Regime Switching Correlation Hedging

Transcription

1 Regime Swiching Correlaion Hedging Hsiang-Tai Lee Associae Professor Deparmen of Banking and Finance,, Universiy Rd., Puli, Nanou Hsien, Naional Chi Nan Universiy, Taiwan 5456 (886) # 4648 sagerlee@ncnu.edu.w Absrac The aricle invesigaes he hedging effeciveness of commodiy fuures when he correlaions of spo and fuures reurn series are subjec o muli-sae regime shifs. An independen swiching dynamic condiional correlaion GARCH (IS-DCC) which is free from he pah-dependency and recombining problems is proposed o model muli-regime swiching correlaions. Resuls of hedging exercises show ha in general, IS-DCC ouperforms sae-independen DCC GARCH and hree-sae IS-DCC exhibis superior hedging effeciveness when full sample period is applied, illusraing imporance of modeling higher-sae swiching correlaions for fuures hedging. Keywords: Dynamic Fuures Hedging, Hedge raio, GARCH model, Markov regime swiching

2 Regime Swiching Correlaion Hedging Absrac The aricle invesigaes he hedging effeciveness of commodiy fuures when he correlaions of spo and fuures reurn series are subjec o muli-sae regime shifs. An independen swiching dynamic condiional correlaion GARCH (IS-DCC) which is free from he pah-dependency and recombining problems is proposed o model muli-regime swiching correlaions. Resuls of hedging exercises show ha in general, IS-DCC ouperforms sae-independen DCC GARCH and hree-sae IS-DCC exhibis superior hedging effeciveness when full sample period is applied, illusraing imporance of modeling higher-sae swiching correlaions for fuures hedging. Keywords: Dynamic Fuures Hedging, Hedge raio, GARCH model, Markov regime swiching

3 I. Inroducion I is widely known ha when hedging a spo posiion wih a posiion in he fuures marke, he minimum variance hedge raio (MVHR) is equal o he raio of he covariance of spo and fuures reurns o he variance of fuures reurns. A common approach o condiional minimum-variance hedging is o model he ime-varying condiional variance-covariance marix of reurns using a mulivariae GARCH model, and use forecass from his model o consruc a forecas of he condiional MVHR (Baillie and Myers, 99; Kroner and Sulan, 993; Park and Swizer, 995; Gagnon and Lypny, 995; Brooks e al., 00; and Bysröm, 003). Recen sudies recognize ha he relaionship beween spo and fuures reurns may be characerized by regime shifs (Sarno and Valene, 000, 005a, 005b). The implicaion is ha o improve he fuures hedging effeciveness, he sae-dependen propery beween spo and fuures series should also be aken accoun in developing dynamic hedging sraegies. Alizadeh and Nomikos (004), Lee e al. (006), Lee and Yoder (007a), Lee and Yoder (007b) and Alizadeh e al. (008) respecively, propose regime swiching leas square model, regime swiching sae space model, regime swiching Varying Correlaion GARCH (VC-GARCH) model, regime swiching BEKK-GARCH model and regime swiching vecor error correcion model for fuures hedging and find ha he hedging effeciveness are improved compared o sae-independen sraegies. To furher incorporae he effecs of unanicipaed news evens in deermining of opimal hedge raio, Lee (009a) develops a Markov regime swiching Generalized Orhogonal GARCH model wih condiional jump dynamics for esimaing he opimal hedge raio. Furher exension is o release he assumpion of join normaliy beween spo and fuures reurn 3

