Overestimation in the Traditional GARCH Model During Jump Periods. Abstract

Transcription

1 Overesimaion in he Tradiional GARCH Model During Jump Periods Wan-Hsiu Cheng Nanhua Universiy Absrac The radiional coninuous and smooh models, like he GARCH model, may fail o capure exreme reurns volailiy. Therefore, his sudy applies he bivariae poisson (CBP)-GARCH model o sudy jump dynamics in price volailiy of crude oil and heaing oil during he pas 0 years. The empirical resuls indicae ha he variance and covariance of he GARCH and CBP-GARCH models were found o be similar in low jump inensiy periods and o diverge during jump evens. Significan overesimaions occur during high jump ime periods in he GARCH model because of assumpions of coninuiy, and easily leading o excessive hedging and overly measuring risk. Neverheless, in he CBP-GARCH model, he specific shocks are assumed o be independen of normal volailiy and o reduce he persisence of abnormal volailiy. Therefore, he CBP-GARCH model is appropriae and necessary in high volailiy markes. The auhors would like o hank he Naional Science Council of he Republic of China, Taiwan for financially supporing his research under Conrac No. NSC H Ciaion: Cheng, Wan-Hsiu, (008) "Overesimaion in he Tradiional GARCH Model During Jump Periods." Economics Bullein, Vol. 3, No. 68 pp. -0 Submied: June 3, 008. Acceped: Ocober 8, 008. URL: hp://economicsbullein.vanderbil.edu/008/volume3/eb-08c30068a.pdf

2 . INTRODUCTION Crude oil no only is he world s mos acively raded commodiy, bu also leads o he larges-volume fuures conrac rading for a physical commodiy on he New York Mercanile Exchange (NYMEX). Given heir excellen liquidiy and price ransparency, hese fuures conracs are used as a key inernaional pricing benchmark. Noably, heaing oil is a produc refined from crude oil. Wih he lifing of U.S. price conrols on heaing oil in he mid 970's, he NYMEX began developing a heaing oil fuures conrac and, in 978, inroduced he world's firs successful energy fuures conrac. Heaing oil fuures has become one of he premier disillae conracs in fuures rading. In 983, crude oil fuures conracs were inroduced. In is early years, he NYMEX Division heaing oil conrac mainly araced wholesalers and large consumers of heaing oil in he New York Harbor area. Today, a wide variey of businesses, including oil refiners, wholesale markeers, heaing oil reailers, rucking companies, airlines, and marine ranspor operaors, as well as oher major consumers of fuel oil, have embraced his conrac as a risk managemen vehicle and pricing mechanism. The dominan price componen is he cos of crude oil; hence, heaing oil prices should be closely linked o he cos of crude oil prices. Observing spo prices over he pas 0 years (Fig. ) reveals similar rends in crude oil and heaing oil spo prices, bu here are occasional surges in heaing oil spo prices, such as a he end of 989 and in early 000. These surges resul from rapid supply and demand shifs caused by weaher, refinery shudowns, or poliical insabiliy. As is seen in Panel B of Fig., jumps in reurns are occasional. These phenomena can be described by he jump model ha focuses on he abnormal volailiy. However, almos no sudies have invesigaed jumps in crude oil and heaing oil in energy markes. Accordingly, his sudy employs he correlaed bivariae poisson - generalized auoregressive condiional heeroscedasiciy model (CBP-GARCH model) o demonsrae he need o consider such jumps. The GARCH model remains he widely used model for researching he volailiy behavior of energy asses (Lin and Tamvakis, 00; Ewing e al., 00; Sadorsky, 00; Hammoudeh e al., 003). However, radiional coninuous and smooh models such as he GARCH may fail o capure exreme reurns volailiy. Accurae esimaion of he variance and covariance of he asses can improve performance in forecasing, hedging, risk managemen, ec. Volailiy esimaion and forecasing have been he main ask in financial markes during he pas wo decades, and hey are fundamenal o mos areas of finance -- for example, asse pricing, porfolio selecion, volailiy relaionship, hedging, risk, ec. Mos sudies assume ha ime series daa follow a smooh and According o EIA s (Energy Informaion Agency) Peroleum Markeing Monhly (00), he final price o consumers of home heaing oil can be broken down in percenage erms as follows: 4% crude oil purchase cos, % refining coss, and 46% disribuion and markeing coss.

