HIGH FREQUNCY ANALYSIS ON JUMPS AND LONG MEMORY VOLATILITY IN COMMODITY FUTURES PRICES*

Transcription

1 HIGH FREQUNCY ANALYSIS ON JUMPS AND LONG MEMORY VOLATILITY IN COMMODITY FUTURES PRICES* by Youg Wook Ha + Departmet of Ecoomics Hallym Uiversity Korea ad Jeogseok Sog Departmet of Iteratioal Trade Chug-Ag Uiversity Korea This versio: November 006 *Youg Wook Ha s work is fiacially supported from Hallym Uiversity Research Grat. The authors are also grateful to Robert J. Myers ad the Istitute for Fiacial Markets for makig available their real time commodity futures prices. This is a very prelimiary versio. Please do ot cite without authors permissio. + Correspodece author: Youg Wook Ha, Departmet of Ecoomics, Hallym Uiversity, Chucheo, Gagwo-Do, Korea ywha@hallym.ac.kr. Phoe: Fax:

2 HIGH FREQUNCY ANALYSIS ON JUMPS AND LONG MEMORY VOLATILITY IN COMMODITY FUTURES PRICES* Abstract We cocer the high frequecy returs of 15 miute commodity futures prices. The basic FIGARCH model with the usual ormality assumptio is foud to be iappropriate i represetig the high frequecy commodity futures returs ad the rejectio appears to be due to the jumps which are occurred i the high frequecy returs. Hece, this paper relies o the geeralized FIGARCH model combied with the Beroulli distributio that allows for jumps i the high frequecy commodity futures returs. This paper shows that the geeralized FIGARCH-Beroulli distributio model performs quite well ad that the jumps spuriously icrease the log memory persistece i the volatility process of the high frequecy commodity futures returs. Keywords: High frequecy commodity futures, Jumps, FIGARCH, Beroulli distributio, Log memory property. EMF classificatio code: 40 1

3 INTRODUCTION We cosider the high frequecy 15 miute commodity futures prices of cattle, cor, hogs ad gasolie. 1) I particular, we focus maily o fidig a appropriate model of the high frequecy commodity futures prices. Sice there has bee the apparet lack of ay ecoomic theory explaiig the dyamics of the first two coditioal momets i asset prices icludig commodity futures prices, may ecoometricias have commoly used the exteded models of the traditioal ARMA models for the meas ad the ARCH models for the variaces to describe ad represet the dyamic process of the asset prices. These traditioal models have usually bee estimated by the approximate Quasi Maximum Likelihood (QMLE) method uder the assumptio that the iovatios are ormally distributed. Ad, the ormality assumptio has bee justified by Bollerslev ad Wooldridge (199). Thus, we characterize the process of the high frequecy commodity futures returs by applyig some recet developmets i modelig the volatility process. Firs we use the relatively recet FIGARCH model of Baillie et al. (1996) with the Gaussia ormality assumptio i order to represet the high frequecy commodity futures returs. The primary results from the Maximum Likelihood Estimatio (MLE) of the basic FIGARCH model idicate that the FIGARCH model uder the Gaussia ormality assumptio geerally seems to match the dyamics of high frequecy futures returs ad is a satisfyig startig poit for studyig the

4 uderlyig features of the high frequecy futures returs data. O the other had, usig the usual FIGARCH model with the ormal distributio assumptio leads to excess kurtosis, which may be related to coditioal mea jumps i the high frequecy futures returs. Frases ad Ghijlsels (1999) have proposed that the estimated residuals from GARCH model have excess kurtosis due to eglected additive outliers (AOs). These jumps might lead to outliers i the level ad volatility process that caot be take accouted for by the simple ormal distributio model (Hotta ad Tsay, 1998). Accordigly, this paper aalyzes jumps i the coditioal mea process of the high frequecy futures retur series. Jumps i the coditioal mea process are of sigificat iteres ad the log memory property i the coditioal variace process caot be extracted without a appropriate specificatio of the coditioal mea process. The basic FIGARCH model assumig a ormal distributio is ulikely to represet the process of high frequecy futures returs with a mixture of distributios. For this purpose, it appears more useful to apply the jump diffusio process proposed by Press (1967) i order to properly accout for the jumps. Sice the statistical ad ecoomic explaatios for the jumps ad the log memory property are quite differe this paper employs a ormal mixture distributio model, the FIGARCH model combied with Beroulli jump process to accout for the jumps i the coditioal mea process ad the log memory property i the coditioal variace process. Thus, i order to cosider the existece of the jumps, we adopt the geeralized 3

