International Review of Financial Analysis 18 (2009) 58–65

Contents lists available at ScienceDirect

International Review of Financial Analysis

Short-run deviations and time-varying hedge ratios: Evidencefrom agricultural futures markets☆

Taufiq Choudhry ⁎School of Management, University of Southampton, Highfield, Southampton SO17 1BJ, UK

☆ The author thanks two anonymous referees andcomments and suggestions. The author also thanks theFinancial Management Association conference 2006comments and suggestions on an earlier draft of the paomissions are the author's responsibility alone.⁎ Tel.: +44 2380599286; fax: +44 2380593844.

E-mail address: T.Choudhry@soton.ac.uk.1 Brorsen and Fofana (2001) provide and discuss seve

to the success or failure of agricultural commodities fut

1057-5219/$ – see front matter © 2008 Elsevier Inc. Aldoi:10.1016/j.irfa.2008.11.003

a b s t r a c t

a r t i c l e i n f o

Article history:

This paper investigates th Received 28 March 2007Received in revised form 17 July 2008Accepted 21 November 2008Available online 3 December 2008

JEL classification:G1G13G15

Keywords:Hedge ratioGARCHBEKK GARCHGARCH-XBEKK GARCH-X and variance

e hedging effectiveness of time-varying hedge ratios in the agriculturalcommodities futures markets using four different versions of the GARCH models. The GARCH modelsapplied are the standard bivariate GARCH, the bivariate BEKK GARCH, the bivariate GARCH-X and thebivariate BEKK GARCH-X. Futures data for corn, coffee, wheat, sugar, soybeans, live cattle and hogs areapplied. Comparison of the hedging effectiveness is done for the within sample period (1980–2004), and twoout-of-sample periods (2002–2004 and 2003–2004). Results indicate superior performance of the portfoliosbased on the GARCH-X model estimated hedge ratio during all periods.

© 2008 Elsevier Inc. All rights reserved.

1. Introduction

Transfer of risk is one of the main functions of the futures markets.Risks are transferred to those willing to bear them, as hedgers reducetheir risk by paying a premium to speculators. For agriculturalcommodities, risk may occur due to drought, near record production,an increase in demand, a decrease in international production, etc.Hedging by the agricultural producers generally involves selling thecommodity futures because producers of the commodity want to lockin a price floor. Simultaneously speculators and investors looking tolock in a price ceiling are buying the contract. The commodity futuresmarkets thus provide ameans to transfer risk betweenpersons holdingthe physical commodity (hedgers) and investors speculating in themarket.1 This paper empirically investigates the hedging effectivenessin the agricultural commodity futures' market. Our paper is motivatedby Yang and Awokuse (2003) who indicated that knowledge of how

the editor for several usefulparticipants of the EuropeanMadrid, Spain for valuable

per. Any remaining errors and

ral characteristics as importantures contracts.

l rights reserved.

effective hedging function performs on the commodity futuresmarketis essential to understanding these markets.

This paper tries to expand the knowledge in this area by in-vestigating and comparing the risk-reducing ability of differentoptimal time-varying hedge ratios for the futures of seven agriculturalcommodities: corn, coffee, wheat, sugar, soybeans, live cattle and hogs.An optimal hedge ratio is defined as the proportion of a cash positionthat should be covered with an opposite position on a futures market.Corn, coffee, wheat, sugar, and soybeans are storable commodities andlive cattle and hogs are non-storable commodities.2 According toCovey and Bessler (1995) commodity futures markets with differentstorability characteristics may perform in different manners. Yang andAwokuse (2003) provide some proof of it by showing that hedgingeffectiveness is stronger for storable agricultural commodities thannon-storable commodities.

The traditional constant hedge ratio obtained by means of theordinary least square (OLS) has been discarded as being inappropriate,because it ignores the heteroskedasticity often encountered in priceseries. Baillie and Myers (1991) further claim that if the jointdistribution of cash price and futures prices is changing over time,estimating a constant hedge ratio may not be appropriate. In thispaper time-varying hedge ratios are estimated and employed. They

2 According to Covey and Bessler (1995) no asset is perfectly storable or non-storable.An asset with minimum storage costs is an asset which does not easily spoiland can be stored cheaply relative to its value.

59T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

are estimated using four different types of the generalized autore-gressive conditional heteroskedasticity (GARCH) models: the stan-dard bivariate GARCH, bivariate BEKK GARCH, the bivariate GARCH-X,and the bivariate BEKK GARCH-X.3 Haigh and Holt (2002), Bera, Gar-cia, and Roh (1997), Sephton (1993), Baillie and Myers (1991) andMyers (1991) using agricultural commodities futures show thatGARCH hedge ratios are superior to the ones based on the traditionalregressions.

The GARCH-X and the BEKKGARCH-Xmodels applied in this paperare different from the other two GARCHmodels because they take intoconsideration the effects of the short-run deviations from the long-run relationship between the cash and futures prices on the con-ditional variance and covariance (second conditional moments of thebivariate distribution) of log difference of the cash and the futuresprices. The BEKK GARCH and the BEKK GARCH-X models are alsounique because they allow time variation in the conditional correla-tions as well as the conditional variance. To our knowledge, no otherpaper applies the GARCH-X and/or the BEKK GARCH-X in the es-timation and comparison of time-varying hedge ratios for agriculturalfutures market. All GARCH methods applied take into considerationthe effects of the short-run deviations on the first moment (mean) ofthe bivariate distributions of the variables.

If the four time-varying hedge ratios are different, then more thanone interesting question arises: first, which method is more effective?And second, does taking into consideration the effects of the short-rundeviations make the hedge ratio more effective? A further inquiryand contribution of this paper is comparing the strength of thehedging effectiveness for storable commodities against the non-storable commodities, especially talking into consideration the short-run deviations.

The short-run deviations are represented by the error correctionterm from a cointegration relationship between the commodities cashand the futures prices.4 Long-run relationship between the commod-ities cash price and the futures price is determined by means ofthe Engle and Granger (1987) cointegration test. Yang, Bessler, andLeatham (2001) claim that prevalent cointegration between cashand futures prices on commodity markets suggest that cointegrationshould be incorporated into commodity hedging decisions.5 Evenwhen the GARCH effect is considered, allowance for the existenceof cointegration is argued to be an indispensable component whencomparing ex-post performance of various hedging strategies. Tocheck for the effects of cointegration on hedging effectiveness inagricultural futures markets via the GARCH-X and BEKK-X is one of themain objectives of the paper.