3 coninuous volailiy process, and GARCH is now widely used in his field (see he survey in Poon and Granger, 003; Bauwens e al., 006). However, he exisence of jumps implies ha diffusion models are misspecified saisically. Jorion (988) conended ha ime-varying volailiy and occasional jumps are possibly he wo mos noable feaures of daily financial ime series. Moreover, Park (00) menioned ha he sandardized residuals of he GARCH model sill have excess kurosis, albei less han for he raw financial reurns (also see Bollerslev, 987; Baillie and Bollerslev, 989; Hsieh, 989). Furhermore, Chan (003) noed ha alhough mulivariae GARCH models adequaely accoun for heeroskedasiciy, hey do no fully capure he lepokurosis in uncondiional disribuions ha is frequenly observed in financial daa. Consequenly, financial economerics furher invesigaes volailiy wih jumps (e.g., Chang and Kim, 00; Pan, 00; Eraker, Johannes and Polson, 003; Chan and Maheu, 00; Johannes, 004; Maheu and McCurdy, 003). Mos jump models have been successfully applied o analyze foreign exchange and sock marke reurns, and hese models can improve performance in capuring price behavior in physical commodiies. In meal markes, Chan and Young (006) found ha he jump model closely fis copper spo and fuures daa. Furhermore, his sudy inroduced he jump GARCH model o he energy marke and invesigaed he price behavior of crude oil and heaing oil, he mos liquid rading asses in NYMEX. The radiional GARCH model is clearly misapplied o energy asses wih high volailiy, paricularly in siuaions involving large jumps. The problems wih esimaion ha resul from he radiional GARCH model in such scenarios are eliminaed by he bivariae jump GARCH model, because his model successfully reduces he limiaions associaed wih he coninuous volailiy process and he univariae jump assumpion from previous sudies. Jumps represen a response o unusual news evens as par of he laen news process and have he poenial o capure boh smooh and sudden price volailiy movemens (Chan and Young, 006). Press (967), who inroduced an independen jump process in which jump arrival was governed by a Poisson disribuion, was he firs o apply he Poisson jump model o financial markes. Alhough jumps canno be observed, an ex-pos filer can always be consruced o infer heir probabiliy. Tucker and Pond (988), Akgiray and Booh (988) and Hsieh (989) all found ha he Poisson jump model provides an effecive saisical characerizaion of daily exchange raes. The basic jump models have been furher exended in various direcions, and combining hem wih he ARCH/GARCH model is an essenial applicaion (Jorion, 988; Vlaar and Palm, 993). Several sudies emphasize ha he ime-varying jump fis closely wih realiy (Beas, 99; Eraker e al., 00; Das, 003; Chan and Maheu, 003; Maheu and McCurdy, 004). However, all of he above

4 models are limied by he use of he univariae seing for capuring he price volailiy of specific asses. Mos researchers now accep ha financial volailiies move ogeher over ime across asses and markes. Recognizing his feaure via a mulivariae modeling framework yields more relevan empirical models han working wih separae univariae models. Therefore, Chan (003) devised a bivariae jump model ha combined he Correlaed Bivariae Poisson (CBP) funcion and he GARCH model o analyze jump dynamics. The CBP-GARCH model was applied o energy markes o examine price volailiy for crude oil and heaing oil. This sudy has hree aims. Firs, his invesigaion aims o accuraely model volailiy for high-volailiy energy marke asses. This sudy provides a complee analysis of he price volailiy of crude oil and heaing oil over he pas 0 years and examines wheher he jump model has beer performance. Second, his sudy aemps o use bivariae jump models o accuraely esimae he volailiies of wo closely relaed asses. This model is applied in his way no only because of volailiy spillovers beween markes and asses, bu also because of he imporance of he covariance beween series. Therefore, following Chan (003), his sudy discusses he volailiy characerisics of crude oil and heaing oil using he correlaed bivariae jump model. We furher relax he srong resricions on jump size and inensiy in he simple CBP-GARCH model. Boh he asymmeric ime-varying assumpion in jump size and he auoregressive erm in jump inensiy are added in he model. Third, his sudy aaches imporance o he problem of overesimaion and invesigaes wheher i occurs in he radiional GARCH model when considering jump evens. Overesimaion of variance and covariance will bias furher applicaions of he model, such as hedging, value of risk, porfolio consrucing, ec. Hedging is paricularly imporan during high-volailiy periods. Overesimaion will cause excessive hedging ogeher wih increased coss and reduced hedge effeciveness. Moreover, overesimaion of volailiy increases he value a risk and leads o he loss of poenial profi. The remainder of his paper is organized as follows. Secion presens he mehodology of he GARCH and CBP-GARCH models. Secion 3 hen explains he daa and descripive saisics. Nex, Secion 4 describes he empirical resuls. Finally, he las secion presens conclusions.. METHODOLOGY GARCH model The GARCH model has been widely used in volailiy esimaion since being inroduced by Bollerslev (986). The sandard VEC bivariae model proposed by Thanks for he anonymous reviewers suggesion. These exensions provide significan improvemen in volailiy forecasing over he simple CBP-GARCH model. 3