5 FIGARCH model combied with the Beroulli distributio which allows for a jump possibility istead of the usual ormal distributio assumig the jump probability is costat ad is exogeously determied. We fid that the FIGARCH-Beroulli distributio model performs quite well ad that a ormal mixture distributio model, the FIGARCH model combied with the Beroulli jump process, ca improve estimates of the log memory parameter. Specificatio of the coditioal mea process without cosiderig the jumps seems to cause distorted higher estimates of the log memory parameter i the volatility process of the high frequecy futures returs. This is quite uderstadable give that the jumps which otherwise may be spuriously associated with additioal volatility are fully accouted for i the mixture distributio. The results of this paper ca provide importat implicatios for the uderstadig of the itraday dyamics of the high frequecy commodity prices ad hece for empirical applicatios such as optimal hedge ratio estimatio, tests for futures market efficiecy, tests for the risk maageme optio valuatio ad portfolio maagemet. The pla of the rest of this paper is as follows. The ext sectio describes the 15 miute commodity futures returs of cattle, cor, hog ad gasolie ad the basic properties of the high frequecy commodity futures returs data. I particular, a strog itraday periodicity ad a log memory property are foud to be very sigificat i the high frequecy commodity futures returs. This is followed by the applicatio of the log memory volatility, FIGARCH model to 4

6 represet the high frequecy commodity futures returs. For the aalysis of the high frequecy commodity returs, we first apply the Flexible Fourier Form (FFF) proposed by Gallat (1981, 198) to elimiate the itraday periodicity i the high frequecy commodity futures returs ad the uses the basic FIGARCH model of Baillie et al. (1996) with a ormal distributio to estimate the log memory property i the volatility process of the high frequecy filtered commodity futures returs. The FIGARCH model is foud to be ecoometrically superior to the model of the regular stable GARCH model. But the primary results show excess kurtosis which caot be accouted for by the ormal distributio model. Ad, the ext sectio the aalyzes jumps i the coditioal mea process of the high frequecy futures returs usig a ormal mixture distributio model. The FIGARCH model combied with Beroulli process is to represet the coditioal mea jumps ad the log memory volatility process of the high frequecy commodity futures returs. I particular, the Beroulli jump process is foud to be quite appropriate for accoutig for the coditioal mea jumps ad i capturig the effects of the jumps o the high commodity frequecy commodity futures returs data. The fial sectio provides a brief coclusio. BASIC ANALYSIS OF HIGH FREQUENCY FUTURES RETURNS We examie four high frequecy commodity futures data; cattle, cor, hogs, ad 5

7 gasolie, which are obtaied from the Futures Idustry Istitute data ceter. The high frequecy commodity futures prices are for real-time trasactio records, which we iitially covert to 15- miute price itervals by usig the last price quoted before the ed of every 5-miute iterval over the tradig day. ) Cattle ad hogs are both importat livestock commodities i U.S. agriculture but their differet life cycles mea differet iheret price dyamics, eve though they are foud to have a lot of similarity i the stochastic properties of prices for these two livestock commodities as preseted by Baillie et al. (007). Cor is a major aual crop that is of critical importace to U.S. agriculture sice it is used heavily as aimal feed, ad Gasolie is icluded to see if results are markedly differet for a atural resource based commodity. The returs of the 15-miute commodity futures prices are defied i the covetioal maer as cotiuously compouded rates of retur ad calculated as the first differece of the atural logarithm of prices. The -th iterval retur durig day t is defied as R = 100 [l( Pt, ) l( Pt, 1 )] (1) where t = 1,..,T (tradig days), = 1,,K (itraday itervals) ad Pt, is the futures price for the -th itraday iterval durig tradig day t. The details of the basic statistics ad the sample periods used for the raw (uadjusted) 15-miute futures returs are provided i Table 1. For example, the sample mea of the 15-miute cor futures retur is foud to be which are very close to zero ad idistiguishable at the stadard sigificace level give the sample 6

