Forecasting Volatility of Returns for Corn using GARCH Models

Transcription

1 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 4 Agriculural Consorium of exas Forecasing Volailiy of Reurns for Corn using GARCH Models Naveen Musunuru * Mark Yu Arley Larson Deparmen of Agriculural Sciences, Norhwes Missouri Sae Universiy, Maryville, MO Deparmen of Environmenal and Agriculural Managemen, arleon Sae Universiy, Sephenville, X 7640 ABSRAC he purpose of his paper is o model and forecas volailiy of reurns for corn fuures prices using GARCH models. Non-linear models from he GARCH family, specifically GARCH and EGARCH are employed o assess he role of asymmeries and o analyze he ime varying volailiy of corn fuures prices. he resuls reveal ha he corn reurn series reac differenly o good and bad news. he presence of leverage effec would imply ha he negaive news has bigger impac on volailiy han posiive news of he same magniude. he esimaed volailiy models were compared using symmeric measures for heir forecasing accuracy. I is found ha he EGARCH model provides he bes ou of sample forecass for corn among all he GARCH specificaions. KEY WORDS: volailiy, forecasing, GARCH models, corn fuures INRODUCION Financial marke volailiy analysis has garnered he aenion of academics as well as marke paricipans across he world for he las wo decades. Volailiy can be defined as flucuaions in he sandard deviaion of daily reurns for he seleced asse or commodiy. Volailiy analysis is imporan as a risk managemen ool for hedging effeciveness, as well as, aiding in he selecion and managemen of asse porfolios (Jondeau and Rockinger 003). Commodiy prices flucuae coninuously hroughou he year due o changes in he underlying supply and demand variables. Analyzing he volailiy behavior of an agriculural commodiy, like corn, has implicaions for boh farmers and marke paricipans. For example, marke prices of agriculural commodiies ypically increase before harves and fall afer harves, hereby causing volailiy swings. Any surprising USDA crop repors, wheher hey be he condiion of curren crop progress or changes in he invenory of grain socks (eiher surpluses or shorages), immediaely pu he commodiy markes ino an acceleraion mode. Undersanding volailiy helps farmers in managing heir producion risks and making proper markeing decisions. his also helps * Corresponding auhor: NAVEEN@nwmissouri.edu

2 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 43 Agriculural Consorium of exas farmers in minimizing heir marke exposure during periods of higher volailiy. Volailiy analysis can also be helpful in developing an effecive hedge agains adverse price movemens. Marke invesors can also benefi from hese sudies in properly selecing and managing heir invesmen porfolio. Periods of excess volailiy help commodiy raders, especially day raders, o gain significan profis hrough rading sraegies ailored o volailiies. Knowledge abou he source of price volailiy can be useful o risk managers in making decisions abou he iming of heir decisions (Evans e al. 99). Price limis and conrac margins imposed by commodiy exchanges, also in par, depend upon he volailiy of corresponding commodiies. Commodiy raders who wrie opions also need o forecas he volailiy of he price process over he life ime of he opion (Alexander 00). Volailiy also has an imporan effec on he macro economy of a counry. For example, increased volailiy, beyond a cerain hreshold will increase he risk of losses o invesors and raise concerns regarding he sabiliy of a paricular marke and he overall economy (Pan and Zhang 006). Previous research on volailiy analysis has been mosly concenraed on he financial indices. Volailiy research in he commodiy markes ypically focused on undersanding he sources of volailiy and lile aenion has been paid o forecasing he volailiies. he purpose of he presen paper is o model and forecas volailiy of reurns for corn using differen ypes of GARCH models. We are also ineresed in examining wheher posiive and negaive shocks have an asymmeric effec on reurn volailiy and hereby provide evidence for any leverage effec in corn. he paper uses hree differen ypes of Generalized Auoregressive Condiional Heeroscedasiciy (GARCH) specificaions: he sandard GARCH, Exponenial GARCH (EGARCH), and he hreshold GARCH (GARCH) specificaions o model and forecas he volailiy (condiional variance). hese models are known o capure he characerisics of financial ime series such as ime varying volailiy, non-lineariy dependence, and volailiy clusering (See Pagan 996; Enders 004). he specifics of he ARCH model formulaions are discussed in deail in he nex secion. A quick review of recen lieraure shows various sources for volailiy and is applicaion in differen areas. For example, Bernanke and Gerler (999) discussed he role of volailiy of financial markes and is effec on moneary policy. Crao and Ray (000) sudied he volailiy of commodiy markes and concluded ha he volailiy is more persisen for energy markes han he currency markes. Bajpai and Mohany (008) used EGARCH model wih normal and non-normal errors o esimae he volailiy of exchange rae. heir resuls indicae a negaive relaionship beween exchange rae volailiy and U.S. coon expors o major counries. Brorsen and Irwin (987) invesigaed if here is a significan relaionship beween he echnical rading and increased volailiy of en differen commodiies. heir resuls show ha echnical rading is no a significan facor in conribuing o he volailiy of commodiies. According o Irwin e al. (008), recen surges in he volailiy of agriculural commodiies are due o srucural changes in he markes and srong linkages wih he energy complex. Crain and Lee (996) suggesed ha he grain price volailiy is influenced by changes in governmen programs and according o he auhors, volailiy ypically ransfers from fuures markes o cash markes. Wih regard o he forecasing abiliy, Cao and say (99) poin ou ha he GARCH model produces beer forecass han GARCH, EGARCH, and ARMA models on he U.S. sock exchange. Balaban (00) argues symmeric GARCH models provide relaively good forecass of monhly exchange rae volailiy in comparison wih asymmeric models.