5 Bollerslev, Engle and Wooldridge (988) is briefly described by R = μ + ε () where R is a vecor of reurns, μ is a drif coefficien, and he error erm of he reurns ε follows he normal disribuion wih mean 0 and variance H. The condiional variance equaion is H = C + Aε ε + BH () where C is a 3 column vecor and A and B are 3 3 marices. Furhermore, assuming normally disribued errors in he esimaion process implies he following simple log-likelihood funcion, T L (θ) = (ln H + ε H ε ) (3) = where θ represens he vecor of parameers o be esimaed and T is he number of observaions. Since he log-likelihood funcion in his case is non-linear, we use numerical maximizaion echniques o esimae he model. CBP-GARCH model The CBP-GARCH model is a combinaion of he GARCH (Bollerslev, 986) and he Poisson Correlaed funcion (M Kendrick, 96; Campbell, 934). The model is defined as follows: R = Rˆ + ε + J, (4) where R is a vecor of reurns consising of a mean equaion Rˆ, a random disurbance ε, and a jump componen J. The random disurbance follows a bivariae normal disribuion wih zero mean and variance covariance marix H ~. In a bivariae framework, he jump componen J has a bivariae normal disribuion wih zero mean and variance covariance marix Δ. The normal disurbance and he jump componens are assumed o be independen, defined as: n n Y,i E ( Y,i ) = i= i= J n n (5) Y, j E ( Y, j) j= j= Here, Y i is a random variable called jump size. The sum of he Y i means ha he reurn may experience n number of jumps depending on he news conen ha eners he marke wihin any single ime period. Each of hese jump sizes is governed by a normal disribuion wih mean θ and variance δ. In oher words, he jump size for he wo spos (crude oil and heaing oil) can be characerized as: 4

6 Y ~ N( θ, ) and Y ~ N( θ, ). (6),i δ, j δ The mean of he disribuion is allowed o vary asymmerically over ime as a funcion of he size and sign of recen reurns in each marke: θ = θ + + θ R ( I(R )) + θ R I(R ) 0 + θ = θ + θ ( I(R )) + θ R I(R ) (7) 0 R where I (x) = if x > 0 and 0 oherwise. In equaion (5), wo discree couning variables n and n conrol he arrival of jumps and hey are consruced by hree independen Poisson variables, namely, n, n, and n 3. Each one of hese variables has a probabiliy densiy funcion given by P(n i = j Φ e ) = λi λ j i j!. (8) The expeced value and he variance of n i are boh equal o λ i, which is also referred o as he jump inensiy. The correlaed jump inensiy couners (M Kendrick, 96; Campbell, 934) are defined as n and = n + n 3 n. (9) = n + n 3 By consrucion, each of hese couning variables ( n and n ) is capable of generaing independen jumps ( n and n ) and correlaed jumps ( n 3 ). The laer conribue jumps o boh series. Using he change of variables mehod and inegraing ou n 3 yields he join probabiliy densiy for n and n, given as: min(i, j) j k j k k ( λ +λ +λ λ λ λ 3 ) 3 P(n = i, n = j Φ ) = e. (0) k= 0 (i k)!(j k)!k! The expeced number of jumps is equal o E(n i ) = λ i + λ 3. () The definiion of he ime varying jump inensiies is given by Chan (003), ha is, λ = λ + η + γ λ r λ = λ + η + γ λ r 5

