Modelling Long Memory Volatility in Agricultural Commodity Futures Returns*

Transcription

1 Modelling Long Memory Volailiy in Agriculural Commodiy Fuures Reurns* Chia-Lin Chang Deparmen of Applied Economics Deparmen of Finance Naional Chung Hsing Universiy Taichung, Taiwan Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Deparmen of Quaniaive Economics Compluense Universiy of Madrid and Insiue of Economic Research Kyoo Universiy, Japan Roengchai Tansucha Faculy of Economics Maejo Universiy Chiang Mai, Thailand Revised: May 0 * For financial suppor, he firs auhor is mos graeful o he Naional Science Council, Taiwan, he second auhor wishes o hank he Ausralian Research Council, Naional Science Council, Taiwan, and he Japan Sociey for he Promoion of Science, and he hird auhor acknowledges he Faculy of Economics, Maejo Universiy.

2 Absrac This paper esimaes a long memory volailiy model for 6 agriculural commodiy fuures reurns from differen fuures markes, namely corn, oas, soybeans, soybean meal, soybean oil, whea, live cale, cale feeder, pork, cocoa, coffee, coon, orange juice, Kansas Ciy whea, rubber, and palm oil. The class of fracional GARCH models, namely he FIGARCH model of Baillie e al. (996), FIEGARCH model of Bollerslev and Mikkelsen (996), and FIAPARCH model of Tse (998), are modelled and compared wih he GARCH model of Bollerslev (986), EGARCH model of Nelson (99), and APARCH model of Ding e al. (993). The esimaed d parameers, indicaing long-erm dependence, sugges ha fracional inegraion is found in mos of agriculural commodiy fuures reurns series. In addiion, he FIGARCH (,d,) and FIEGARCH(,d,) models are found o ouperform heir GARCH(,) and EGARCH(,) counerpars. Keywords: Long memory, agriculural commodiy fuures, fracional inegraion, asymmeric, condiional volailiy. JEL: Q4, Q, C, C5.

3 . Inroducion Accurae modelling of volailiy in asse reurns is one of he major issues of concern in financial economics. Poon and Granger (003) have menioned ha, even hough volailiy is no he same as risk, when i is inerpreed as uncerainy i becomes a key inpu o many imporan financial applicaions, such as invesmen, porfolio consrucion, opion pricing, hedging and risk managemen. Research on volailiy models has focused on such differen properies of he reurns series as is ime-varying condiional momens, volailiy clusering, asymmeric paerns and long persisence, among ohers. Derivaive markes, paricularly commodiy fuures markes, have become more sophisicaed since he Chicago Broad of Trade commenced fuures rading in 848. The fuures price depends on he flow of informaion around he world. Small changes in prices could have remendous effecs on rading resuls across fuures markes. This disincion implies ha he fuures marke is more volaile and has high risk. This feaure is also paricularly imporan in he agriculural commodiy fuures marke, where facors such as drough, naural disaser, deforesaion, and deb defaul can have a major impac on demand and supply of commodiies, and hence on he presen and fuures prices of he commodiy. In modern ime series modelling, following he seminal work of Engle (98), a group of ime series models named Auoregressive Condiional Heeroskedasiy (ARCH), and laer generalized by Bollerslev (986) as Generalized Auoregressive Condiional Heeroskedasiy (GARCH), has been used o model ime-varying condiional volailiy. The ARCH and GARCH models explain ime series behaviour by allowing he condiional variance o evolve dynamically over ime and o respond o previous price changes. These models consider nonlineariy in he condiional mean equaion, and are also able o explain volailiy clusering and volailiy persisence. A considerable empirical lieraure in commodiy cash and fuures markes has used a variey of GARCH models o esimae expeced price and reurns volailiy. The GARCH model assumes ha negaive and posiive shocks of equal magniude have idenical impacs on he condiional variance. In order o accommodae differenial impacs on condiional variance beween posiive and negaive shocks of equal magniude, Glosen e al. (99) proposed he asymmeric GARCH, or GJR model. As he posiive and negaive 3

4 shocks on condiional volailiy, called leverage effec, are asymmeric, Nelson (99) proposed he Exponenial GARCH (EGARCH) model. In erms of volailiy persisence, a GARCH model feaures an exponenial decay in he auocorrelaion of condiional variances. However, a shock in he volailiy series seems o have very long memory and impacs on fuure volailiy over a long horizon. Baillie e al. (007) explained ha he long memory refers o he presence of very slow hyperbolic decay in he auocorrelaions and impulse response weighs. Therefore, economerically, he long memory is beween he usual exponenial raes of decay associaed wih he class of saionary and inverible ARMA models, and he alernaive exreme of infinie persisence associaed wih inegraed, uni roo processes. Therefore, Baillie, Bollerslev and Mikkelsen (996) (hereafer denoed BBM) proposed he FIGARCH(p,d,q) model, and Bollerslev and Mikkelsen (996) proposed he FIEGARCH(p,d,q) model, where a full descripion of he properies of he process and he appropriae quasi-maximum likelihood esimaion (QMLE) mehod can be found. Several previous papers have observed and provided applicaion of fracional inegraed models in many fields, namely sock reurns (Bollerslev and Mikkelsen (996), Degiannakis (004) and Niguez % (007), Lux and Kaizoji (007), Kang and Yoon (007), Jefferis and Thupayagale (008), Ruiz and Veiga (008)); exchange rae (Baillie e al. (996), Davidson (004), and Conrad and Lamla (007)) and inflaion rae (Baillie e al. (00)). However, in he lieraure o dae, here have been few applicaions of he fracionally inegraed GARCH class models o commodiy fuures markes. Barkoulas e al. (997) examined he fracional srucure of commodiies spo prices, namely aluminum, cocoa, coffee, copper, rice and rubber. They found ha some commodiy spo price ime series display a fracional srucure, and he fracional orders vary among hese commodiies because he processes involved in he price movemens of each commodiy varies. Crao and Ray (000) invesigaed long-erm memory in he reurns and volailiy of commodiy fuures marke, namely five currencies, welve agriculural commodiies, hree meals and heaing oil, and five currencies fuures markes. They found ha commodiy fuures volailiies are ypically more persisen han currency fuures volailiies. However, hey do no explicily esimae he FIGARCH model. Jin and Frechee (004) examined he 4