The risk-reducing effectiveness of the time-varying hedge ratios isinvestigated by checking performance of the ratios in the withinsample period (1980–2004) and two out-of-sample periods (2002–2004 and 2003–2004). The hedging effectiveness is estimated andcompared by checking the variance of the portfolios created usingthese hedge ratios. The lower the variance of the portfolio, the higheris the hedging effectiveness of the hedge ratio.

The structure of the paper is as follows: Section 2 describes anddiscusses the optimal hedge ratio and the four GARCH models: thedata and its basic statistics are described in Section 3: the empiricalresults are presented in Section 4: and Section 5 is the conclusion.

3 See Choudhry, 2004; Moschini and Myers, 2002; Baillie and Myers, 1991; Kronerand Sultan, 1993 for application of the ARCH and GARCH models in estimation of time-varying hedge ratios.

4 Baillie and Myers (1991), Covey and Bessler (1992), Fortenbery and Zapata (1993,1997) provide studies of cointegration between commodities spot and future prices.

5 Ghosh (1995), Ghosh and Clayton (1996) and Kroner and Sultan (1993) haveshown that hedge ratios and hedging performance may change considerably ifcointegration between the cash and futures prices is omitted from the statisticalmodels and estimations.

2. Optimal hedge ratios and the GARCH models

2.1. The hedge ratio

The returns on the portfolio of an investor trying to hedge someproportion of the cash position in a futures market can be representedby:

rt = rct − βt−1rft ð1Þ

where rt is the return on holding the portfolio between t−1 and t; rtc

is the return on holding the cash position for the same period; rtf is thereturn on holding the futures position for the same period; and βt−1

is the hedge ratio. The variance of the return on the hedged portfolio isgiven by

Vt = Vct − β2

t − 1Vft − 2βt−1V

cft ð2Þ

where Vt, V tc, V t

f, represent the conditional variances of the portfolio,cash and futures positions respectively, while Vt

cf represents condi-tional covariance between the cash and futures position.

The value of βt−1, which minimises the conditional variance of thehedged portfolio return (Eq. (2))), is the optimal hedge ratio (Baillieand Myers, 1991).6 It is given by:

βt−1 = Vcft = V f

t ð3Þ

Commonly, the value of the hedge ratio is less than unity, so thatthe hedge ratio that minimises risk in the absence of basis risk turnsout to be dominated by βwhen basis risk is taken into consideration.7

Time-varying optimal hedge ratio can also be based on utilitymaximization. Based on Myers (1991), under this scenario an in-dividual investor wants to determine the optimal allocation of initialwealth between two investment opportunities: purchase of a riskyasset, and purchase of a risk-free asset. There is a futures market in therisky asset and the investor can therefore hedge by selling contractswhich mature at or after the period. Using the von Neumann–Morgenstern utility function and a time-varying conditional covar-iance, Myers (1991) is able to show that optimal hedge ratio is equal tothe one presented by Eq. (3). In this model, it is assumed that optimalhedge ratio is preference-free but the demand for the asset dependsupon investor risk preferences, as well as on the probability dis-tribution of asset price.

Brorsen (1995) presents a different theory of hedging involvingagricultural commodities which assumes that producers are riskneutral, forward pricing is costly, and that borrowing costs are non-linear. With nonlinear borrowing costs, highly leveraged farm firmshedge more than low leveraged firms and optima hedge ratios in-crease as output price variability increases. Thus, according to Brorsen(1995) the presence of risk-averse preferences is not required forfuture markets to exist. Turvey and Baker (1989) provide an expectedutility model of optimal hedging that explicitly takes into considera-tion that capital structure of the farm firm. They show that hedgingincreases as farms debt relative to assets increases. This is because thefarmer's use of the futures, which decreases business risk, offsets tosome extent the increased financial risk due to leverage.

6 As indicated by Cecchetti et al. (1988), the return on a hedged position willnormally be exposed to risk caused by unanticipated changes in the relative pricebetween the position being hedged and the futures contract.This ‘basis risk’ ensuresthat no hedge ratio completely eliminates risk.

7 According to Cecchetti et al. (1988), the optimal hedge ratio β can be expressed asρσ c/σ f, where ρ is the correlation between futures price and cash price, σ c is thecash standard deviation, and σ f is the futures standard deviation. Thus, if the futureshave the same or higher price volatility than the cash, the hedge ratio can be no greaterthan the correlation between them, which will be less than unity.

60 T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

2.2. Bivariate GARCH

The following bivariate GARCH(p,q) model may be used torepresent the log difference of the cash (spot) and futures prices:

yt = μ + δ zt−1ð Þ + et ð4Þ

et =Xt−1fN 0;Htð Þ ð5Þ

vech Htð Þ = C +XP

j=1

Ajvech et− j

� �2+

Xq

j=1

Bjvech Ht− j

� �ð6Þ

where yt=(rtc, rtf) is a (2×1) vector containing the log difference ofthe cash (rtc) price and futures (rtf) prices; Ht is a (2×2) conditionalcovariance matrix; C is a (3×1) parameter vector (constant); Aj and Bjare (3×3) parameter matrices; and vech is the column stackingoperator that stacks the lower triangular portion of a symmetricmatrix. The error correction term (zt) from the cointegrationrepresents the short-run deviations from a long-run relationshipbetween the cash price and the futures price.8 A significant andpositive coefficient (δ) on the error term implies that an increase inshort-run deviations raises the log difference of cash and/or futureprices. The opposite is true if the error term coefficient is negative andsignificant.

Given the bivariate GARCH model of the log difference of the cashand the futures prices presented above, the time-varying hedge ratiocan be expressed as:

βt = H12;t = H22;t ð7Þ

Where H12,t is the estimated conditional covariance between thelog difference of the cash and futures prices, and H22,t is the estimatedconditional variance of the log difference of the futures prices from thebivariate GARCH model. Given that conditional covariance is time-dependent, the optimal hedge ratio will be time-dependent.