7 3 = λ 3 + η3r + η4r λ + γ λ, () 3 3 where ri is he rae of reurn for asse i a ime ( ) and r i is an approximaion of he las period s volailiy. The jump inensiies are assumed o be relaed o marke condiions, which are relaed in volailiy. In he same way, he covariance is governed by variaions in he las period s volailiies from boh series. The parameric srucure no only inroduces addiional jump dynamics ino he model bu also allows a ime-varying correlaion beween he couning variables n and n. The correlaion is calculaed as follows: Corr(n,n λ 3 ) =. (3) ( λ + λ )( λ + λ ) 3 3 Combining he GARCH model wih he CBP funcion, he probabiliy densiy funcion for R given i and j jumps in spo and spo is defined by f (R n [ u H u ] / = i,n = j, Φ ) = H N / ij, exp ij, ij, ij,, (4) (π) where u ij, is he usual error erm wih he jump componen ij, J represening he effec of i and j jumps: r rˆ iθ + ( λ + λ3 ) θ u ij, = R Rˆ Jij, =. (5) r rˆ jθ + ( λ + λ3 ) θ The variance covariance marix H ij, can be separaed ino wo pars: he variance covariance marix for he normal random disurbance componens Δ. ij, H ~ and for he jump Firs, he variance and covariance marix for he normal random disribuion can be defined as in equaion (), in which he erm ~ ε refers o he sum of a disurbance and a jump componen. Second, he variance covariance marix for he jump componens is iδ ρ ij δδ Δ ij, =, (6) ρ ij δδ jδ where ρ is he correlaion coefficien beween Y and Y. The variance covariance marix for he CBP-GARCH model is hen a sum of Δ. H ~ and ij, 6

8 Finally, he condiional densiy of reurns is defined by Φ ) = f (R n = i, n = j, Φ )P(n = i, n = j, Φ ) i= 0 j= 0 P (R. (7) The log likelihood funcion is simply he sum of he log condiional densiies: N ln L = ln P(R ). (8) = Φ Numerical maximizaion echniques are used o esimae he model. DATA and DESCRIPTIVE STATISTICS This sudy analyzed he price volailiy of he WTI crude oil spo price and he New York Harbor No. heaing oil spo price using he CBP-GARCH model. The sample period ran from June, 986 o July 3, 007 and conained 5,39 observaions. All daa was obained from he U.S. Deparmen of Energy (DOE). Table liss he descripive saisics for spo reurns 3. Crude oil and heaing oil displayed similar reurns, while he sandard deviaion of heaing oil slighly exceeded ha of crude oil. However, he ime series plos in Fig. reveal ha he higher variance of heaing oil resuled from he occasional large price changes. Hisorically, heaing oil prices are generally higher during he winer monhs when demand is sronger, bu specific evens in recen years have led o wha seems o be a weakening of his seasonal effec on average (Fig. ). More noably, jumps of heaing oil during he sample period occur a he end of 989 and in he early pars of 000, 003 and 005. The bigges heaing oil crisis occurred in February in response o a reducion in supply; cold weaher was responsible for driving up demand a he end of 989; and a combinaion of low invenories, high winer demand and he specer of war looming caused he high prices in early 003. In lae Augus and Sepember 005, he heaing oil price was a a near record high, up o 7.67, because of hurricanes Karina and Ria. As for crude oil, he larges jump evens are relaed o he Gulf War. The firs war was in 990 and 99, and he second one began in 003. Lasly, he upward rends of boh crude oil and heaing oil drop due o he crude invenory having jumped far more han expeced and he unusually mild winer weaher. Furhermore, boh he crude oil and heaing oil reurns exhibi negaive skewness and are lepokuric. The skewness of crude oil and heaing oil is and , respecively, and he excess kurosis is and , respecively, wih all he values being significan a he % level. Table also liss he covariance/correlaion 3 Boh he series are saionary in he Dickey-Fuller es and he Phillips-Perron es. 4 According o he DOE, consumers paid an average of $. per gallon hroughou he winer in 999; however, during lae January o early February 000, he prices quickly wen from $. o $.99 per gallon, an increase of 64%. The DOE esablished he Norheas Heaing Oil Reserve in July 000 o guard agains poenial shorfalls and price spikes. 7