5 presence of fracional inegraion in he volailiy of foureen agriculural fuures prices series using daa from 970 o 000. The resuls show ha he volailiy series exhibi srong longerm dependence, which is an indicaor of fracional inegraion. In addiion, FIGARCH(,d,) performs significanly beer han a radiional volailiy model, GARCH(,), in modelling agriculural price volailiy. Baillie e al. (007) examined long memory in volailiy properies of boh daily and highfrequency inraday fuures reurns for six imporan commodiies. They found ha he volailiy processes were found o be accuraely described by FIGARCH models, wih saisically significan long memory parameer esimaes. Recenly, Hyun-Joung (008) explored a long memory condiional volailiy model on inernaional grain markes, namely whea, corn and soybeans, and compared he performance of he models in capuring dependence of he price volailiy, and also emphasized suiabiliy of he suden- densiy inended o accoun for non-normal, fa-ailed properies of he daa. The empirical resuls showed ha grain cash price volailiies exhibi long memory and ha he memory is adequaely modelled by a fracionally inegraed process and implemened by FIGARCH models. In addiion, he suiabiliy of he FIGARCH models is under he suden- disribuion and he compeiiveness of he parsimonious FIGARCH(,d,0) model. Therefore, i is desirable o use long memory condiional variance models for analysis of grain price volailiy dynamics. The fracionally inegraed mulivariae condiional volailiy model of Brunei and Gilber (000) applied he univariae volailiy (FIGARCH) model o mulivariae GARCH models by esimaing and esing coinegraed bivariae FIGARCH models using NYMEX and IPE crude oil markes. They found a common order of fracional inegraion for he wo volailiy process, and confirmed ha hey are fracionally coinegraed. An esimaed error correcion FIGARCH model indicaed ha he predominan adjusmen is he IPE oward NYMEX. Coakley e al. (008) explored he relaionship beween basis long memory and hedging effeciveness measures wih error-correcion and he mulivariae GARCH (FIEC-BEKK) model, employing spo daily daa and heir corresponding fuures conracs for five commodiies, namely soybeans, cocoa, heaing oil, gold and live cale. The resuls 5

6 presened a long memory componen ha should heoreically affec hedging effeciveness. Recenly, Sephon (009) reexamined he findings in Jin and Frechee (004) and used he same daase o provide evidence of fracional inegraion using he FIGARCH and FIAPGARCH models. The updaed empirical resuls generally confirm he presence of long memory in condiional variances, wih some commodiy fuures displaying significan leverage effecs. The aims of he paper are o analyse agriculural commodiy fuures reurns using several condiional volailiy models, namely GARCH, EGARCH and APARCH, and fracionally inegraed condiional volailiy models, namely FIGARCH, FIEGARCH and FIAPARCH, as an exension of exising resuls. The paper differs from exising sudies in hree respecs. Firs, due o changes in he financial and economic environmen, such as he global financial crisis, an increasing number of marke paricipans, produc yield uncerainy, changes in he demand and supply posiion of agriculural commodiies and growing inernaional compeiion, agriculural commodiy fuures markes have maured considerably over he las decade. An exension in he sample period from 000 o 009, giving an addiional,500 observaions, is inended o allow a suiable analysis of hese issues. Second, none of he preceding papers has used a variey of fracionally inegraed GARCH models for purposes of comparison wih convenional GARCH models. This paper esimaes five fracionally inegraed GARCH models, namely FIGARCH of Baillie e al. (996), FIGARCH of Chung (999), FIEGARCH of BBM (996), FIAPGARCH of Ding, Granger and Engle (993), and FIAPGACH of Tse (998), and compares he esimaes wih convenional GARCH models. Four imporan agriculural commodiy fuures are considered, namely coon, orange juice, and wo ropical rain plans in palm oil and rubber. These agriculural commodiy fuures have no ye been examined using long memory models. The empirical findings in his paper should make a useful conribuion o all agens involved in he sale, purchase and disribuion of agriculural commodiies, including relaed indusries. This empirical analysis given below indicaes ha, on he basis of he EGARCH and APGARCH models, mos agriculural commodiy fuures reurns have asymmeric effecs, wih only a few displaying leverage effecs. Thus, i would appear ha he GARCH model is no appropriae for analyzing agriculural commodiy fuures reurns. Moreover, evidence of 6