3 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 44 Agriculural Consorium of exas he srucure of he paper is organized as follows: Secion II describes he economeric mehodology employed in he paper, Secion III describes he daa, Secion IV discusses he resuls obained from he analysis, and finally, he las secion summarizes he paper. MEHODOLOGY Our analysis of volailiy forecasing begins wih he calculaion of coninuously compounded daily reurns for corn based on he following equaion r p p () ln Where r represens he daily log reurns for corn, p denoes he daily selemen price for he commodiy, while p represens he selemen prices wih one lag. Random Walk Model. he behavior of asse prices relaing o is random naure has araced he aenion of researchers worldwide. Proponens of Efficien Marke Hypohesis (EMH) argue ha he asse prices ypically behave in a random fashion and any aemp o forecas fuure values based on is pas values is fuile (Fama 965, 970; Cooper 98). he basic model for esimaing he volailiy of reurns using OLS is he naïve random walk (RW) model and is given by: r () Where is he mean value of reurns, which is expeced o be insignificanly differen from zero under EMH, and is he error erm. he drawback of he above model is ha i can be used only o characerize he mean reurns. radiional economeric models such as ordinary leas squares are buil upon he assumpion of consan variance. he error variances may no be consan over ime. he assumpion of consan variance of he error erm is inconsisen wih financial ime series where he variance is heeroskedasic and ime-varying. In order o accoun for he ime varying volailiy which canno be capured hrough linear models like OLS, his sudy uses GARCH models. GARCH Specificaions. he Generalized Auoregressive Condiional Heeroscedasiciy (GARCH), was developed independenly by Bollerslev (986) and aylor (986), was used in he presen sudy o invesigae he effec of volailiy of corn fuures prices. he appeal of he GARCH model is ha i akes ino consideraion boh mean and volailiy in modeling he financial reurns, and has an advanage over he radiional regression models. I also has he abiliy o capure volailiy clusering, a characerisic of financial ime series, where large reurns are followed by large reurns, small reurns followed by small reurns, leading o coniguous periods of volailiy and sabiliy (Mandelbro 963). Rarely, any higher order model han GARCH (,) is needed o capure volailiy clusering (Alexander 00; Brooks 008). he GARCH model is based on he assumpion ha forecass of ime varying variance depend upon he lagged variance of he asse. he analysis of he model involves esimaion of wo disinc specificaions: one for he condiional mean and he oher for condiional variance.

