The stock index futures hedge ratio with structural changes

Transcription

1 Invesmen Managemen and Financial Innovaions Volume 11 Issue Po-Kai Huang (Taiwan) The sock index fuures hedge raio wih srucural changes Absrac This paper esimaes he opimal sock index fuures hedge raio for S&P 500 FTSE 100 and Nikkei 225 sock indexes using bivariae GARCH model wih srucural changes (bivariae ICSS-GARCH). The aricle uses ICSS (Ieraed Cumulaive Sums of Squares) algorihm proposed by Inclan and Tiao (1994) o idenify ime poins of srucural changes in he financial ime series. The resuls show ha excep for FTSE 100 he bivariae ICSS-GARCH model does no ouperform he OLS and OLS-CI models. However he bivariae ICSS-GARCH model has beer performance han he bivariae GARCH model for all hree markes. The finding suggess he necessiy o incorporae srucural changes in GARCH models and shows he imporance o consider srucural changes when esimaing he hedge raios. Keywords: hedge raio srucural changes ICSS algorihm. JEL Classificaion: G11 C32. Inroducion Sock is one of he major invesmen insrumens for invesors. Since sysem risk canno be diversified away by porfolio formaion sock invesors would bear he sysem risk. Wih he exisence of sock index fuures markes invesors can hedge heir sock posiions wih sock index fuures o preven he value of heir sock porfolios from being affeced by he sysem risk. Thus he nex imporan issue is how o calculae he opimal hedge raio. One of he ways o hedge risk is using a naïve hedging sraegy. Invesors esablish sock index fuures posiions equal in magniude bu opposie sign o he sock porfolio. The hedge raio is equal o one. However only in he absence of basis risk on he spo commimen day he naïve hedge can fully reduce risk. In order o ge he opimal hedge raio under he exisence of basis risk Ederingon (1979) uses ordinary leas squares (OLS) o esimae he hedge raio and noices ha he hedge raio is less han one in mos cases. Even hough he OLS echnique accouns for basis risk i ignores he fac ha spo and fuures prices ofen have a uni roo and are coinegraed (Wahab and Lashgari 1993). Hence inegraing an error correcion erm ino OLS can improve he hedging performance (Ghosh 1993). The hedge raios esimaed by OLS wihou and wih an error correcion erm are boh consan over ime. However he hedge raio should be imevarying. Chang Chou and Nelling (2000) poin ou ha when sock marke volailiy increases he demand for hedging will increase. Their resul implies ha he hedge raio should vary wih spo volailiy. Therefore many sudies such as Park and Swizer (1995) and Yeh and Gannon (2000) esimae ime-varying hedge raios using bivariae ARCH and GARCH models. Po-Kai Huang The hedge raio may also depend on he level of price volailiy. When here are srucural changes in volailiy which are caused by poliical social or economic evens and unknown in advance he opimal hedge raio can vary. Lamoureux and Lasrapes (1990) consider ha because of a failure o ake accoun of srucural changes in he financial ime series an ARCH/GARCH model may overesimae he persisence in variance which means ha he impac of shocks on volailiy does no die ou quickly under he model. Wilson Aggarwal and Inclan (1996) also sugges ha if hedgers accoun for srucural changes in variance he porfolio will be correcly hedged. Furhermore ignoring srucural changes may significanly overesimae he degree of volailiy ransmission (Ewing and Malik 2005; Arago- Manzana and Fernandez-Izquierdo 2007; Marcelo Quiros and Quiros 2008). Therefore he hedge raio esimaed by eiher a bivariae ARCH or GARCH model wihou considering srucural changes may no be opimal. Mansur Cochran and Shaffer (2007) use a GARCH model wih srucural changes o compue he opimal hedge raio of exchange raes and show ha he raio can improve hedging performance. This is because hose srucural changes have influence on he volailiy srucure of he join spo and fuures disribuion. Sock markes ofen experience srucural changes (Aggarwal Inclan and Leal 1999). However few sudies have explored ha wheher srucural changes affec he hedge raio of sock markes. Our paper fills he gap in he lieraure by esimaing he opimal sock index fuures hedge raio based on a bivariae GARCH model wih srucural changes. Our resuls show ha excep for FTSE 100 he bivariae ICSS-GARCH model does no ouperform he OLS and OLS-CI models. However hedged porfolios consruced by he bivariae ICSS-GARCH model have beer performance han hose by he bivariae GARCH model across all he hree markes. 97

2 Invesmen Managemen and Financial Innovaions Volume 11 Issue Our finding provides a jusificaion for incorporaing srucural changes in GARCH models and shows ha i is imporan o consider srucural changes when esimaing he hedge raio. Our resuls are consisen wih he Mansur Cochran and Shaffer (2007) who show ha ICSS-GARCH model ouperforms he basic GARCH model for exchange raes markes. The remainder of his paper is organized as follows. Secion 1 describes he heory of fuures hedging. Secion 2 provides lieraure review of hedge raio esimaion secion 3 lays ou he mehodology o esimae hedge raio. Secion 4 describes he daa. Secion 5 provides he resuls of he hedge raio esimaion. The final secion concludes. 1. Theory of fuures hedging The concep of fuures hedging has been proposed in he early sages. Keynes (1930) assers ha hedgers owning spo endowmen and execuing shor hedge are willing o pay risk premium o speculaors o be exemped from price risk. The nex quesion for hedgers is how o ge he opimal hedge raio. There have been many differen heoreical approaches o deriving he opimal hedge raio depending on objecive funcions. The objecive funcion of he naïve hedging sraegy is risk avoidance. Working (1953) however argues ha he objecive funcion of hedgers is expeced profi maximizaion. In paricular because of basis risk shor hedgers would hedge if he basis is expeced o fall and would no hedge if he basis is expeced o rise. Johnson (1960) and Sein (1961) propose porfolio heory o incorporae risk avoidance of he naïve hedging sraegy wih Working s expeced profi maximizaion. The objecive funcion is o minimize he variance of a hedged porfolio. Alhough he minimum variance (MV) hedge raio ignores he expeced reurn of he hedged porfolio i is ofen used in many sudies because i is simple o undersand and o esimae. In his paper we adop he MV approach o deriving he opimal hedge raio. Neverheless following Kroner and Sulan (1993) we sar wih he objecive funcion wihin he mean-variance framework o show ha he opimal mean-variance hedge raio is he same as he MV hedge raio under he assumpion ha he fuures price follows a maringale process 1. 1 Pok Poshakwale and Ford (2009) invesigae hedging effeciveness of dynamic and consan models in he emerging marke of Malaysia. Paricularly hey use boh minimum variance and expeced uiliy maximum mehod o measure in-sample and ou-of-sample hedging performance. Assume ha an invesor holds a porfolio including one uni in he spo marke and a shor posiion of -b unis in he fuures marke. The payoff of his porfolio r is: r s bf (1) where s and f are he price changes of spo and fuures respecively. Assume furher ha he expeced uiliy funcion of he invesor can be expressed in he mean-variance framework E[ U()] r E() r Var() r (2) where is he degree of risk aversion > 0. The objecive funcion of he invesor isas follows: b Max E U r Max E s be f b b (3) [ ( )] ( ) ( ) 2 s f sf b By solving he firs-order condiion he opimal hedge raio b * is: * E() f b. sf 2 2 f 2 f (4) The firs erm of he righ hand side is he opimal hedge raio under he objecive funcion of variance minimizaion and he second erm is he hedge raio under he objecive funcion of payoff maximizaion. Shor hedgers would increase shor fuures posiion if fuures prices are expeced o fall and decrease oherwise. We assume ha fuures prices follow a maringale process so equaion (4) can be rewrien as: * sf b. 2 f (5) b * is a consan hedge raio and is ofen esimaed by he ordinary leas squares esimaor from a imeseries regression of changes in spo prices on changes in fuures prices. However he join disribuion of spo and fuures price changes is ime varying so he consan hedge raio may no be suiable for pracice use. Consider he dynamic process below. r s b f (6) 1 where s and f are spo and fuures price changes from ime -1 o respecively. b -1 is he shor fuures posiion a ime -1. The expeced uiliy funcion of he invesor is as follows: EUr [ ( )] Er ( ) ( r ). (7) The subscrip under he expecaion operaor and he variance symbol are used o emphasize ha hey 98