7 long memory is found for each agriculural commodiy fuures reurns using boh FIGARCH and FIAPGARCH of he BBM and Chung specificaions. In addiion, asymmeric and leverage effecs are found for some agriculural commodiies using FIEGARCH and FIAPGARCH, which suggess ha he FIGARCH model is no appropriae for modelling agriculural commodiy fuures reurns. The remainder of he paper is organized as follows. Secion discusses he mehodological approach used in he paper. Secion 3 describes he commodiy fuures prices ime series. Secion 4 presens he resuls from empirical modelling, and Secion 5 provides some concluding commens.. Economeric Models. Univariae Condiional Volailiy Models This secion presens he volailiy models in commodiy fuures reurns, namely he GARCH model of Bollerslev (986), EGARCH model of Nelson (99) and APARCH model of Ding e al. (993), and fracionally inegraed condiional volailiy models, namely FIGARCH model of Baillie e al. (996), FIEGACH model of Bollerslev and Mikkelsen (996), and FIAPARCH model of Tse (998). Following Engle (98), consider he ime series, y = E ( y ) + ε, where E ( y ) 7 is he condiional expecaion of y a ime and ε is he associaed error. The generalized auoregressive condiional heeroskedasiy (GARCH) model of Bollerslev (986) is given as follows: ε = hη, η N(0,) () p q = + j j + j j = + ( ) + ( ) j= j= h ω αε βh ω α L ε β L σ () where ω > 0, α 0 and β 0 are sufficien condiions o ensure ha he condiional variance 0 ( L) L L β = β + β +K j j h >, L is he lag operaor, ( ) shor-run persisence of shocks o reurns, p α L = α L+ α L + K+ α L and + β L q. In () he parameer q α j represens he ARCH effec, or he β j represens he GARCH effecs, and ( α j + β j ) p

8 measures he persisence of he conribuion of shocks o reurn i o long-run persisence. If he roos of α( L) β ( L) and ( L) β lie ouside he uni circle, hen { } sabiliy and covariance saionariy. The volailiy shocks decay a a geomeric rae. ε exhibis Equaion () assumes ha he condiional variance is a funcion of he magniudes of he lagged residuals and no heir sign, such ha a posiive shock ( 0) on he condiional variance as a negaive shock ( 0) ε > has he same impac ε < of equal magniude. In order o accommodae differenial impacs on he condiional variance of posiive and negaive shocks, Glosen e al. (99) proposed he asymmeric GARCH, or GJR model, as given by where r ( ( )) h = ω+ α + γ I η ε + β h (3) j j j j j j j= j= I i s 0, εi 0 = (4), εi < 0 is an indicaor funcion o differeniae beween posiive and negaive shocks. Bollerslev (986) showed he necessary and sufficien condiion for he second-order saionariy of GARCH is r s j j= j= j ( ) α + β <. For he GARCH(,) model, Nelson (990) obained he log- momen condiion for he sric saionary and ergodiciy as ( ) imporan in deriving he saisical properies of he QMLE. E log αη + β < 0, which is In an alernaive model ha accommodaes asymmery beween posiive and negaive shocks, and possibly also leverage, Nelson (99) proposed he Exponenial GARCH (EGARCH) model, inerpreing as ARMA-ype models for he logarihm of he condiional variance, namely: log h p p q = ω+ α η + γη + β log h. (4) i i i i j j i= i= j= In (4), η i and i η capure he size and sign effecs, respecively, of he sandardized shocks. Unlike he GARCH model, EGARCH in (4) uses he sandardized residual raher han he uncondiional shocks. As EGARCH also uses he logarihms of condiional volailiy, here are no resricions on he parameers in (4). As he sandardized shocks have 8

9 finie momens, he momen condiional of (4) are sraighforward. The disincions beween EGARCH and he previous wo GARCH models are discussed in McAleer (005) and McAleer e al. (007) Alernaively, Bollerslev and Mikkelsen (996) proposed expressing he EGARCH model as follows: ( σ ) = ω+ β( L) + α( L) g( z ) ln. (5) The value of g( z ) depends on several elemens. Following Nelson (99), in order o accommodae he asymmeric relaion beween reurns and volailiy changes, he value of g( z ) mus be a funcion of boh he magniude and sign of z, which yields he funcion g( z ) expressed as ( ) { g z = γ z + γ z E z (6) sign effec magniude effec The parameer γ capures he leverage effec. If γ < 0, he fuures condiional variances will increase proporionally more as a resul of a negaive shock han for a posiive shock of he same absolue magniude. Ding, Granger and Engle (993) proposed an asymmeric power GARCH (APARCH) model, whereby he power of he sandard deviaion, esimaed. The APARCH(p,q) is definded as: σ, where δ > 0, is a parameer o be δ and < < γ, ( i =,..., ) i p q δ δ δ = + j( i i i) + j j j= j= σ ω α ε γ ε β σ (7) q. This model ness a leas seven ARCH-ype models, namely he ARCH model of Engle (98), GARCH model of Bollerslev (986), Taylor/Schwer GARCH in sandard deviaion of Taylor (986) and Schwer (990), GJR model of Glosen e al. (993), TARCH of Zakoian (994), NARCH of Higgins and Bera (99), and log- ARCH of Geweke (986) and Panula (986). Following Ding e al. (993), if ω > 0 and p αie( z γiz) δ + β j <, a saionary soluion for equaion (7) exiss and is given by i= j= q 9

