Modeling the joint dynamics of spot and futures markets with. a regime switching long memory volatility process

Transcription

1 Modeling he join dynamics of spo and fuures markes wih a regime swiching long memory volailiy process JONATHAN DARK Deparmen of Finance, The Universiy of Melbourne, Ausralia SUMMARY The mehods used o model he dynamics of spo and fuures markes ignore he effecs of changes in he cos of carry (COC) on he dynamics of he basis and is rae of convergence. This is of paricular imporance given he hisorically low shor erm ineres raes currenly experienced in many counries. This paper proposes a bivariae model ha exhibis long memory in volailiy, basis convergence and regime swiches ha occur via a laen markov process. Mone carlo simulaion reveals ha he model parameers can be well esimaed via maximum likelihood. The proposed model is suppored by applicaions o equiy and currency markes. I is found ha regime swiches in basis dynamics occur beween high and low volailiy saes associaed wih high and low absolue values of he COC. In and ou of sample forecass generally suppor he proposed model, however minimum variance hedge raio esimaion provides mixed suppor. Keywords: long memory, regime swiching, dynamic fuures hedging, cos of carry Correspondence o: Jonahan Dark, Deparmen of Finance, The Universiy of Melbourne, Vicoria 31, Ausralia. jdark@unimelb.edu.au, (Ph) ; (fax)

2 The las few decades have produced a voluminous lieraure ha examines he dynamics of fuures markes and heir underlying asses. These dynamics are imporan for a number of applicaions including arbirage pricing heory (Brennan and Schwarz, 199; Fung and Draper, 1999), price discovery and volailiy ransmission (Koumos and Tucker, 1996; Tse, 1999, Bhar, 1) and hedge raio esimaion (Kroner and Sulan, 1993; Park and Swizer, 1995). Convenional approaches o modeling he dynamics of spo and fuures markes ypically employ a model from he error correcion GARCH family of processes. Chen e al (1999) and Dark (7) are criical of his approach given ha i reas he fuures and he spo as any wo asses, ignoring basis convergence over he life of he conrac. 1 They herefore model he spo and he basis where is convergence is explicily incorporaed ino he dynamics. This paper exends heir approach by proposing a model of he spo and basis ha allows for long memory in volailiy, basis convergence and swiches in basis dynamics ha occur via a laen markov process. The consideraion of regime swiches is imporan, given ha significan changes in he cos of carry (COC) will affec he magniude of he basis and produce poenially differen dynamics and raes of convergence. This is likely o be imporan in he curren environmen, where expansionary moneary policies (designed o comba he effecs of he GFC) have produced record low shor erm ineres raes in many counries. Furher, decreases in he absolue value of he basis are associaed wih lower volailiy in he spo and fuures and higher levels of spo fuures correlaion (Lee, 1994; Ng and Pirrong, 1994; 1 Lien and Yang (8) are also criical of he sandard approach and develop an error correcion GARCH model ha includes he lagged basis in he spo fuures correlaion and he spo and fuures volailiies. Their approach however fails o impose basis convergence. Given ha a decrease in shor erm ineres raes may decrease he absolue value of he basis, his is consisen wih he posiive relaion beween equiy marke variance and shor erm ineres raes (Campbell, 1987; Glosen e al 1993).

3 Lien and Yang, 8). This suggess ha decreases in he absolue value of he COC will resul in higher and poenially less volaile minimum variance hedge raios (MVHRs). Recen lieraure ha joinly models spo fuures dynamics has used eiher swiching models beween he spo and he fuures (Alizadeh and Nomikos, 4; Lee and Yoder, 7) or long memory models (Dark, 7). 3 These approaches however ignore he lieraure which acknowledges he presence of occasional breaks and long memory in volailiy (see for example Kilic, 9). In he presence of regime swiches, a sligh misclassificaion (from failure o forecas he regime or esimaion error) may mean ha forecass from a regime swiching model will have a higher mean square error han forecass from a random walk (Dacco and Sachell, 1999). The developmen of regime swiching models ha more adequaely capure he underlying dynamics is herefore an imporan area for research. The paper herefore develops a bivariae long memory volailiy process wih swiches driven by a laen markov chain. This is imporan given ha his is he firs aemp o allow for regime swiches in a long memory volailiy process in his way. The paper herefore considers he finie sample properies of he maximum likelihood esimaor (MLE) via mone carlo simulaion. Applicaions of he model o he S&P5, he Serling/USD and he Canadian TSX 6 demonsrae ha regime swiches in he basis occur beween high and low volailiy saes associaed wih high and low absolue values of he COC. In and ou of sample forecass generally suppor he proposed model and demonsrae is abiliy o 3 Alizadeh and Nomikos (4) and Lee and Yoder (7) allow for high and low volailiy saes in he spo and he fuures. Dark (7) employs a long memory model of he spo and basis ha allows for basis convergence. Whils decreases in he absolue value of he basis are associaed wih lower volailiy in he spo and fuures, he relaionship is no srong (Ng and Pirrong, 1994). A regime swiching model beween he spo and he fuures is herefore likely o idenify high and low volailiy saes ha do no coincide wih high and low COC saes. As noed above however, he proposed approach more adequaely capures he spo fuures dynamics given ha he convenional mehods ignore basis convergence. 3