8 deviatios of 0.01 ad 0.1. Ad, Figure 1 displays correspodig picture of the 15 miute cor future returs represetig that the returs are cetered o zero but there exist several jumps (large chages) ad obvious volatility clusterig i the high frequecy futures retur series. However, the high frequecy cor returs appear ot to be ormally distributed sice the sample skewess ad kurtosis are ad 5.148, which are all foud to be statistically sigificat. I particular, the estimated kurtosis statistics for the high frequecy cor futures returs is foud to be relatively large, which implies the rejectio of a Gaussia ormal distributio assumptio. The high excess kurtosis may be due to the occurreces of umerous jumps that have take place i the high frequecy cor futures returs as preseted i Figure 1. These jumps could lead to the level ad volatility outliers that the ormal distributio caot take ito accout (Hotta ad Tsay, 1998). Actually, the high frequecy cor futures returs are characterized by several large jumps or shifts followed by ostesibly radom movemet. The jumps i the high frequecy cor futures returs may be caused by several fiacial ad ecoomic evets i the cor futures markets. The correspodig graphs for the other commodities are ot show to reserve space but they all exhibit the quite similar patter. The volatility processes i the high frequecy commodity futures returs are further aalyzed. Figure plots the sample autocorrelatios of high frequecy cor futures returs for lags of up to 10 tradig days i 15-miute itervals displayed i the horizotal axis for the 7

9 returs, the squared returs ad the absolute returs of the raw 15-miute high frequecy cor futures returs series. I particular, Figure shows that there geerally exists a small egative but sigificat first-order autocorrelatio i returs, which may be due to the o-sychroous tradig pheomeo while higher order autocorrelatios are ot sigificat at covetioal levels. The autocorrelatio fuctios of the absolute returs exhibit a proouced U shape suggestig substatial itraday periodicity ad decay very slowly at the hyperbolic rate, which is a typical feature of a log memory property. These are i lie with the fidigs of Cai et al. (001) who characterized similar itraday periodicity i the 5-miute high frequecy gold prices. To coserve space the correspodig graphs for the other commodities are ot show but are available upo request to the authors. However, it is observed that the similar shapes ad the amplitudes of the itraday periodicity i the autocorrelatios of absolute returs exist i the other commodities. This seasoal patter seems to be closely related to the itraday tradig activity i commodity futures markets as preseted by Muller et al. (1990) ad Cai et al. (001). FIGARCH MODEL WITH A NORMAL DISTRIBUTION As with may aalyses of high frequecy asset price returs like stocks, bods ad exchage rates (Aderse et al., 005), it is foud that the high frequecy commodity futures returs display cosiderable itraday periodicity, which is usually attributed to istitutioal 8

10 tradig features. This periodicity is removed usig the FFF filterig method of Gallat (1981, 198), which is explaied i detail i Baillie et al. (007). Thus, the filtered high frequecy 15- mimute commodity futures returs is defied as, y R / s = where s is the itraday periodicity estimated from FFF. Figure 3 represets the correlograms of the filtered high frequecy cor futures returs while the correlograms of the other commodities are ot icluded i this paper to reserve space but they are available upo the request to the authors. It shows that the FFF filter seems to remove much of itraday periodicity preseted i the raw absolute returs successfully as i Baillie et al. (007). The filtered high frequecy cor futures returs (y )are virtually foud to be statioary with small autocorrelatios at the first few lags. O the other had, the volatility processes of the filtered high frequecy cor futures returs are foud to be very persistetly autocorrelated with log memory hyperbolic decay. The log memory patters i the volatility process of the high frequecy cor futures returs are almost same as the patter i the high frequecy gold futures returs i Cai et al. (001). 3) The correlograms of the other high frequecy futures returs are also foud to exhibit the similar patters. A model that is cosistet with these stylized facts is the MA()-FIGARCH(p, d, q) process, y R / s = μ + θ ( L) ε = () ε = (3) z σ 9