4 series and using regime swiching copula GARCH model for fuures hedging (Lee, 009b). All hese elaboraions are found o improve fuures hedging effeciveness. Alhough hese regime swiching GARCH models have capured much of he observed behavior in he spo and fuures reurn series, hey possess some limiaions. Firsly, hese models allow mean, volailiy, and correlaion equaions o be saedependen simulaneously and as a consequence, discussion of he number of regimes is limied o wo due o he poenial problems of overparameerizaion and convergence for higher regimes. As poined ou by Caporin and Billio (005), a full Markov swiching model is highly unsable given he huge number of swiching parameers. To he auhor s knowledge, no muli-regime mulivariae GARCH model has been applied for fuures hedging. Secondly, all hese models are subjec o he well-known pah-dependency problem, (Cai, 994; Hamilon and Susmel, 994; Gray, 996; and Lee e al., 007a, 007b). Recombining procedures are required o approximae he residuals, variances and correlaion a each ime poin and hese procedures ineviably creae compuaional burden and as poined ou by Hass, e. al. (004), he analyical racabiliy of he dynamic process is problemaic. This sudy aemps o invesigae if allowing he correlaion of spo and fuures reurn series o be subjec o muli-sae swiching improves he fuures hedging effeciveness by proposing an independen swiching dynamic condiional correlaion GARCH (IS-DCC) model. There are several reasons ha we argue for an independen swiching model for he correlaion. Firsly, ime-varying correlaions risks are widely noiced in recen finance lieraure. For insance, Krishnan e al. (009) find ha he correlaion of reurns beween asses has varied subsanially over ime and invesors 4

5 would pay a premium for securiies ha perform well in regimes in which he correlaion is high. Ang and Chen (00) find ha Correlaions beween U.S. socks and he aggregae U.S. marke are much greaer for downside moves han for upside moves. Ledoi e al. (003) find ha he level of correlaion for inernaional sock markes depends on he phase of he business cycle. All hese findings sugges a sae-dependen ime-varying correlaion for modeling financial ime series. Secondly, limiing he swiching only for he correlaion miigaes he problems of overparameerizaion and convergence and he discussion of regime swiching effec on fuures hedging wih more han wo saes is possible. Lee e al. (007a) model explicily he sae-dependen imevarying correlaion process. Their model, however, limis he number of saes o wo. Finally, he proposed IS-DCC avoids he problem of pah-dependency and is free from he requiremen of recombining procedure. This reduces he burden of compuaion and avoids he analyical inracabiliy problem. The remainder of he aricle is organized as follows. The proposed IS-DCC is presened in secion II. Secion III addresses he esimaion issue encounered for he IS- DCC. The minimum variance hedge raio under regime swiching and measuring hedging performance are discussed in secion IV. This is followed by daa descripion and empirical resuls. A conclusion ends he aricle. II. Independen Swiching Dynamic Condiional Correlaion GARCH Model (IS-DCC) The independen swiching dynamic condiional correlaion GARCH (IS-DCC) is a modificaion of he Markov regime swiching dynamic condiional correlaion GARCH 5

6 (MS-DCC; Caporin and Billio, 005) such ha no problems of pah-dependency will occur and recombining procedure is no required. The specificaion of IS-DCC is given below: Suppose ha he observed -dimensioned economic process R is given by R μ, () e μ D ε, () μ c f is a vecor of condiional means, T sands for ranspose, where T T e e, e, D ε is assumed o be normally disribued c f H e ~ N 0,, (3) wih ime-dependen variance-covariance marix H. ε D R μ is he normalized residual vecor, N sands for normal disribuion and is he informaion se up o ime. The ime-varying variance-covariance marix H is given by H D Γ D, (4) where diag, D, i c, f h i is a diagonal marix wih he volailiies of spo and fuures reurns on he i h elemen. The condiional variances dynamic are assumed o follow a sae-independen GARCH(,) process T γ diagα e e diagβ D D diag i i i where is Hadamard produc and γ i, α i, and i, (5) β, i c, f are GARCH coefficiens. In he sae-independen dynamic condiional correlaion GARCH model (Engle, 00), Γ is he correlaion marix and is defined as Γ Q / Q diagq / diag, (6) 6

7 where Q is he condiional sandardized residual covariance marix and for a resricive case, is given by Q Q ε ε Q, (7) where Q is he uncondiional covariance marix of he sandardized residuals and can be replaced by he sample covariance marix T Q ε ε o simplify he esimaion T i i i (Bauwens, Lauren, and Rombous, 006). To incorporae regime shif ino Engel s DCC model, Caporin and Billio (005) inroduce a Markov regime swiching dynamic condiional correlaion (MS-DCC) model wih he condiional sandardized residual covariance marix specified as Q s s Q s ε ε s Q, (8) where, s is he sae variable following a firs-order, wo-sae Markov process. Compared o equaion (7), parameers driving he sysem dynamics are sae-dependen. The regime dependen srucure is resriced o he correlaion excluding any effec on variance. As poined ou by Caporin and Billio (005), a full Markov swiching model is highly unsable given he huge number of swiching parameers. Given he join presence of regime swiching and ime-varying correlaion in each regime in equaion (8), recombining procedure is required o solve he well-known pahdependency problem (Cai, 994; Hamilon and Susmel, 994; Gray, 996; and Lee e al., 007a and 007b). Analog o Kim s filer (994), Caporin and Billio (005) propose a modified Hamilon filer for esimaing MS-DCC. In heir proposed filering algorihm, he condiional sandardized residual covariance marix following dynamic i Q evolves according o he 7