2.3. Bivariate BEKK GARCH

In the BEKK model of Engle and Kroner (1995) the conditionalcovariance matrix is parameterized as

vech Htð Þ = CVC +XK

K=1

Xq

i=1

AVKiet− ieVt− iAki +XK

K=1

Xp

i=1

BVKjHt− jBkj

ð8ÞEqs. (4) and (5) also apply to the BEKK model and are defined asbefore. In Eq. (8) Aki, i=1,…, q, k=1,… K, and Bkj j=1, … p, k=1,…,K are all N×N matrices.9 The time-varying hedge ratio based on theBEKK GARCH model is also expressed as Eq. (7).

2.4. Bivariate GARCH-X and BEKK-X

The GARCH-Xmodel provides an extension of the standard GARCHmodel linked to an error-correction model of cointegrated series on

8 The following cointegration relationship is investigated by means of the Engle andGranger (1987) method:

St = η + γFt + zt

St and Ft are log of cash index and futures index respectively. The residuals zt are testedfor unit root(s) to check for cointegration between St and Ft. The error correction term,which represents the short-run deviations from the long-run cointegrated relation-ship, has important predictive powers for the conditional mean of the cointegratedseries (Engle & Yoo, 1987). Cointegration is found between the log of cash and futuresprices for all seven commodities. These results are available on request.

9 This formulation has the advantage over the general specification of themultivariate GARCH that conditional variance (Ht) is guaranteed to be positive for allt (Bollerslev et al., 1994).

the secondmoment of the bivariate distributions of the variables (Lee,1994). Given that short-run deviations from the long-run relationshipbetween the cash and futures prices may affect the conditionalvariance and conditional covariance, then they will also influence thetime-varying optimal hedge ratio, as defined in Eq. (7). In the GARCH-X model conditional heteroskedasticity may be modelled with afunction of the lagged error correction term (Lee, 1994).

The following bivariate GARCH(p,q)-X model may be used torepresent the log difference of the cash prices and the futures prices:

vech Htð Þ = C +Xp

j=1

Ajvech et− j

� �2+

Xq

j=1

Bjvech Ht− j

� �+

Xk

j=1

Djvech Zt−1ð Þ2

ð9Þ

Once again, Eqs. (4) and (5) (defined as before) also apply to theGARCH-X model. The squared error term (zt−1) in the conditionalvariance and covariance equation (Eq. (9)) measures the influences ofthe short-run deviations on conditional variance and covariance. Asignificant positive effect may imply that the further the series deviatefrom each other in the short run, the harder they are to predict (Lee,1994).

As advocated by Lee (1994), the square of the error-correctionterm (z) lagged once should be applied in the GARCH(1,1)-X model.The parameters D11 and D33 indicate the effects of the short-rundeviations between the cash and the futures prices from a long-runcointegrated relationship on the conditional variance of the residualsof the log difference of the cash and futures prices, respectively. Theparameter D22 shows the effect of the short-run deviations on theconditional covariance between the two variables. If D33 and D22 aresignificant, then H12 (conditional covariance) and H22 (conditionalvariance of futures returns) are going to differ from the standardGARCH model H12 and H22 implying a different time-varying hedgeratio. A similar extension can be made to the standard BEKK GARCHlinked to an error-correction model of cointegrated series on thesecond moment of the bivariate distributions of the variables. Such amodel is known as the BEKK-X.

3. Data and basic statistics

Daily cash (spot) and the futures prices of corn, coffee, wheat,sugar, soybeans, live cattle and hogs are used in the empirical tests.Live cattle and hogs are non-storable commodities while rest arestorable. The futures price for a storable asset is considered equal tothe asset's cash price plus the asset's cost-of-carry (Covey & Bessler,1995). A futures price of a non-storable asset is considered the futuresmarket's forecast of the asset's cash price which will be obtainedduring the delivery period of a particular futures contract at one of itsspecified delivery location (Covey & Bessler, 1995). All the data rangefrom August 1980 to July 2004. All futures price indices are continuousseries.10

The coffee and sugar #11 future prices are from the Coffee, Sugarand Cocoa Exchange (CSCE), the corn, soybeans and wheat futuresprices are from the Chicago Board of Trade (CBOT) and the live cattleand hogs future prices are from Chicago Mercantile Exchange (CME).Thus all futures data are from the US markets. The futures com-modities were picked because of their popularity among the investors.During the last several years, the three highest volume of trade inCBOTare of corn, soybeans andwheat respectively. Similarly live cattle

10 The continuous series is a perpetual series of futures prices. It starts at the nearestcontract month, which forms the first values for the continuous series, either until thecontract reaches its expiry date or until the first business day of the actual contractmonth. At this point, the next trading contract month is taken. As indicated by one ofthe referees, splice bias is introduced when the nearby futures contract are used forestimation changes.

Table 1Basic statistics of the total period (1980–2004).

Variables Mean Variance Kurtosis Skewness Jarque–Bera

Log difference of cash priceCorn −0.00003 0.00023 6.4595a −0.1707a 10458.58a

Wheat −0.00002 0.00030 19.4547a −0.3470a 94725.77a

Coffee −0.00024 0.00062 24.104a 1.0591a 146352.07a

Sugar −0.00010 0.00067 16.1421a −0.1610a 65156.76a

Soybean 0.00000 0.00023 9.6940a −0.3537a 23614.45a

Cattle 0.00007 0.00007 47.0164a −0.1156a 552559.0a

Hog 0.00005 0.00055 89.5296a −0.1717a 2003582.1a

Log difference of futures priceCorn −0.00005 0.00021 35.1361a −1.9629a 312436.96a

Wheat −0.00005 0.00026 35.1583a −2.0543a 313194.2a

Coffee −0.00017 0.00067 8.4794a 0.0589 17975.68a

Sugar −0.00019 0.00080 10.1253a 0.2661a 25696.93a

Soybean −0.00002 0.00019 7.9557a −0.6270a 16213.67a

Cattle 0.00045 0.00012 11.4588a −1.3622a 34676.1a

Hog 0.00048 0.00046 33.3667a −0.0228 278288.9a

Square of log difference of cash priceCorn 0.00023a 0.00000 135.5173a 9.6557a 4683683.3a

Wheat 0.00030a 0.000002 1085.92a 29.0089a 295596594a

Coffee 0.0006a 0.00001 779.899a 22.957a 152562103.7a

Sugar 0.00067a 0.000008 1226.732a 30.2520a 377070362a

Soybean 0.00022a 0.000001 653.500a 20.6059a 107171564a

Cattle 0.00071a 0.00000 1679.40a 37.3566a 721570409a

Hog 0.00055a 0.000027 2457.55a 47.7152a 1511916320a

Square of log difference of futures priceCorn 0.00021a 0.000002 2219.1217a 42.2562a 1232705434.3a