3 Invesmen Managemen and Financial Innovaions Volume 11 Issue are calculaed condiional on available informaion a ime. By solving he firs-order condiion he opimal hedge raio b is: * E ( f ) b. * s1f f 2 1 f1 (8) We furher assume ha fuures prices follow a maringale process so equaion (8) can be rewrien in he following manner: s1f1 2 f1 * b. (9) The difference beween b * and b * comes from he replacemen of he consan uncondiional variance by he ime varying condiional variance. b * varies wih ime o capure he price change caused by new informaion. 2. A review of hedge raio esimaion 2.1. Consan hedge raio. I is convenien for hedgers o esimae he hedge raio by ordinary leas squares (OLS) echnique. In addiion he OLS echnique accouns for basis risk. Ederingon (1979) uses OLS o esimae he hedge raio of GNMA T- Bill whea and corn fuures markes. They noice ha hedge raios are less han one in mos cases conrary o he naïve hedging sraegy. However he OLS echnique ignores he fac ha spo and fuures ofen share a uni roo and are coinegraed. Using Sandard and Poor 500 (S&P 500) index and he Financial Times 100 index Wahab and Lashgari (1993) show ha cash and fuures markes are coinegraed and i is appropriae o represen each series as an error correcion process. Therefore inegraing an error correcion erm ino OLS (OLS- CI) can improve he hedging performance. For example Ghosh (1993) documens ha hedge raios esimaed by an error correcion model have smaller forecas errors han hose by an OLS model Time-varying hedge raio. The hedge raios esimaed by OLS and OLS wih an error correcion erm are boh consan over ime. However he hedge raio should be ime-varying. Baillie and Myers (1991) argue ha since opimal hedge raios depend on he condiional disribuion of price movemens hey will almos cerainly vary over ime as his condiional disribuion changes. In oher words hey argue ha he ime-invarian opimal hedge raio is inappropriae. In addiion Chang Chou and Nelling (2000) poin ou ha when sock marke volailiy increases he demand for hedging will increase. Their resuls also imply ha hedge raios should vary wih spo volailiy. Many sudies have used ARCH/GARCH models o esimae ime-varying hedge raios for commodiy ineres rae and sock index fuures. For example Cecchei Cumby and Figlewski (1988) noe ha as expecaions abou risk and reurn changed hedge raios of Treasury bonds fuures esimaed by he univariae ARCH framework vary from 0.52 o over Myers (1991) noes ha bivariae GARCH models have heoreical advanages over OLS models. Using whea fuures conracs raded a he Chicago Board of Trade he shows ha he bivariae GARCH model provides superior hedging performance o he OLS model. Baillie and Myers (1991) use a bivariae GARCH model o esimae hedge raios for six commodiies fuures conracs such as beef coffee corn coon gold and soybeans and find ha a consan hedge raio is quie cosly for some commodiies. Choudhry (2004) compares he hedging effeciveness of an OLS model wih ha of a bivariae GARCH model. Using Ausralian Hong Kong and Japanese sock fuures markes his resuls show ha he ime-varying GARCH hedge raios ouperform he consan raios in mos of he cases. Pok Poshakwale and Ford (2009) invesigae hedging effeciveness of dynamic and consan models in he emerging marke of Malaysia. The resuls show ha ou of sample hedging performance of dynamic GARCH models in he Malaysian emerging marke is as good as he one repored for he highly developed markes in he previous lieraure. Kroner and Sulan (1993) and Park and Swizer (1995) furher apply a bivariae GARCH error correcion model o esimae ime-varying hedge raios. The error correcion erm of he model is used o capure he long-run relaionship beween spo and fuures prices. These wo sudies examine he issue using foreign currency fuures and sock index fuures respecively and boh find ha he bivariae GARCH error correcion model improves he hedging performance over several oher models such as he naïve OLS and OLS-CI models Time-varying hedge raio and srucural changes. Due o srucural changes which are caused by facors unknown a priori such as poliical social and economic evens he hedge raios esimaed by a bivariae ARCH or GARCH model may no be opimal. Lamoureux and Lasrapes (1990) consider ha failing o ake accoun of srucural changes in he financial ime series may cause an ARCH/GARCH model o overesimae he persisence in variance. Tha is he model assumes ha impacs of shocks on volailiy do no die ou quickly. This model misspecificaion in persisence may has influence on he hedging performance. However previous few sudies have explored wheher a GARCH model wih srucural changes 99