9 marix. The saic correlaion coefficien and covariance are and 4.989, respecively, and indicae a srong and posiive relaionship beween crude oil and heaing oil during he pas 0 years. EMPIRICAL RESULTS Esimaion resuls of GARCH and CBP-GARCH model The empirical resuls of he GARCH and CBP-GARCH models are lised in Table, and volailiy and covariance are illusraed in Fig. 3. As for he GARCH model, all he parameers in he condiional variance equaion are significan under he % level. This indicaes ha he volailiy of boh crude and heaing oil ( h and h ) are direcly affeced by pas innovaion and volailiy. Higher levels of hisorical condiional volailiy are associaed wih increased curren condiional volailiy. Condiional covariance ( h ) also exhibis he characerisics of volailiy clusering. Nex, in he CBP-GARCH model, all he parameers in he GARCH volailiy erms are significan under he % level, as in he GARCH model. Furhermore, he jump componens of jump size and inensiy are discussed below. The jump size means are significanly negaive a he % level ( θ 0 and θ 0 ), and he asymmeric effecs only exis in he heaing oil marke. The jump sizes of heaing + oil are significanly posiive in relaion o las period s posiive reurns ( θ ). This indicaes ha he mean of he jump size is increasing while he las period s heaing oil prices go up. However, he negaive informaion of he down spo prices does no affec he jump size mean. The variance of he jump size (δ ) is and for crude oil and heaing oil, respecively, and boh are significan under he % level, indicaing ha jump variance is higher for crude oil. The jump correlaion is up o beween crude oil and heaing oil, revealing ha he bivariae jump seing is essenial in his sudy. Boh jump inensiies ( λ and λ ) are significanly relaed o he hisorical volailiy ( η) and he auoregressive erms ( γ ). The former relaionship is sronger for crude oil ( η ), and he laer one is sronger for heaing oil ( γ ). Moreover, he characerisics of he jump inensiy covariance ( λ ) are he same wih he specific jump inensiy of crude oil and heaing oil. The parameers, η 3, η4 and γ 3, are all significan under he % level. Figure 4 shows he jump inensiies. Figure 5 also plos he average monhly jump inensiy; a clear seasonal effec can be observed for heaing oil. On average, he jump inensiy of heaing oil is highes in February, followed by January, March and hen December 5. The jump inensiy is lowes in July, June and May. Addiionally, he jump inensiy of crude oil is more sable and higher 3 5 By observing he pas 0 years, he jump inensiy is no a high level excep in March 000. Therefore, his paper re-compues he average values wihou he abnormal March 000, and he average jump inensiy in March is lower han December. 8

10 han ha of heaing oil, excep in February. Oherwise, he correlaion beween he number of jumps is as shown in Fig. 6. The average correlaion over he las 0 years is , and he correlaion coefficien is lower when he range beween crude oil and heaing oil is widening, paricularly when heaing oil prices diverge from crude oil prices. Overesimaion in he GARCH model during he jump periods Table 3 liss he descripive saisics of he variance. The average variances for crude oil and heaing oil are 6.57 and under he GARCH model, compared o and under he CBP-GARCH model. Moreover, he covariance is using he GARCH model and using he CBP-GARCH model. These analyical resuls demonsrae ha he CBP-GARCH model is characerized by smooh esimaion resuls and a relaively narrow variaion range. Furhermore, in regard o Figure 4, which illusraes jump inensiy, four periods wih exremely high jump inensiies were seleced o analyze in deph, including wo jump evens in heaing oil which were previously described (Panel A o B in Fig. 7) and wo jump evens in crude oil resuling from he wo Gulf Wars (Panel C and D in Fig. 7). The variances, found o be similar using he GARCH and CBP-GARCH models in peaceime 6, diverged during hese specific periods. The variances are higher for he GARCH model han he CBP-GARCH model 7. Take Panel B as an example: shrinking supply rapidly drove up heaing oil prices during Feb. 000, and he wo models clearly displayed differen esimaions during ha period. Furhermore, when he jumps in he price of crude oil during he wo Gulf wars are examined, esimaions of variance in he GARCH model are higher as well. The findings in his sudy are also suppored by he evidence lised in Table 4 and Table 5. Boh ables repor he forecasing errors of he measured volailiy ha comparing wih he sandard deviaion of price differences 8 using mean absolue percenage error (MAPE) 9. The forecasing errors are grouped by wo differen rules. Table 4 liss he daily forecasing errors boh in he peaceime and high jump ime periods, ha defined as he lowes 5% and highes 5% fraciles of jump probabiliies respecively, and able 5 liss he monhly average forecasing errors owing o srong seasonal effecs. Firs, all he MAPEs are lower in he CBP-GARCH model han in he GARCH model, and all he difference values are significanly 6 Peaceime in his paper indicaes low jump probabiliy period, during which he series daa are smooh and coninuous wihou sudden evens such as he Gulf Wars. 7 All he characerisics of covariance are similar wih variance. To save space in his paper, we do no repor he covariance figures. 8 Regnier (007) indicaes ha he sandard deviaion of log price differences is he bes general measure of volailiy and an indicaor of changes in volailiy over ime. Therefore, volailiy in his paper is measured as sandard deviaion over a 3-year period of log differences. 9 MAPE is he average of he absolue errors expressed in percenage erms. 9