Wheat 0.00026a 0.000002 2681.57a 46.2491a 1799542364a

Coffee 0.0006a 0.000004 301.141a 14.057a 22865312.5a

Sugar 0.0008a 0.000008 545.721a 18.2399a 74773152a

Soybean 0.00019a 0.00000 1091.792a 25.241a 298589677a

Cattle 0.00045a 0.00000 220.690a 13.029a 12343724a

Hog 0.00046a 0.000007 405.957a 17.6081a 41502730a

Log difference of cash price× log difference of futures priceCorn 0.00001a 0.00000 276.6449a 8.9428a 19209876.08a

Wheat 0.00002a 0.00000 48.2379a 0.4954a 581871.87a

Coffee 0.00001 0.000001 232.473a −2.1527a 13513278.8a

Sugar 0.00007a 0.000001 239.94a 8.024a 14454723.5a

Soybean 0.00014 0.000004 134.935a 4.6708a 4572949.12a

Cattle 0.000007a 0.00000 116.413a 4.4295a 3407022a

Hog 0.000013a 0.00000 227.671a 5.9541a 12991861.4a

a Implies significantly different from zero at 1% level.

12 Basic statistics from the two out-of-sample periods provide similar results and areavailable on request.13 Bollerslev (1988) provides a method of selecting the length of p and q in a GARCH

61T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

and hogs have the two highest volume of trade in CME. Coffee andsugar #11 futures also have the highest volume of trade in CSCE.

The wheat cash prices are from the CBOT; the corn cash price arethe price of Corn No. 2 yellow in Central Illinois; the cash price ofSantos coffee from New York Board of Trade (NYBOT)11 represent thecash price for coffee; soybeans cash price is the price of soybeans inSoutheast Iowa; sugar cash prices are reported by the CSCE; the livecattle cash price is taken from the Commodity research bureau andICX index and the hog cash price are taken from IHX hog index. Alldata are obtained from Global Financial Data.

Tomek (1997, 2000) raises concern regarding the quality of thecash price in any agricultural commodity. According to Tomek thechanging nature of cash markets makes it difficult to observe aconsistent cash price series and also, the local cash markets may berelatively less efficient than futures markets. Further according toWang and Tomek (2007), agricultural commodity cash prices will beauto-correlated convergent series. This is because of the biologicalnature of commodity production and storage and the cost of arbitrageover time. Similarly cash prices for livestock are auto-correlatedbecause of the dynamics inherent in managing herds and flocks. Mostcommodity prices have seasonality, and commodity prices in cash

11 In 1998, the New York Board of Trade became the parent company of the New YorkCotton Exchange (NYCE) and the Coffee, Sugar and Cocoa Exchange (CSCE).

markets may not adjust instantly to new information (Tomek, 1997,2000). This paper employs the best available data.

Table 1 presents the basic statistics of four series: log difference ofthe cash prices and the futures prices, square of the first two series andthe cross product of the first two series for the within sample period(1980–2004).12 Results show almost all series are significantly skewedand, as expected, all series are found to have significant and positivekurtosis, implying higher peaks and fatter tails. Thus, the Jarque–Berastatistic shows all series to be non-normal. The mean and variance ofall four series seem to stay similar across the three periods. This mayimply a lack of structural breaks in the different series.

4. Empirical results

4.1. Bivariate GARCH, BEKK, GARCH-X and BEKK-X results

The results from the standard bivariate GARCH(1,1), BEKK(1,1),GARCH-X(1,1) and BEKK-X(1,1)models for thewithin sample period arequite standard.13 In order to preserve space only the results for corn areprovided inTable2but the rest are available on request. Inmost tests, theARCHcoefficients are all positive and significant, thus implying volatilityclustering both in the log difference of cash price (A11) and futures price(A33). The ARCH coefficients are also less than unity in all significantcases. In all four models for all commodities the GARCH coefficients (B11and B33) are significant and positive, implying GARCH effect. A largecoefficient of the GARCH term indicates that shocks to conditionalvariance take a long time to die out and volatility persists. The sign andsignificance of the covariance parameters indicate positive andsignificant interaction between the two prices in most cases.

The short-run deviations from a long-run relationship between thecash price and future prices have significant effect on both themean ofcash returns (δ1) and the log difference of futures prices (δ2) in mostof the cases. For the majority of the commodities, the effect on themean of the cash returns is negative and significant. In the case of logdifference of futures prices, the effect is mostly positive andsignificant. Thus, an increase in short-run deviations lowers the cashreturns but increases the log difference of future prices.

For GARCH-X and BEKK-X, the parameters measuring the effects ofthe short-run deviations on the conditional variance of cash returns(D11) and the log difference of the futures prices (D33) are found to bepositive and significant in all tests. A positive and significant effect ofthe short-run deviations on the conditional variance implies that asthe deviation between the cash and future prices gets larger, thevolatility of log difference of the cash and futures prices increases, andprediction becomes more difficult. The parameter D22 measures theaffect of the short-run deviations on the conditional covariancebetween the two variables. For GARCH-X, D22 is found to be significantand positive only in the case of sugar and corn. Using the BEKK-X theparameter D22 is not significant for any commodity.

Specification adequacy of the first two conditional moments isverified through the serial correlation test of white noise. These testsemploy the Ljung–Box Q statistics on the standardized (normalised)residuals (εt/Ht

1/2), standardized squared residuals (εt/Ht2), and the

cross-standardized residuals. The latter are the cross product betweenthe standardized residuals of cash and futures. All series are found tobe free of serial correlation (at the 5% level). Absence of serialcorrelation in the standardized squared residuals implies the absenceof need to encompass a higher order ARCH process (Giannopoulos,1995).

model. Tests in this paper were also conducted with different combinations of p and q,with p=q=2 being the maximum lag length. Results based on log-likelihoodfunction and likelihood ratio tests indicate that the best combination is p=q=1.These results are available on request.

Table 2GARCH model results.