4 capure shifs in basis dynamics over an ou of sample period ha includes he global financial crisis (GFC). Relaive o convenional hedging approaches however, he effecs on porfolio risk reducion are mixed. 4 The srucure of he paper is as follows. Secion 1 presens he daa and illusraes he need o allow for long memory in volailiy and regime swiches in he basis. Secion presens a mone-carlo simulaion sudy, which reveals ha he parameers for he proposed model can be well esimaed via maximum likelihood. Secion 3 presens he applicaion of he model and Secion 4 concludes. 1. DATA Three daily daa ses are used o illusrae he imporance of regime swiches in basis dynamics and long memory in volailiy. All daa ses end on July 6, 11. The firs consiss of spo and fuures prices on he S&P5 from April 6, 198. The second consiss of he Serling/USD spo and fuures from January, 198. The hird consiss of he Canadian equiy index, he TSX 6 and is fuures from Sepember 8, For all daa ses he in sample period ends on he las rading day of December 6. This means ha he ou of sample period is dominaed by he global financial crisis. All daa is obained from Daasream where fuures prices are linked by rolling over on he firs day of he conrac expiraion monh. 5 Preliminary invesigaions examine proxies for he differenials ha are inpus ino he COC model as well as he exisence of long memory in volailiy. Le F represen he spo and fuures prices a ime, and le he basis be B F C C and. I is 4 Appendix A derives an expression for he bias in he MVHR when regime swiches are presen bu ignored. The reasons for he mixed hedging resul are discussed in he conclusion. 5 Daasream links nearby conracs. Alizadeh and Nomikos (4) creae a perpeual fuures conrac wih consan mauriy by weighing near and disan conracs. This approach however is no appropriae for mos paricipans who only hold he nearby conrac. For he conracs seleced, rollover occurs approximaely 3 weeks prior o expiraion. 4

5 well known ha he COC model can be expressed as F Ce i d T for equiy indices and F i if T Ce for currencies, where i ( i f ) is he coninuously compounded domesic (foreign) ineres rae, d is he annualized coninuously compounded dividend yield and T is he ime o mauriy in years. Given ha each of he fuures conracs expire quarerly, ineres raes over 3 monh horizons are obained. The (i d ) differenial for he S&P5 is calculaed via he US 3 monh Treasury bill rae a consan mauriy and he S&P5 dividend yield is sourced from Daasream. The Serling/USD differenial ( i i ) employed he 3 monh middle rae for UK Serling cerificaes and he 3 monh middle rae for US Treasury Bills in he secondary marke. The Canadian TSX 6 (i d ) differenial employed he 3 monh middle rae for he Canadian Treasury bill and he TSX 6 dividend yield. 6 Figures 1, and 3 plo he normalized basis B C, is change B C 1 (where boh basis series are muliplied by 1) and he COC differenials for he S&P5, he Serling/USD and he TSX 6 respecively. 7 The spikey paern in B C is caused by he jump in he basis on rollover o he nex conrac and he subsequen convergence o zero (subjec o ransacion coss) over he life of he conrac. High (low) absolue COC differenials are associaed wih high (low) levels of basis volailiy. The low volailiy saes for he S&P5 basis from approximaely November 1991 o Augus 1994, f 6 The yield on he TSX 6 was only available afer January 8,. Given ha yields from o he middle of 8 averaged approximaely % and were sable, his value was assumed for he yields from he beginning of he sample. These variables are no used in he esimaion and are only used for graphical purposes below. 7 Noe ha he basis series have had some ouliers removed from he graphs (in order o enhance readabiliy). Ouliers for he S&P5 normalised basis were on Ocober 19, 1987 (-1.4) and Ocober, 1987 (-8.7). For he S&P5 change in he basis he ouliers were on Ocober 19, 1987 (-8.1) and Ocober 1, 1987 (8.6). For he Canadian TSX 6 he ouliers were on Ocober 5, (7.5) for he normalised basis and on Ocober 5 and 6, for he change (6.8 and -6.7 respecively). The Serling/USD had no ouliers. The models below include a dummy variable in he mean equaion on hese daes. 5

6 Augus 1 o Augus 5 and Sepember 9 o July 11 coincide wih a low absolue value for he COC differenial. For Canada, he high volailiy during he year and from Augus 8 o April 9 is also associaed wih large absolue values of he COC differenial. The visual idenificaion of basis volailiy saes for he Serling/USD is less obvious, which is consisen wih he volailiy in he absolue value of he COC differenial. Imporanly by modeling he fuures, raher han he basis, convenional mehods ignore basis convergence and are less likely o capure he changing dynamics caused by he swiches beween hese volailiy saes. Furher, for all markes, he ou of sample period experiences a leas one shif in he basis volailiy dynamics/he absolue value of he COC differenial. The abiliy of he proposed model o predic hese shifs over he GFC and he implicaions for hedging will be examined in Secion 3. (Inser Figures 1 o 3) Table 1 ess he presence of long memory in he cash, fuures and basis volailiies via Lo s (1991) rescaled range saisic, he KPSS es and he specral densiy esimae of Robinson (1994). The resuls are consisen wih he exensive lieraure ha suppors long memory in financial marke volailiy, wih sronger evidence using absolue (raher han squared) reurns (Baillie, 1996; Andersen and Bollerslev, 1997b; Elder and Jin, 7). 8 8 Granger (198) demonsraed ha he cross-secional aggregaion of a small number of N dependen AR processes will resul in he aggregae series following an ARMA(N,N-1) process. If he AR coefficiens have a bea disribuion, hen in he limi he aggregae process is I(d). Ding and Granger (1996) exend his o he second momen and Lippi and Zaffaroni (4) relax he bea disribuion assumpion. Long memory in volailiy may arise from he aggregaion of muliple volailiy componens caused by heerogeneous informaion flows (Andersen and Bollerslev, 1997a) or heerogeneous raders (Muller e al, 1997). Long memory in sock indices may be due o he aggregaion of individual socks which are only weakly dependen (Lobao and Savin, 1998), i may also arise from markov processes (Chen e al, 1) or a heavy ailed regime swiching sochasic volailiy process (Liu, ). The inabiliy of Lo s and he KPSS es o idenify long memory in he S&P5 fuures squared reurns is due o he ess low power (Giriais e al, 3). 6