11 posiive in he sign es. In Table 4, he differences are 364 and 476 in peaceime and 0.74 and in he high jump ime period for crude oil and heaing oil, respecively. In oher words, he measuremen errors of he CBP-GARCH models are lower han hose of he GARCH model by a leas 55% in he high jump period. The greaer he difference, he beer he performance of he CBP-GARCH model. We have similar resuls in Table 5. Combined wih he monhly jump inensiy in Figure 5, he percenage error is much lower for he CBP-GARCH model han he GARCH model during he monhs of higher jump inensiy, whereas he difference values in Table 5 are higher. Take heaing oil as an example: he highes and lowes difference values are and 39 in February and July respecively, which is in conras o he highes and lowes jump inensiy. Tha is, he measuremen errors of he CBP-GARCH models are lower han hose of he GARCH model, wih 59% and %, respecively, in February and July. The same obvious resuls appear in crude oil; he difference drops along wih he drop in jump inensiies. In he highes jump inensiy monh, January, he CBP-GARCH model ouperforms he GARCH model, and he measuremen error is 45% lower in he CBP-GARCH model. The measuremen error is only 6% lower during he lowes jump inensiy monh, July. All he resuls imply ha he CBP-GARCH model performs much beer in high jump inensiy monhs, while his performance decreases as he jump inensiy decreases. This sudy argues ha, because of he assumpion of coninuiy, variance may be overesimaed in he radiional GARCH model during high volailiy periods. Tha is, overall shocks canno be disinguished as normal or abnormal shocks, hus moving he volailiy o a high level in he nex period. Neverheless, he CBP-GARCH model assumes ha he specific shock akes he form of a jump, independen of normal volailiy, and reduces he persisence of abnormal volailiy. Accordingly, he variances in he GARCH model exceed hose in he CBP-GARCH model when facing specific evens or he asses wih high volailiy. Furher applicaions can herefore be easily biased based on he overesimaion of variance and covariance. CONCLUSIONS This sudy examines he price volailiy of crude oil and heaing oil during he pas 0 years using he CBP-GARCH model. Boh feaures of jump and bivariae are considered in he CBP-GARCH model for fiing he daa accuraely, especially during jump periods. The empirical resuls indicae ha he CBP-GARCH model ouperforms he GARCH model; however, he performance decreases as he jump inensiy decreases. The measuremen errors of he CBP-GARCH models are lower han hose of he GARCH model, wih 55% and 7% for heaing oil and crude oil, 0

12 respecively, in a high jump period. These measuremen errors are 4% and 3% lower in peaceime. In monhly resuls, he measuremen errors of he CBP-GARCH models are lower han hose of he GARCH model in he highes jump inensiy monh, wih 59% and 45% for heaing oil and crude oil, respecively. These measuremen errors are % and 6% lower in he lowes jump inensiy monh. Tha is, he CBP-GARCH model can capure he volailiy reasonably accuraely, especially during high jump ime periods, during eiher occurrences of specific jump evens or high jump inensiy monhs (usually in winer monhs). Furhermore, he variance and covariance of he GARCH and CBP-GARCH models were found o be similar in peaceime, bu divergen when jump evens such as he Gulf Wars occurred. Due o he assumpion of coninuiy in he radiional GARCH model, boh he variance and covariance of he GARCH model are overesimaions. Tha is, he overall shocks canno be disinguished as normal or abnormal shocks, and hey move he volailiy o a high level in he nex period. Furher applicaions can be easily biased based on his overesimaion. Neverheless, he CBP-GARCH model provides more informaion for regulaing he defecs in he GARCH model. In he CBP-GARCH model, he specific shocks are assumed o be independen of normal volailiy and o reduce he persisence of abnormal volailiy. Therefore, he CBP-GARCH model is appropriae and necessary in high volailiy markes. The overesimaion of variance and covariance will bias furher applicaions of he GARCH model and can lead o, for example, excessive hedging. For his reason, his paper is useful o raders, speculaors and oher paricipans in markes seeking o reduce ransacion coss and maximize profis.