GARCH BEKK GARCH-X BEKK-X

µ1×10−4 1.6603 (1.1435) −1.0851 (−0.5497) 1.4442 (0.9521) 0.0001 (0.6117)δ1 −0.0237a (−6.8022) −0.0062c (−1.7814) −0.0215a (−5.0417) −0.0217a (−4.4844)µ2×10−4 −0.0886 (−0.7428) −5.3107c (−1.7663) −0.9223 (−0.7154) −0.0001 (−0.8175)δ2 0.0724a (24.3000) 0.0293a (7.0810) 0.0759a (17.3608) 0.0787a (13.9011)C1 84.0000a (14.1400) 0.0028a (4.7433) 77.000a (13.5068) 0.0025a (7.9917)A11 0.1027a (18.3218) 0.2977a (8.6963) 0.0973a (18.6137) 0.2940a (14.2381)B11 0.8629a (126.3859) 0.9452a (78.3185) 0.8629a (128.592) 0.9396a (112.095)D11 _ _ 0.0007a (5.1611) 0.0008c (1.6522)C3 72.0000a (27.8602) 0.0037a (9.8381) 79.0000a (22.1063) 0.0024a (4.1001)A33 0.1160a (30.2647) 0.3273a (9.1173) 0.0983a (18.3065) 0.2842a (10.0155)B33 0.8599a (242.0081) 0.9205a (73.9110) 0.8288a (163.1952) 0.9257a (49.0385)D33 _ _ 0.0043a (27.0836) 0.0044b (2.3193)C2 1.0000b (1.9675) _ 1.0000c (1.8848) _A22 0.0215a (10.1367) _ 0.0198a (9.5410) _B22 0.9694a (388.3794) _ 0.9672a (350.255) _D22 _ _ −0.00004 (−1.3942) 0.0003 (0.7596)C12 _ 0.0007a (3.1666) _ 0.0002 (1.5054)L 45857.056 33006.140 45935.072 34872.835

LB(9) test for serial correlation in the residualsεt/ht1/2 — cash 10.9987 10.0391 5.3175 5.0870εt2/ht — cash 10.6232 3.5348 3.9901 8.4105εt/ht1/2 — futures 5.8925 11.2525 3.1948 2.3841εt2/ht — futures 3.3340 5.1296 3.4460 6.4514CSR 8.0457 9.9979 8.0883 8.2632LR – – 156.356⁎⁎⁎ 176.760⁎⁎⁎

Notes:a, b and c imply significance at the 1%, 5% and 10% level, respectively. t-statistics in the parentheses; L= log likelihood function value. LB = Ljung–Box statistics for serial correlationof the order 9. εt2/Ht = standardized squared residuals. εt/Ht

1/2 = standardized residuals. Cross standardized residuals (CSR) = standardized residuals (cash)×standardizedresiduals (futures). LR = likelihood ratio test. The null hypotheses of the LR test is tested by means of the χ2 statistics.⁎⁎⁎ Implies rejection of the null at the 1% level.

62 T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

Further, the likelihood ratio test (LR) is applied to assess thestatistical significance of the incremental explanatory power asso-ciated with the general model. In other words, in the LR tests arelatively more complex model is compared to a simpler model to seeif it fits a particular dataset significantly better. Thus the LR test isconducted between the standard GARCH and GARCH-X and alsobetween standard BEKK and BEKK-X. The null hypothesis in the LR testis that both models perform the same while the alternate null is thatthe complex model outperforms the standard model. In all cases, theLR test significantly rejects the null at the 1% level (by means of the χ2

statistics), indicating that the complex model GARCH-X fits betterthan the standard GARCH, and the BEKK-X fits better than thestandard BEKK. This is true for all periods.

4.2. Within sample period hedge ratios comparison

Comparison between the effectiveness of different hedge ratios ismade by constructing portfolios implied by the computed ratios, andthe change in the variance of these portfolios indicates the hedgingeffectivenessof thehedge ratios. Theportfolios are constructedas (rtc−βt⁎rt

f), where as defined before rtc is the log difference of the cash (spot)prices, rtf is the log difference of the futures prices, and βt⁎ is theestimated optimal hedge ratio. The variance of these constructedportfolios is estimated and compared. For example, for comparisonbetween the GARCH and GARCH-X based portfolios, the change invariance is calculated as (VarGARCH-VarGARCHX)/VarGARCH. Comparisonis also provided between the four time-varying hedge ratio-orientedportfolios and an unhedged portfolio. Variance of an unhedgedportfolio is presented by the variance of the returns in the cashmarket.

Table 3 presents the variance of the portfolios and the comparisonresults for thewithin sample period (January 1980–July 2004). The tableshows the variance of the portfolios estimated using the different typesof hedge ratios and the percentage change in the variance of theportfolios constructed. Portfolios createdusing thehedge ratios fromtheGARCH-X model outperform other portfolios in the large majority of

the cases. This is true for both storable and non-storable commodities.The differences in the percentage change are quite small, usually lessthan 5%, except in the case of soybeans. For soybeans, the GARCH-Xtime-varying hedge-ratio portfolios outperform the unhedged portfoliowith a variance reduction of 13.90%, the BEKK-X portfolio by 9.86%, thestandard BEKK portfolio by 10.70% and the standard GARCH portfolio by10.28%. As indicated by one of the referee, the soybeans results could bedue to the boom in soybeans production in Brazil during the 1980s andthe recent bio-fuel boom. As Frechette (1997) explains the US and Brazilproduce two third of the soybeans in the world and the timing of theharvest in the two countries is different. In the US they are harvested inOctober/November and, in Brazil theyare harvested inMarch/April. Theoff-seasoning timing of northern and southern harvest complicatesprice dynamics of soybeans and the shape of the futures price profile(Frechette, 1997).

The results for BEKK-X-oriented portfolios aremixed. It does worsethan the standard bivariate GARCH for all commodities except forsoybeans, and it is equal for hogs. Once again, the differences aresmaller than 5% (even for soybeans). The BEKK-X does better than thestandard BEKK and the unhedged portfolio for most commodities. Thestandard GARCH also does better than the standard BEKK, except forcorn and also does better than the unhedged portfolios for allcommodities. The standard BEKK does better than the unhedgedportfolios for all commodities except for corn, and coffee where theyare the equal. Usually, the percentage differences in the portfoliovariances are smaller than 5%.

Overall, the GARCH-X portfolios provide the strongest and thestandard BEKK the weakest results among the GARCH models. Thisresult is true for both the storable and non-storable commodities. TheGARCH-X result also backs the claim of Yang et al. (2001) and otherpapers that cointegration between cash and futures prices oncommodity markets should be incorporated into commodity hedgingdecisions. This turns to be the main contribution of the results. Allmodels seem to perform better than the unhedged model, justifyingthe use of futures contracts.