4 Invesmen Managemen and Financial Innovaions Volume 11 Issue can improve hedging performance. The sudy of Mansur Cochran and Shaffer (2007) is an excepion. They argue ha srucural changes have influence on he volailiy srucure of he join disribuion of spo and fuures prices. Therefore hey use a GARCH model considering srucural changes o compue he hedge raios in currency fuures and show ha he model can improve hedging performance. Sock markes ofen experience srucural changes (Aggarwal Inclan and Leal 1999). However few sudies have explored ha wheher srucural changes affec he hedge raio of sock markes. Our paper fills he gap in he lieraure by esimaing he sock index fuures hedge raios using a bivariae GARCH model wih srucural changes. Recenly some sudies have emphasized ha GARCH model wih srucural changes can improve hedging performance. Lien and Yang (2010) sugges ha daily currency risk can be beer hedged wih currency fuures when conrolling for uncondiional variance breaks in he bivariae GARCH model. Arago and Salvador (2011) employ several mulivariae GARCH models o esimae he opimal hedge raios for he Spanish sock marke. They show ha more complex models including sudden changes in volailiy ouperform he simpler models in hedging effeciveness boh wih in-sample and ou-of-sample analysis. Many approaches have been proposed o find srucural changes. Lamoureux and Lasrapes (1990) arbirarily assume ha srucural changes in he uncondiional variance occur a every 302 observaions over a range of 4228 observaions. Similar o Lamoureux and Lasrapes (1990) Kearns and Pagan (1993) rim he daa by separaely omiing observaions on reurns whose absolue value exceeded x per cen where x equals o and 1. Diebold (1986) argue ha persisen movemens in variance may be due o a failure o include policy regime dummies for he condiional variance inercep. Lasrapes (1989) confirms he argumen of Diebold (1986) improves he ARCH model finess and reduces volailiy persisence by accouning for moneary policy regime shifs in he model. Acker and Racine (1997) direcly consider Ocober 1989 crash even as a srucural change. However hese approaches are eiher arbirary or biased. Therefore a saisical procedure ha can effecively deec he acual srucural changes would be required. The ICSS (Ieraed Cumulaive Sums of Squares) algorihm proposed by Inclan and Tiao (1994) is ofen used o idenify ime poins of srucural changes in he financial ime series. For example he approach has been used in sudies such as Wilson Aggarwal and Inclan (1996) Aggarwal Inclan and Leal (1999) Malik (2003) Malik and Hassan (2004) and Malik Ewing and Payne (2005). They all conclude ha when srucural changes deeced by he ICSS algorihm are incorporaed ino ARCH/GARCH models he persisence of financial asse volailiy overesimaed by ARCH/GARCH models decreases dramaically. 3. Mehodology 3.1. A bivariae GARCH model. Kroner and Sulan (1993) propose a bivariae GARCH (11) * error correcion model o esimae b : s ( S F ) (10) 0s 1s 1 1 s f ( S F ) (11) 0f 1f 1 1 f s 1 ~N(0 H) f hss hsf hs 0 H hsf h ff 0 h f 1 hs h f s 0s 1 s s 1 2 s s 1 (12) (13) h v v v h (14) h v v v h (15) b f 0f 1 f f 1 2 f f 1 ĥ * sf (16) ĥff where S -1 and F -1 are he naural logarihm of spo and fuures prices a ime -1 respecively. -1 is he informaion se a ime -1. The erm (S -1 F -1 ) is he error correcion erm which capures he comovemen and long-run sable equilibrium beween sock and fuures prices. The error correcion erm should be added ino he bivariae ime series model if wo variables are coinegraed (Engle and Granger 1987). b is he ime-varying hedge raio. * 3.2. The ICSS algorihm. The mehodology we will use o deec srucural changes in he variance of an observed ime series is based on he ICSS (Ieraed Cumulaive Sums of Squares) algorihm proposed by Inclan and Tiao (1994). The analysis assumes ha variance of a ime series is saionary over an iniial period unil he occurrence of a srucural change caused by an exogenous shock. The variance is hen saionary again unil he nex shock occurs. The process is repeaed hrough ime and yields an unknown number of srucural changes in he variance. 100

5 Invesmen Managemen and Financial Innovaions Volume 11 Issue Le {a } be a series wih zero mean and wih 2 uncondiional variance and = 1 T. The 2 variance wihin each inerval is denoed by j j = 0 1 N T and given as follows: 1 k k1 k2 2 2 NT NT k T (17) where k 1 k 2 k N are he se of ime poins when T srucural changes occur. Inclan and Tiao (1994) use a cumulaive sum of squares approach o esimae he number of changes in variance and ime poins of srucural changes. Le k k (18) 1 C a k T be he (mean-cenered) cumulaive sum of he squares from he firs observaion of he series o he kh poin in ime. Define he saisic D k as follows: C k k Dk k 1... T CT T wih D 0 D 0. T (19) If a ime series has no srucural changes in variance he D k saisics will oscillae around zero. On he conrary if he series conains one or more srucural changes he D k saisics will drif eiher upward or downward away from zero. Criical values used o deec a significan srucural change in variance are obained from he disribuion of D k saisics under he null hypohesis of homogeneous variance. The null hypohesis is rejeced if he maximum absolue value of D k is greaer han he criical value. Define k * o be he value of k a which max k D k is aained. k * is aken as an esimae of he srucure-change poin if max k T 2 Dk exceeds a predeermined boundary. The erm T 2 is required for sandardizing he disribuion. Under he null hypohesis of homogeneous variance T 2 D k behaves like a Brownian bridge asympoically. The criical value of 95h percenile of he asympoic disribuion of max k T 2 Dk is Following Aggarwal Inclan and Leal (1999) we also se he criical value o be However if he analyzed series conains muliple srucurechange poins D k saisics is insufficien o find hese due o masking effecs. To solve he problem Inclan and Tiao (1994) propose an ieraive scheme ha uses D k saisics o sysemaically idenify any possible srucure-change poins a differen pieces of he series A bivariae GARCH model wih srucural changes. In he univariae ARCH/GARCH model when srucural changes deeced by he ICSS algorihm are incorporaed direcly ino condiional variance in form of dummy variables he overesimaed persisence of variance decreases dramaically (Wilson Aggarwal and Inclan 1996; Aggarwal Inclan and Leal 1999; Malik 2003; Malik and Hassan 2004; and Malik Ewing and Payne 2005). Srucural changes also affec he volailiy srucure of he join spo and fuures disribuion. Therefore following Mansur Cochran and Shaffer (2007) we use a bivariae GARCH model wih srucural changes o esimae he hedge raios for sock index fuures. The model is as follows: s ( S F ) (20) 0s 1s 1 1 s f ( S F ) (21) 0f 1f 1 1 f s 1 ~N(0 H) f hss hsf hs 0 H hsf h ff 0 h f 1 hs h f n s 0s 1 s s 1 2 s s 1 si si i2 (22) (23) h v v v h d D (24) m f 0f 1 f f 1 2 f f 1 f j f j j2 h v v v h d D (25) b * ĥsf (26) ĥ ff where i and j are srucural changes deeced by he ICSS algorihm and n and m are he number of srucural changes for spo and fuures reurns respecively. D si (D fj ) are dummy variables. If he ICSS algorihm deecs a shif in he volailiy of spo or fuures reurns D si (D fj ) ake a value of one from each poin of srucural change onwards zero elsewhere. As a resul of inercep specificaion in he condiional variance we add only m-1 and n-1 srucural changes in he condiional variance o avoid he problem of mulicollineariy. b * is he ime-varying hedge raio. 101