11 d [ 1 β ( L)] σ = ω + [1 β ( L) φ( L)(1 L) ] ε (4) where s is the itraday periodicity estimated from FFF, ad ~ i. i..(0, 1), P t is the asset z t, d price, μ ad ω are scalar parameters, ad β(l) ad φ(l) are polyomials i the lag operator to be defied later. The polyomial i the lag operator associated with the movig average process is θ ( L) = 1+ θ1 L + θ L θ L, ad d cotiues to represet the log memory parameter. The FIGARCH model i equatio (4) is motivated by otig that the stadard GARCH (p, q) model of Bollerslev (1986) ca be expressed as σ = ω+ α( L) ε + β( L) σ, t t t where the polyomials are α α α α q ( L) 1L+ L ql, β β β β ( L) 1L+ L p pl. The GARCH(p, q) process ca also be expressed as the ARMA[max(p, q), p] process i squared iovatios [ 1 ( L) ( L) ] [ 1 ( L) ] α β ε = ω + β υ where t t υ ε σ ad is a zero mea, t t t, serially ucorrelated process which has the iterpretatio of beig the iovatios i the coditioal variace. Similarly, the FIGARCH(p, d, q) process ca be writte aturally as [ ] φ ε = ω + β υ, (5) ( L)(1 L) d t 1 ( L) where φ( L) = [1 α( L) β ( L)] is a polyomial i the lag operator of order max(p, q). t Equatio (5) ca be easily show to trasform to equatio (4), which is the stadard represetatio for the coditioal variace i the FIGARCH(p, d, q) process. Further details cocerig the FIGARCH process ca be foud i Baillie et al. (1996). The parameter d characterizes the log memory property of hyperbolic decay i volatility because it allows for 10

12 autocorrelatios decayig at a slow hyperbolic rate. The above model (), (3), ad (4) is estimated for futures returs o our four commodities of iterest by maximizig the Gaussia log likelihood fuctio, T k T 1 l( L; Θ ) = ( )l(π ) ( ) [l( σ, + ε σ ] (6) t= 1 = 1 where Θ is a vector cotaiig the ukow parameters to be estimated. However, it has log bee recogized that most asset returs are ot well represeted by assumig z t i equatio () is ormally distributed; for example see McFarlad et al.(198). Cosequetly, iferece is usually based o the QMLE of Bollerslev ad Wooldridge (199), which is valid whe z t is o- Gaussia. Deotig the vector of parameter estimates obtaied from maximizig (6) usig a sample of T observatios o equatios (), (3) ad (4) with z t beig o-ormal by ^ T Θ, the the limitig distributio of ^ T Θ is ^ 1/ 1 1 T T ( Θ Θ ) N[0,A( Θ ) B( Θ )A( Θ ) ], (7) where A(.) ad B(.) represet the Hessia ad outer product gradiet respectively, ad Θ 0 deotes the vector of true parameter values. Equatio (7) is used to calculate the robust stadard errors that are reported i the subsequet results i this paper, with the Hessia ad outer product gradiet matrices beig evaluated at the poit ^ T Θ for practical implemetatio. Cosiderable previous work i the literature has examied high frequecy returs i stock, equity ad foreig exchage markets, but to date very little aalysis has bee doe o 11

13 high frequecy commodity returs (Cai et al., 001; Martes ad Zei, 004; Baillie et al. 007). This sectio of the paper represets a extesive aalysis of the volatility properties of high frequecy commodity futures returs usig the FIGARCH model with the ormal distributio of Baillie et al. (1996). The orders of the MA ad GARCH polyomials i the lag operator are chose to be as parsimoious as possible but still provide a adequate represetatio of the autocorrelatio structure of the high frequecy data. The exact parametric specificatio of the model that best represets the degree of autocorrelatio i the coditioal mea ad coditioal variace of high frequecy commodity returs are foud to be the MA(1)-FIGARCH(1, d, 1) model for cattle, hogs ad cor ad the MA(1)-FIGARCH(0, d, 1) model for gasolie. 4) Table presets results of applyig the above model to high frequecy commodity futures returs for the four commodities discussed earlier. All the models have small but sigificat MA(1) parameter estimates, which is usually attributed to the o-sychroous tradig pheomeo. The estimated log memory volatility parameters, d, are i the rage betwee 0.0 ad 0.35 for most of the commodities cosidered ad are geerally statistically sigificat idicatig the sigificat log memory characteristics i the volatility of the high frequecy returs. Thus, the hypotheses that d = 0 (statioary GARCH) ad also d =1 (itegrated GARCH) are cosistetly rejected for all commodities usig stadard sigificace levels. 1