8 i Q ε ε Q i, j Q, j, j, j, j, (9) given S possible values of Q a ime, here will be ime, i, j,,, S S possible values for Q a, S is he number of saes. The recursive naure of he regime swiching process produces an S -fold increase in he number of cases o consider in each ieraion of he filer and make he model inracable. To make he evoluion of he process racable, he correlaion marixes are collapsed based on he following condiional expecaions: S i, j Ps j, s i Q j i Q Ps j, (0) where Ps j, s i is he condiional regime probabiliy of being in sae i a ime and in sae j a ime. Alhough his recombining mehod solves he problem of esimaion difficulies, i creaes compuaional burden and is analyical inracabiliy is a serious drawback. Consider he following correlaion dynamic Q Q E Q, () where E ε ε. If, Q can be expressed as Q i Q E i, () i where reflecs he magniude of a uni shock s immediae impac on he nex period s Q, is a parameer of ineria and indicaes he memory in Q, and he oal impac of a uni shock o fuure Q is. In he regime swiching GARCH model, he relaionship beween he paern wih which Q responds o shocks and he parameers 8

9 and is far from obvious if recombining mehod is used because he lagged Q is replaced wih he recombined variances. Moreover, i is possible ha he covariance of one regime will sill be affeced by shocks even if in ha regime is zero. A more deail discussion of his problem is given in he appendix A. Analog o he independen swiching idea proposed by Hass e al. (004) ha is aimed o solve he problem of univaraie pah-dependency problem in he variance process, an independen swiching covariance process is suggesed below: s s Q s ε ε s Q s Q s, (3) and he corresponded correlaion dynamic is s diag Q s / Q s diagq s / Γ. (4) Compared o equaion (8), his specificaion allows he covariance process in each regime o evolving independenly and avoid he pah-dependency problem. Furhermore, because we have S covariance process o evolve in parallel according o differen se of parameers, he specificaion preserves he economic significan of he covariance dynamics in each regime and we refer his model as he independen swiching dynamic condiional correlaion GARCH (IS-DCC). Equaions ()-(5) and (3)-(4) consiue he specificaion of he IS-DCC model and he i sae IS-DCC model is denoed as IS DCC(i) in his aricle. Under his noaion, IS DCC() will be he saeindependen DCC GARCH proposed by Engle. III. Esimaion and Hamilon Filer for he IS-DCC model The esimaion of parameers is performed wih maximum likelihood approach. To maximize he likelihood one has o evaluae 9

10 L T log f R, (5) where is he vecor of unknown parameers o be esimaed, T is he oal number of observaions, and f R is he mixure disribuion weighed by regime probabiliy. To do his we have o use Hamilon filer (Hamilon, 989, 994) o evaluae he regime probabiliy because he sae variable is unobserved. The Hamilon filering procedure for he IS-DCC is depiced below: (i) Given he filered probabiliies ξ ˆ projecs he sae probabiliies ξ ˆ, (6) Pξ ˆ where p s ˆ p s ξ, p s S p s ˆ p s ξ, (7) p s S and P is he ransiion probabiliy marix wih he i, j elemen p s j s i defined as p s j s i exp exp i, j exp exp i, i, i, S, j,,, S exp exp i, i, exp i, S, j S (8) where ' s are unresriced parameers o be esimaed. (ii) Evaluae he regime dependen likelihood i i i Q i ε ε iq i Q i, (9) Q i / Q i diag Q i / Γ diag, i,,, S, (0) 0

