demeaning. The null hypothesis is the presence of r -1 cointegration relationships. A test statistic that is greater than the critical value leads to the rejection of the null hypothesis in favor of the alternative, i.e., the existence of r cointegration relationships. In addition, the estimated cointegration coefficients are provided by the eigenvectors associated with eigenvalues and converge to their real values. Therefore, by using both [START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF] and [START_REF] Nielsen | Nonparametric cointegration analysis of fractional systems with unknown integration orders[END_REF] cointegration approach, we can analyze the effect of the long-term estimation bias on the hedge ratio efficiency.

The Ms-VECM with Gjr-MGarch11 error structure can be expressed by

ΔX t = c + Γ st ΔX t-1 + Π st X t-1 + t,st t,st = s,t,st f,t,st |Ω t-1 IN (0, H t,st ) (9)
where ΔX t = (Δs t , Δf t ) (resp. X t-1 = (s t-1 , f t-1 ) ) is the vector of log-returns (resp. log-price) and c is a vector of constant. Γ st and Π st are coefficient matrices related to short-and long-term dynamics, respectively.12 These (2 × 2) matrices depend on the regime st, st = 1, 2. t,st is a regime-dependent Gaussian white noise vector. With our multivariate Garch error structure, the error covariance matrix, H t,st , is time-and regime-dependent.

As mentioned by [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF], two steps are necessary to estimate this model. Firstly, we check the existence of a cointegrating relationship between spot and futures prices. Considering a linear process, we apply the Johansen (1988)'s test. The λ max and λ trace statistics allow us to check the rank of the matrix Π. Under the alternative hypothesis, there is at least one cointegrating relationship. If the rank of the long-term adjustment is non-null, Π can be decomposed such as Π = αβ . The vectors α and β are (2 × 1) coefficient vectors referring to the error correction coefficients, i.e., characterizing the adjustment process to the long-term equilibrium, and the long-term coefficients, describing the long-term equilibrium, respectively. In addition, we apply the likelihood ratio test from [START_REF] Johansen | Likelihood-Based Inference in Cointegrated Vector Autoregressive Models[END_REF] to check the existence of unitary long-term coefficients between spot and futures prices. The non-reject of the null hypothesis of unit coefficient will favor the [START_REF] Garbade | Price movements and price discovery in futures and cash markets[END_REF] model against that proposed by Figuerola-Ferretti and [START_REF] Figuerola-Ferretti | Modelling and measuring price discovery in commodity markets[END_REF].

Secondly, we introduce regime shifts depending on an unobserved state variable st. The latter can takes two values, st = 1, 2, corresponding to two different regimes. This variable follows a first order Markov process with the transition probability matrix,

P
where P 12 (resp. P 21 ) is the probability that the system will shift from state 1 (resp.

2) to state 2 (resp. 1). P 11 (resp. P 22 ) is the probability that the system will stay in regime 1 (resp. regime 2). We obviously have P 11 + P 12 = 1 and P 21 + P 22 = 1.

All the coefficients depend on the regime st except for the long-term coefficients, β. Indeed, variables with a nonlinear cointegrating relationship do not admit an error correction model [START_REF] Gonzalo | Threshold Effects in Multivariate Error Correction Models[END_REF]. In the presence of a cointegrating relationship, the Π st matrix is decomposed as Π st = α st β .

The conditional covariance matrix of error terms, H t,st , is regime-dependent, time-varying, and follows a multivariate Garch specification with a [START_REF] Baba | Multivariate simultaneous generalized arch[END_REF] framework, i.e., BEKK, as

H t,st = C st C st + A st t-1 t-1 A st + B st H t-1 B st + D st η t-1 η t-1 D st (11)
with t-1 and H t-1 being the vector of mean equation residuals and the global covariance matrix for the past period, respectively. η t-1 is negative past shocks, i.e., η t-1 = min( t-1 , 0). C st is a (2 × 2) lower triangular matrix containing regimedependent coefficients. A st , B st and D st are (2 × 2) diagonal matrices of coefficients measuring the past shock effects on the conditional covariance matrix, their persistence and the additional effect of a past negative shock, respectively. However, the conditional covariance matrix depends on the sequence of all previous regimes through H t-1 . With this path-dependence problem, the estimation by the maximum likelihood method is numerically infeasible. To overcome this problem, we follow the formulations of [START_REF] Gray | Modeling the conditional distribution of interest rates as a regime-switching process[END_REF] and [START_REF] Lee | A bivariate markov regime switching garch approach to estimate time varying minimum variance hedge ratios[END_REF] concerning the conditional variances, h ss and h f f , and the conditional covariance, h sf , respectively, as