7 (Inser Table 1) An alernaive view however argues ha if occasional breaks are ignored, a shor memory process may easily be confused wih a long memory process (Diebold and Inoue, 1; Granger and Hyung, 4, Mikosch and Sarica, 4). This ension has spawned a lieraure ha suppors occasional breaks and long memory in volailiy. Lobao and Savin (1998) removed he effecs of srucural breaks by esing for long memory on sub-samples of daily daa. Andersen and Bollerslev (1997b) esed for long memory in shor spans of high frequency daa. Granger and Hyung (4) and Belrai and Morana (6) removed he effecs of breaks and esed for long memory on he break free series. Morana and Belrai (4), Baillie and Morana (9), Marens e al (9), Haldrup and Nielsen (6) and Kilic (9) esimaed models ha joinly capure srucural breaks and long memory. Given ha he daa considered in his sudy appears o exhibi srucural breaks and long memory in volailiy, a model ha capures hese dynamics is a worhwhile pursui. The previous research has no specified a model wih long memory in volailiy and regime swiches driven by a laen markov chain. Furher, he previous long memory models ha capure breaks and long memory have only considered univariae specificaions. The developmen of a markov swiching bivariae long memory volailiy model is he subjec of he nex secion.. MONTE CARLO SIMULATION The asympoic heory for he long memory volailiy models and heir mixing properies remains elusive. Via mone-carlo simulaion, Baillie e al (1996) consider he finie sample properies of MLE for he univariae FIGARCH process and show ha i 7

8 provides reasonable esimaes. This secion herefore considers he finie sample properies of MLE for he proposed model. To define he model le 1 be he informaion se a ime -1 and le s 1, denoe he unobserved sae variable a ime, which follows a firs order wo sae markov process. The FIGARCH (Baillie e al, 1996) and FIAPARCH (Tse, 1998) processes are considered. 9 The FIGARCH(1,d,1) model is specified as follows C C 1 (1) c, B bs,, j () C-1 s, j 1 s, j ~ N, H (3) d L1 L c 1 L 1 1 c, c c c c c, 1 db 1 1 L1 L 1 L 1 1 bs,, j bs, j bs, j bs, j bs, j b, (4) (5) j 1, (6) cb,, s j cb, s j c, b,, s j where,,, j,, s c bsj.the FIAPARCH(1,d,1) model replaces Equaions 4 and 5 wih 1 d c L L L c, c c c c c, c c, c (7) 1 d 1 1 b 1 1 L1 L 1 L bs,, j bs, j bs, j bs, j bs, j b, bs, j b, Three poins are worh highlighing. Firs, i is only he basis ha is subjec o regime swiches via he b, b, b, b and b parameers. 1 A consan fracional differencing bs,, j (8) 9 Oher long memory models ha could be implemened include FIEGARCH (Bollerslev and Mikkelsen, 1996) and HYGARCH (Davidson, 4). 1 A model where boh asses are subjec o regime swiching is an area for furher research. 8

9 parameer ( d b ) is consisen wih oher models ha incorporae srucural change ino he FIGARCH process (Baillie and Morana, 9; Kilic, 9). Second, he model does no impose basis convergence. This is because he main objecive of his secion is o examine he abiliy of MLE o esimae a model wih long memory and regime swiching via a laen markov chain. Basis convergence will be inroduced in secion Third, o keep he model racable, each regime exhibis consan correlaion. 1 To overcome he pah dependency problem, he recombining mehod is employed (Gray, 1996; Lee and Yoder, 7), where b, 1, bs,, 1 1, bs,, p 1 p and (9) p Pr s 1 1, 1 f p f 1 p 1, 1 1, 1, 1 1, 1 P 1Q f1, 1p1, 1 f, 1 1 p1, 1 f1, 1p1, 1 f, 1 1 p1, 1 (1) where P and Q represen he consan sae ransiion probabiliies 1 11 PPr s 1 s 1 p g (11) 1 QPr s s p h and (1) f f r s j, j, ' 1 s, j s, j s, j s, j H exp.5 H j 1, (13) 11 Furher, assume var F F 1 ~ I d f and var C C 1 ~ Idc where dc, d f 1. While similar markes may have common orders of fracional inegraion, he correlaion beween spo and fuures reurns is less han uniy. Therefore, given var B var F F 1 var C C 1 cov F F, C C var F var C I d i is highly likely ha 1 1 hen if and are var B Id I d = Id and 1, var B ~ I. If however f c cf. 1 Time varying correlaions are herefore capured via changes beween saes. A model ha allowed for ime varying correlaions wihin saes could for example employ he specificaion of Tse and Tsui (). This is no pursued here. 9