11 H f R i D i D Γ, () s i, m / H i / exp ' R μ H ir μ, () where m is he number of dimension and is equal o wo for our hedging applicaion. Define η f R f R f R he densiy of s s s, (3) S R condiional on pas observaions and being in regime i,,, S a ime. (iii) Evaluae he mixure likelihood R ξ ˆ η f, (4) where is an m vecor of ones and denoes elemens-by-elemens muliplicaion. (iv) Updae he join probabiliies The sae-probabiliy is updaed wih he following equaion ξˆ ξˆ ξˆ η η (5) (v) Ierae (i) o (iv) unil he end of he sample and he likelihood is obained as a by-produc of his filer L T log ξˆ η (6)

12 Differen from Billio and Caporin s filer, he sep of approximaion for he covariance marix is no required in his filering algorihm since IS-DCC is pahindependen. To iniialize he filer, he regime probabiliies are se equal o he uncondiional probabiliies. Define he seady sae probabiliies vecor as s p s p s S π p. (7) These probabiliies are he soluion of he sysem of equaions T Pπ π and π, which can be shown as I S P π AA Aν S, where A, and ν 0S S, S I is an S S ideniy marix and 0 S is an S zero vecor. IV. Sae-dependen MVHR and Measuring Hedging Performance I is well known ha he esimaed ime-varying minimum variance hedge raio denoed as for sae-independen hedging is given by given by Cov rc, r, r f,. (8) Var f, Lee (009b) derives a formula for wo-sae regime swiching hedge raio which is rc,,, rf,, p, Covrc,,, rf,, Varr p Varr p, Cov, (9) p, f,,, f,, where p, is he regime probabiliy of being in sae one a ime. Because he focus of his aricle has been invesigaing he effecs of muli-regime swiching in correlaion on

13 fuures hedging, a formula for muli-sae regime swiching hedge raio is required. The S sae regime swiching hedge raio can be generalized as where S pi, i S i Cov rc, i,, rf, i,, (30) p Var r i i, p,, i,,, S f, i, are he sae probabiliies of being in sae i and Covr c r and, i,, f, i, Var are respecively he condiional covariance of r f, i, spo and fuures reurns and condiional variance of fuures reurns in sae i. Noice ha, when here is no regime shifs, S and equaion (30) collapses o he convenional sae-independen hedge raio given in equaion (8). Hedging performance is evaluaed from boh a risk-minimizaion and a uiliy sandpoin. From a risk-minimizaion sandpoin, a hedger chooses a hedging sraegy o minimize he variance of he hedged porfolio reurn or equivalenly o maximize he variance reducion of a hedging sraegy compared o he unhedged posiion. The variance of he hedged porfolio reurn is where Var r c, r f,, (3) is defined in equaion (30) and esimaed from he proposed IS DCC model. This can be proved as follows. Le r p be he hedging porfolio reurn which is given by r p p p p p S porfolio reurn in sae porfolio reurn in sae porfolio reurn in sae S s p r s S χ r s χ r r s S c f S c f, Deriving he variance of his sae-dependen hedging porfolio reurn p r wih respec o χ and using he assumpion of independen swiching gives equaion (30). 3

14 Because dynamic hedging sraegies are poenially more cosly implemen han saic models since frequen rebalancing of he hedged porfolio is required, hedging effeciveness is more appropriaely assessed by considering he economic benefis measured wih uiliy funcions. Consider a hedger wih a mean-variance expeced uiliy funcion (Kroner and Sulan, 993; Gagnon e al., 998; Lafuene and Novales, 003; Alizadeh and Nomikos, 004; and Lee e al., 006): E U r Er Varr p, p, p,, (3) where is he coefficien of absolue risk aversion, E sands for expecaion operaor and r p, is he reurn from he hedged porfolio. A dynamic hedging sraegy is considered o be superior o a saic ordinary leas square (OLS) mehod if i has higher expeced uiliy ne of ransacion coss. In addiion o measuring he economic significance of dynamic hedging sraegies wih uiliy funcion, i is also ineresing o es if he bes IS DCC model saisically significanly ouperforms OLS. According o Sullivan e al. (999) and Whie (000), daa snooping occurs when a given se of daa is used more han once for purposes of inference or model selecion. To avoid daa snooping problem, Whie s realiy check (Sullivan e al., 999 and Whie, 000) is also performed o es he hypohesis ha he bes performing IS DCC model has no predicive superioriy over he benchmark, saic OLS model. Whie s realiy check is based on he following l performance saisic: f N T f R, (33) 4