h ss,t = π 1,t (r 2 s,1,t +h ss,1,t )+(1-π 1,t )(r 2 s,2,t +h ss,2,t )-[π 1,t r s,1,t +(1-π 1,t )r s,2,t ] 2 (12) h f f,t = π 1,t (r 2 f,1,t +h f f,1,t )+(1-π 1,t )(r 2 f,2,t +h f f,2,t )-[π 1,t r f,1,t +(1-π 1,t )r f,2,t ] 2 (13) h sf,t = π 1,t (r s,1,t r f,1,t + h sf,1,t ) + (1 -π 1,t )(r s,2,t r f,2,t + h sf,2,t ) (14) -[π 1,t r s,1,t + (1 -π 1,t )r s,2,t ][π 1,t r f,1,t + (1 -π 1,t )r f,2,t ]
In equations ( 12), ( 13) and ( 14), π st,t is the probability of being in the state st at the time t. h ss,st,t (resp. h f f,st,t ) is the regime-dependent variance concerning the spot (resp. futures) price at the time t and is contained in H t,st . Similarly, h sf,st,t is the state-dependent covariance at the time t and is an element of the same matrix. r s,st,t (resp. r f,st,t ) is the regime-dependent conditional mean of the spot (resp. futures) price equation at the time t. These latter are calculated from the following equations:

s,t = Δs t -[π 1,t r s,1,t + (1 -π 1,t )r s,2,t ] (15) f,t = Δf t -[π 1,t r f,1,t + (1 -π 1,t )r f,2,t ] (16)
This Ms-VEC model is estimated by maximizing of the likelihood function. Each state-dependent error follows a N-dimensional normal distribution with zero mean and H t,st covariance matrix. The global density function is a mixture of these distributions weighted by the probability of being in each regime:

f (X t , θ) = π 1,t 2π |H t,1 | -1 2 exp(- 1 2 t,1 H -1 t,1 t,1 ) (17)
+ π 2,t 2π |H t,2 | -1 2 exp(- 1 2 t,2 H -1 t,2 t,2 ) L(θ) = T t=1 logf (X t , θ) (18)
with θ denoting the parameter vector. The log-likelihood function (equation ( 18)) is maximized using the expectation-maximisation algorithm proposed by [START_REF] Dempser | Maximum likelihood from incomplete data via the em algorithm[END_REF] under constraints like π 1,t + π 2,t = 1, π 1,t 0 and π 2,t 1.

With our specification, we can compute the dynamic hedge ratio as

δ t |Ω t-1 = h sf,t-1 h f f,t-1 (19)
where h sf,t-1 et h f f,t-1 are defined in equations ( 14) and ( 13), respectively.

In order to analyze the hedging strategies' performance of each specification 13 we compute hedged portfolios each week and their returns variance over the samples chosen as

V AR(Δs t -δ t Δf t ) (20) 13 We estimate 22 specifications including 8 linear and 14 nonlinear models. Specifications vary about inclusion, or not, of error correction and autoregressive terms in mean equation, asymmetry in variance equation, as well as parameters allowed to switch. In addition, we use an OLS model and a naive model, i.e., with a unit hedging ratio.

In addition, as in [START_REF] Kroner | Time-varying distributions and dynamic hedging with foreign currency futures[END_REF] or [START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF] among others, we compute the hedger's utility function as

E t-1 U (Δs t -δ t Δf t ) = E t-1 (Δs t -δ t Δf t ) -k × V AR t-1 (Δs t -δ t Δf t ) ( 21
)
where k is the degree of risk aversion. This utility function represents economic benefits from the hedging strategy. Another way to consider this benefit is the value-at-risk (VaR) exposure and is computed as

V aR = W 0 [E(Δs t -δ t Δf t ) + Z α V AR(Δs t -δ t Δf t )] ( 22
)
where W 0 is the initial value of the portfolio and Z α is the normal distribution quantile.