6 Invesmen Managemen and Financial Innovaions Volume 11 Issue Daa descripion Daa for he sudy are obained from Daasream daabase. The daily prices used in his sudy are S&P 500 index and fuures FTSE 100 index and fuures and Nikkei 225 index and fuures. The hree fuures conracs are raded in Chicago Mercanile Exchange (CME) London Inernaional Financial Fuures Exchange (LIFFE) and Osaka Securiies Exchange (OSE) respecively. A any ime S&P 500 index fuures has eigh conracs ousanding wih delivery in he March quarerly cycle. For FTSE 100 index fuures he hree neares quarerly monhs (March June Sepember and December) will be lised. The conrac monhs of Nikkei 225 Fuures are 5 near conracs in he March quarerly cycle. The prices of conracs wih he neares expiraion dae are used for he sudy. The daily prices are ransformed ino weekly raes of reurns based on Wednesday prices. When here is no rading on a given Wednesday he las rading day before Wednesday is used o compue reurns. The sample period is from January o December including 939 weekly observaions 1. Reurns are defined as changes in he logarihmic prices R ln( p ) ln( p 1). The reason for using weekly raher han daily daa is wofold. The firs one is ha weekly reurns conain less noise han daily measures. The oher one is weekly hedging adjusmens would incur lower hedging cos han he daily adjusmen sraegy. Summary saisics of price and reurn series for he hree financial markes are presened in Table 1 (see Appendix). Panel B repors he resul of he reurn series. The sample mean for he S&P 500 and FTSE 100 sock index spo and fuures reurns are significanly differen from zero bu hose for Nikkei 225 are close o zero. The sample kurosis and Jarque-Bera saisics show ha he reurn series are no normally disribued. There is evidence of serial correlaion in S&P 500 and FTSE 100 sock index and fuures reurn series. Thus a condiional-mean equaion o ake accoun of he serial correlaion in he reurns is required 2. In addiion he Q 2 (8) and Q 2 (16) saisics indicae significan serial correlaion in he squared reurns across all he markes which suggess he need o model he condiional heeroscedasiciy. 1 I have no access righs o Daasream due o conrac expiraion so he sample period sops in December We idenify he bes-fiing specificaion of condiional-mean equaion by Box-Jenkins echniques for S&P 500 and FTSE 100 sock index and fuures. The parial auocorrelaion funcion suggess ha he ARMA ( 1 7 0) model would be appropriae for S&P 500 spo and fuures reurn series and ARMA( 1 0) model would be appropriae for FTSE 100 spo and fuures reurn series. The saisics are shown in Table 5 Table 6 and Table Empirical resuls 5.1. Uni roo es. To es for he saionariy of prices and reurns he augmened Dickey-Fuller es (ADF) is used. The null hypohesis is ha a ime series has a uni roo. We consider hree differen specificaions as follows. p Y Y Y (27) 1 i i1 i2 p Y Y Y (28) 0 1 i i1 i2 p Y Y Y (29) i i1 i2 where Y can denoe eiher price or reurn for spo and fuures. The appropriae number of lagged differences p is deermined by BIC crierion. Specifically we choose a number as p from zero o 20 o minimize he BIC crierion. The difference beween he hree equaions is he presence of drif erm or ime rend. Equaion (27) is like a pure random walk model equaion (28) adds a drif erm ( 0 ) and equaion (29) exends boh a drif and a ime rend (). In all cases if he parameer is significanly differen from zero series Y does no conain a uni roo indicaing ha series Y is saionary. Table 2 (see Appendix) presens he resuls of uni roo ess conduced on prices and reurns of spo and fuures. The resuls confirm he presence of a uni roo in he logarihmic price indices bu here is no evidence of a uni roo in heir firs differences i.e. reurns. Uni roo ess indicae ha spo and fuures series for each index are boh nonsaionary in prices and reurns are saionary. Because ha spo and fuures series are inegraed of order one I(1) ess for coinegraion can be underaken. Therefore he nex sep is o es wheher spo and fuures prices are coinegraed Coinegraion es. Two ime series are said o be coinegraed if hey share a common rend. Tha is here is a long-erm equilibrium relaionship beween he wo variables. Engle and Granger (1987) iniiae he coinegraion es echnique. They propose he Engle-Granger wo-sep procedure o es wheher wo nonsaionary ime series are coinegraed. The main advanage of heir mehod is is simpliciy. Based on he Engle-Granger wo-sep procedure he firs sep is regressing spo prices on fuures prices. The equaion is as follows: S F e (30) where S and F are spo and fuures prices in he logarihmic form respecively. Residual e can be 102

7 Invesmen Managemen and Financial Innovaions Volume 11 Issue regarded as emporary deviaions from he long-run equilibrium. The second sep is esing residuals e for uni roos. The null hypohesis is ha spo and fuures are non-coninegraed. If (as shown in equaions (27) (28) and (29)) is significanly differen from zero e residual is saionary indicaing ha spo and fuures prices are coinegraed. Table 3 (see Appendix) presens he Engle-Granger coinegraion resuls for ADF uni roo ess on residuals from he coinegraing regression. The coinegraing parameers are highly significan and are approximaely uniy. are all significanly differen from zero providing evidence of coinegraion in spo and fuures markes. The resuls are also consisen wih Wahab and Lashgari (1993) who show ha he cash and fuures markes are coinegraed. Therefore he error correcion erm (S -1 F -1 ) which capures he co-movemen and long-run sable equilibrium beween sock and fuures prices can be included in he model o esimae he opimal hedge raio Srucural changes deeced by he ICSS algorihm. Table 4 (see Appendix) repors he number and daes of srucural changes in variance idenified by he ICSS algorihm for weekly sock index spo and fuures reurns. For each period Table 4 also provides he level of annualized sandard deviaion for weekly reurns and numbers of observaions per period. The numbers of srucural changes are no he same in spo and fuures for each marke wih he excepion of Japan markes. The S&P 500 and FTSE 100 sock index have eigh break poins in variances and heir fuures have six break poins; Nikkei 225 sock index spo and fuures boh have nine break poins. Excep he sevenh srucural change in Japan change poins of fuures markes eiher lead or synchronize hose of spo markes. The regimes of differen variance srucures are shown in Figures 1 and 2 for S&P 500 Figures 3 and 4 for FTSE 100 and Figures 5 and 6 for Nikkei 225. Boundaries are se o be ± 3 sandard deviaions where he sandard deviaion is he uncondiional volailiy calculaed wihin each regime period Noes: Doed lines specify boundaries of ± 3 sandard deviaions. Srucural changes are deeced using he ICSS algorihm. Sample period is from January o December Fig. 1. S&P 500 sock index weekly reurns Noes: Doed lines specify boundaries of ± 3 sandard deviaions. Srucural changes are deeced using he ICSS algorihm. Sample period is from January o December Fig. 2. S&P 500 sock index fuures weekly reurns 103