13 REFERENCE Akgiray, V., Booh, G. G., Mixed jump-diffusion process modeling of exchange rae movemens. Review of Economics and Saisics 988; 70; Andersen, T. G., Bollerslev, T., Diebold, F. X., Ebens, H., The disribuion of sock reurn volailiy. Journal of Financial Economics 00; 6; Baillie, R. T., Bollerslev, T., The message in daily exchange raes: A condiional variance able. Journal of Business and Economic Saisics 989; 7; Bauwens, L., Lauren, S., Rombous, J. V. K., Mulivariae GARCH models: A survey. Journal of Applied Economerics 006; ; Beas, D. S., The crash of 87: Was i expeced? The evidence from he opions markes. Journal of Finance 99; 46; Bollerslev, T., Generalized auoregressive condiional heeroskedasiciy. Journal of Economics 986; 3; Bollerslev, T., A condiionally heeroskedasic ime series model for speculaive prices and raes of reurn. Review of Economics and Saisics 987; 69; Campbell, J. T., The poisson correlaion funcion. Proceedings of he Edinburgh Mahemaical Sociey 934; Series ; 8-6. Chan, W. H., Young, D., Jumping hedges: An examinaion of movemens in cooper spo and fuures markes. Journal of Fuures Markes 006; 6; Chan, W. H., Maheu, J. M., Condiional jump dynamics in sock marke reurns. Journal of Business & Economic Saisics 00; 0; Chan, W. H., A correlaed bivariae poisson jump model for foreign exchange. Empirical Economics 003; 8; Chang, K. H., Kim, M. J., Jumps and ime-varying correlaions in daily foreign exchange raes. Journal of Inernaional Money and Finance 00; 0; Das, S. R., Poisson-gaussian processes and he bond marke. NBER working paper: 663; 998. Eraker, B., Johannes, M., Polson, N., The impac of jumps in volailiy and reurns. Journal of Finance 003; 63; Ewing, B. T., Malik, F., Ozfidan, O., Volailiy ransmission in he oil and naural gas markes. Energy Economics 00; 4; Hammoudeh, S., Li, H., Jeon, B., Causaliy and volailiy spillovers among peroleum prices of WTI, gasoline and heaing oil in differen locaions. Norh American Journal of Economics and Finance 003; 4; Hsieh, D., A. Tesing for nonlinear dependence in daily foreign exchange rae changes. Journal of Business 989; 6; Johannes, M., The saisical and economic role of jumps in coninuous-ime ineres rae models. Journal of Finance 003; 59; 7-60.

14 Jorion, P., On jump processes in he foreign exchange and sock markes. Review of Financial Sudies 988; ; Lin, S. X., Tamvakis, M. N., Spillover effecs in energy fuures markes. Energy Economics 00; 3; M Kendrick, A. G., Applicaions of mahemaics o medical problems. Proceedings of he Edinburgh Mahemaical Sociey 96; 44; Maheu, J. M., McCurdy, T. H., News arrival, jump dynamics and volailiy componens for individual sock reurns. Journal of Finance 004; 59; Pan, J., The jump-risk premia implici in opions: evidence from an inegraed ime-series sudy. Journal of Financial Economics 00; 63; Park, B. J., An oulier robus GARCH model and forecasing volailiy of exchange rae reurn. Journal of Forecasing 00; ; Poon, S-H., Granger, C. W., Forecasing volailiy in financial markes: A review. Journal of Financial Lieraure 003; 4; Press, S. J., A compound evens model for securiy prices. Journal of Business 967; 40; Regnier, E., Oil and energy price volailiy. Energy Economics, 007; 9; Sadorsky, P., Time-varying risk premiums in peroleum fuures prices. Energy Economics 00; 4; Tucker, A. L., Pond, L., The probabiliy disribuion of foreign exchange raes: ess of candidae process. Review of Economics and Saisics 988; 70; Vlaar, P. J. G., Palm, F. C., The message in weekly exchange raes in he European Moneary Sysem: mean reversion, condiional heeroskedasiciy and jumps. Journal of Business and Economic Saisics 993; ;