Empirical results

Futures contracts on corn-based ethanol were launched on floor-based trading in March 2005. The CBOT launched the ethanol contract on the electronic platform in 2006 contributing to an increase in liquidity within the market. In 2007 options contracts were also launched in the market. For the first time, the volume reached 1,000 contracts in July 2006 and it really took off after 2009 with a sharp increase in the spot prices. During previous decades, and especially in the initial phase of construction of the ethanol futures market, the main objective was to attract and concentrate the liquidity required for commercial traders to achieve hedging activities. Nevertheless, the rise in transaction volumes has been accompanied by a concentration of traders' liquidity on the shortest maturity contracts exchanged on commodity markets. This factor has been observed and studied, for example on the WTI market in the US [START_REF] Hache | Speculative trading and oil price dynamic: A study of the wti market[END_REF].14 For ethanol futures prices, we observed a decrease in transaction volumes between 2008 and 2016 as contract terms grew longer (Figure 1), and a virtual absence of liquidity for long-term contracts (compared to short-term maturity). In fact, the inadequate information available at any given moment t on contracts whose maturity period is greater than one year does not give traders the incentives to trade in the market. As a consequence, the liquidity for distant contracts at a maturity greater than five months decreases sharply. Moreover the maturity greater than two months registered a sharp decline in transaction volumes after 2012.

On the one hand, by studying available data from 2008 to 2016, we observed a marked rise in transaction volumes for each maturity. Measured in batches of 29,000 gallons (a standard financial contract for ethanol on the CBOT), these transactions have risen, for two-month term contracts, from around 78,864 in 2008 to 404,133 in 2016, i.e., multiplied by a factor of 5 (Figure 2). On the other hand, the share of noncommercial players increased from around 15% before 2008 to over 35% on average since 2014 (Figure 3). However, both the increase in the volume of transactions on financial trading floors and the growing share of non-commercial players should be kept in perspective. As mentioned previously, during the previous three decades and especially in the initial phase of construction of the commodities markets, the main objective of the different derivatives marketplaces was to attract and concentrate the liquidity required for commercial traders to achieve hedging activities. In October 1974, the NYMEX launched the first energy contracts for industrial fuel oil. [START_REF] Simon | Les marchés à terme de produits pétroliers[END_REF] explains the failure of this first attempt by the under-development of the financial markets and because of the very specific contract specifications (the delivery point of the futures contracts was Rotterdam which held no appeal for the American commercial players). A contract for heating oil in the NYMEX was also launched in 1978 and was abandoned because of inadequate liquidity's volume. During the 1980s in the context of deregulation put in place by the Reagan administration, the NYMEX decided a simultaneous launch of energy contracts: gasoline (1981), crude oil (1983), and heating oil (1990). In Europe the International Petroleum Exchange (IPE) launched its first fuel oil contract in 1981. Since then, financial markets have registered an increase in transactions volume and in the share of noncommercial players in the exchange markets. In the petroleum sector, competition between the two main exchanges (i.e., the NYMEX in New York and the Intercontinental exchange [ICE] in London) led to a strong deregulation process. In the US, for example, the introduction at the end of December 2000 of the law modernizing commodities markets, the Commodity Futures Modernization Act (CFMA), triggered market instability in the crude oil market [START_REF] Medlock | Who is in the oil futures market and how has it changed[END_REF][START_REF] Hache | Speculative trading and oil price dynamic: A study of the wti market[END_REF]. [START_REF] Garbade | Price movements and price discovery in futures and cash markets[END_REF] and [START_REF] Figuerola-Ferretti | Modelling and measuring price discovery in commodity markets[END_REF] to explain the ethanol market, we apply Johansen(1988)'s cointegration tests. The results in Table 2 confirm the presence of a long-term relationship between spot and futures ethanol prices regardless of the cointegration test used. The Likelihood Ratio test does not reject the null hypothesis of unit coefficient at a 10% significant level. Thus, the [START_REF] Garbade | Price movements and price discovery in futures and cash markets[END_REF]'s theory is a valid explanation of the long-term relationship between spot and futures prices in the ethanol market. Finally, the long-term causality tests conclude in favor of a price discovery process from futures to spot prices, at a 10% significant level. These findings are in line with the informational efficiency of the US ethanol market.