8 Invesmen Managemen and Financial Innovaions Volume 11 Issue Noes: Doed lines specify boundaries of ± 3 sandard deviaions. Srucural changes are deeced using he ICSS algorihm. Sample period is from January o December Fig. 3. FTSE 100 sock index weekly reurns Noes: Doed lines specify boundaries of ± 3 sandard deviaions. Srucural changes are deeced using he ICSS algorihm. Sample period is from January o December Fig. 4. FTSE 100 sock index fuures weekly reurns Noes: Doed lines specify boundaries of ± 3 sandard deviaions. Srucural changes are deeced using he ICSS algorihm. Sample period is from January o December Fig. 5. Nikkei 225 sock index weekly reurns 104

9 Invesmen Managemen and Financial Innovaions Volume 11 Issue Noes: Doed lines specify boundaries of ± 3 sandard deviaions. Srucural changes are deeced using he ICSS algorihm. Sample period is from January o December Fig. 6. Nikkei 225 sock index fuures weekly reurns 5.4. Consan hedge raio OLS and OLS-CI. Table 5 (see Appendix) repors he resuls from consan hedge raio models OLS and OLS-CI. The OLS and OLS-CI models are obained by imposing v 1s = v 2s = v 1f = v 2f = 1s = 1f = 0 and v 1s = v 2s = v 1f = v 2f = 0 in equaions (10) o (16). The hedge raios esimaed by hese wo models are consan because condiional variance is assumed o be ime-invarian. As expeced he coefficien of he error correcion erm is significan for each ime series. In addiion he likelihood raio (LR) es saisics under he null hypohesis H 0 : 1s = 1f = 0 presened a he boom of Table 5 are all significan a he 1% level. These resuls show ha he OLS-CI model fis he daa beer han he OLS model. Consisen wih he empirical resuls of Kroner and Sulan (1993) and Mansur Cochran and Shaffer (2007) hedge raios esimaed by he OLS model are smaller han hose esimaed by he OLS-CI model across he financial markes excep for S&P Time-varying hedge raio bivariae GARCH. Table 6 (see Appendix) repors he resuls from he bivariae GARCH model. The bivariae GARCH model is obained by imposing he resricion d si = d fj = 0 in equaions (20) o (26). The GARCH coefficiens (v 1s v 2s v 1f and v 2f ) are all significan a he 1% level across all he sock indices implying ha i is appropriae o include he GARCH specificaion in he hedge raio esimaion model. The likelihood raio es saisics LR 1 under he null hypohesis H 01 : v 1s = v 2s = v 1f = v 2f = 1s = 1f = 0 is significan a he 1% level. These resuls show ha he bivariae GARCH model fis he daa beer han he OLS model. In addiion he likelihood raio es saisics LR 2 under he null hypohesis H 02 : v 1s = v 2s = v 1f = v 2f = 0 is also significan a he 1% level. This shows ha he bivariae GARCH model fis he daa beer han he OLS-CI model Time-varying hedge raio ICSS-GARCH. Table 7 (see Appendix) repors he resuls from he bivariae ICSS-GARCH model. Many coefficiens of srucural change dummy variables are a leas significan a he 10% level for each sock index. Afer including srucural changes in he model some GARCH coefficiens (v 1s v 2s v 1f and v 2f ) are insignifican for each markes. I implies ha i is also appropriae o include he GARCH specificaion in he hedge raio esimaion model. However he GARCH effec is only parially capured by he srucural changes deeced by he ICSS algorihm. Therefore a more complee analysis should allow for boh he GARCH effec and he srucural change effec. In addiion he likelihood raio es saisics LR 1 LR 2 and LR 3 which compare he OLS OLS-CI and bivariae GARCH models wih he bivariae ICSS-GARCH model show ha he bivariae ICSS-GARCH model fis he daa well for each sock index Hedging performance comparison. Considering ha an invesor holds a porfolio including one uni in he spo marke and a shor posiion of -b unis in he fuures marke he following basic profi model is used. s b f (31) where is he rae of reurn. The objecive funcion of minimum variance (MV) hedge raio is o minimize he variance of he hedged porfolio in oher words o minimize he variance of. In order o compare he hedging performance of he four esimaion models we use equaion (31) o consruc a porfolio and calculae he variance of. 105