10 where is he normal cumulaive densiy funcion, and ' r C C, B C. To 1 1 begin he recursion for he regime probabiliy 1 Q Pr s 1. (14) P Q The FIGARCH model ses he pre-sample values of o b, N 1 Pr s 1 b,, s 1 Pr s 1 1 b,, s N, where N is he sample size.,, The FIAPARCH model ses he pre-sample values of b, bs,, bs j j b o N 1,,1 bs,, Pr s 1 1 Pr s 1 N bs bs,, 1 bs, 1 bs,, 1 bs,, bs, bs,, A runcaion lag of 1 observaions is employed and non-negaiviy imposed via he Bollerslev and Mikkelsen (1996) condiions. The likelihood funcion is equal o N log 1, 1, 1 1,,. (15) 1 LL p f p f Three experimens are performed wih sample sizes of 3 and 5 observaions. Samples of his size are realisic given ha his ype of model is suied o daily daa. Table presens he parameer values, heir bias and roo mean square error (RMSE). Figure 4 presens kernel densiy esimaes for seleced parameers. For experimen 1, he average duraion of sae 1 and is 15 periods ( 1 p ). For experimen, he average duraion of sae 1 () is 167 (43) periods. Experimen 1 considers asses ha are srongly posiively correlaed while Experimen considers asses ha have posiive bu weaker levels of correlaion. Experimen 3 inroduces volailiy asymmeries ino Experimen 1 via FIAPARCH. (Inser Table and Figure 4) 1

11 The resuls sugges ha for he sample sizes considered, MLE provides reasonable esimaes of he parameers. The esimaes appear roo n consisen and he inroducion of asymmeries in Experimen 3 only slighly increases he bias and RMSE of he esimaed parameers. The b and b esimaes are a lile unsable bu comparable o he spo parameers ha are no subjec o swiching. On less han.5% of he simulaions, he d b esimae for he FIAPARCH process converged o he upper boundary of 1. Despie his he kernel esimaes are similar o hose for he FIGARCH process. 3.1 Models esimaed 3. APPLICATION This secion applies he model in Secion o he daa ses in Secion 1. The model is augmened o allow for condiional mean dynamics and basis convergence. For robusness and comparaive purposes, four alernaive variance specificaions for he spo and basis equaions are considered: FIGARCH(1,d,1), FIAPARCH(1,d,1), GARCH(1,1) and GJR(1,1) (Glosen e al, 1993). For each variance specificaion, hree models are esimaed, giving a oal of 1 models. Model 1 is he proposed regime swiching model (he RS model). The condiional mean dynamics are specified as C C C a b (16) 1 c c c, 1 C B B B a I m b c m C C C, 1 1 s j bs, j os, j 1, 1, sj bs, j bs, j bs,, j (17) 1 1 where I 1, is an indicaor variable equal o one on conrac rollover, oherwise zero, m represens he days o mauriy of he fuures conrac (divided by 1) and B 1 C 1 acs like an error correcion erm ensuring ha he spo and fuures coinegrae 11

12 (provided b bs, is significan). Any remaining serial correlaion is capured by he j auoregressive erms. Given ha he basis residuals are a power funcion of s j and, Var B C m s, j s, j 1 bs,, j Cov C C B C m. s j s, j 1 s, j 1 cbs,, j m, Model is nesed wihin model 1 and does no allow for regime swiching (he No RS model). This model is similar o Chen e al (1999) and Dark (7). Model 3 represens he convenional approach and is a vecor error correcion model (VECM) beween he spo and he fuures wih condiionally heeroscedasic variances. Informaion crieria (AIC and SIC) and residual diagnosics indicaed ha one lag in he reurns for he VECM was required. Given reurn non normaliy, all models employ quasi MLE Model resuls Table 3 presens he key parameers for he bes model he RS FIAPARCH process. The resuls for he remaining 11 models are available on reques. (Inser Table 3) Long memory is presen in he spo and basis volailiies for all markes. Esimaes of he fracional differencing parameers (d) are saisically significan and reasonably close o he specral densiy esimaes. Asymmeries are presen in he spo volailiies for all markes ( c ). I is only he Serling/USD basis volailiy ha exhibis volailiy asymmeries. Posiive shocks o he Serling/USD in sae 1 impar higher volailiy han negaive shocks of equal magniude. The asymmeric effec in 13 Given ha consan ransiion probabiliies may be oo resricive (Gray 1996; Perez-Quinos and Timmerman, ), a fourh model ha allows for ime varying ransiion probabiliies could allow he P Pr s 1 s 1 g g COC and ransiion probabiliies o evolve according o 1 1 Q Pr s s h h COC where COC is he COC differenial a ime. This raises 1 1 quesions of endogeneiy and is an avenue for furher research. 1

13 sae is no significan. 14 The lack of volailiy asymmeries in he S&P5 and TSX 6 basis volailiies mean ha hey were modeled via he FIGARCH specificaion. 15 The mauriy effec parameers o, 1, and generally suppor an increase in he basis on rollover wih convergence in he condiional mean and variance over he life of he conrac. A comparison of he sae 1 and basis parameers suppors regime swiching, wih differen dynamics and levels of spo basis correlaion beween regimes. This is suppored by he AIC which srongly suppors he RS models over he No RS models. 16 The b b esimaes (no repored) suppor coinegraion beween he spo and fuures prices which is consisen wih he VECMs. Figure 5 plos he probabiliy of being in sae 1 ( p 1, ) from he RS-FIAPARCH models. The plos include he probabiliy implied by he model esimaion as well as he ou of sample sae forecas a ime, condiional on he informaion se a ime -1. The probabiliy of being in sae 1 represens he sae associaed wih he high absolue basis value/volailiy for he S&P5 and he TSX 6. For he Serling/USD, sae 1 denoes he sae associaed wih a low absolue value for he basis. The probabiliy plos are quie plausible and do remarkably well in he ou of sample period racking he absolue COC differenials and basis volailiies. The forecasing performance of he models will be considered formally in he nex secion. (Inser Figure 5) 14 The definiion of he basis is arbirary (F S or S -F ) and herefore asymmeries may be due o shocks of eiher sign. As a resul, on esimaing he GJR models, differen specificaions were ried where he indicaor variables swiched on when he shock was eiher negaive or posiive. The RS-GJR coefficiens ha capure he asymmeric volailiy response in he Serling/USD are also significan a he 5% level for sae The Engle and Ng (1993) es for asymmeries repored in he Table produced a p value of.9 for boh he S&P5 and TSX 6 basis volailiies. Whils his is significan a 1%, he FIAPARCH specificaions for he basis eiher experienced convergence difficulies or he b esimaes were no significan. 16 The AIC can be used o deermine he number of regimes (Psaradakis and Spagnolo, 3). 13