We estimate the Ms-VEC model with two states applied to both the mean and the variance equations. These two states refer to low and high volatility regimes. Table 3 presents results with the Nielsen's cointegration specification. 15 In each 15 Table 3 presents results for the best models in terms of explanatory power and hedging strategy.

state, only futures prices adjusts to equilibrium (α f,st ). This result highlights the minor role of futures prices in the discovery process in the short term. Note that the adjustment process is faster during the low volatility regime (st = 2), compared to the high volatility regime (st = 1). Concerning the short-run dynamics (γ ij,st ), these two markets (spot and futures) seem to be disconnected during normal periods and only past changes of futures prices have a significant impact on spot prices for the high volatility state (γ sf,1 ). This last result highlights the fact that futures market can help in understanding the ethanol price dynamics during periods of instability. Furthermore, the relationship is regime-dependent, confirming the ability of our Markov-switching specification to describe it. Figure 4 presents the probability of being in the regime of high volatility. 16 Two main periods of high volatility are observed in 2008-2009 and 2013-2014. Market volatility during these two periods could be explained by the low liquidity during the first one (Figure 1) and by few positions taken by commercial agents for the second period (Figure 2). [START_REF] Schwarz | Estimating the dimension of a model[END_REF]'s Information Criterion. The two P-value columns refer to the P-value of two tests mentioned. P-value inferior to 0.05 leads to the null hypothesis reject of zero cointegrating relationship against one. Cointegration vector column mentions coefficients estimated with βs normalized to unity. The LR test checks the existence of a one-to-one relationship between spot and futures prices. We mention the P-value of the test. The next two lines present the Nielsen (2010) test results with the test statistic and the critical value associated at a 5% significance level. The chosen specification is constant and without trend. The null hypothesis is rejected when the test statistic is superior to the critical value. Note that constant is not estimated with this procedure. The causality test refers to the [START_REF] Toda | Statistical inference in vector autoregressions with possibly integrated processes[END_REF] test whose null hypothesis is the absence of long-term causality.

Turning to the conditional variance equation, as expected, we note a high persistence degree (a 2 ii,st + b 2 ii,st for st = 1, 2) during volatile periods. This feature is common to oil and gasoline markets [START_REF] Fong | A markov switching model of the conditional volatility of crude oil prices[END_REF][START_REF] Alizadeh | A markov regime switching approach for hedging energy commodities[END_REF]. In addition, our specification captures well the leverage effect especially during volatile periods with high and significant coefficients d ii,st . Losses in players' portfolio, i.e., negative shocks, have a greater impact on future volatilities than gains, i.e., positive shocks. Finally, the probability of switching from high to low variance states (P 12 ) is

We interpret results only for the more explanatory model. Results concerning the 22 specifications are available upon request. 16 We represent the smoothed probability which provides the best estimation of the states at each time using full-sample information. See [START_REF] Krolzig | Markov-Switching Vector Autoregressions: Modelling, Statistical Inference, and Application to Business Cycle Analysis[END_REF] for further details on its calculation as well as on other existing probabilities. This figure concerns the Ms-VECM N -Gjr-MGarch. The figure for the VECM J -Ms-MGarch is similar and available upon request to the authors.

greater compared to the probability of switching from low to high variance regimes (P 21 ). This result indicates a shorter duration for high volatility regimes and is confirmed by the average expected state duration calculation proposed by [START_REF] Hamilton | A new approach to the economic analysis of nonstationary time series and the business cycle[END_REF].17 This latter result is interesting as it supports the idea of a certain efficiency of the financial market in the short run through the arbitrage process realized by the different players. These durations are nine and twenty weeks for high and low volatility regimes, respectively. Our different model specifications allow us to compute the dynamic hedge ratios. We also compute the naive (δ = 1) and OLS hedge ratios of [START_REF] Ederington | The hedging performance of the new futures markets[END_REF]. We provide information about a non-hedged strategy for comparison purpose. In addition, we compute cross-hedge ratios with gasoline futures markets estimating from our different specifications. These latter will allow us to compare hedging with the ethanol futures market and cross-hedging with the gasoline futures market. 18The gasoline market could be used by ethanol commercial agents for risk hedging [START_REF] Franken | Cash ethanol cross-hedging opportunities[END_REF]. Table 4 provides variance, utility and value-at-risk for main specifications and each market,19 as well as the variance improvement of the best strategy compared to each other. During the final period of our sample, i.e., Panel A, the optimal specification is a VAR-Gjr-MGarch. The lack of high volatility (Figure 4) during this period explains this result, as well as a possible lack of a cointegration relationship. In addition, all cross-hedging strategies underperform both direct hedging strategies and the situation without a hedging strategy. [START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF] and Nielsen (2010)'s cointegration estimation, respectively. For each parameter, we mention the estimated coefficients and the P-value of the Student test in bracket. The coefficient is significant at the 10%, 5% or 1% if P-value is less than 0.10, 0.05 or 0.01, respectively. LogL, JB, Q(6) and Q 2 (6) are the log-likelihood, the [START_REF] Jarque | Efficient tests for normality, homoscedasticity and serial independence of regression residuals[END_REF] test for normality, the [START_REF] Ljung | On a measure of lack of fit in time series models[END_REF] test for autocorrelation and the ARCH test [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variances of the united kingdom inflation[END_REF] for heterosckedasticity, respectively.