Table 3With-in period portfolio variance and percentage change in the variance.

Hedge ratio Corn Wheat Coffee Sugar Soybeans Cattle Hogs

GARCH 0.000229 0.000296 0.000615 0.000647 0.000214 0.000320 0.000520BEKK 0.000234 0.000300 0.000616 0.000653 0.000215 0.000330 0.000528GARCH-X 0.000229 0.000295 0.000615 0.000646 0.000192 0.000316 0.000518BEKK-X 0.000232 0.000297 0.000636 0.000654 0.000213 0.000335 0.000520No hedge 0.000234 0.000304 0.000616 0.000673 0.000223 0.000337 0.000530

Percentage change in the portfolio variance between GARCH-X and other methodsGARCH 0.000 0.338 0.000 0.155 10.280 1.250 0.349BEKK 2.137 1.667 0.162 1.072 10.700 4.242 1.894BEKK-X 1.293 0.673 3.302 1.223 9.859 5.672 0.349No hedge 2.137 2.961 0.162 4.012 13.901 6.231 2.264

Percentage change in the portfolio variance between BEKK-X and other methods (excluding GARCH-X)GARCH −1.131 −0.338 −3.414 −1.082 0.467 −4.689 0.000BEKK 0.850 1.000 −3.246 −0.153 0.930 1.493 1.515No hedge 0.850 2.303 −3.246 2.823 4.484 0.594 0.377

Percentage change in the portfolio variance between BEKK GARCH and other methods (excluding GARCH-X and BEKK-X)GARCH 2.137 −1.351 −0.163 −0.927 −0.467 −3.125 −1.515no hedge 0.000 1.316 0.000 2.978 3.587 2.121 0.377

Percentage change in the portfolio variance between GARCH and no hedgeNo hedge 2.137 2.632 0.163 3.863 4.036 5.313 1.887

Notes:The change in the variance between GARCH and GARCH-X is estimated as (VarGARCH−VarGARCHX)/VarGARCH. The change in the variance between GARCH and BEKK is estimated as(VarGARCH−VarBEKK)/VarGARCH. The change in the variance between GARCH and BEKK-X is estimated as (VarGARCH−VarBEKKX)/VarGARCH. The change in the variance betweenGARCH-X and BEKK-X is estimated as (VarBEKK-X−VarGARCHX)/VarBEKK-X. The change in the variance between GARCH-X and BEKK is estimated as (VarBEKK−VarGAARCH)/VarBEKK.The change in the variance between BEKK and BEKK-X is estimated as (VarBEKK−VarBEKKX)/VarBEKK.

63T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

4.3. Out-of-sample periods hedge ratios comparison

Baillie and Myers (1991) further claim that the more reliablemeasure of hedging effectiveness is the hedging performance ofdifferent methods for out-of-sample periods. This paper compares thehedgingeffectiveness of the differentmethods during two different out-of-sample time periods: fromAugust 2002 to July 2004 (two years), andfrom August 2003 to July 2004 (one year). Two different lengths of out-of-sample periods are applied to checkwhether changing the length hasany significant effect on the hedging effectiveness of the hedge ratios. Inorder to investigate the out-of-sample hedging effectiveness of thehedging methods, all GARCH models are estimated for the periodsJanuary 1980 to July 2002, and January 1980 to July 2003, and then theestimated parameters are applied to compute the hedge ratios and theportfolios for the two out-of-sample periods.14 Once again, the varianceof these portfolios is compared, and the change in the variance indicatesthe hedging effectiveness of the hedge ratios.

Table 4 shows the variance of the out-of-sample portfolios and thepercentage change in variance of the portfolios from August 2002 toJuly 2004. The set-up of Table 4 is the same as for Table 3. For mostcommodities the GARCH-X based portfolio again provides the lowestvariance. The GARCH and the standard BEKK do better in the case ofhogs, and the BEKK-X in the case of corn. Once again, in most cases thedifferences are smaller than 5% and sometimes less than 1%. TheBEKK-X outperforms the standard BEKK (except for hogs) and theunhedged portfolios (except for coffee), but not the standard GARCHportfolios (in most cases). The BEKK-X does better than the standardGARCH for corn and soybeans only. Similarly, the standard BEKK alsodoes better than the standard GARCH for corn and live cattle only. Theunhedged portfolio underperforms the standard BEKK for most of thecommodities. The standard GARCH does better than the unhedgedportfolio for all commodities except for corn. In the case of soybeans,the differences are large when involving the unhedged portfolio.

14 The GARCH estimations for the period 1980–2002 and 1980–2003 are notprovided, in order to save space, but are available on request. These parameters aresimilar to the ones estimated for the whole sample period. Once again, cointegration isalso found during these periods.

Table 5 shows the results from the shorter out-of-sample (August2003–July 2004) period. The results are quite similar to the previoustwo results. Among the GARCH models, portfolios based on theGARCH-X model again perform best, and the standard BEKK doesworst. For wheat, the standard GARCH model-based portfolios dobetter than other models. But in most cases, the differences are quitesmall. The soybeans results are also small in size.

Changing the length of the out-of-sample period does not affect theperformance of the hedge ratios by much. The GARCH-X and thestandard GARCH provide similar performances (this result is similar tothe within sample period results). They provide the lowest varianceportfolios, and the standard BEKK and the unhedged portfolios providethe highest. The BEKK-X performs better than the standard BEKK. Thereduction in the variances are quite small in large majority of the testsbut this is expected given daily data has been applied. Sephton (1993)also indicates small changes in the portfolio variances while applyingdaily data. As shown by Kroner and Sultan (1993) small size im-provements do not imply that the economic viability of the proposedstrategy is questionable. The GARCHbased portfolio should be applied ifit makes the investor's utility greater than the reduction in the returncaused by the transaction cost incurred. Comparing between storableand non-storable commodities, results show thatGARCH-Xusually doeswell for both commodities. We do not find much difference in thehedging effectiveness of the two different types of commodities thuscontradicting the results of Yang and Awokuse (2003). The strongGARCH-X again promotes the application of cointegration whenestimating the hedging effectiveness in the futures markets.