10 Invesmen Managemen and Financial Innovaions Volume 11 Issue Table 8 (see Appendix) presens he resuls. Surprisingly he bivariae ICSS-GARCH model does no ouperform he OLS and OLS-CI models excep for FTSE 100. The hedge raio esimaed by he bivariae ICSS-GARCH model yields he lowes variance for FTSE 100 markes. However for S&P 500 and Nikkei 225 markes heir variance of and are higher han hose of he OLS and OLS-CI models. The bes model for S&P 500 marke is eiher OLS or OLS-CI. The bes model for Nikkei 225 marke is OLS. In paricular he bivariae ICSS-GARCH model has beer performance han he bivariae GARCH model for all he hree markes. I may imply ha srucural changes idenified by he ICSS algorihm indeed have influence on he volailiy srucure of he join spo and fuures disribuion. In order o increase he hedging performance i is necessary o include he srucural changes in he bivariae GARCH model. Our resuls are consisen wih he Mansur Cochran and Shaffer (2007) who show ha ICSS-GARCH ouperforms he basic GARCH model for exchange raes markes. Conclusion Calculaing hedge raios for sock index fuures is an imporan issue o preven he value of a sock porfolio from being affeced by sysemaic risk. Many sudies use bivariae ARCH and GARCH models o esimaed ime-varying hedge raios. However Lamoureux and Lasrapes (1990) sugges ha failing o ake accoun of srucural changes in he financial ime series can cause ARCH/GARCH References 106 models o overesimae he persisence in variance. A marke wih srucural changes in variance experiences imporan informaion shocks and hese shocks can be incorporaed ino he volailiy generaing process of he oher marke. Therefore ignoring srucural changes may significanly overesimae he degree of volailiy ransmission (Ewing and Malik 2005; Arago-Manzana and Fernandez-Izquierdo 2007; Marcelo Quiros and Quiros 2008). Furhermore hese srucure changes can have influence on he volailiy srucure of he join spo and fuures disribuion (Mansur Cochran and Shaffer 2007). In his sudy we use he bivariae GARCH model wih srucural changes (bivariae ICSS-GARCH) o esimae he sock index fuures hedge raios for S&P 500 FTSE 100 and Nikkei 225 sock indexes. We use he ICSS (Ieraed Cumulaive Sums of Squares) algorihm proposed by Inclan and Tiao (1994) o idenify ime poins of srucural changes in he financial ime series. Our resuls show ha excep for FTSE 100 he bivariae ICSS-GARCH model does no ouperform he OLS and OLS-CI models. However he bivariae ICSS-GARCH model has beer performance han he bivariae GARCH model for all he hree markes. Our findings sugges he necessiy o incorporae srucural changes in GARCH models and show he imporance o consider srucural changes when esimaing he hedge raios. Our resuls are consisen wih he Mansur Cochran and Shaffer (2007) who show ha he ICSS- GARCH model ouperforms he basic GARCH model for exchange raes markes. 1. Aggarwal R. Inclan C. and Leal R. (1999). Volailiy in Emerging sock markes Journal of Financial and Quaniaive Analysis Vol. 34 No. 1 pp Aragó V. and Salvador E. (2011). Sudden Changes in Variance and Time Varying Hedge Raios European Journal of Operaional Research Vol. 215 No. 2 pp Arago-Manzana V. and Fernandez-Izquierdo M.A. (2007). Influence of Srucural Changes in Transmission of Informaion beween Sock Markes: A European Empirical Sudy Journal of Mulinaional Financial Managemen Vol. 17 No. 2 pp Baillie R.T. and Myers R.J. (1991). Bivariae GARCH Esimaion of he Epimal Commodiy Fuures Hedge Journal of Applied Economerics Vol. 6 No. 2 pp Cecchei S.G. Cumby R.E. and Figlewski S. (1988). Esimaion of he Opimal Fuures Hedge Review of Economics and Saisics Vol. 70 No. 4 pp Chang E. Chou R.Y. and Nelling E.F. (2000). Marke Volailiy and he Demand for Hedging in Sock Index Fuures Journal of Fuures Markes Vol. 20 No. 2 pp Choudhry T. (2004). The Hedging Effeciveness of Consan and Time-varying Hedge Raios Using Three Pacific Basin Sock Fuures Inernaional Review of Economics and Finance Vol. 13 No. 4 pp Ederingon L.H. (1979). The Hedging Performance of he New Fuures Markes Journal of Finance Vol. 34 No. 1 pp Engle R.F. Granger C.W.J. (1987). Co-Inegraion and Error Correcion: Represenaion Esimaion and Tesing Economerica Vol. 55 No. 22 pp Ewing B.T. and Malik F. (2005). Re-Examining he Asymmeric Predicabiliy of Condiional Variances: The Role of Sudden Changes in Variance Journal of Banking and Finance Vol. 29 No. 10 pp Ghosh A. (1993). Hedging Wih Sock Index Fuures: Esimaion and Forecasing wih Error Correcion Model Journal of Fuures Markes Vol. 13 No. 7 pp

11 Invesmen Managemen and Financial Innovaions Volume 11 Issue Inclan C. and Tiao G.C. (1994). Use of Cumulaive Sums of Squares for Rerospecive Deecion of Changes of Variance Journal of he American Saisical Associaion Vol. 89 No. 427 pp Johnson L.L. (1960). The Theory of Hedging and Speculaion in Commodiy Fuures Review of Economic Sudies Vol. 27 No. 3 pp Keynes J.M. (1930). A Treaise on Money Vol. 2 UK: Macmillan. 15. Kroner K.F. and Sulan J. (1993). Time-Varying Disribuions and Dynamic Hedging wih Foreign Currency Fuures Journal of Financial and Quaniaive Analysis Vol. 28 No. 4 pp Lamoureux C.G. and Lasrapes W.D. (1990). Persisence in Variance Srucural Change and he GARCH Model Journal of Business and Economic Saisics Vol. 8 No. 2 pp Lien D. and Yang L. (2010). The Effecs of Srucural Breaks and Long Memory on Currency Hedging Journal of Fuures Markes Vol. 30 No. 7 pp Ljung G.M. and Box G.E.P. (1978). On a Measure of Lack of Fi in Time Series Models Biomerika Vol. 65 No. 2 pp Malik F. Ewing B.T. and Payne J.E. (2005). Measuring Volailiy Persisence in he Presence of Sudden Changes in he Variance of Canadian Sock Reurns Canadian Journal of Economics Vol. 38 No. 3 pp Mansur I. Cochran S.J. and Shaffer D. (2007). Foreign Exchange Volailiy Shifs and Fuures Hedging: An ICSS-GARCH Approach Review of Pacific Basin Financial Markes and Policies Vol. 10 No. 3 pp Marcelo J.L.M. Quiros J.L.M. and Quiros M.M.M. (2008). Asymmeric Variance and Spillover Effecs: Regime Shifs in he Spanish Sock Marke Journal of Inernaional Financial Markes Insiuions and Money Vol. 18 No. 1 pp Myers R.J. (1991). Esimaing Time-Varying Opimal Hedge Raios on Fuures Markes Journal of Fuures Markes Vol. 11 No. 1 pp Park T.H. and Swizer L.N. (1995). Time-Varying Disribuions and he Opimal Hedge Raios for Sock Index Fuures Applied Financial Economics Vol. 5 No. 3 pp Pok W.C. Poshakwale S.S. and Ford J.L. (2009). Sock Index Fuures Hedging in he Emerging Malaysian Marke Global Finance Journal Vol. 20 No. 3 pp Sein J.L. (1961). The Simulaneous Deerminaion of Spo and Fuures Prices American Economic Review Vol. 51 No. 5 pp Wahab M. and Lashgari M. (1993). Price Dynamics and Error Correcion in Sock Index and Sock Index Fuures Markes: A Coinegraion Approach Journal of Fuures Markes Vol. 13 No. 7 pp Wilson B. Aggarwal R. and Inclan C. (1998). Deecing Volailiy Changes Across he Oil Secor Journal of Fuures Markes Vol. 16 No. 3 pp Working H. (1953). Fuures Trading and Hedging American Economic Review Vol. 43 No. 3 pp Yeh S.C. and Gannon G.L. (2000). Comparing Trading Performance of he Consan and Dynamic Hedge Models: A Noe Review of Quaniaive Finance and Accouning Vol. 14 No. 2 pp Appendix Table 1. Descripive saisics Panel A shows he naural logarihm of sock index spo and fuures prices. Panel B shows reurns of spo and fuures. Reurns are defined as he change in logarihmic prices i.e. r = ln(p ) ln(p -1 ). J-B is he Jarque-Bera es for normaliy. (j) denoe he jh order auocorrelaion of index and reurns. Q 1 (j) represen he Ljung and Box (1978) es saisic for no serial correlaion in he series up o lag j. Q 2 (j) represen he Ljung and Box (1978) es saisics for no serial correlaion in he squares of he series up o lag j. Mean Sd dev Skewness Kurosis J-B (8) (16) Q1(8) Q1(16) Q2(8) Q2(16) Panel A: The naural logarihm of sock index spo and fuures prices S&P 500 FTSE 100 Nikkei 225 Spo *** *** *** 99.63*** *** *** *** *** Fuures *** *** *** 99.48*** *** *** *** *** Spo *** *** *** 70.32*** *** *** *** *** Fuures *** *** *** 69.68*** *** *** *** *** Spo *** *** *** *** *** Fuures *** * *** *** *** *** Panel B: Spo and fuures sock index reurns S&P 500 FTSE 100 Nikkei 225 Spo ** ** *** *** *** 34.53*** *** *** Fuures ** *** *** *** *** 35.92*** *** *** Spo * * *** *** *** 27.84** *** *** Fuures * ** *** *** *** 29.40** *** *** Spo *** 70.12*** *** *** Fuures *** 57.22*** *** *** Noes: *** ** and * indicae significance a he 1% 5% and 10% level respecively. 107