Table 5 displays results of the main hedging strategies for two other panels,20 i.e., the first half of 2010 and 2012. The VECM-Ms-MGarch with Johansen's cointegration provides the best strategy for both periods. This result confirms the suitability of Markov-switching and Johansen's cointegration specifications for the hedging strategy on the ethanol market. Note that coefficients of this model are consistent with the previous specification presented (Table 3). Hedgers can decrease $1,268 and $2,453 of their average weekly value-at-risk with an initial portfolio value of $1,000,000 compared to the simple OLS specification. These weekly decreases correspond to $9,144 and $17,689 annualized decreases, that is to say, only 0.09% and 1.77% of the initial portfolio value. Finally, cross-hedging strategies outperform the non-hedged situation for each period with the OLS and Ms-VECM N -Gjr-MGarch for panels B and C, respectively. This last result highlights the ability of the Nielsen procedure to provide a good hedging strategy.

Table 6 present results for the out-of-sample simulation concerning the period from December 28, 2016, to June 21, 2017, i.e., 25 observations. Concerning non-linear specifications, we estimate the model at each point of time to forecast states' probabilities as well as state-dependent conditional mean and variance-covariance matrix. We then compute the prediction of the hedge ratio after recomposition of the global variance-covariance matrix coming from equations ( 13) and ( 14). The optimal hedging strategy is the linear multivariate Garch specification with a variance improvement of 80.5% and 22.9% compared to a no-hedged situation and the OLS-based hedge ratio, respectively. However, this strategy does not significantly outperform most of the specifications studied especially the naive strategy consisting of a unit hedge ratio. In addition, cross-hedging with the gasoline market is not efficient compared to the direct hedging strategy. This result is also valid compared to the no-hedged situation for most of the strategies. Furthermore, nonlinear specifications do not seem efficient for hedging in the ethanol market. This result could be explained by the difficulty in well-forecasting the states' probability or by the absence of high volatility periods. Finally, the Johansen (1988) cointegration procedure outperforms the nonparametric approach of Nielsen (2010) for 10 strategies against eight with ethanol markets but out-performs for eight against 10 strategies with cross-hedging in gasoline futures market. 21 ) and Utility (Util.) are presented in 10 -4 and 10 -3 , respectively. Variance improvement (V. Impr.) measures the incremental variance reduction of the best strategy versus the other strategies with the formula: [V ar(Strategy i ) -V ar(Best)]/V ar(Strategy i ). VaR is in US dollars for an initial investment of $1 million and k = 4. J and N refer to [START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF] and Nielsen (2010)'s cointegration estimation, respectively. 21 These results come from the comparison between [START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF] and Nielsen (2010)'s approach for each specification including those not presented in Table 4, 5 and 6.

Conclusion

In this paper, we analyze the ethanol prices dynamics in the US from 2008 to 2016. For this purpose, we use a Markov-switching vector error correction model with an asymmetric Garch error structure. This specification allows us to study the short-term, long-term and variance dynamics across different volatility regimes. From the cointegration test we could not reject the hypothesis of a long-term equilibrium relationship between spot and futures prices. Two distinct states (low and high volatility) should be distinguished for the short-term dynamics. We provide several dynamic hedge ratios and we examine their performance through in-sample and out-sample simulations.