10 E δ ( σ ) α = α γ + β 0 p q δ ( z z) i i j i= j= In order o esimae he parameers of model ()-(7), maximum likelihood esimaion is used wih a join normal disribuion of η. However, when he process for η does no follow a normal disribuion, or when he condiional disribuion is no known, he soluion o maximizing he likelihood funcion is he quasi-mle (QMLE) approach.. Univariae fracional inegraed condiional volailiy models The long memory propery can be defined hrough he properies of he auocorrelaion funcion, which is defined as ρ cov ( x, x ) var ( x ) = for ineger lag k. A covariance k saionary ime series process is expeced o have auocorrelaions such ha lim ρ = 0. Mos of he well-known class of saionary and inverible ime series processes have k k auocorrelaions ha decay a he relaively fas exponenial rae, so ha ρ k m k, where m <, and his propery is rue, for example, for he well-known saionary and inverible ARMA(p,q) process. For long memory processes, he auocorrelaions decay a an hyperbolic rae which is consisen wih ρ k ck and d is he long memory parameer. d as k increases wihou limi, where c is a consan In applicaions, i ofen occurs ha he esimaed sum of parameers α and β in GARCH(,) is close o uniy, ha is p q α + β, or for GARCH(p,q), he process i i= j= j exhibis srong persisence. If p q α + β <, he process ( ε ) is second-order saionary, i i= j= j and a shock o he condiional variance h has a decaying impac on h + h, when h increases, and is asympoically negligible. However, if p q α + β j, he effec on h + hdoes no die i i= j= ou asympoically. 0

11 This propery is called persisence. Under he resricion p q α + β =, Engle and i i= j= j Bollerslev (986) developed he inegraed GARCH (IGARCH) model, meaning ha curren informaion remains of imporance when forecasing he volailiy for all horizons: ( L)( L) = + ( L) φ ε ω β υ (8) where υ = ε σ is he innovaion in he condiional variance process or maringale difference process wih respec o σ or h, and has mean 0 and no serial correlaion, φ( L ) = α( L ) β ( L ) ( L ), and all he roos of φ ( L) and ( L) uni circle. β lie ouside he However, volailiy ends o change quie slowly over ime and, as shown in Ding e al. (993), he effecs of a shock can ake a considerable ime o decay. Therefore, he disincion beween I(0) and I() processes seems o be oo resricive. Indeed, he propagaion of shock in an I(0) process shocks dies ou a an exponenial rae (so ha i only capures shor memory) and, for an I() process, he persisence of shocks is infinie and here is no mean reversion, whereas 0< d < shocks die ou a a slow hyperbolic rae. Baillie e al. (996) inroduced he fracionally inegraed GARCH (FIGARCH) model in order o capure he long memory effec in volailiy, allows a hyperbolic decay of he coefficien, β j, which is posiive, summable, and saisfies he uni roo condiion. This model mimics he ARFIMA framework of he condiional mean equaion. The FIGARCH(p,d,q) process is defined as: where all he roos of φ ( L) and ( L) d ( L)( L) = + ( L) { } φ ε ω β υ, (9) { } FIGARCH process can be represened as: or h = + ( L i ) = + ( L) β lie ouside he uni circle. Analogously o (9), he { } ( ) ( ) ( )( ) d h = ω β L + β L φ L L ε, (0) i ω ω λ ε ω λ ε, when 0 d λ( L) < <. ( ) d L, where 0< d <, is he fracional differencing operaor, and is value depends on he decay rae of a shock o condiional volailiy. I is also mos convenienly expressed in erm of he hypergeomeric

12 funcion: ( L) k = 0 Γ ( d + ) ( k ) ( d k ) d = Γ + Γ + L k () d L L dl d d L d d d L k = 0 k 6 3 = = d k or ( ) ( ) ( ) ( )( ) I is easy o show ha ω > 0, d d β φ and d ( + d) K () d φ β φ β are sufficien o ensure ha he condiional variance of he FIGARCH(,d,) is posiive almos surely for all. FIGARCH ness he GARCH model when d = 0, and he IGARCH model when d =. Approximae maximum likelihood esimaes of he parameers of he FIGARCH(p,d,q) process in (9) can be obained by QMLE. Chung (999) argued ha he mehod of parameerizaion of he FIGARCH model of Baillie e al. (996) may have a specificaion problem, and underscores some drawbacks in he BBM model. There may be a srucural problem in he BBM specificaion in parallel wih he ARFIMA framework of he condiional mean equaion, hereby leading o difficul inerpreaions of he esimaed parameers. Indeed, he fracional differencing operaor applies o he consan erm in he mean equaion (ARFIMA), while i does no do so in he variance equaion (FIGARCH). Therefore, Chung (999) redefines he FIGARCH model as: where σ is he uncondiional variance of d ( L)( L) ( ) = ( ) ( L ) φ ε σ β ε σ, (3) (3), we can formulae he condiional variance as: ε. If we reain he same definiion of ( L) { ( ) d ( )( ) }( ) h = + L L L λ as in σ β φ ε σ (4) or h = + ( L)( ) σ λ ε σ. (5) In order o accommodae asymmeries beween posiive and negaive shocks, called he leverage effec, Bollerslev and Mikkelsen (996) exend he FIGARCH process o FIEGARCH, o correspond wih Nelson s (99) Exponenial GARCH model o allow for asymmery. The FIEGARCH(p,d,q) model is given as d ( h ) = + ( L) ( L) + ( L) g( z ) ln ω φ α, (6)