12 Invesmen Managemen and Financial Innovaions Volume 11 Issue Table 2. ADF uni roo es Panel A shows sock index spo and fuures prices in he naural logarihmic form. Panel B shows reurns of spo and fuures. Reurns are defined as he change in he logarihmic prices i.e. r = ln(p ) ln(p -1 ). ADF models are as follows: Model 1: Y Y Y. p 1 i i1 i2 p Model 2: Y Y Y. 0 1 i i1 i2 Model 3: Y Y Y i i1 i2 p where Y denoes he series of ineres. The appropriae number of lagged differences p is deermined by he BIC crierion. Specifically we choose a number as p from zero o 20 o minimize he BIC. Panel A: Spo and fuures sock index S&P 500 FTSE 100 Nikkei 225 Panel B: Spo and fuures sock index reurns S&P 500 FTSE 100 Nikkei 225 Noe: * indicae significance a he 1% level. Model 1 Model 2 Model 3 p p p Spo Fuures Spo Fuures Spo Fuures Spo * * * Fuures * * * Spo * * * Fuures * * * Spo * * * Fuures * * * Table 3. Engle and Granger s wo-sep co-inegraion es The firs sep is regressing spo prices on fuures prices. The equaion is as follow. S F e where S and F are spo and fuures prices in he logarihmic form. The second sep is esing residuals e for uni roos. The ADF models are he same as hose shown in Table 2. S&P 500 FTSE 100 Nikkei *** (-3.56) *** (-17.05) *** (18.82) *** ( ) *** ( ) *** ( ) Model 1 Model 2 Model 3 p p p *** *** *** *** *** *** *** *** *** Noes: *** ** and * indicae significance a he 1% 5% and 10% level respecively. Table 4. Srucural changes in volailiy Panel A: S&P 500 No. of srucural changes Time period Sd Observaions per period Jan Aug Aug Feb Feb Apr Apr Dec Spo 8 Dec Jul Jul Jun Jun Mar Mar Mar Apr Dec

13 Invesmen Managemen and Financial Innovaions Volume 11 Issue Table 4 (con.). Srucural changes in volailiy Panel A: S&P 500 No. of srucural changes Time period Sd Observaions per period Jan Aug Aug Feb Feb Apr Fuures 6 Apr Mar Apr Oc Oc Mar Mar Dec Panel B: FTSE 100 No. of srucural changes Time period Sd Observaions per period Jan Apr Apr Sep Sep Dec Dec Apr Spo 8 May Jul Aug May May Jun Jun Mar Mar Dec Jan Dec Dec Apr May Jul Fuures 6 Aug May May Jun Jun Mar Mar Dec Panel C: Nikkei 225 No. of srucural changes Time period Sd Observaions per period Jan Feb Feb Dec Jan Mar Apr Sep Spo 9 Oc Mar Mar Jan Jan Sep Sep Sep Sep Dec Dec Dec Jan Feb Feb Dec Jan Mar Mar Sep Fuures 9 Sep Mar Mar Jan Jan Sep Sep Sep Sep Dec Dec Dec

14 Invesmen Managemen and Financial Innovaions Volume 11 Issue The OLS-CI model is as follows: s ( S F ) 0s 1s 1 1 s f ( S F ) 0f 1f 1 1 f hss hsf H hsf h ff ĥ * sf b. ĥ ff Table 5. Consan hedge raios OLS and OLS-CI models The OLS model is obained by imposing he resricion 1s = 1f = 0. L represens he log-likelihood values. LR is he likelihood raio es for model superioriy beween he OLS and OLS-CI models and is 2 disribued wih 2 degrees of freedom. The numbers in he parenheses are -saisics compued using Whie (1980) heeroscedasic consisen sandard errors under he null hypohesis ha he coefficien is zero. 0s (10 3 ) s-1 s-7 S&P 500 FTSE 100 Nikkei 225 OLS CI OLS CI OLS CI *** (3.34) *** (-7.76) ** (-2.29) 1s - 0f (10 3 ) f-1 f *** (3.33) *** (-8.14) ** (-2.25) 1f *** (19.10) *** (-20.16) *** (-13.68) *** (-7.72) *** (18.75) *** (-19.84) *** (-13.30) *** (6.11) ** (2.27) *** (-6.35) *** (11.14) *** (-15.17) *** (-3.41) *** (-3.85) ** (2.12) *** (-6.97) *** (8.38) *** (10.40) *** (-15.98) *** (-3.46) *** (-11.59) *** (-3.92) *** (13.67) *** (10.45) hss (10 3 ) hsf (10 3 ) hff (10 3 ) sf *** (751.57) *** ( ) *** (685.19) *** ( ) *** ( ) *** ( ) b * L LR 34.96*** 9.52*** *** Noes: *** ** and * indicae significance a he 1% 5% and 10% level respecively. 110