4 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 45 Agriculural Consorium of exas he basic GARCH (,) can be represened as: r 0, (3) ; r r h h h (4) Where 0, 0, 0 are required o ensure ha he condiional variance is never negaive. he variance h is a funcion of an inercep, a shock from he prior period and he variance from he las period h. he ARCH erms indicaes he shor run persisence of shocks whereas he GARCH erm represens he conribuion of shocks o long run persisence. is a measure of persisence of volailiy clusering. If is very close o, i shows high persisence in volailiy clusering. he GARCH (,) is weak saionary if ( ). he above GARCH model assumes a symmeric volailiy response o marke news. According o GARCH specificaion, posiive and negaive shocks have he same effec on volailiy, as he unexpeced reurn always eners he condiional variance as a square. I has been suggesed in he financial lieraure ha negaive shocks in he marke have a larger impac on volailiy han posiive shocks of he same magniude (Aseriou and Hall 0; Brooks 008; Zivo 008; Bollerslev e al. 99; Engle and Ng 993). As a resul, Asymmeric GARCH models are more appropriae. wo Asymmeric GARCH models (GARCH and EGARCH) have been employed in he presen paper o sudy he possible asymmeries ypically aribued o leverage effecs for corn fuures reurns. Asymmery can be inroduced in he ARCH models by weighing differenly for posiive and negaive residuals, hus, r ; 0, (5) h h h I (6) his model is called GARCH, following he works of Zakoian (994) and Glosen e al. (993) where α, β, and γ are consan parameers and I is an indicaor dummy variable ha akes he value of if 0 and zero oherwise. When is posiive, he oal conribuion o volailiy is and when is negaive, he oal conribuion o he volailiy is. he GARCH (,) model is asymmeric as long as 0. he GARCH models can be exended o higher order specificaions by including more lagged erms. he GARCH (p,q) model is defined by adding p erms o he righ side of equaion (6), so ha h p i i ii i jh j q j (7)

5 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 46 Agriculural Consorium of exas he parameers in he model usually consrained by 0, 0, 0 and 0. he EGARCH specificaion of condiional volailiy due o Nelson (99) may be expressed as: r 0, (8) ; r ln h lnh h h h (9) As he name indicaes, EGARCH assumes condiional variance as exponenial, whereas GARCH reas condiional variance as quadraic. he above model has several advanages over he radiional GARCH specificaion. As h is modeled in log form, even if he parameers are negaive, h becomes posiive. Anoher advanage is allowance of asymmeries in he EGARCH model formulaion. In EGARCH, capures he asymmerical effec and herefore any non-zero values shows he impac of any exernal even being asymmeric. For deailed informaion on GARCH models readers may refer o Bollerslev e al. (99, 994). Forecasing Mehodology. he random walk and GARCH models are evaluaed in erms of heir abiliy o forecas fuure reurns. he forecasing performance of each model is evaluaed by using sandard symmeric measures: he roo mean square error (RMSE), he mean absolue error (MAE), he mean absolue percen error (MAPE), and he heil inequaliy coefficien (IC). he forecasing saisics are given as follows: RMSE ˆ Where is one sep ahead volailiy forecas, number of forecass. MAE ( ˆ ) (0) is he acual volailiy and is he ˆ () MAPE ˆ () IC ( ˆ ) ( ˆ ) ( ) (3)

6 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 47 Agriculural Consorium of exas he heil inequaliy coefficien is he scaled measure ha always lies beween 0 and where zero indicaes a perfec fi. he bes model for forecasing is he one wih he smalles value for ha measure. he daa used in he presen paper is he daily selemen prices for corn, covering he period of January 3, 995 o June 6, 0, excluding public holidays. In order o eliminae price disorions caused by price gaps locaed beween expiring conracs and subsequen fuures conracs, his sudy used coninuous corn fuures conrac developed from he selemen prices. he oal sample comprises 3954 observaions spanning approximaely seveneen years of daily daa. Corn is raded on he Chicago Board of rade (CBO) and is he mos acively raded (liquid) conrac among all he agriculural commodiies. As of June 0, he average daily volume for December 0 corn is 37,33 conracs wih an open ineres of 40,8. In order o make forecass, he full sample is divided ino wo pars: an in sample of 3954 observaions (January 03, 995 o Sepember 6, 00) and an ou of sample of 439 observaions (Sepember 7, 00 o June 6, 0). he las 0% of observaions are reserved for forecasing purposes. RESULS Figure represens he price index for corn (panel a) and he ime series of daily reurns calculaed from he selemen prices (panel b) for he sudy period. Visual inspecion of he reurn series shows ha he mean reurns are consan bu he variances change over ime. he commodiy exhibis volailiy clusering propery indicaing periods of high volailiy (urbulence) and low volailiy (ranquiliy). From he figure, i is eviden ha he volailiy of corn had increased significanly during he recen imes when compared o he iniial periods. Periods of high volailiy show large posiive and negaive reurns when compared o he low volailiy periods. he boom par of figure consiss of hisogram of reurns (panel c) and a Gaussian QQ plo (panel d). he disribuion of reurns is characerized by a high peak a he cener, which is considered o be a sylized fac of financial ime series. For a deailed discussion of sylized facs, please see aylor (005) and Kovacic (008). he QQ plo plos he quaniles of wo disribuions: he empirical disribuion of corn reurns and he hypohesized Gaussian disribuion. he QQ plo clearly shows ha he disribuion ails for corn are heavier han he ails of he Gaussian disribuion.