The ethanol market is characterized by its efficiency and a price-discovery process from futures to spot prices in the long term. The cost-of-carry model from [START_REF] Garbade | Price movements and price discovery in futures and cash markets[END_REF] is able to well-explain the long-term relationship. In addition, the ethanol futures market can well-explain the spot prices dynamics during the periods of high volatility. Furthermore, hedging strategies based on ethanol futures contracts always outperform the cross-hedging strategy based on the use of gasoline futures contracts. Markov-switching specification and Johansen (1988)'s cointegration procedure are able to provide an efficient hedging strategy for two-third of the periods analyzed. Then , a simple multivariate Garch model is the best hedging strategy during the first half of 2017 according to the out-of-sample simulation. Finally, while Nielsen (2010)'s nonparametric tool provides a clear explanation power for the price dynamics, it cannot be used as a hedging strategy in the ethanol market.

In order to get a full understanding of the different hedging strategies in the financial ethanol market, this paper could be extended in various ways. The methodology used with RBOB gasoline market could be applied to other commodities futures markets such as crude oil, corn or sugar. More globally compared to mature futures market (crude oil, sugar, etc.), the ethanol market was launched in 2005 and the traders' behaviour could have been influenced by many factors such as a lack of information regarding the physical production, commercial strategy (anti-dumping), fiscal policy in producing countries (taxes in Brazil, in the US, etc. and more global uncertainties regarding energy and environmental policies all around the world, etc.). It could explain the fact that the ethanol futures market has not appealed to the traders since the beginning of the last decade but it could become a key market in the near future with the environmental and regulatory constraints of a 2 • C scenario.

Figure 1 :

 1 Figure 1: Open interest by contracts maturity

Figure 2 :

 2 Figure 2: Position (number of contracts) by actors

Figure 4 :

 4 Figure 4: Smoothed probabilities of being in a high volatility state

Table 1 :

 1 Summary statistics and unit root tests

	Variables	Spot	Log Futures	First-log differences Spot Futures
	Mean	/	/	0.000	0.000
	Std. errors	/	/	0.050	0.040
	Skewness	/	/	0.047	-0.283
	Kurtosis	/	/	6.039	4.288
	ADF	0.047*	0.297	0.001*	0.001*
	PP	0.099*	0.306	0.001*	0.001*
	KPSS	0.010	0.010	0.100*	0.100*
	Perron	-1.148 -3.8	-1.229 -3.8	/ /	/ /
	Q(6)	0.001	0.001	0.001	0.681
	Q 2 (6)	0.001	0.001	0.001	0.001

Note: This table reports descriptive statistics and the p-value of the unit root tests applied, i.e., Augmented

[START_REF] Dickey | Likelihood ratio statistics for autoregressive time series with a unit root[END_REF]

's test (ADF),

[START_REF] Phillips | Testing for a unit root in time series regression[END_REF]

's test (PP) and

[START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root[END_REF]

's test (KPSS). The Perron's line refers to the

[START_REF] Perron | Testing for a unit root in a time series with a changing mean[END_REF]

test with the test's statistic and the critical value at a 5% significance level in the first and second line, respectively. The critical value comes from

[START_REF] Perron | Testing for a unit root in a time series with a changing mean: Corrections and extensions[END_REF]

. The null hypothesis of unit root with break is rejected if the test statistics is greater than the critical value. The star mentions the stationarity of the variable at a 10% significance level. Q(6) and Q 2 (6) are the p-value of the

[START_REF] Ljung | On a measure of lack of fit in time series models[END_REF]

's test and ARCH test

[START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variances of the united kingdom inflation[END_REF]

for 6 th order autocorrelation, respectively.