15 where l is he number of alernaive models considered and f is he observed performance measure for period. The k h elemen of f ^ is defined as: r r ˆ ˆ k, rc, Bes IS DCC, rf, c, OLS f,, (34) fˆ where ˆ and ˆ OLS are he esimaes of hedge raios from he bes IS DCC Bes IS DCC, model and saic OLS, respecively. The null hypohesis ha he bes performing superioriy over he saic OLS is given by * Ef 0 H : max 0 k,,, l k IS DCC has no predicive, (35) where * f is he rue performance value for each model applied o he daa. k Because DCC(i) IS, i,,, S IS DCC(i) ouperforms DCC( i ) are nesed models, o invesigae if IS, i,,, S, he Diebold-Mariano (995) and Wes (996) (DMW) es is performed. To consruc he DMW saisic, le d f f i, i,, and d N d ˆ, hen he DMW es saisic, i,,, S T R is compued as follows, DMW d, (36) N V T where V N dˆ d R, R denoes he lengh of esimaion period, N is he lengh of he predicion period, T is he sample size, f is he square error loss funcion, and Poliis and Romano s (994) saionary boosrap resampling mehod is used for implemening he Whie s realiy check wih 000 boosrap simulaions and a smoohing parameers of q=0.5 (Lee e al., 006; Lee and Yoder, 007a). 5

16 , ˆ and v i, rc, ˆ i, rf, wih ˆ i, he hedge raios esimaed from he v i rc, i, rf, IS DCC(i) model. The criical values of DMW es for nesed models have o be adjused o produce correc ess (McCracken, 007). McCracken s criical values depend on he N / R raio and he number of addiional esimaed parameers in he unresriced model. The es is one-sided wih he null hypohesis ha he predicive abiliy of an unresriced model is no superior o is nesed model which is given by H f f 0 0 E i, i, while he alernaive is H A E, (37) f f 0. (38) i, i, Rejecion of he null hypohesis implies ha he predicive abiliy of an unresriced model is superior o is nesed model. Alhough he of ineres in his sudy, is hedging performance is also compared wih MS DCC model is no IS DCC. Since MS DCC is no nesed wihin he IS DCC model, regular criical values for DMW saisics are applied. V. Daa Descripion and Empirical Resuls The proposed IS DCC is applied o nearby fuures conracs of whea and corn raded in he Chicago Board of Trade (CBOT), cocoa and coffee raded in he New York Board of Trade (NYBOT), and crude oil, naural gas, heaing oil, and plainum raded in he New York Mercanile Exchange (NYMEX) for he period January 99 o December 008. Spo and Fuures prices are Wednesday prices obained from Daasream and he Energy Informaion Adminisraion (US Deparmen of Energy). Tuesday s closing price 6

17 is used when a holiday occurs on Wednesday. The reurns of each price series are compued as he changes in he naural logarihms of prices muliplied by 00. Esimaion of all models was conduced using daa from January 99 o December 007; he remaining daa are used for ou-of-sample analysis. The sub-period hedging effeciveness is also invesigaed in his sudy. The sample is furher spli ino wo periods: pre-000 (from January 99 o December 999) and pos-000 (from January 000 o December 008). The las year daa in each sub-period are used for ou-of-sample analysis. Table I provides summary saisics of he reurns series for each commodiy over he full sample period and wo sub-sample periods. For he full sample period, all reurns are posiive and small. The larges mean reurns are 0.4% and 0.09% for spo and fuures daa, respecively and he smalles mean reurns are 0.00% for boh spo and fuures daa, respecively. The uncondiional volailiies indicae ha in general, he pos- 000 period is more volaile han pre-000 period. According o he Skewness, lepokurosis, and significan Jarque-Bera saisics, he uncondiional disribuions of spo and fuures reurns for all commodiies are asymmeric, fa-ailed, and non-gaussian. Parameer esimaes from alernaive models are presened in able II. The parameers are esimaed by maximizing he log-likelihood funcions in equaion (5) using numerical consrained opimizaion procedure in GAUSS. Shown in he las row of able II, LRT repors he saisics of likelihood raio es of IS DCC(i) and is nesed model IS DCC( i ). The number of sae i is increased unil ha he IS DCC(i) does no show significan increase in likelihood value compared o IS DCC( i ) and he criical values a % for i,3, 4 and 5 are 3.8, 6.8, 0.09 and 3., respecively. The number of parameers in DCC(i) IS is equal o 8 ii i. 7