7 seriesdaa(corn.s) he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 48 Agriculural Consorium of exas Daily Closing Prices for Corn Corn Daily Reurns (a) Hisogram of Corn Daily Reurns (b) QQ Plo of Corn Daily Reurns (c) - 0 (d) Figure. Corn Daily Reurns and ail Disribuion. he descripive saisics for he ime series of daily reurns for corn are presened in able. his able includes minimum, maximum, average daily reurns, sandard deviaion, skewness, kurosis, and Jarque-Bera saisics of he reurns. As expeced of financial ime series, he mean of reurns is close o zero. Posiive mean reurns show ha he price series of corn has increased hrough ime. he sandard deviaion of he daily reurns is.847% which is equivalen o an annualized volailiy of 9.3%. Corn shows high sandard deviaion and herefore considered o be a volaile commodiy. he saisics also show a subsanial difference beween maximum and minimum reurns for his commodiy. he presence of sligh negaive skewness indicaes ha he lower ail of he disribuion was hicker han he upper ail and decline in reurns are more common han is increases. he kurosis for he ime series is 7, which is above he normal value of 3, and is considered as lepokuric in naure. Generally, eiher a very high or very low kurosis value indicaes lepokuric or playkuric disribuion of he sample daa. he Jarque-Bera saisics indicae ha he reurn series is non-normal and significan as evidenced by is p-value. hese findings are consisen wih earlier discussion relaed o he hisogram of reurns and QQ plo.

8 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 49 Agriculural Consorium of exas able. Descripive Saisics of Daily Reurns for Corn. Mean Skewness Maximum Kurosis Minimum Jarque-Bera Sd. Deviaion Probabiliy able shows he esimaion resuls for he mean and variance equaions for random walk (RW), GARCH, GARCH, and EGARCH models of volailiy for corn. he z saisics are also repored in he parenheses for each model. he resuls for RW model sugges ha he mean of he reurn series is no significanly differen from zero, which is consisen wih he random walk hypohesis. he Ljung-Box Q saisics of he sandard residuals (9.9), squared residuals (40.58) and ARCH-LM ess (5.99) are significan and show he presence of significan ARCH effecs in he model. Since he OLS esimae of RW is an inadequae model o capure he financial reurn characerisics such as ime varying volailiy and volailiy clusering, GARCH models were furher used o undersand he naure of commodiy daa. he model rankings also sugges ha he RW model is he leas preferred model among all he specificaions. Columns 3, 4, and 5 in able show he mean reurns and variance equaion of he GARCH (,), GARCH (,), and EGARCH (,) models respecively for he volailiy esimaion. Preliminary analysis suggess ha he condiional mean equaion for corn was bes modeled as an auoregressive process, especially, an AR (). he recen lieraure also suggess he inclusion of AR () is useful in order o remove any serial correlaion in he reurns which may be caused by non-synchronous rading (Lo and MacKinlay 988; Campbell e al. 997; say 00). hus he mean equaion in all he GARCH specificaions includes an AR () erm for his sudy. he z saisics indicae he significance of he inercep and coefficiens a 5% significance level. he mean daily reurns range from % o % for all he GARCH specificaions, whereas only GARCH (,) coefficien proved o be significan a 5% level. From he mean equaion in he GARCH models, we also observe ha he lagged value ( ) is significan for corn for all he specificaions indicaing ha he reurns of his commodiy exhibi serial correlaion and reflecs inefficiency during he period of sudy. he coefficiens of he condiional variance equaion, α and β, are posiive and significan for all he GARCH models suggesing srong suppor for ARCH and GARCH effecs. he GARCH coefficien (β) can be used o undersand he impac of pas volailiy on curren volailiy. he GARCH coefficien is significan a 5% level suggesing ha he curren volailiy is affeced by pas volailiy for corn. As ypical of GARCH models for financial reurns, he sum of he coefficiens on lagged squared error (α) and lagged condiional variance (β) is very close o one implying ha shocks o he condiional variance will be highly persisen for corn. A high persisence indicaes ha he shocks are likely o die slowly. If here is a new price shock, i will have implicaion on reurns for a longer period. he only excepion here is EGARCH model where sum of boh α and β coefficiens are greaer han one and parameers are overesimaed. he asymmeric (leverage) coefficien γ capures he impac of negaive versus posiive shocks on volailiy. Leverage coefficien (γ) when greaer han zero under he GARCH model, indicaes ha he negaive shocks cause more volailiy han posiive shocks. Accordingly, γ is posiive and significan for corn suggesing he presence of leverage effec. For his commodiy, negaive shocks end o cause more volailiy han