14 Finally, diagnosic ess indicae ha hese models mus be viewed wih cauion. Ljung Box diagnosics idenify some remaining dynamics in he squared sandardized residuals. Whils he GJR and FIAPARCH models improve he join es for asymmeries of Engle and Ng (1993), some weak asymmeric effecs remain, paricularly in he S&P5 spo equaion. Jarque-Bera ess indicae ha he esimaes are consisen bu no efficien and Nyblom (1989) ess indicae ha some parameer insabiliy remains in some models. 3.3 Forecasing performance This secion compares he in and ou of sample forecasing performance of he bes model (RS-FIAPARCH) o he remaining RS and No RS models. One sep ahead forecass condiional on he informaion se are considered for hree reasons. Firs, regime swiching models have difficuly idenifying regime swiches many periods ahead and should herefore only be used for shor erm forecasing. Second, whils long memory models are useful for long erm forecasing, hey may sill ouperform shor memory models over 1 day horizons (Crao and Ray, 1996; Degiannakis, 4; Grau, ). Third, muli-period MVHRs which require condiional forecass over he life of he hedge perform no beer han he sraegy in Secion 3.4 which only requires one sep ahead forecass (Dark, 7, 11). 17 To assess forecasing performance, le H ( H ˆ ) denoe he acual (forecas) covariance marix and le e be he forecas error marix defined as he elemens of e H Hˆ, where H are calculaed as, C C 1 B C and 1 17 Forecass from he RS models are obained via a probabiliy weighed average of he forecass from boh saes. To illusrae, he forecas of m is consruced as, s 1, s cb, 1m1 p1, 1 cb, 1, sm p1, 1 cb, 1, s m 1 cb, 1 1 ˆ ˆ ˆ ˆ ˆ where pˆ 1, is he one sep ahead forecas of p 1 1, 1. 14

15 C C B C. Lauren e al (8) show ha squared marix norms, namely 1 1 he p norm wih p = 1 and p = (he Frebenius norm) of he forecas error marix e provide a consisen ranking of he forecasing performance of alernaive models. 18 Le he loss differenial ( d ) be calculaed via,, d e e (18) al bench where e and al, e are he squared p norms of he error marix for an, bench alernaive model and he benchmark (he RS-FIAPARCH model). Null hypoheses of zero mean and median loss differenials are esed via he procedures in Diebold and Mariano (1995). A posiive loss differenial ha is saisically significan suppors he long memory model over is shor memory counerpar. 19 I should be noed ha whils he loss differenials provide a consisen ordering, hey may lack power given ha condiional variances place oo much emphasis on he errors caused by new shocks, paricularly large ones. This is problemaic because condiional volailiy models are based on pas informaion and are no designed o predic new shocks. The confidence inervals on mean error saisics can herefore be wide and reduce he abiliy of esing procedures o disinguish beween models (Poon, 8). As a consequence slighly lower levels of confidence will be applied when esing he null of zero mean loss differenials. (Inser Table 4) 18 This is performed by exending he condiions in Hansen and Lunde (1996) o a mulivariae seing. The p norm of a real m x n marix A is defined as m n p A a i1 j1 ij 1 p. 19 Auocorrelograms and he BDS es indicaed ha he difference series have very lile if any serial correlaion. Noneheless ess of he null of zero mean loss differenial employed a Newey-Wes correcion. The convenional Sign es was used o es he null of zero median loss differenial. 15

16 Table 4 presens in and ou of sample mean and median loss differenials for he hree markes. Given he low power of he mean error saisics he p values are repored for each differenial. The in sample resuls suppor he RS-FIAPARCH model for he S&P5 and Serling/USD forecass, however he No RS FIAPARCH model ends o perform slighly beer for he TSX 6. The ou of sample S&P5 resuls vary according o wheher a mean or median differenial is employed. The RS-FIAPARCH ouperforms a number of models wih and wihou RS. However depending on wheher he mean or median loss is considered, i may be saisically indisinguishable from or perform slighly worse han models wihou RS. The resuls for he Serling/USD srongly suppor he RS-FIAPARCH model, wih i clearly ouperforming all No RS models. The performance relaive o he RS FIGARCH and RS GJR models is comparable. Finally, he ou of sample resuls for he TSX 6 are a lile mixed bu generally suppor he RS-FIAPARCH model. The model ouperforms he oher RS models as well as he No RS GARCH, GJR and FIGARCH models. The performance however appears comparable o he No RS FIAPARCH which is consisen wih he in sample resuls. In summary, he in and ou of sample forecasing resuls generally suppor he proposed model and idenify he imporance of regime swiching when forecasing he spo basis dynamics. The implicaions for hedging are considered in he nex secion. I should be noed ha in order o avoid any bias from he runcaion lag of 1 observaions when esimaing he FIGARCH and FIAPARCH processes, he firs 5 observaions are no considered when forecasing in sample. 16