14 Table also reports the robust Wald test statistics, deoted by W, for testig the ull hypothesis of GARCH versus a FIGARCH data geeratig process. Uder the ull, W will have a asymptotic χ distributio ad, from Table, the GARCH model is rejected for every 1 commodity at stadard sigificace levels. This robust Wald test supports the coclusio obtaied both here ad i Ji ad Frechette (004) that FIGARCH is superior to GARCH for modelig the coditioal variaces of the high frequecy commodity futures returs. Evidetly, the log memory property is the characteristic feature of high frequecy commodity futures returs, ad FIGARCH represets a sigificat improvemet over GARCH. Thus, the estimated MA-FIGARCH models appear to describe the futures retur data quite well so that it may be a satisfyig startig poit to aalyze the ature of the uderlyig distributios i the high frequecy commodity returs. O the other hads, the focus of this paper is primarily directed at the assumptio of the Gaussia ormal distributio. Uder the ormality assumptio, the estimated excess kurtosis are foud to be 4.7, 6.6, 6.1 ad 4.7 for the high frequecy futures returs of the cattle, cor, hog ad gasolie respectively, which are eough to reject the ormal distributio. The ormal distributio assumptio seems to lead to excess kurtosis, ad the excess kurtosis may be resulted from the jump (large chages) i the high frequecy commodity futures prices caused by several fiacial ad ecoomic evets as preseted i Figure 1 for the cor futures returs. These evets 13

15 cocerig expected future flows ca result i price chages well above ormal ad might be better captured by jumps rather tha ormal iovatios. These jumps might lead to the level ad volatility outliers which ca ot be take ito accout for by the simple ormal distributio as Hotta ad Tsay (1998) preseted. Thus, the assumptio of the ormal distributio seems to be iappropriate to represet the high frequecy commodity returs series properly due to the jumps. FIGARCH-BERNOULLI DISTRIBUTION MODEL WITH JUMPS Sice the presece of the jumps is primarily resposible for the rejectio of the usual ormality assumptio, it seems to call for the use of aother model. Oe model to be cosidered is to itroduce jumps through the use of a ormal mixture distributio. We employ a ormal mixture distributio model, the jump diffusio model proposed by Press (1967), i order to accout for the coditioal jumps i the high frequecy futures returs. Iitially, Press (1967) proposed a jump diffusio model for stock prices uder the assumptio that the logarithm of the stock price follows a Browia motio process o which i.i.d. ormal distributed jumps are superimposed. I particular, we aalyze the impact of jumps i the coditioal mea process o the log memory property i the coditioal variace process of the high frequecy commodity futures returs series by usig a ormal mixture distributio model. Efficiet estimatio of the parameters of 14

16 cotiuous time processes is geerally challegig, I order to give a alterative perspective o the cotiuous time formulatio, it is cosidered iterestig to fit a ormal mixture model i discrete time, takig advatage of the relatively simple formulatio. Hece, i order to model the jumps occurred i the high frequecy commodity futures returs appropriately, we rely o a jump-diffusio FIGARCH model that assumes the high frequecy commodity returs are draw from a mixture of ormal distributio ad jump process. I particular, we cosider this model i the cotext of a Beroulli distributio. The Beroulli distributio models the stochastic jumps i the 15-miute high frequecy commodity futures returs series. The mai characteristic of the Beroulli process is that over a fixed time period, oe relevat iformatio arrives i foreig exchage markets ad a jump occurs i the high frequecy commodity futures returs with probability (λ) which is draw from a Beroulli distributio ad is forced i the (0,1) iterval. The jump size is give by the radom variable ν, which is assumed to be NID(ν, δ ). The combied MA(1)-FIGARCH (1,d,1) model with Beroulli distributio is, y μ + λν + ε + bε = (8) ε ~ (1 λ) N ( λν, δ ) + λn( ν λν, σ + ) (9) δ σ d ω + βσ 1 + [ 1 βl (1 φl)(1 L) ] ε = (10) The high frequecy commodity futures returs are still specified as followig the MA(1) 15

17 process, with a jump probability (λ) which is costat ad is draw from a Beroulli distributio (0< λ <1) ad ν is the mea of the jump distributio while δ captures the variace of the jump distributio implyig the additioal volatility related to the jumps. The volatility process is the FIGARCH(1,d,1) model as developed earlier. The log likelihood fuctio for the combied model has the followig form, T k T (1 λ) ( ε + λν ) l( ξ ) = ( )l(π ) {[ ] exp[ h h ( ε (1 λ) ν ) exp[ ( h + δ ) ]} ] + [ ( h 1 t= 1 = 1 + δ ) λ ] (11) The form of the likelihood fuctio for the Beroulli-ormal mixture distributio is basically similar to that proposed by Vlaar ad Palm (1993) which studied foreig exchage rates i the EMS (Europea Moetary System) usig a GARCH framework. Ad, the ormalized residuals are used for the statistical iferece istead of the usual stadardized residuals sice the stadardizatio may ot lead to i.i.d. residuals i the mixture distributio model with time depedet variace as suggested by Vlaar ad Palm (1997) ad Beie ad Lauret (003). Hece, this paper ivestigates the high frequecy futures returs by combiig the FIGARCH model with the Beroulli jump diffusio models to cosider jumps i the coditioal mea ad capture the log memory property i the coditioal variace. Jump process is icluded i a attempt to reduce the ifluece of the coditioal mea jumps o the MA-FIGARCH specificatio. The estimated parameters for the high frequecy futures returs series over 16