9 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 50 Agriculural Consorium of exas posiive news. Under EGARCH model, when he leverage coefficien is less han zero, hen he posiive shocks (good news) generae less volailiy han negaive shocks (bad news). Accordingly, wih a negaive and significan γ, he resuls indicae ha negaive news caused more volailiy for corn confirming he earlier resuls of GARCH model. able. Volailiy Models and heir Corresponding Resuls. Parameer RW GARCH (,) GARCH (,) EGARCH (,) Mean Equaion (0.7) * (.4) 0.036* (.6) Variance Equaion 3.44E-06* (9.7) 0.069* (7.6) 0.94* (380.87) (.64) 0.040* (.38) 3.77E-06* (8.40) 0.055* (.58) 0.99* (358.66) 0.038* (5.06) LB 0 LB 0 ARCH- LM es 9.9* (0.03) 40.58* (0.00) 5.99* (0.0) 8.8 (0.50) 3.5 (0.95) (0.93) 8.33 (0.50) 3.43 (0.94) (0.98) (.53) 0.044* (.74) * (-4.68) 0.6* (6.4) 0.98* (48.37) -0.04* (-.4) 7.74 (0.56) 3.5 (0.95) 0.05 (0.90) AIC LL is AR() coefficien; *denoes significance a 5% level. Numbers in parenheses below coefficien esimaes are z saisics. AIC, LL are Akaike informaion crieria, and log likelihood respecively. LB 0 and LB 0 are he Ljung-Box saisics for he sandardized and squared sandardized residuals using 0 lag, respecively. Numbers in parenheses below he LB saisics and arch coefficiens are he p-values. Superscrip denoes he rank of model. Finally, o deermine which GARCH model provides a reasonable explanaion of behavior of commodiy reurns, some diagnosic ess are performed. he diagnosic ess resuls show ha he GARCH models are correcly specified and here are no remaining ARCH effecs in all he esimaed GARCH models. he Ljung-Box Q saisics for he sandard residuals and squared residuals are insignifican, suggesing ha all he GARCH models are correcly specified (able ). Overall, using he minimum AIC, maximum log likelihood values as model selecion crieria (Alagidede and Panagioidis 006) for he GARCH specificaions, he model rankings indicae ha he