18 Namely, he number of parameers o be esimaed for DCC, IS DCC() o IS DCC(5) are 0, 4, 0, 8, and 38, respecively. 3 The LRT of crude oil is sill significan when he number of saes is increased o five. However, we do no proceed o IS DCC(6) since here are fify parameers o be esimaed ( ) and empirically, increasing he number of saes o five no longer creae significan gains compared o four saes. As shown in able II, all condiional mean ' s esimaed are small which is consisen wih he small average reurn repored in he summary saisics able. For he volailiy equaion, heaing oil daa has he larges volailiy persisence and whea daa has he smalles volailiy persisence among all commodiies invesigaed in his aricle. Taking DCC model for insance, heaing oil daa has he larges which is equal o and for spo and fuures reurns, respecively and whea daa has he smalles which is equal o and for spo and fuures reurns, respecively. In he correlaion equaion, reflecs he memory in correlaion. In saeindependen DCC, coffee and corn have he larges and smalles memory in correlaion wih equal o 0.94 and 0.3, respecively. For he sae-dependen DCC models, he memory is no a consan bu regime-dependen. For example, in he IS DCC(5) for corn is decomposed ino five possible memory srenghs, 0.00, 0.03, 0.379, and 0.78 in five differen regimes. Mos of he parameers in he correlaion equaion are significan implying he imporance of modeling he regime-swiching ime-varying correlaion of spo and fuures reurns. The oal impac of a uni shock o fuure 3 To save space, esimaion resuls of MS-DCC and parameers are no repored here bu are available from he auhors upon reques. ' s for he ransiion probabiliies 8

19 correlaion is. For he sae-dependen DCC models, he oal impac of a uni shock of naural gas and corn have he larges and smalles shock o he fuure correlaions wih equal o 0.94 and 0.06, respecively. The oal impac in he regime-independen model is somewhere in beween he larges impac and he smalles impac in he regime-dependen model. Taking naural gas for insance, he saedependen impac srenghs are 0, 0.69, 0.958, and and he regime-independen impac 0.94 ( he smalles impac ) is somewhere in beween he larges impac and Table III repors he ou-of-sample hedging effeciveness of alernaive hedging sraegies. Ou-of-sample hedging effeciveness is considered because for he hedger, wha maers mos is he hedging performance in he fuure no in he pas. I is found ha in general IS DCC() ouperforms DCC. The only excepion is coffee. The percenage variance reducion of IS DCC() is 64.3% which is lower han ha of DCC wih a 64.3% variance reducion. 4 This is consisen wih mos findings in he previous regime swiching hedging sudies ha allowing he hedge raio o be saedependen increases he hedging effeciveness. This aricle invesigaes if allowing he number of regime o be more han wo can furher improve he hedging effeciveness. Empirical resuls reveal ha when he number of regimes is increased from wo o hree, IS DCC(3) ouperforms IS DCC() for all commodiies considered in his sudy. Compared o IS DCC(), he larges and smalles improvemens of IS DCC(3) are.55% and 0.03% for cocoa and naural gas, respecively. The resuls, however, are no 4 Percenage variance reducions are calculaed as he differences of variance of unhedged posiion and esimaed variance of aleraive models over variance of unhedged posiion muliplied by 00. 9

20 promising when he number of saes is increased from hree o four. Only IS DCC(4) of corn creaes a 0.6% significanly improvemen compared o IS DCC(3). I is also found ha increasing he number of saes from four o five as appeared in he corn, crude oil and heaing oil daa does no always provide furher hedging benefi. Only IS DCC(5) for crude oil provides a 0.0% improvemen compared o IS DCC(4) and IS DCC(5) is inferior o IS DCC(3) wih a 0.07% less in variance reducion. Generally speaking, allowing he correlaion o be subjec o regime swiching improves he hedging effeciveness compared o a model wih saeindependen correlaion and hree-sae correlaion hedging exhibis superior performance when full sample daa is invesigaed. Due o he frequenly rebalancing requiremen of dynamic hedging sraegies, hey are more cosly han saic OLS hedging. Following oher empirical sudies (Lafuene and Novales, 003; Alizadeh and Nomikos, 004; and Lee e al., 006), he economics value of hese dynamic hedging mehods are also invesigaed by comparing he uiliy improvemens of hese mehods relaive o saic OLS hedging. The hedger is assumed o have an expeced uiliy funcion given by equaion (3) wih he coefficien of absolue risk aversion equal o 4. As shown in able III, aking whea daa for example, he average weekly variance of he reurns from hedged porfolio for OLS and IS-DCC3 hedging are.753 and 8.039, respecively. Alhough no repored here, he hedged porfolio reurns of OLS and IS DCC(3) hedging are -0.66% and 0.78%, respecively. Based on equaion (3), if an invesor adops OLS hedging, he average weekly uiliy is U -0.66% Wih IS DCC(3), he average weekly uiliy is OLS 0