15 The bivariae GARCH model is as follows: s ( S F ) 0s 1s 1 1 s f ( S F ) 0f 1f 1 1 f h h h 0 1 h 0 H ss sf s s hsf h ff 0 h f 1 0 h f h v v v h s 0s 1 ss 1 2 s s 1 h v v v h f 0 f 1 f f 1 2 f f 1 ĥ * sf b. ĥ ff Invesmen Managemen and Financial Innovaions Volume 11 Issue Table 6. Hedge raio bivariae GARCH model L represens he log-likelihood values. LR 1 and LR 2 denoe he likelihood raio es for he bivariae GARCH model fiing he daa beer han he OLS model and OLS-CI model respecively and are 2 disribued wih 6 and 4 degrees of freedom respecively. Q 1 (j) and Q 2 (j) are he Ljung-Box (1978) es saisics for no serial correlaion in he sandardized and squared sandardized residuals up o lag j. Q 1 (j) and Q 2 (j) are asympoically 2 disribued. The criical values a he 5% level are for j = 8 and for j = 16. The numbers in he parenheses are -saisics compued using Whie (1980) heeroscedasic consisen sandard errors under he null hypohesis ha he coefficien is zero. 0s (10 3 ) s-1 s-7 1s 0f (10 3 ) f-1 f-7 1f v0s (10 3 ) v1s v2s v0f (10 3 ) v1f v2f sf S&P 500 FTSE 100 Nikkei *** (4.21) *** (-6.97) * (-1.71) (-0.55) *** (4.13) *** (-6.77) * (-1.90) (0.68) ** (2.33) *** (3.87) *** (33.84) ** (2.51) *** (4.36) *** (39.09) *** (835.44) *** (3.40) * (-1.72) (1.32) ** (2.27) *** (3.38) * (-1.78) *** (2.77) ** (2.18) *** (4.29) *** (57.53) ** (2.05) *** (4.32) *** (65.97) *** (565.36) (-0.65) (1.35) (1.25) *** (3.42) *** (4.19) *** (32.78) *** (3.49) *** (4.34) *** (32.81) *** (791.72) Average b * Max b * Min b * L LR *** *** *** LR *** *** *** Q1(8) [spo] [fuures] [5.58] [7.08] [5.20] [5.71] [5.97] [4.42] Q1(16) [spo] [fuures] [18.23] [18.37] [7.71] [8.90] [13.93] [11.75] Q2(8) [spo] [fuures] [11.25] [9.50] [7.18] [7.34] [18.65**] [23.55***] Q2(16) [spo] [fuures] [14.94] [13.23] [13.54] [13.13] [25.72*] [28.81**] Noes: *** ** and * indicae significance a he 1% 5% and 10% level respecively. 111

16 Invesmen Managemen and Financial Innovaions Volume 11 Issue The bivariae ICSS-GARCH model is as follows: s ( S F ) 0s 1s 1 1 s f ( S F ) 0f 1f 1 1 f h h h 0 1 h 0 H ss sf s s hsf h ff 0 h f 1 0 h f n s 0s 1 s s 1 2 s s 1 si si i1 h v v v h d D m f 0f 1 f f 1 2 f f 1 f i f i i1 h v v v h d D ĥ * sf b. ĥ ff Table 7. Hedge raios he bivariae ICSS-GARCH model L represens he log-likelihood value. LR 1 and LR 2 denoe he likelihood raio es for model superioriy beween he bivariae ICSS- GARCH model and he OLS and OLS-CI models respecively and are 2 disribued wih 6 and 4 degrees of freedom respecively. LR 3 es saisics es for model superioriy beween he bivariae ICSS-GARCH model and he bivariae GARCH model and are 2 disribued wih 6 4 and 12 degrees of freedom for S&P 500 FTSE 100 and Nikkei 225 respecively. Q 1 (j) and Q 2 (j) are he Ljung- Box (1978) es saisics for no serial correlaion in he sandardized and squared sandardized residuals up o lag j. Q 1 (j) and Q 2 (j) are asympoically 2 disribued. The criical values a he 5% level are for j = 8 and for j = 16. The numbers in he parenheses are -saisics compued using Whie (1980) heeroscedasic consisen sandard errors under he null hypohesis ha he coefficien is zero. 0s (10 3 ) s-1 s-7 1s 0f (10 3 ) f-1 f-7 1f v0s (10 3 ) v1s v2s v0f (10 3 ) v1f v2f ds2 (10 3 ) ds3 (10 3 ) ds4 (10 3 ) ds5 (10 3 ) S&P 500 FTSE 100 Nikkei *** (3.72) *** (-3.41) * (-1.68) (-0.46) *** (3.60) *** (-3.28) * (-1.73) (0.45) *** (10.77) *** (2.89) *** (23.62) *** (8.70) *** (2.77) *** (27.36) *** (-2.94) (0.31) * (1.89) (1.48) *** (3.28) (-1.29) (0.53) * (1.71) *** (3.20) (-1.21) (-0.90) (0.48) *** (2.70) *** (9.29) (1.06) *** (11.78) *** (8.97) (1.30) *** (7.30) (-1.12) (-0.12) (1.25) (0.86) (1.26) *** (8.87) (1.60) *** (13.60) *** (7.65) * (1.75) *** (16.42) (0.25) * (1.81) *** (-3.13) ** (-2.23) 112

17 ds6 (10 3 ) ds7 (10 3 ) ds8 (10 3 ) Invesmen Managemen and Financial Innovaions Volume 11 Issue Table 7 (con.). Hedge raios he bivariae ICSS-GARCH model S&P 500 FTSE 100 Nikkei (0.87) ** (-2.15) (-0.68) *** (-4.02) *** (3.22) *** (-4.06) ds9 (10 3 ) - - df2 (10 3 ) df3 (10 3 ) df4 (10 3 ) df5 (10 3 ) df6 (10 3 ) *** (-2.86) (0.62) * (1.86) (0.83) ** (-2.41) (0.63) (0.34) *** (-4.05) *** (3.04) *** (-3.84) df7 (10 3 ) - - df8 (10 3 ) - - df9 (10 3 ) - - sf *** (814.98) *** (765.54) ** (2.09) ** (-2.10) *** (5.46) *** (-4.51) (0.27) ** (2.19) *** (-4.49) ** (-1.99) * (1.75) * (-1.76) *** (4.80) *** (-4.15) *** ( ) Average b * Max b * Min b * L LR *** *** *** LR *** *** *** LR *** *** *** Q1(8) [spo] [fuures] [4.27] [6.10] [4.75] [5.22] [4.70] [3.22] Q1(16) [spo] [fuures] [16.75] [16.29] [8.09] [8.37] [17.24] [14.99] Q2(8) [spo] [fuures] [14.38*] [11.05] [23.77***] [27.33***] [21.55***] [24.16***] Q2(16) [spo] [fuures] [22.71] [17.78] [30.27**] [35.33***] [28.37**] [31.25**] Noes: *** ** and * indicae significance a he 1% 5% and 10% level respecively. Table 8. Hedging performance comparison of differen models Hedging performance is measured by he variance of where = s b f. The hedge raio b is esimaed by OLS OLS-CI bivariae GARCH and bivariae ICSS-GARCH models. The numbers in he able are muliplied by S&P 500 FTSE 100 Nikkei 225 OLS OLS-CI GARCH ICSS-GARCH