18 differet commodities are reported i Table 3. The estimated parameters (j) for the probability of a jump are all sigificat at the covetioal level of sigificace across differet commodities, implyig that the jumps are quite sigificat i the coditioal mea process for the high frequecy commodity futures returs. The jumps itesities (λ) calculated from the estimated (j) are 0.18, 0.13, 0.15 ad 0.11 for the high frequecy commodity futures returs of cattle, cor, hog ad gasolie respectively, which idicate the probability of jumps i the high frequecy commodity futures returs occurrig durig the sample period. Oe iterestig issue cocers the iterpretatio of the jumps ad whether or ot they correspod to ecoomic ad fiacial evets i the commodity futures markets. Without more detailed iformatio, it is difficult to distiguish these effects. We would leave this issue for a future study. The estimated parameters (υ) which represet the impacts of the jumps o the mea process are foud to be isigificat for the high frequecy commodity futures returs. This may be due to a geeral patter of very quick ad effective exchage rate coditioal mea adjustmet after the jumps. However, the effects of the jumps o the volatility process (δ ) of the high frequecy futures returs are estimated to be very sigificat ad much greater tha those o the mea process. The effects of jumps o volatility process appear to be more importat ad more sigificat tha the effects o the mea process. These results are geerally similar to the 17

19 case of the high frequecy gold futures returs i Cai et al. (001) I particular, the estimated log memory parameters of the high frequecy returs are , , ad for the high frequecy commodity futures returs of cattle, cor, hog ad gasolie respectively, ad they are all very sigificat. The log memory parameters are foud to be much lower tha those estimated from the basic MA-FIGARCH model without cosiderig the jumps. This suggests that the log memory property of the high frequecy commodity futures returs may be sigificatly affected by jumps i the coditioal mea process ad higher values of the log memory parameters ca be iduced whe jumps i the coditioal mea process are ot accouted for. This result is quite uderstadable give that the jumps which otherwise may be spuriously associated with additioal volatility are fully accouted for i the mixture distributio model. 5) Ad, the estimatio results show that the kurtosis statistics are reduced sigificatly for the various commodity futures returs after the jumps are accouted for. Thus, the greater log memory property ad the excess kurtosis seem to be related to asymmetric adjustmets to coditioal variace i respose to the jumps, which is much more gradual ad persistet tha the coditioal mea adjustmets. I particular, the jumps appear to be the drivig force behid the log memory property i the volatility process of the high frequecy futures returs. This cofirms that the mixture distributio geerally outperforms the 18

20 simple ormal distributio ad that hat accoutig for o-uiform flows of iformatio ca sigificatly improve the fit of the model. CONCLUSION We examie the properties of high commodity frequecy returs of 15-miute cattle, cor, hog ad gasolie futures prices. The strog itraday periodicity ad the log memory property are foud to have a sigificat impact o the volatility process of the high frequecy futures returs. Firs after elimiatig the itraday periodicity by FFF method, we apply the FIGARCH model with a usual ormality assumptio ad fid that the FIGARCH model provides a better represetatio of the log memory property i the volatility process of the high frequecy commodity futures retur series tha the usual GARCH model. The geeral appropriateess ad robustess of the FIGARCH model persist for differet high frequecy commodity futures returs. Bu there still exists high excess kurtosis i the high frequecy commodity futures returs implyig the rejectio of the ormality assumptio, which may be caused by jumps i the coditioal meas of the high frequecy commodity futures returs. Jumps i the coditioal mea process of the high frequecy returs data may be caused by ecoomic ad fiacial evets i commodity futures markets. These features ca be better modeled by usig the FIGARCH model combied with the 19