17 3.4 Hedging Oucomes This secion examines he relaive performance of he 1 models when esimaing dynamic MVHRs. The RS and No RS models forecas in and ou of sample MVHRs from ime o +1 condiional on he informaion se a ime ( 1 ) via c, 1 cb, c, 1 b, 1m1 cb, 1m1 m. (19) As he conrac approaches mauriy ( m ), he MVHR converges o uniy which is opimal given ha a hedge held o mauriy has no basis risk (see Chen e al, 1999 and Dark, 7). The four VECMs esimae he dynamic MVHRs via condiional one sep ahead forecass of 1 cf, 1 f, 1. This represens he convenional mehod of dynamic MVHR esimaion and ignores basis convergence. 1 To assess wheher he approach resuls in saisically significan reducions in risk, wo measures are considered he variance and he lower parial momen (LPM). The LPM is he area in he ail of he hedged porfolio reurn disribuion below a specified arge and is calculaed as, max, LPM R E R R df R () where is he arge reurn, R is he hedged porfolio reurn, is he order of he LPM (a measure of risk aversion) and df R is he cumulaive disribuion funcion of he hedged porfolio. 1 See Chen e al (3) and Lien and Tse () for reviews of he hedging lieraure. The LPM is no aligned wih he objecive of variance reducion which underpins he MVHR. Noneheless, he paper follows Coer and Hanly (6) who esimae MVHRs bu evaluae heir performance via ail risk minimizaion crieria. See Lien and Tse (1998) and Power and Vedenov (1) for ail risk minimizing fuures hedge raios. 17

18 Table 5 presens porfolio variances for all models plus hose obained from he naïve and OLS approach. 3 The differences in variance are measured relaive o he RS- FIAPARCH model for all markes excep for he S&P5 in sample. Here he RS FIGARCH model is he benchmark given ha i provides significanly lower porfolio variances han RS FIAPARCH. 4 The in sample resuls for he S&P5 indicae ha he RS-FIGARCH process ouperforms all approaches excep for he No RS FIGARCH and No RS FIAPARCH processes which provide comparable performance. These resuls herefore indicae ha long memory in volailiy and basis convergence are imporan when hedging, however here is no benefi o be gained from allowing for regime swiching. The ou of sample resuls are mixed bu indicae ha he RS-FIAPARCH and RS-FIGARCH model do no even ouperform he naïve hedge raio. For he Serling/USD, he RS-FIAPARCH model ouperforms he naïve, OLS and VECM models, however he No RS models perform equally well. This suggess ha basis convergence is imporan, however long memory and regime swiching do no improve hedging oucomes. The ou of sample resuls indicae ha he dynamic MVHRs ouperform he naïve and OLS MVHRs, however all dynamic approaches provide comparable performance. The resuls for he TSX 6 suppor he RS- FIAPARCH model over he alernaives in sample. This good resul however does no hold ou of sample where he performance of each of he approaches is comparable. 3 The naïve hedge raio ses 1. The OLS MVHR is he slope coefficien from he OLS regression C C F F The significance of he differences is calculaed using he moving block boosrap of Kunsch (1989). The following es saisic is calculaed, f X 1 a X b n where X a and X b represen he boosrapped porfolio reurns from he alernaive and he benchmark respecively. Each boosrapped densiy is based on 5 replicaions. The moving block mehod is chosen given ha i provides lower variances han he circular block and saionary boosraps (see Lahiri, 1999). Following Hall e al (1995), 1/5 he opimal block size is l n. Where nl did no generae an ineger, he number of blocks was rounded up o he neares ineger and observaions were removed from he end of he series. 18

19 (Inser Table 5) Tables 6 o 8 repor he LPMs for orders of 3 which is consisen wih risk aversion and he order employed by Coer and Hanly (6). Given he negaive skew in all hedged porfolio reurn disribuions, he absolue values of he lower ail arges are larger han he upper ail arges. 5 The resuls are mixed bu generally provide he following conclusions. Firs, he resuls ofen differ according o he order of he LPM and wheher he lower or upper ail is considered. Second, like he variance measures, he ou of sample resuls generally provide very lile difference beween he alernaive approaches. Third, he S&P5 and Serling/USD provide some suppor for models ha allow for basis convergence. Long memory and regime swiching however appears unimporan when hedging. Fourh, he RS models provided superior in sample risk reducion when seeking o minimize lef ail risk on he TSX 6. 6 This resul however did no apply o he ou of sample period. (Inser Tables 6 o 8) 4. CONCLUSION This paper proposed a model of he spo and basis ha allows for long memory and asymmeries in volailiy, basis convergence, and swiches in basis dynamics ha occur via a laen markov process. Mone carlo simulaion revealed ha he model can be reasonably well esimaed via maximum likelihood. An applicaion of he model o equiy and currency markes suppored he proposed specificaion and demonsraed ha basis dynamics exhibi regime swiches beween high and low volailiy saes ha are associaed wih high and low absolue values of he basis/cos of carry. In and ou of sample forecass generally suppor he proposed model and demonsrae is abiliy o 5 The significance of he differences is esed using he block boosrap procedure above. 6 Noe ha he lower ail measures for reurns <-1.5% are zero because here were no reurns less han his amoun. 19

20 capure shifs in basis dynamics over an ou of sample period ha includes he global financial crisis (GFC). The mixed hedging resuls are consisen wih Lee and Yoder (7) and Kofman and McGlenchy (5) who employ alernaive approaches ha allow for regime swiching. These mixed resuls may be due o a sligh misclassificaion from failure o forecas he regime or esimaion error (Dacco and Sachell, 1999). The findings sugges ha even if he proposed model provides improved forecass of he covariance marix, he effecs on he MVHR may wash ou given ha he MVHR is a combinaion of forecass. The proposed model noneheless hrows doub over he sandard approaches o modeling spo fuures dynamics and he esimaion of MVHRs. By modeling he spo and he fuures direcly, he sandard approaches run he risk of ignoring he rich dynamics and insighs ha can be obained when direcly modeling he basis.