15 Table Table. Descripive saisics of reurn Mean Sd. deviaion Min. Max. Skewness Excess kurosis Crude oil *** *** Heaing oil *** *** Covariance/Correlaion Marix Crude oil Heaing oil Crude oil Heaing oil Noes: *** represens significance under % level. The covariance/correlaion marix has he covariance on and below he diagonal and he correlaion above i. Table. Empirical resuls of GARCH and CBP-GARCH model GARCH model CBP-GARCH model Mean equaion μ μ *** Variance equaion c *** 4 *** Jump size c *** 75 *** c *** *** a 0.43 *** 380 *** a 0.69 *** 448 *** a 0.35 *** 569 *** b *** *** b *** *** b *** *** θ *** θ θ 8 θ *** θ θ *** δ *** δ *** ρ *** Jump inensiy λ 008 λ 008 λ 003 ** 3 η 487 *** η *** η 34 *** 3 η 054 *** 4 γ *** γ *** γ *** Log-likelihood value Noes: *, **, *** represen significance under 0%, 5% and % levels, respecively. Please 4

16 coordinae equaions () and () wih he par of mean and variance equaions in he able, equaions (6), (7), and (6) wih jump size, and equaion () wih jump inensiy. Table 3. Descripive saisics of measured volailiies Mean Sd. deviaion Min. Max. Crude oil GARCH CBP-GARCH Heaing oil GARCH CBP-GARCH Covariance GARCH CBP-GARCH Correlaion GARCH CBP-GARCH Noes: *, **, *** represen significance under 0%, 5% and % levels, respecively. Table 4. Forecasing errors in peace and high jump periods Crude oil Heaing oil () () ()-() () () ()-() GARCH CBP- Difference GARCH CBP- Difference GARCH GARCH The peace ime * * The high jump ime * * Noes: * represens significance under 5% levels in he sign es. The lowes 5% and highes 5% fraciles of jump probabiliies are defined as he peaceime and high jump ime periods, respecively. Table 5. Forecasing errors sored by monh Crude oil Heaing oil () () ()-() () () ()-() GARCH CBP- GARCH Difference GARCH CBP- GARCH Difference Jan * Jan * Feb Feb * Mar * Mar * Apr * Apr * May * May * Jun * Jun * Jul * Jul * Aug * Aug * Sep * Sep * Oc * Oc * Nov * Nov Dec * Dec * Noes: * represens significance under 5% levels in he sign es. 5

17 Figures 5 Crude oil Heaing oil Par A. Time series plos of spo prices (cens per gallon) 3 Crude oil Heaing oil Par B. Reurns Figure. Time series plos and reurns of crude oil and heaing oil 85.0 Crude oil Heaing oil Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oc. Nov. Dec. Figure. The average monhly spo price from June 986 o July GARCH CBP-GARCH Par A. The condiional variance of crude oil 6

18 GARCH CBP-GARCH Par B. The condiional variance of heaing oil 5 0 GARCH CBP-GARCH Par C. The covariance beween crude oil and heaing oil Figure 3. The condiional variance and covariance under GARCH and CBP-GARCH model GARCH CBP-GARCH Figure 4. The jump inensiy of crude oil and heaing oil 7

19 9 Crude oil Heaing oil Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oc. Nov. Dec. Figure 5. The average monhly jump inensiy Figure 6. The correlaion beween he number of jumps of crude oil and heaing oil 5 GARCH CBP-GARCH Reurn Par A. Heaing oil variance (July 989-June 990) GARCH CBP-GARCH Reurn Par B. Heaing oil variance (July 999-June 000) 8

20 30 5 GARCH CBP-GARCH Reurn Panel C. Crude oil variance (Gulf War I) 8 GARCH CBP-GARCH Reurn Panel D. Crude oil variance (Gulf War II) Figure 7. Variance in each model during he specific periods 9