While the GARCH-X model shows potential for improved riskmanagement, the implementation of it can require frequent and costlyposition changes in the futures market. According to Park and Switzer(1995), the trade-off between the risk reduction and the transactioncost will determine the practicality of any of the GARCH hedgingmethod.15 Furthermore according to Myers (1991), since the differentGARCH models are more complex to estimate, and since the continual

15 Park and Switzer (1995) suggest an alternate strategy method that involves lessfrequent rebalancing, such as rebalancing only when the hedge ratio changes by a fixedamount.

Table 4Out-of-sample period (2 years) portfolio variance and percentage change in the variance.

Hedge ratio Corn Wheat Coffee Sugar Soybeans Cattle Hogs

GARCH 0.000275 0.000653 0.00130 0.000587 0.000208 0.000329 0.000360BEKK 0.000274 0.000683 0.00136 0.000593 0.000211 0.000335 0.000366GARCH-X 0.000271 0.000650 0.00130 0.000585 0.000208 0.000326 0.000367BEKK-X 0.000269 0.000660 0.00135 0.000590 0.000210 0.000330 0.000371No hedge 0.000271 0.000670 0.00130 0.000591 0.000260 0.000339 0.000384

Percentage change in the portfolio variance between GARCH-X and other methodsGARCH 1.455 0.459 0.000 0.341 0.000 0.912 −1.944BEKK 1.095 4.832 4.441 1.349 1.422 2.687 −0.273BEKK-X −0.743 1.515 3.704 0.847 0.952 1.212 1.078No hedge 0.000 2.985 0.000 1.015 20.000 3.835 4.427

Percentage change in the portfolio variance between BEKK-X and other methods (excluding GARCH-X)GARCH 2.182 −1.072 −3.846 −0.511 0.952 −0.003 −3.056BEKK 1.825 3.368 0.735 0.506 0.474 1.149 −1.366No hedge 0.743 1.493 −3.846 0.169 19.230 2.655 3.385

Percentage change in the portfolio variance between BEKK GARCH and other methods (excluding GARCH-X and BEKK-X)GARCH 0.363 −4.594 −4.615 −1.022 −1.442 1.791 −1.667No hedge −1.110 −1.941 −4.615 −0.338 −1.442 1.180 4.688

Percentage change in the portfolio variance between GARCH and no hedgeNo hedge −1.476 2.537 0.000 0.677 20.000 2.950 6.250

See note end of Table 3.

64 T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

futures adjustments required entail extra commission charges, theextra cost of workingwith any GARCHmodel may only bewarranted ifthe investor is extremely risk averse.

5. Conclusion

One of the main functions of the futures market is to provide ahedging (risk transfer) mechanism. It is also awell-documented claimin the futures market literature that the optimal hedge ratio should betime-varying and not constant. An optimal hedge ratio is defined asthe proportion of a cash position that should be covered with anopposite position on a futures market. Lately, different versions of theGARCH models have been applied to estimate time-varying hedgeratios for different futures markets. This paper investigated thehedging effectiveness of GARCH estimated time-varying hedge ratiosin seven agricultural commodities futures: corn, wheat, coffee, sugar,

Table 5Out-of-sample period (1 years) portfolio variance and percentage change in the variance.

Hedge ratio Corn Wheat Coffee

GARCH 0.000169 0.000528 0.000860BEKK 0.000168 0.000548 0.000903GARCH-X 0.000167 0.000530 0.000850BEKK-X 0.000170 0.000540 0.000890No hedge 0.000174 0.000550 0.000900

Percentage change in the portfolio variance between GARCH-X and other methodsGARCH 1.183 −0.379 1.163BEKK 0.595 3.285 5.869BEKK-X 1.765 1.851 4.494No hedge 4.023 3.636 5.555

Percentage change in the portfolio variance between BEKK-X and other methods (excludinGARCH −0.592 −2.272 −3.488BEKK −1.190 1.460 1.440No hedge 2.300 1.818 1.111

Percentage change in the portfolio variance between BEKK GARCH and other methods (excGARCH 0.592 −3.788 −5.000No hedge 3.448 0.364 −0.333

Percentage change in the portfolio variance between GARCH and no hedgeNo hedge 2.874 4.000 4.444

See note end of Table 3.

soybeans, live cattle and hogs. Live cattle and hogs are non-storableagricultural commodities, the others are storable. In this way thispaper also provided a comparison between the hedging effectivenessof storable and non-storable futures commodities. Our paper wasmotivated by Yang and Awokuse (2003) who indicated that knowl-edge of how effective hedging function performs on commodityfutures market is essential to understanding these markets.

The time-varying hedge ratios were estimated by means of fourdifferent types of GARCH models: the standard bivariate GARCH,bivariate BEKK, bivariate GARCH-X, and bivariate BEKK-X. The GARCH-X and the BEKK-X are unique among the GARCHmodels in taking intoconsideration the effects of the short-run deviations from a long-runrelationship between the cash and the futures price indices on thehedge ratio. The long-run relationship between the price indices isestimated by the Engle–Granger cointegration method. One of themain objectives of the paper is to test the effects of the cointegration

Sugar Soybeans Cattle Hogs

0.000460 0.000202 0.000220 0.0002920.000464 0.000205 0.000210 0.0002960.000458 0.000201 0.000216 0.0002910.000460 0.000200 0.000217 0.0002980.000463 0.000200 0.000228 0.000310

0.435 0.495 1.818 0.3431.293 1.951 −0.021 1.6890.435 −0.500 0.461 2.3491.092 −0.500 5.263 6.129

g GARCH-X)0.000 0.990 1.364 −2.0550.862 2.440 −3.333 −0.67560.652 0.000 4.825 3.8710

luding GARCH-X and BEKK-X)−0.870 −1.463 4.545 −1.370−0.216 −2.500 7.895 4.516

0.652 −1.000 3.5088 5.807

65T. Choudhry / International Review of Financial Analysis 18 (2009) 58–65

between the cash and the futures prices on the hedging effectivenessin the agricultural futures markets. The hedging effectiveness is es-timated and compared by checking the variance of the portfolioscreated using these hedge ratios. The lower the variance of theportfolio, the higher is the hedging effectiveness of the hedge ratio.