21 U ISDCC % 4. The hedger s ne benefi from using IS DCC(3) hedging over OLS hedging is equal o U U C ISDCC C, where C sands for he ransacion cos from dynamic rebalancing. This implies ha if C 5. 3, he IS DCC(3) hedging is preferred o OLS hedging. Since he ypical round rip ransacion cos is around 0.0% o 0.04%, a mean-variance expeced uiliy-maximizing hedger will benefi from hedging wih IS DCC(3) even afer aking accoun of hese ransacion coss. I is found ha DCC does no creae uiliy gain for cocoa, heaing oil and naural gas daa and all sae-dependen IS DCC hedging generae uiliy gains compared o OLS hedging. To es he saisical significance of he hedging effeciveness of he bes IS DCC over he benchmark, saic OLS hedging, Whie s realiy check is performed. As repored in able III, based on Whie s realiy check p-values, he no improvemen null hypohesis of he bes OLS IS DCC over OLS is rejeced a leas a 0% significan level for mos of he commodiies. Excepions are cocoa and naural gas daa wih realiy check p-values equal o 0.43 and 0.67, respecively. Because DCC(i) IS, i,,, S IS DCC(i) significanly ouperforms DCC( i ) are nesed models, o invesigae if IS, i,,, S, he Diebold- Mariano (995) and Wes (996) (DMW) es is performed wih adjused criical values repored by McCracken (007). McCracken s criical values depend on he N / R raio and he number of addiional esimaed parameers in he unresriced model. When he full daa sample is applied, he N / R raio is equal o 0.06 and he number of addiional esimaed parameers for IS DCC(i), i=,3,4 and 5 are four, six, eigh, and en, respecively. The criical values are abulaed for N / R 0 and 0., and we consruc he

22 values for N / R by inerpolaion. As repored in able IV, I s found ha IS DCC() is superior o DCC a 0% significan level for corn, crude oil and heaing oil bu no he res of he commodiies. IS DCC(3) is superior o IS DCC() a 5% level significan level for cocoa and coffee and a 0% level for whea and crude oil. Alhough IS DCC(3) does no provide significan improvemen over IS DCC() for corn, naural gas, heaing oil and plainum, all DMW saisics are posiive implying ha IS DCC(3) is no inferior o IS DCC() and has a endency o be superior o IS DCC(). When IS DCC(3) is compared wih DCC, again, all DMW saisics are posiive and IS DCC(3) is superior o DCC a 5% level significan level for whea and crude oil and a 0% level for corn, cocoa, coffee and heaing oil. When he number of saes is increased from hree o four, IS DCC(4) significanly ouperforms IS DCC(3) only for corn. The DMW saisics for coffee and naural gas are negaive and significan a 0% level, implying ha IS DCC(3) ouperforms IS DCC(4) for hese wo commodiies. The performances are no significanly differen for crude oil, heaing oil and plainum. When he number of saes is furher increased from four o five, all DMW saisics are no significan indicaing ha he performance of IS DCC(5) is saisically indifferen o IS DCC(4). Overall, IS DCC(3) exhibis superior performance when full sample daa is invesigaed. Alhough he MS DCC model is no of ineres in his paper due o is requiremen of recombining procedure and he problem of analyical inracabiliy, a comparison of he proposed IS DCC and MS DCC is also performed and repored in able V. I s found ha IS DCC() is superior o MS DCC a 0% significan level