21 Beroulli jump process. Such model is costructed to ivestigate the effects of coditioal mea jumps o the log memory property for differet high frequecy commodity futures returs data. The combied FIGARCH-Beroulli model appears to be quite appropriate for describig jumps i the coditioal mea process ad the log memory property i the volatility process of the high frequecy futures returs series. I particular, the log memory parameters ad the values of kurtosis estimated from the combied models are foud to be much lower tha those from the basic FIGARCH model without cosiderig the coditioal mea jumps. This costitutes strog evidece that the specificatio of the coditioal mea process without cosiderig the jumps may spuriously distort the estimates of the log memory parameters. It is hoped that these results may be helpful i deepeig our uderstadig the dyamics of commodity futures prices ad i developig empirical applicatios such as optimal hedge ratio estimatio, tests for futures market efficiecy, tests for the aoucemet effect of market ews, optio valuatio, risk maagemet ad portfolio maagemet.. 0

22 Edotes 1. Cai etal. (001) have used 5-miute gold futures prices to fid the effects of US ews o the high frequecy gold futures returs. Ad Martes ad Zei (004) have used high frequecy oil prices data to ivestigate the realized daily volatility measures.. The origial commodity futures prices data i this paper is the same as i Baillie et al. (007). For more iformatio ad data availability see 3. The log memory patter is foud i the volatility process of daily commodity futures prices. See Baillie et al. (007), Ji ad Frechette (004), Martes ad Zei (004), Bruetti ad Gilbert (000) ad Crato ad Ray (000). 4. The Box-Pierce portmateau statistics show that the models specified for each commodity do a good job of capturig the autocorrelatios i the mea ad volatility of the commodity retur series. I each case there is o evidece of additioal autocorrelatio i the stadardized residuals or squared stadardized residuals, idicatig that the chose model specificatio provides a adequate fit. It is iterestig to ote that the autocorrelatio i the mea teds to persist more for cattle ad hogs tha for the other commodities (i.e. more MA terms i the mea required for a adequate fit). Furthermore, these commodities also seem to require more flexible models to capture their autocorrelatio i volatility as well (i.e. more GARCH terms required for a adequate fit). 1

23 5. Several recet papers preset that the decrease of the persistece of shocks i foreig exchage rates whe accoutig for jumps appropriately. See Diebold ad Ioue (1999), Grager ad Hyug (1999) ad Beie ad Lauret (003).

24 Refereces Aderse, T. G., Bollerslev, T., Diebold, F. X., & Vega, C. (005). Real-time price discovery i stock, bod ad foreig exchage markets. NBER Workig Paper No. W1131. Baillie, R. T., Bollerslev, T., & Mikkelse, H.-O. (1996). Fractioally itegrated geeralized autoregressive coditioal heteroskedasticity. Joural of Ecoometrics, 74, Baillie, R. T., Ha, Y.-W., Myers, R. J.& Sog, J. (007). Log memory models for daily ad high frequecy commodity futures prices. Joural of Futures Marke i press. Beie, M., & Laure S. (003). Cetral bak itervetios ad jumps i double log memory models of daily exchage rates. Joural of Empirical Fiace, 10, Bollerslev, T. (1986). Geeralized autoregressive coditioal heteroscedasticity. Joural of Ecoometrics, 31, Bollerslev, T., & Wooldridge, J. M. (199). Quasi-maximum likelihood estimatio ad iferece i dyamic models with time varyig covariaces. Ecoometric Reviews, 11, Bruetti, C., & Gilber C.L. (000). Bivariate FIGARCH ad fractioal coitegratio. Joural of Empirical Fiace, 7, Cai, J., Cheug, Y.-L., & Wog, M. C. S. (001). What moves the gold market? Joural of 3