13 where g( z ) = θz + γ z E z, he firs erm ( ) ( γ z E z ) θ is he sign effec, and he second erm is he magniude effec. All he roos of φ ( L ) and ( L) z λ are an auoregressive polynomial and a moving average polynomial in he lag operaor L and lie ouside he uni circle, and boh polynomials do no have a common roo. When d = 0, he FIEGARCH(p,d,q) process reduces o EGARCH of Nelson (99), and when d =, he process becomes inegraed EGARCH (IEGARCH). Bollerslev and Mikkelson (996) presened evidence on he efficiency of QMLE applied o esimae he parameers of he FIEGARCH process. Tse (998) proposed a model which combines he fracionally inegraed GARCH formulaion of Baillie e al. (996) wih he asymmeric power ARCH specificaion of Ding, Granger and Engle (993) (see Ling and McAleer (00) for he heoreical properies of he model). This model increases he flexibiliy of he condiional variance specificaion by allowing: (a) an asymmeric response of volailiy o posiive and negaive shocks; (b) he daa o deermine ha power of reurns for which he predicable srucure in he volailiy paern is he sronges; and (c) long range volailiy dependence. The FIAPARCH(p,d,q) model can be wrien as: { ( ) ( )( d L L L) }( ) δ σ = ω+ β φ ε γε, (7) where γ is he leverage coefficien, and δ is he parameer for he power erm ha akes (finie) posiive values. When d = 0, he FIAPARCH(p,d,q) process reduces o APARCH of Ding e al. (993). When γ = 0 and δ =, he process in (3) reduces o he FIGARCH(p,d,q) specificaion, which includes Bollerslev s (986) model when d = 0, and he inegraed specificaion when d =, as special cases. δ 3. Daa The daa are daily synchronous closing fuures prices of agriculural fuures on differen major US commodiy fuures markes, specifically, he Chicago Broad of Trade (CBOT) for corn, oas, soybeans, soybean meal, soybean oil and whea; he Chicago Mercanile Exchange (CME) for cale feeder, live cale and pork, he New York Broad of Trade (NYBOT) for cocoa, coon, coffee and orange juice, and Kansas Ciy Broad of Trade (KCBOT) for whea. 3

14 The 7,889 price observaions from 4 January 979 o 6 April 009 are obained from he DaaSream daabase service. This paper also focuses on wo imporan commodiy fuures prices of ropical rain plans, namely rubber (RSS3), rading on he Tokyo Sock Exchange (TOCOM), which 5,0 price observaions sared from 3 January 990 o 8 April 009, and palm oil, rading on he Malaysia Derivaives Exchange (MDEX), which 7,45 price observaions sared from 3 Ocober 980 o 8 April 009. These wo commodiy fuures prices are expressed in local currencies and are obained from Reuers. The reurns of agriculural fuures prices i of commodiy j a ime in a coninuous compound basis are calculaed as rij, log ( Pij, Pij, ) =, where P ij, and Pij, are he closing prices of he agriculural fuures prices i of commodiy j for day and, respecively. The descripive saisics for he agriculural commodiy fuures reurns series are summarized in Table. The sample mean is quie small, bu he corresponding variance of reurns is much higher. Surprisingly, 4 of 6 reurn series have negaive average reurns, namely cocoa, coffee, coon and orange juice. The normal disribuion has a skewness saisic equal o zero and a kurosis saisic of 3, bu hese agriculural commodiy fuures reurns have high kurosis, suggesing he presence of fa ails, and 9 of 6 reurn series have negaive skewness, signifying he series have a longer lef ail (exreme loss) han righ ail (exreme gain). The Jarque-Bera (J-B) es Lagrange muliplier saisics of he agriculural commodiy fuures reurn series are saisically significan, hereby signifying ha he disribuions for hese reurns are no normal, which may be due parly o he presence of exreme observaions. [Inser Table here] Figures - presen he plos of synchronous agriculural commodiy fuures reurns. These indicae volailiy clusering, or periods of high volailiy followed by periods of ranquiliy, such ha agriculural commodiy fuures reurns flucuae in a range smaller han under he normal disribuion. However, here are some circumsances where agriculural commodiy fuures reurns oscillae in a much wider scale ha is permied by a normal disribuion. 4