Table 2 :

 2 Cointegration and causality tests βss t + f t + β 0 = u t The two first lines present the Johansen (1988)'s test results. The lags column mentions the number of lags in the VEC Model. Lag length choice is based on

	Lags	H 0	P-value λmax test λ trace test	Cointegration vector (βs 1 β 0 )	LR test H 0 : βs=-1 H 0 : β 0 =0
	1	r=0	0.001	0.001	(-1.044 1 0.109)	0.078	0.001
	-	H 0	Test stat	Critical Value	Cointegration vector	-	-
	-	r=0	3.78	3.57	(-1.010 1 -)	-	-
				Causality test		P-value
			Spot to futures prices		0.867
			Futures to spot prices		0.087
	Note:

Table 3 :

 3 Estimation results

		Ms-VECM N -Gjr-MGarch		VECM J -Ms-MGarch
	β s		-1.011	(-)			-1.044	(-)
	β f		1	(-)			1	(-)
	β 0		-	(-)			0.109	(-)
		st = 1	st = 2
	cs,st	-0.005	(0.417)	0.002	(0.429)		0.001	(0.791)
	c f,st	-0.011	(0.049)	0.006	(0.017)		0.003	(0.203)
	α s,st	-0.001	(0.960)	-0.001	(0.977)		-0.001	(0.923)
	α f,st	-0.083	(0.048)	-0.119	(0.001)		-0.076	(0.001)
	γss,st	0.037	(0.801)	-0.145	(0.390)		-0.114	(0.140)
	γ sf,st	0.317	(0.029)	0.230	(0.214)		0.260	(0.002)
	γ f s,st	0.065	(0.552)	0.072	(0.651)		0.051	(0.445)
	γ f f,st	0.025	(0.822)	-0.071	(0.686)		-0.027	(0.733)
						st = 1	st = 2
	c 11,st	0.034	(0.001)	0.011	(0.016)	0.030	(0.001)	0.013	(0.001)
	c 21,st	0.037	(0.001)	0.029	(0.001)	0.034	(0.001)	0.029	(0.001)
	c 22,st	0.043	(0.001)	0.029	(0.001)	0.049	(0.001)	0.030	(0.001)
	a 11,st	0.626	(0.001)	0.292	(0.037)	0.815	(0.001)	0.376	(0.001)
	a 22,st	0.309	(0.060)	0.189	(0.144)	0.536	(0.001)	0.256	(0.027)
	b 11,st	0.001	(0.971)	0.214	(0.266)	0.001	(0.887)	0.149	(0.458)
	b 22,st	0.427	(0.021)	0.001	(0.945)	0.391	(0.006)	0.001	(0.869)
	d 11,st	0.657	(0.025)	0.320	(0.053)	-	-	-	-
	d 22,st	0.571	(0.014)	0.442	(0.005)	-	-	-	-
	P 11		0.890	(0.001)			0.880	(0.001)
	P 12		0.110	(0.009)			0.120	(0.001)
	P 21		0.050	(0.001)			0.056	(0.001)
	P 22		0.950	(0.001)			0.944	(0.001)
	LogL		1.951×10 3			1.936×10 3
			Spot	Futures			Spot	Futures
	JB		0.001	0.001			0.001	0.001
	Q(6)		0.160	0.665			0.004	0.834
	Q 2 (6)		0.001	0.001			0.001	0.001
	Note: J and N refer to

Table 4 :

 4 In-sample hedging simulation

			Ethanol spot and futures		Ethanol spot and gasoline futures
		Var.	V. Impr.	Util.	VaR	Var.	V. Impr.	Util.	VaR
	Panel A
	No Hedged	12.82	56.5%	-5.127	59,073	12.82	56.5%	-5.127	59,073
	Naive	5.650	1.44%	-2.260	39,219	36.67	84.8%	-14.67	99,912
	OLS	6.053	8.02%	-2.423	40,596	18.16	69.3%	-7.266	70,322
	MGarch	5.640	1.28%	-2.256	39,187	18.96	70.6%	-7.583	71,841
	VAR-Gjr-MGarch	5.568	-	-2.227	38,933	18.18	69.3%	-7.271	70,348
	VECM J -MGarch	5.592	0.43%	-2.237	39,019	18.92	70.5%	-7.569	71,774
	VECM N -Gjr-MGarch	5.611	0.76%	-2.244	39,084	18.24	69.4%	-7.295	70,465
	Ms-MGarch	6.054	8.03%	-2.422	40,599	18.13	69.2%	-7.250	70,247
	VAR-Ms-Gjr-MGarch	6.467	13.9%	-2.587	41,960	18.36	69.6%	-7.346	70,708
	VECM J -Ms-Gjr-MGarch	6.596	15.5%	-2.638	42,376	18.17	69.3%	-7.267	70,327
	VECM N -Ms-MGarch	6.620	15.8%	-2.648	42,453	18.00	69.0%	-7.201	70,007
	Ms-VAR-MGarch	5.626	1.02%	-2.250	39,135	18.06	69.2%	-7.222	70,112
	Ms-VECM J -Gjr-MGarch	5.598	0.53%	-2.239	39,038	18.37	69.7%	-7.347	70,712
	Ms-VECM N -MGarch	5.855	4.91%	-2.342	39,927	18.36	69.7%	-7.345	70,705
	Note: Panel A refers to 5/25/16-12/21/16. Variance (Var.