10 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 5 Agriculural Consorium of exas EGARCH (,) is he preferred model for corn and capures mos of he ime series characerisics of he reurns during he sudy period. he models were also evaluaed in erms of heir abiliy o forecas volailiy of fuure reurns. he measures of forecas evaluaion used in he presen sudy include roo mean square error (RMSE), mean absolue error (MAE), mean absolue percen error (MAPE) and heil s inequaliy coefficien (IC). able 3 repors he forecas performance values and he corresponding ranking for all he GARCH models. he resuls indicae ha he relaive differences among forecasing performance measures are quie small and he larges relaive difference beween he bes and wors performing models for ou of sample daa using IC is approximaely 4%. Figure presens he ou of sample volailiy forecas and variance forecas of he corn reurns. he forecasing resuls show ha EGARCH (,) model is he mos preferred among all he models and he naïve RW model performed worse in forecasing he volailiy of reurns for corn. hus he EGARCH model was found o be he bes model o sudy he volailiy behavior and he corresponding forecasing of reurns. able 3. Forecas Performance of he Esimaed GARCH Models. GARCH GARCH Forecas Crieria RW (,) (,) EGARCH (,) Roo Mean Square Error (RMSE) Mean Absolue Error (MAE) Mean Absolue % Error (MAPE) heil Inequaliy Coefficien (IC) Overall Rank 4 3 Forecas sample: Sepember 7, 00 o June 6, 0; superscrip indicaes he rank of he model

11 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 5 Agriculural Consorium of exas GARCH Forecas ± S.E. Forecas of Variance GARCH Forecas ± S.E. Forecas of Variance EGARCH Forecas ± S.E. Forecas of Variance Figure. Volailiy Forecas and Forecas of Variance Graphs. CONCLUSIONS his paper conribues o he exising body of lieraure in wo aspecs: firs, mos of he volailiy sudies seen in he financial lieraure are focused on sock

12 he exas Journal of Agriculure and Naural Resources 6:4-55 (03) 53 Agriculural Consorium of exas exchanges and agriculural commodiies were no explored in deail. By focusing on he mos liquid member of agriculural commodiy group, his sudy aemps o undersand he volailiy behavior for corn. Second, we analyzed alernaive group of GARCH models in order o find he bes model ha can be used o undersand and forecas he commodiy reurns. he significance has been esed using a radiional OLS model, a non-linear symmeric GARCH (,) model, and wo non-linear asymmeric models, GARCH (,) and EGARCH (,). Under GARCH models, he resuls indicaed ha he sum of he coefficiens on he lagged squared error and lagged condiional variance is close o uniy for corn indicaing ha he shocks o he condiional variance will be highly persisen. he leverage effec erm in boh he GARCH and EGARCH specificaions for corn is saisically significan indicaing negaive shocks imply a higher nex period variance han posiive shocks of he same magniude. From he overall resuls, i is eviden ha he EGARCH model performs well wih he daase and seems o capure he dynamics of he corn marke including ime varying volailiy. Agriculural commodiies ypically exhibi periods of high volailiy semming from boh posiive and negaive shocks of new informaion. Marke paricipans adjus o volailiies caused by new informaion as quickly as possible and ry o profi from such inefficiencies. he empirical resuls of his paper sugges, ha by properly analyzing he volailiy of agriculural commodiies, marke paricipans, wheher hey be farmers or invesors, are beer prepared for shifs in marke momenum and in managing heir marke decisions. REFERENCES Alagidede, P. and. Panagioidis Calendar Anomalies in he Ghana Exchange. Working Paper, Deparmen of Economics, Loughborough Universiy, Loughborough, UK. Alexander, C. 00. Marke Models: A Guide o Financial Daa Analysis. John Wiley & Sons, New York. Aseriou, D. and S. Hall. 0. Applied Economerics. Palgrave MacMillan. New York. Bajpai, S. and S. Mohany Impacs of Exchange Rae Volailiy on he U.S. Coon Expors. Paper Presened a he Souhern Agriculural Associaion Annual Meeings, February -6, 008. Dallas, exas. Balaban, E. 00. Comparaive Forecasing Performance of Symmeric and Asymmeric Condiional Volailiy Models of an Exchange Rae. Universiy of Edinbergh, Working Paper Series, 0-06: -4. Bernanke, B. and M. Gerler Moneary Policy and Asse Price Volailiy, In New Challenges for Moneary Policy. A Symposium Sponsored by he Federal Reserve Bank of Kansas Ciy, Bollerslev, Generalized Auoregressive Condiional Heeroscedasiciy. Journal of Economerics 3, Bollerslev,., Chou, R.Y., and K.F. Kroner. 99. ARCH Modeling in Finance. Journal of Economerics 5: Bollerslev,., Engle, R.F., and D.B. Nelson ARCH Model, In Handbook of Economerics IV, , Ed. Engle, R.F.,and D.C. McFadden. Amserdam. Elsevier Science.

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