15 [Inser Figures - here] The uni roo es for all commodiy fuures reurns are summarized in Table. All uni roo ess are conduced wih EViews 6 economeric sofware package. The Augmened Dickey- Fuller (ADF) and Phillips-Perron (PP) ess were used o explore he exisence of uni roos in he individual reurns series. The ADF es accommodaes serial correlaion by specifying explicily he srucure of serial correlaion in he error, bu he PP es allows fairly mild assumpions ha do no assume a specific ype of serial correlaion and heeroskedasiy in he disurbances, and can have higher power han he ADF es under a wide range of circumsances. These resuls are checked by also performing he KPSS es. The null hypohesis of he ADF and PP ess is ha he series have a uni roo, while he null hypohesis of he KPSS es is ha he series are saionary. In Table, based on he ADF and PP es resuls, he large negaive values in all cases indicae rejecion of he null hypohesis a he % level. In addiion, based on he KPSS es, he resuls indicae ha he null hypohesis is no rejeced a he % level, such ha all agriculural commodiy fuures reurns series are saionary. [Inser Table here] 4. Empirical Resuls This secion invesigaes a relevan framework of he condiional variance model hrough comparison among differen specificaions. The univariae condiional volailiy model, namely he GARCH model of Bollerslev (986), EGARCH model of Nelson (99) and APARCH model of Ding e al. (993), and fracionally inegraed class of models, namely he FIGARCH model of Baillie e al. (996), FIGARCH model of Chung (999), FIEGARCH model of Bollerslev and Mikkelsen (996), FIAPARCH model of Tse (998), and FIAPARCH model of Chung (996) wih Gaussian errors, are esimaed by QMLE, which allows for asympoically valid inferences when he sandardized innovaions are no normally disribued. Corresponding esimaes are obained using he BFGS algorihm. The compuaions are performed using he Ox/G@RCH 4. economerics sofware package of Lauren and Peers (006). 5

16 [Inser Table 3 here] The univariae esimaes of he condiional volailiies, GARCH(,), EGARCH(,) and APARCH(,) of each agriculural commodiy fuures reurns are given in Tables 3 o 5. Their respecive esimaes and robus -raios of each parameers are presened including informaion crieria, namely AIC and SIC. Table 3 presens he esimaes of he GARCH(,) models from equaion () and () for commodiy fuures reurns. The coefficiens in he condiional variance equaion are all significan, bu wih corn, coon and whea (Kansas Ciy whea), only in he long run. The deails of he univariae esimaes relaing o he srucural properies, namely he second momen and log-momen condiions, based on agriculural commodiy fuures reurns, are available from he auhors upon reques. [Inser Table 4 here] Table 4 shows he esimaes of he EGARCH(,) models from equaions (5) and (6) for commodiy fuures reurns. Mos commodiy fuures reurns show ha he esimaes of γ are saisically significan, meaning ha hese reurns have an asymmeric effec of negaive and posiive shocks on he condiional variance. Surprisingly, only for calef (cale feeder) and pork are boh esimaes of γ and γ saisically significan, and γ < 0, indicaing ha he condiional variance has a leverage effec. However, for he remainder, namely cale (live cale), coon, soybeans and soy bean oil), he esimaes of γ and γ are no saisically significan, meaning ha an asymmeric effec of negaive and posiive shocks on condiional variance is no presen. Therefore, he GARCH model is preferred o EGARCH for live cale, soybeans, soybean oil and palm oil. [Inser Table 5 here] Table 5 presens he esimaes of he APARCH(,) model from equaion (7) for agriculural commodiy fuures reurns. The power ( δ ) erm esimaed for APARCH is saisically significan for each of hese commodiy fuures reurns, ranging from for Chicago whea o.779 for soybean oil. The asymmeric ( γ ) erm in he APARCH(,) model is 6

17 saisically significan in 7 of 6 cases, whereas only 5 commodiies, namely coffee, soybeans, soybean meal, soybean oil, and Kansas Ciy whea, has γ < 0, which means ha hese condiional volailiies have leverage effecs. These resuls sugges ha GARCH may no be appropriae for commodiy fuures reurns. [Inser Table 6 here] The parameers esimaed for he FIGARCH-BBM(,) and FIGARCH-Chung(,) models are summarized in Tables 6-7. Table 6 presens he esimaed FIGARCH-BBM(,) from equaion (9). The ARCH effecs are saisically significan in 0 of 6 agriculural commodiy fuures reurns, while he GARCH effecs are saisically significan in 5 of 6 agriculural commodiy fuures reurns. However, he sum of he ARCH() and GARCH() effecs is greaer han one in 6 commodiies, namely live cale, cale feeder, cocoa, coffee, corn and coon, which indicaes nonsaionariy. The esimaed d parameers in FIGARCH in all commodiy fuures reurns lie beween 0 and, indicaing he sabiliy of he process, bu for coon he esimaed d parameer is no saically differen from 0, so i reduces o he GARCH model. [Inser Table 7 here] The resuls for he FIGARCH-Chung model from equaion (3) are repored in Table 7, and mirror hose in Table 6. The GARCH effecs are saisically significan for 5 of 6 commodiy fuures reurns, bu he ARCH effecs are saisically significan for of 6 commodiy fuures reurns. There are 6 commodiies, namely live cale, cale feeder, cocoa, coffee, corn and coon, for which he sum of he ARCH() and GARCH() effecs is greaer han, indicaing nonsaionariy. However, he esimaed d parameers in he FIGARCH- Chung model in all agriculural commodiy fuures reurns are saisically significan, and lie beween 0 and, hereby indicaing he sabiliy of he process. Therefore, he FIGARCH- Chung model is preferred o FIGARCH-BBM. [Inser Table 8 here] Table 8 presens he esimaes of he FIEGARCH(,) model from equaion (6). The 7