Table 5 :

 5 In-sample hedging simulation with panel B and C

			Ethanol spot and futures		Ethanol spot and gasoline futures
		Var.	V. Impr.	Util.	VaR	Var.	V. Impr.	Util.	VaR
	Panel B
	No Hedged	9.774	48.3%	-3.910	51,584	9.774	48.3%	-3.910	51,584
	Naive	5.797	12.9%	-2.319	39,728	17.87	71.7%	-7.148	69,751
	OLS	5.395	6.50%	-2.158	38,325	7.752	34.9%	-3.101	45,939
	Gjr-MGarch	5.606	10.0%	-2.242	39,065	7.917	36.2%	-3.167	46,426
	VAR-MGarch	5.612	10.1%	-2.245	39,088	8.072	37.5%	-3.229	46,879
	VECM J -MGarch	5.617	10.2%	-2.247	39,105	8.033	37.2%	-3.213	46,764
	VECM N -Gjr-MGarch	5.645	10.6%	-2.258	39,201	8.069	37.4%	-3.228	46,869
	Ms-MGarch	5.104	1.17%	-2.042	37,278	7.916	36.2%	-3.167	46,425
	VAR-Ms-MGarch	5.129	1.65%	-2.052	37,369	7.848	35.7%	-3.139	46,224
	VECM J -Ms-MGarch	5.044	-	-2.018	37,057	7.883	36.0%	-3.153	46,327
	VECM N -Ms-MGarch	5.160	2.24%	-2.064	37,480	7.818	35.4%	-3.127	46,134
	Ms-VAR-Gjr-MGarch	5.153	2.11%	-2.061	37,454	7.973	36.7%	-3.189	46,590
	Ms-VECM J -MGarch	5.251	3.94%	-2.100	37,808	7.822	35.5%	-3.129	46,148
	Ms-VECM N -Gjr-MGarch	5.105	1.19%	-2.042	37,280	8.232	38.7%	-3.293	47,339
	Panel C

In particular, Johansen's procedure could lead to an estimation bias due to the restrictions imposed on the short-run dynamics which are supposed to be linear.

Cross-hedging occurs when the asset underlying the contract is different than the asset whose price is being hedged[START_REF] Hull | Options, Futures, and Other Derivatives[END_REF].

See Routledge et al. (2000) or[START_REF] Heaney | Approximation for convenience yield in commodity futures pricing[END_REF] for more details on the convenience yield.

In view of the conflicting results for the spot log-price series, we apply the Perron (1990)'s unit root test which confirms its non-stationarity with a break in mean on March 12 2014. We choose this test in view of series characteristics, i.e., the absence of trend and a potential break in the mean. We present results with innovational-outlier model for break date determination. Results with additional-outlier model are similar.

For more details on the testing procedure, see[START_REF] Nielsen | Nonparametric cointegration analysis of fractional systems with unknown integration orders[END_REF].

As mentioned by[START_REF] Nielsen | Nonparametric cointegration analysis of fractional systems with unknown integration orders[END_REF], the choice of d1 = 0.1 maximizes the power of the test.

We estimate a wide range of specifications but only detail the more complex model.

We integrate only one lag in the short-run dynamics according to the information criterion BIC from[START_REF] Schwarz | Estimating the dimension of a model[END_REF] during the Johansen cointegration procedure.

See also the literature review in[START_REF] Lautier | Term structure models of commodity prices: a review[END_REF].

The average expected duration of state 1 (resp. 2) can be calculated by ( P 12 ) -1 (resp. (P 21 ) -1 ).

New York Harbor Reformulated RBOB Regular Gasoline Contract. More details on http://www.cmegroup.com/trading/energy/refined-products/rbob-gasoline.html

A table with all the specifications is available upon request to the authors.

A table with all the specifications is available upon request to the authors.