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24 Granger, C. and Hyung, N. 4. Occasional srucural breaks and long memory wih an applicaion o he S&P5 absolue sock reurns, Journal of Empirical Finance, 11, Grau, T.,, Modelling Daily Value a Risk using FIGARCH ype models, mimeo, Universiy of Alicane. Gray, S Modelling he condiional disribuion of ineres raes as a regime swiching process, Journal of Financial Economics, 4, 7-6. Haldrup, N. and Nielsen, M. 6. A regime swiching long memory model for elecriciy prices, Journal of Economerics, 135, Hall, P., Horowiz, J. and Jing, B On blocking rules for he boosrap wih dependen daa, Biomerika, 8, Kilic, R. 9. Long memory and nonlineariy in condiional variances: A Smooh Transiion FIGARCH model. Journal of Empirical Finance, 18, Kofman, P and McGlenchy, P. 5. Srucurally Sound Dynamic Index Fuures Hedging, The Journal of Fuures Markes, 5 (1), Koumos, G. and Tucker, M Temporal Relaionships and Dynamic Ineracions beween Spo and Fuures Sock Markes. Journal of Fuures Markes, 16, Kroner, K. and Sulan, J Time-Varying Disribuions and Dynamic Hedging wih Foreign Currency Fuures. Journal of Financial and Quaniaive Analysis, 8, Kunsch, H The Jackknife and he Boosrap for General Saionary Observaions, Annals of Saisics, 17,

25 Kwiakowski, D., Phillips, P., Schmid, P. and Shin, Y Tesing he Null Hypohesis of Saionariy agains he alernaive of a Uni Roo. Journal of Economerics, 54, Lahiri, S Theoreical Comparisons of Block Boosrap Mehods, Annals of Saisics, 7, Lee, H. and Yoder, J. 7. A bivariae Markov regime swiching GARCH approach o esimae ime varying minimum variance hedge raios, Applied Economics, 39, Lee, T Spread and Volailiy in Spo and Forward Exchange Raes, Journal of Inernaional Money and Finance, 13, Lien, D. and Tse, Y Hedging Time-Varying Downside Risk, Journal of Fuures Markes, 18, Lien, D. and Tse, Y.. Some Recen Developmens in Fuures Hedging, Journal of Economic Surveys, 16(3), Lien, D. and Yang, L. 8. Asymmeric Effec of Basis on Dynamic Fuures Hedging: Empirical Evidence from Commodiy Markes, Journal of Banking and Finance, 3, Lippi, M. and Zaffaroni, P. 4. Conemporaneous aggregaion of linear dynamic models in large economies. Journal of Economerics, 1, Liu, M.. Modelling Long Memory in Sock Marke Volailiy. Journal of Economerics, 99, Lo, A Long-Term Memory in Sock Marke Prices. Economerica, 59,

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28 Appendix A: Bias in he MVHR from failing o allow for srucural breaks This Appendix derives an analyical expression for he bias in he MVHR when srucural breaks are presen and ignored. For illusraive purposes, a RS model wih wo saes, consan volailiy and consan ransiion probabiliies is employed. The following daa generaing process is assumed C (A.1) cs, j cs,, j C-1 B j 1, (A.) bs, j bs,, m j C-1 cs, 1 cs, 1 cbs, 1 Sae 1 N s, H, 1 s1 (A.3) bs, 1 cb, s 1 b, s 1 cs, cs, cbs, Sae N s, H, s (A.4) bs, cb, s b, s and capures he effec of basis volailiy convergence over he life of he fuures conrac. Consider a hedger ha esimaes a consan MVHR a ime using uncondiional momens. If he hedger allows for regime swiches and correcly idenifies he regime, he MVHR will be 7 s j m cs, j cbs, j cs,, j bsm j cbs, m j. (A.5) Now consider a hedger ha ignores regime swiching. The esimaed covariance marix will be equal o ha obained via a mixure of wo mulivariae normal disribuions. The wo componen mulivariae normal mixure (MNM) disribuion is 7 This assumes ha if a RS model is applied, i will correcly idenify he sae wih he model inferring a 1% probabiliy of being in ha sae. This of course is unrealisic. RS models make probabilisic saemens abou he likelihood of each sae. Therefore even a well specified model will generae covariance forecass ha are a probabiliy weighed average of he covariances in each sae. The acual bias is herefore likely o be less han ha presened here. 8

29 s, s, s s, s, s f x x H x H (A.6) where denoes he mulivariae normal densiy and s 1 and s are he mixing weighs. The MNM densiy has an expeced value and covariance marix equal o s1 s1 s s E X (A.7) ' ' s 1 s 1 s 1 s 1 s s s s ' s s s s s s s s Cov X H H (A.8) Assuming ha he spo and fuures markes are coinegraed ( B C 1, ), he b s j hedger who ignores regime shifs would esimae a MVHR ( ) which is relaed o he underlying DGP as follows m c cb c bm cbm (A.9) where 1, 1, 1,, c s cs cs s cs cs s1 cs, 1 s1 s cs, 1 cs, s cs, (A.1) (A.11) cb s1 cb, s1 s cb, s. (A.1) b s1 b, s1 s b, s 9