18 esimaed γ parameers are saisically significan in 9 of 6 agriculural commodiy fuures reurns, meaning hese reurns have asymmeric effecs of negaive and posiive shocks on he condiional variance. However, only for cale feeder are boh he esimaes of γ and γ saisically significan, and γ < 0, indicaing ha he condiional variance has a leverage effec. For he remainder, namely live cale, coffee, coon, soybeans, soybean oil, palm oil and rubber, he esimaes of γ and γ are no saisically significan, such ha an asymmeric effec of negaive and posiive shocks of equal magniude on he condiional variance is no observed. Thus, for hese agriculural commodiy fuures reurn series, he FIGARCH model is preferred o FIEGARCH. The esimaed d parameers in FIEGARCH are saisically significan in 9 of 6 cases and lie beween 0 and, hereby indicaing he sabiliy of he process. [Inser Tables 9 and 0 here] Tables 9 and 0 show he esimaes of he FIAPARCH(,) model of Tse (998) and FIAPARCH(,) model of Chung (999), respecively. Table 9 presens he esimaes of he FIAPARCH(,) model of Tse (998) for agriculural commodiy fuures reurns. The power parameer esimaes ( δ ) of all agriculural commodiy fuures reurns are saisically significan, and range from for live cale o.84 for orange juice. The asymmeric ( γ ) erm in he FIAPARCH(,) model is saisically significan in 9 of 6 cases, bu only 6 commodiies, namely coffee, soybeans, soybean meal, soybean oil, Chicago whea and Kansas Ciy whea, have γ < 0, so ha hese condiional volailiies have leverage effecs. In addiion, he esimaed d for all agriculural commodiy fuures reurns is saisically significan. Therefore, he FIGARCH model is no appropriae for modelling agriculural commodiy fuures reurns. Table 0 presens he esimaes of he FIAPARCH(,) model of Chung (999) for agriculural commodiy fuures reurns. The power parameer esimaes ( δ ) of all agriculural commodiy fuures reurns are saisically significan, and range from for live cale o.84 for orange juice. The asymmeric ( γ ) erm in he FIAPARCH(,) model is saisically significan in 7 of 6 cases, bu only for 4 commodiies, namely coffee, soybeans, 8

19 soybean oil, and Kansas Ciy whea, is γ < 0, so ha hese condiional volailiies have leverage effecs. In addiion, he esimaed d parameers for all agriculural commodiy fuures reurns are saisically significan, which leads o he same conclusion as FIAPARCH(,)-Chung, namely ha FIGARCH is no appropriae for modelling commodiy fuures reurns. 5. Concluding Remarks The paper esimaed he long memory volailiy model in 6 agriculural commodiy fuures reurns from differen fuures markes, namely CBOT for corn, oas, soybeans, soybean meal, soybean oil and whea; CME for live cale, cale feeder and pork; NBOT for cocoa, coffee, coon, and orange juice; KCBT for whea; TOCOM for rubber (RSS3); and MDEX for palm oil. The class of fracional GARCH models, namely FIGARCH of Baillie e al. (996), FIEGACH of Bollerslev and Mikkelsen (996), and FIAPARCH of Tse (998), were esimaed and compared wih he GARCH model of Bollerslev (986), EGARCH of Nelson (99), and APARCH of Ding e al. (993). The empirical resuls showed ha, following he oucomes of he uni roo ess, all agriculural commodiy fuures reurns series were found o be saionary. The EGARCH (,) model ou-performed GARCH(,), and he APARCH model was also preferred o GARCH(,). The robus saisics of he esimaed d parameers, indicaing long erm dependence, suggesed evidence of fracional inegraion in mos agriculural commodiy fuures markes. Consequenly, he fracionally inegraed models, namely FIGARCH(,d,) and FIEGARCH(,d,), performed significanly beer han radiional condiional volailiy models, such as GARCH(,) and EGARCH(,), for modelling agriculural commodiy fuures reurns. 9

20 References Baillie, R., T. Bollerslev and H. Mikkelsen (996), Fracionally Inegraed Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 73, Baillie, R., Y. Han, R. Myers, and J. Song (007), Long Memory Models for Daily and High Frequency Commodiy Fuures Reurns, Journal of Fuures Markes, 7, Baillie, R., Y. Han and T. Kwon (00), Furher Long Memory Properies of Inflaionary Shocks, Souhern Economic Journal, 68, Barkoulas, J., W. Labys and J. Onochie (997), Fracional Dynamics in Inernaional Commodiy Prices, Journal of Fuures Markes, 7, Bollerslev, T. (986), Generalized Auoregressive Condiional Heeroscedasiciy, Journal of Economerices, 3, Bollerslev, T. and H. Mikkelsen (996), Modeling and Pricing Long Memory in Sock Marke Volailiy, Journal of Economerics, 73, Brunei, C. and C. Gilber (000), Bivariae FIGARCH and Fracional Coinegraion, Journal of Empirical Finance, 7, Chung, C. (999), Esimaing he Fracionally Inegraed GARCH Model, unpublished paper, Naional Taiwan Universiy. Coakley, J., J. Dollery and N. Kellard (008), The Role of Long Memory in Hedging Effeciveness, Compuaional Saisics and Daa Analysis, 5, Conrad, C. and M. Lamla (007), The High-Frequency Response of he EUR-US Dollar Exchange Rae o ECB Moneary Policy Announcemens, KOF Working Papers, No 74. Crao, N. and B. Ray (000), Memory in Reurns and Volailiies of Fuures Conracs, Journal of Fuures Markes, 0, Davidson, J. (004), Momen and Memory Properies of Linear Condiional Heeroscedasiciy Models, and A New Model, Journal of Business and Economic Saisics,, 6-9. Degiannakis, S. (004), Volailiy Forecasing: Evidence from a Fracional Inegraed Asymmeric Power ARCH Skewed- Model, Applied Financial Economics, 4, Ding, Z., C. Granger and R. Engle (993), A Long Memory Propery of Sock Marke Reurns and a New Model, Journal of Empirical Finance,