25 Futures Markets, 1, Crato, N., & Ray, B. K. (000). Memory i returs ad volatilities of futures cotracts. Joural of Futures Markets, 0, Diebold, F. X. & Ioue, A. (1999). Log memory ad structural chages. Mauscrip NYU. Frases, P. H., & Ghijsels, H. (1999). Additive outliers, GARCH ad forecastig volatility. Iteratioal Joural of Forecastig, 15, 1-9. Galla A. R. (1981). O the bias i flexible fuctio forms ad a essetially ubiased form: the Fourier flexible form. Joural of Ecoometrics, 15, Galla A. R. (198). Ubiased determiatio of productio techologies. Joural of Ecoometrics, 0, Grager, C.W.J.& Hyug, N. (004). Occasioal structural breaks ad log memory with a applicatio to the S&P500 absolute stock returs. Joural of Empirical Fiace, 11, Hotta, L.K, & Tsay, R.S. (1998). Outliers i GARCH process, Upublished mauscript. Ji, H. J., & Frechette, D. (004). Fractioal itegratio i agricultural futures price volatilities. America Joural of Agricultural Ecoomics, 86, McFarlad, J. W., Petti R., & Sug, S. K. (198). The distributio of foreig exchage price chages: tradig day effects ad risk measuremet. Joural of Fiace, 37,

26 Martes, M., & Zei, J. (004). Predictig fiacial volatility: high frequecy time-series forecasts vis-à-vis implied volatility. Joural of Futures Markets, 4, Müller, U.A., Dacoroga, M. M., Olse, R.B., Picte O.V., Schworz, M. & Morgeegg, C. (1990). Statistical study of foreig exchage rates, empirical evidece of price chage law ad itraday Aalysis. Joural of Bakig ad Fiace, 14, Press, J. (1967). A compoud evets model for security prices. Joural of Busiess, 40, Vlaar, P.J.G.., & Palm, F. (1993). The message i weekly exchage rates i the Europea Moetary System: mea reversio, coditioal heteroskedasticity ad jumps. Joural of Busiess ad Ecoomic Statistics, 11, Vlaar, P.J.G.. & Palm, F. (1997). Iterest rate differetials ad excess returs i the Europea Moetary System. Joural of Iteratioal Fiacial Markets, Istitutios ad Moey, 7,

27 Table 1: Basic Statistics for Raw High Frequecy Commodity Futures Returs Cattle Cor Hog Gasolie First Obs. 99/05/03 99/05/03 99/05/03 99/0503 9:30 13:15 9:30 10:15 Last Obs. 00/1/8 00/1/8 00/1/8 00/1/8 13:00 13:15 13:00 15:00 Sample Size Mea Variace Skewes Kurtosis

28 Table : Estimated MA-FIGARCH model for Filtered High Frequecy Commodity Futures Returs Cattle Cor Hogs Gasolie μ (0.0015) (0.007) (0.0033) (0.0039) θ (0.0144) (0.0143) (0.0158) (0.018) d (0.0366) (0.0539) (0.060) (0.051) ω (0.0019) (0.0015) (0.001) (0.0049) β (0.3876) (0.081) (0.0816) (0.083) φ (0.380) (0.0968) (0.0964) m m Q(50) Q (50) W Notes: Robust stadard errors based o QMLE are i paretheses below the correspodig parameter estimates. The diagostic statistics Q(50) ad Q (50) are portmateau statistics based o the first 50 autocorrelatios of the stadardized residuals ad the autocorrelatios of the squared stadardized residuals respectively. The statistics m 3 ad m 4 are the sample skewess ad kurtosis respectively of the stadardized residuals. W is the robust Wald statistic for testig the GARCH specificatio agaist FIGARCH. 7

29 Table 3: Estimated MA-FIGARCH-Beroulli jump model with for Filtered High Frequecy Commodity Futures Returs Cattle Cor Hogs Gasolie μ (0.0019) (0.008) (0.0034) (0.0044) j (0.4475) (0.1871) (0.455) (0.3941) λ [0.184] [0.16] [0.15] [0.108] υ (0.0101) (0.06) (0.040) (0.050) δ (0.0073) (0.035) (0.0761) (0.0735) θ (0.0140) (0.016) (0.014) (0.018) d (0.0303) (0.033) (0.00) (0.0170) ω (0.0003) (0.0007) (0.0001) (0.0067) β (0.908) (0.096) (0.085) (0.008) φ (0.3035) (0.0997) (0.04) - m m Q(50) Q (50) Notes: the same as Table except that a jump itesity of λ, where λ = [1 + exp(j)] -1, 0 <λ < 1, ad is specified to be geerated by the Beroulli distributio. The jump size is give by the radom variable v t which is assumed to be NID (ν, δ ). 8

30 Figure 1: 15 miute Cor Futures Returs Figure: Correlograms of Raw 15 miute Cor Futures Returs for 10 tradig days 9

31 Figure3: Correlograms of Filtered 15 miute Cor Futures Returs for 10 tradig days 30