30 Table 1 Long memory in volailiy R R R/S KPSS Specral R/S KPSS Specral S&P5 Spo c.874 a a.74 a.39 Fuures a.95 a.36 Basis 1.95 b a a a.34 Serling/USD Spo a 1.6 a a a.35 Fuures 4.44 a.616 a a a.35 Basis 4.67 a a a 5.88 a.8 TSX 6 Spo.397 a.66 b a.963 a.4 Fuures.484 a.69 b a 1.16 a.41 Basis b 1.9 a a 1.67 a.35 Boh Lo s R/S es and he KPSS es have a null of shor memory dependence. The criical values are Lo s R/S es: 1%.861<Q<1.747, 5%.89<Q<1.86, 1%.71<Q<.98 KPSS es: 1%.37<Q<.461, 5%.3<Q<.581, 1%.5<Q<.743. a,b,c denoes significance a he 1%, 5% and 1% levels respecively. Specral densiy esimaes are performed via he esimaor of Robinson (1994). 3

31 Table : Mone Carlo Experimens Parameer Sample size True value Experimen 1 Experimen Experimen 3 Bias RMSE True value Bias RMSE True value Bias RMSE c d c c c c c bs, d b bs, bs, bs, bs, bs, bs, bs, bs, bs, cb, s cb, s g h

32 Table 3 Bes models S&P5 Serling/USD TSX 6 Coeff sa Coeff sa Coeff sa d.3 (3.99).37 (5.96).5 (4.77) c c.48 (3.14).64 (1.85).69 (8.3).44 (4.61).1 (.13).41 (3.6) c c 1.7 (7.75) 1.99 (1.56) 1.54 (7.85) Sae 1.54 (7.81).7 (1.9) -.33 (-.67).8 (11.76).4 (1.13) 1.83 (.57) 1.15 (3.56).15 (.94).39 (1.89) d.6 (5.78).9 (5.9).7 (6.1) b b.65 (7.79).51 (.66).47 (.17) (-1.85) - - b (4.4) - - b cb.9 (8.6) -.8 (-3.46) -.6 (-.81) Sae. (3.86) -.53 (-8.9).1 (1.91).6 (3.1) -.4 (-8.91).18 (6.63) 1. (3.3).1 (.64).1 (4.11) b.66 (3.4).7 (9.11) -.1 (-1.5) (.93) - - b (6.89) - - b. (.6) -. (-.97) -.5 (-1.86) cb g.18 (13.1) 3. (14.51).69 (.9) h.18 (14.7) 3. (16.14).4 (19.6) LL Nyblom JBc/JBb a 1931 a 35 a 767 a 1385 a 4195 a Asyc/Asyb a 7.8 c c c c Q / b Q a a 61. a a 3.8 a,b,c denoes significance a he 1%, 5% and 1% levels respecively. LL = log likelihood, Nyblom is he parameer sabiliy es of Nyblom (1989), Q is he Ljung Box Pierce saisic on he squared sandardized residuals for he h lag, Asy is he join es for asymmeries of Engle and Ng (1993). 3

33 Table 4 One day ahead forecas loss differenials No RS models RS models Period GAR GJR FIG FIAP GAR GJR FIG S&P5 Mean In.6 (.13) Ou.443 c (.9).37 c (.7).86 (.91).466 (.15).148 c (.9).17 (.15).58 (.).618 (.13).598 c (.9).68 c (.9).91 (.7).356 (.14).165 c (.9) Median In. a. a. a. a. a. a. a Ou b.34 a. a a.35 a Serling/USD Mean In.8 b (.1) Ou.45 a (.).8 a (.7).37 a (.1).4 a (.).54 a (.).4 a (.).44 a (.).5 (.16).4 b (.3) -.5 b (.).4 b (.4) -. (.89).1 (.1) Median In.7 a.6 a.5 a.4 a.5 a.1 a. Ou.4 a.3 a.5 a.6 a.4 a -.1 a. a TSX 6 Mean In.9 (.) Ou.746 c (.7). (.91) 1.4 b (.4).84 (.19) (.15) -.1 a (.4) -.58 (.).15 (.15).713 c (.8).3 (.88) 1.4 b (.5).95 (.14) (.14) Median In a a a.1 Ou b. a -.1 a a. c a,b,c denoes significance a he 1%, 5%, and 1% levels respecively. Loss differenials are calculaed via d e, al e, bench, where, al he alernaive model and e is he squared Frebenius norm of he error marix for e is he squared Frebenius norm of he error marix for he, bench benchmark model (he RS-FIAPARCH model). The benchmark model is he RS-FIAPARCH for he S&P5, Serling/USD and TSX 6. Posiive loss differenials ha are saisically significan indicae ha he benchmark model ouperforms he alernaive. To es for a non zero mean and median loss differenials, he ess of Diebold and Mariano (1995) are performed. Brackeed values for he mean loss differenials represen he p value for he es of a non zero mean. 33

34 Table 5 Porfolio variances from minimum variance hedging S&P5 Serling/USD TSX 6 In sample Ou of sample In sample Ou of sample In sample Ou of sample Var Diff Var Diff Var Diff Var Diff Var Diff Var Diff Naïve a a b a OLS.97.9 c a b b.65.9 a GAR VECM b d c No RS a c d RS d c b GJR VECM c c c No RS a.94.4 a RS a.91.1 d b FIG VECM b b No RS d d d RS d FIAP VECM d d b No RS b d RS a Differences for all models are relaive o he RS-FIAPARCH model, excep for he S&P5 (in sample) where differences are measured relaive o he RS-FIGARCH process. a,b,c,d denoes significance a he 1%, 5%, 1% and % levels respecively 34