21 Engle, R.F. (98), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 55, Engle, R.F. and T. Bollerslev (986), Modeling he persisence of condiional variances, Economeric Reviews, 5, -50. Geweke, J. (986), Modeling he Persisence of Condiional Variances: A Commen, Economeric Reviews, 5, Glosen, L., R. Jagannahan and D. Runkle (99), On he Relaion beween he Expeced Value and Volailiy and of he Nominal Excess Reurns on Socks, Journal of Finance, 46, Higgins, M. and A. Bera (99), A Class of Nonlinear ARCH Models, Inernaional Economic Review, 33, Hyun-Joung, J. (008), A Long Memory Condiional Variance Model for Inernaional Grain Markes, Journal of Rural Developmen, 3, Jefferis, K. and P. Thupayagale (008), Long Memory in Souhern Africa Sock Markes, Souh African Journal of Economics, 73, Jin, H. and D. Frechee (004), Fracional Inegraion in Agriculural Fuures Price Volailiies, American Journal of Agriculural Economics, 86, Kang, S. and S. Yoon (007), Long Memory Properies in Reurn and Volailiy: Evidence from he Korean Sock Marke, Physica A, 385, Lauren S. and J. Peers (006), G@RCH 4., Esimaing and Forecasing ARCH Models, London, Timberlake Consulans. Ling, S. and M. McAleer (00), Necessary and Sufficien Momen Condiions for he GARCH(r,s) and Asymmeric Power GARCH(r,s) Models, Economeric Theory, 8, Lux, T. and T. Kaizoji (007), Forecasing Volailiy and Volume in he Tokyo Sock Marke: Long Memory, Fracaliy and Regime Swiching, Journal of Economic Dynamics and Conrol, 3, McAleer, M. (005), Auomaed Inference and Learning in Modeling Financial Volailiy, Economeric Theory,, 3-6. McAleer, M., F. Chan, and D. Marinova (007), An Economeric Analysis of Asymmeric Volailiy: Theory and Applicaion o Paens, Journal of Economerics, 39, Nelson, D. (99), Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Economerica, 59,

22 Niguez %, T. (007), Volailiy and VaR Forecasing in he Madrid Sock Exchange, Spanish Economic Review, 0, Panula, S. (986), Modeling he Persisence of Condiional Variances: A Commen, Economeric Reviews, 5, Poon, S. and C. Granger (003), Forecasing Volailiy in Financial Markes: A Review, Journal of Economic Lieraure, XLI, Ruiz, E. and H. Veiga (008), Modelling Long-Memory Volailiies wih Leverage Effec: A- LMSV versus FIEGARCH, Compuaional Saisics and Daa Analysis, 5, Schwer, W. (990), Sock Volailiy and he Crash of 87. Review of Financial Sudies, 3, Sephon, P. (009), Fracional inegraion in agriculural fuures price volailiies revisied. Agriculural Economics, 40, 03-. Taylor, S. (986), Modelling Financial Time Series. New York, Wiley. Tse, Y. (998), The Condiional Heeroscedasiciy of he Yen-Dollar Exchange Rae, Journal of Applied Economerics, 93, Zakoian, J. (994), Threshold Heeroskedasiciy Models, Journal of Economic Dynamics and Conrol, 5,

23 Table. Descripive Saisics for Agriculural Commodiy Fuures Reurns Commodiy Mean Max Min S.D. Skewness Kurosis Jarque-Bera calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber

24 Table. Uni Roo Tess for Reurns Commodiy Augmened Dicky-Fuller Phillip-Peron KPSS N C C&T N C C&T C C&T calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noe: All enries are significan a he % level. 4

25 Table 3. Esimaed GARCH(,) Models Commodiy ω α β AIC SIC calef calel cocoa coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level. 5

26 Table 4. Esimaed EGARCH(,) Models Commodiy ω α β γ γ AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level. 6

27 Table 5. Esimaed APARCH(,) Models Commodiy ω α β γ δ AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level 7

28 Table 6. Esimaed FIGARCH(,)-BBM Models Commodiy ω d α β AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level. 8

29 Table 7. Esimaed FIGARCH(,)-Chung Models Commodiy ω d α β AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level. 9

30 Table 8. Esimaed FIEGARCH(,) Models Commodiy ω d α β γ γ AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level. 30

31 Table 9. Esimaed FIAPARCH(,)-BBM Models Commodiy ω d α β γ δ AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level 3

32 Table 0. Esimaed FIAPARCH(,)-Chung Models Commodiy ω d α β γ δ AIC SIC calef calel cocoany coffee corn coon oas orange pork soybean soymeal soyoil wheac wheak palm rubber Noes: () The wo enries for each parameer are heir respecive parameer esimaes and robus - raios. () Enries in bold are significan a he 5% level. 3