The empirical tests were conducted by applying daily data. Theeffectiveness of the hedge ratio was investigated by comparing thewithin sample period (August 1980–July 2004) and out-of-sampleperiod performance of the different hedge ratios for two periods,August 2002–July 2004 (two years) and August 2003–July 2004 (oneyear). The two different lengths of out-of-sample periods were thenapplied to investigate the effect of changing the length on the hedgingeffectiveness of the hedge ratios.

Results show that during the within sample period and the twoout-of-sample periods, the GARCH-X-oriented hedge ratio overallperforms better than the other GARCH methods and the unhedgedportfolio. This result backs the claim by Yang et al. (2001) and othersregarding the importance of the use of cointegration between cashand futures prices in hedging ratios. Among the GARCH modelsapplied, the standard BEKK-oriented hedge ratios provided the worstperformance. Also, changing the length of the out-of-sample perioddoes not change the hedging effectiveness of the GARCH-orientedhedge ratios. This is especially true in the case of standard GARCH andthe GARCH-X method. Results do not show much difference in thehedging effectiveness of storable and non-storable commodities. Thiscontradicts of what was reported by Yang and Awokuse (2003).

As stated by Bera et al. (1997) further study is needed to assess thecosts of GARCHmodels specification and implementation relative to thegains in variance reduction. The implementation of the GARCH modelscan require frequent and costly position changes in the futures market.In addition, the estimation and continual updating of GARCHmodels forpractical use can be time consuming and costly. Future studies of thehedging performance of these models in framework which explicitlyincorporates these costswill provide amuchbetter understandingof theusefulness of these models for managing price risk.

References

Baillie, R., & Myers, R. (1991). Bivariate GARCH estimation of the optimal commodityfutures hedge. Journal of Applied Econometrics, 6, 109−124.

Bera, A., Garcia, P., & Roh, J. (1997). Estimation of time-varying hedge ratios for corn andsoybeans: BGARCH and random coefficient approaches. OFOR working paper number(pp. 97-06).

Bollerslev, T. (1988). On the correlation structure for the generalized autoregressiveconditional heteroscedastic process. Journal of Time Series Analysis, 9, 121−131.

Bollerslev, T., Engle, R., & Nelson, D. (1994). In R. F. Engle & D. L. McFadden (Eds.), ARCHmodelsHandbook of econometrics, Vol. IV. Elsevier Science.

Brorsen, B. W. (1995). Optimal hedge ratios with risk-netural producers and nonlinearborrowing costs. American Journal of Agricultural Economics, 77, 174−181.

Brorsen, B. W., & Fofana, N. (2001). Success and failure of agricultural futures contracts.Journal of Agribusiness, 19, 129−145.

Cecchetti, S., Cumby, R., & Figlewski, S. (1988). Estimation of the optimal futures hedge.Review of Economics and Statistics, 70, 623−630.

Choudhry, T. (2004). Time-varying distribution and hedging effectiveness of threePacific-basin stock futures. International Review of Economics and Finance, 13,371−385.

Covey, T., & Bessler, D. (1992). Testing for Granger's full causality. Review of Economicsand Statistics, 74, 146−153.

Covey, T., & Bessler, D. (1995). Asset storability and the informational content of inter-temporal prices. Journal of Empirical Finance, 1995, 103−115.

Engle, R., & Granger, C. (1987). Cointegration and error correction: representative,estimation, and testing. Econometrica, 55, 251−276.

Engle, R., & Kroner, K. (1995). Multivariate simultaneous generalized ARCH. EconometricTheory, 11, 122−150.

Engle, R., & Yoo, B. (1987). Forecasting and testing in the cointegrated system. Journal ofEconometrics, 35, 143−159.

Fortenbery, T., & Zapata, H. (1993). An examination of cointegration relations betweenfutures and local grain markets. Journal of Futures Markets, 13, 921−932.

Fortenbery, T., & Zapata, H. (1997). An evaluation of price linkages between futures andcash markets for cheddar cheese. Journal of Futures Markets, 17, 279−301.

Frechette, D. (1997). The dynamics of convenience and the Brazilian soybean boom.American Journal of Agricultural Economics, 79, 1108−1118.

Ghosh, A. (1995). The hedging effectiveness of ECU futures contracts. Forecastingevidence from an error correction model. Financial Review, 30, 567−581.

Ghosh, A., & Clayton, R. (1996). Hedging with international stock index futures. Anintertemporal error correction model. Journal of Financial Research, 19, 477−491.

Giannopoulos, K. (1995). Estimating the time-varying components of internationalstock markets' risk. European Journal of Finance, 1, 129−164.

Haigh, M., & Holt, M. (2002). Combining time-varying and dynamics multi-periodoptimal hedging method. Working paper WP 02-08, Department of Agricultural andResource Economics, University of Maryland, USA.

Kroner, K., & Sultan, J. (1993). Time-varying distribution and dynamic hedging withforeign currency futures. Journal of Financial and Quantitative Analysis, 28,535−551.

Lee, T. (1994). Spread and volatility in spot and forward exchange rates. Journal ofInternational Money and Finance, 13, 375−383.

Moschini, G., & Myers, R. (2002). Testing for constant hedge ratios in commoditymarkets: A multivariate GARCH approach. Journal of Empirical Finance, 9, 589−603.

Myers, R. (1991). Estimating time varying hedge ratios on futures markets. Journal ofFutures Markets, 11, 39−53.

Park, T., & Switzer, L. (1995). Time-varying distribution and the optimal hedge ratios forstock index futures. Applied Financial Economics, 5, 131−137.

Sephton, P. (1993). Optimal hedge ratios at the Winnipeg Commodity Exchange. Ca-nadian Journal of Economics, 26, 175−193.

Tomek, W. (1997). Commodity future prices as forecasts. Review of AgriculturalEconomics, 29, 23−44.

Tomek, W. (2000). Commodity prices revisited. Agricultural and Resources Economics,29, 125−137.

Turvey, C., & Baker, T. (1989). Optimal hedging under alternative capital structures andrisk aversion. Canadian Journal of Agricultural Economics, 37, 135−143.

Wang, D., & Tomek, W. (2007). Commodity prices and unit root tests. American Journalof Agricultural Economics, 89, 873−889.

Yang, J., & Awokuse, T. O. (2003). Asset storability and hedging effectiveness incommodity futures markets. Applied Economics Letters, 10, 487−491.

Yang, J., Bessler, D., & Leatham, D. (2001). Asset storability and price discovery incommodity futures markets: A new look. Journal of Futures Markets, 21, 279−300.