23 for whea and a 5% level for naural gas and he performance of IS DCC() is no significanly differen from MS DCC for he res of he commodiies. As for IS DCC(3), i is superior o MS DCC a 5% significan level for whea and naural gas. All DMW saisics are posiive implying ha IS DCC(3) is no inferior o and has a endency o be superior o MS DCC. To check he consisency of he performance of IS DCC over differen hedging periods, he daa is furher spli ino wo sub-samples. In he firs sub-sample (pre-000), in- and ou-of-sample periods are from January 99 o December 998 and from January 999 o December 999, respecively, and in he second sub-sample (pos-000), in- and ou-of-sample periods are from January 000 o December 007 and from January 008 o December 008, respecively. Table VI and VII presen he hedging performances of IS DCC over pos-000 and pre-000 sub-periods, respecively. I is found ha, mos sae-dependen IS DCC hedging ouperform OLS in erms of percenage variance reducion and generae uiliy gains compared o OLS hedging. OLS occasionally ouperforms all dynamic hedging mehods. OLS has he bes performance for coffee in he pos-000 period and for whea and heaing oil in he pre-000 period. 5 For he pos- 000 period, Whie s realiy check p-values show ha he no improvemen null hypohesis of bes IS DCC over OLS is rejeced a he 0% significan level for corn, a he 5% significan level for plainum and a he % significan level for crude oil and naural gas. As for he pre-000 period, he no improvemen null hypohesis of bes IS DCC over OLS is rejeced only for corn a he 5% significan level. 5 This is consisen wih some previous findings ha more elaborae dynamic hedging mehod migh no improve he hedging effeciveness compared o he saic hedging mehod (Bysröm, 003; Lee e al., 006; and Lee and Yoder, 007a). 3

24 For he nesed DCC(i) IS, i,,, S, in he pos-000 period, IS DCC() ouperforms oher models for whea, corn, cocoa, coffee and naural gas. IS DCC(3) has he bes performance for heaing oil and plainum and IS DCC(4) has he bes performance for crude oil. As for he pre-000 period, IS DCC(3) has he bes performance for whea, coffee, heaing oil and plainum and IS DCC(4) has he bes performance for corn and crude oil. IS DCC() ouperforms oher models only for naural gas and he sae-independen DCC has he bes performance for cocoa daa. Overall, wo-sae IS DCC has beer performance for majoriy of he commodiies in he pos-000 period and more han wo-sae majoriy of he commodiies in he pre-000 period. IS DCC has beer performance for The Diebold-Mariano and Wes (DMW) es saisics for he sub-periods are repored in able VIII. The N / R raio for he McCracken s criical values is equal o 0.5 for each sub-period and he number of addiional esimaed parameers for IS DCC(i), i=,3,4 and 5 are four, six, eigh, and en, respecively. The criical values are abulaed for N / R 0. and 0., and we consruc he values for N / R 0. 5 by inerpolaion. In he pos-000 sub-period, IS DCC() is superior o DCC a he 5% significan level for whea, crude oil, heaing oil and plainum. IS DCC(3) provides furher significan improvemen over IS DCC() for heaing oil and IS DCC(4) provides furher significan improvemen over IS DCC(3) for crude oil. As for he pre- 000 sub-period, IS DCC() is superior o DCC a he 0% significan level for plainum and a he % level for whea and heaing oil. IS DCC(3) provides furher significan improvemen over IS DCC() for heaing oil and IS DCC(4) provides 4

25 furher significan improvemen over IS DCC(3) for corn and crude oil. Overall, mos of he DMW saisics of he bes IS DCC are posiive compared o sae-independen DCC in boh sub-periods reveals ha allowing he correlaion o be subjec o regime shifs has a endency o improve he hedging performances. These saisics are significan for whea, crude oil, heaing oil, and plainum in he pos-000 period and for whea and heaing oil in he pre-000 period. Figures shows he hedge raios esimaed by using OLS, DCC, and IS DCC(3) for whea. 6 The condiional hedge raios are very volaile revealing ha adjusmen of he hedge porfolio using dynamic hedging sraegies is highly required. Figure shows he sae-dependen ime-varying correlaions in each regime. The maximum number of saes for whea is hree when full sample is considered. IS DCC(3) decomposes correlaions ino hree differen regimes wih differen volailiies in correlaions. The volailiies of correlaions are equal o 0.006, 0.6 and 0.30 in sae hree, one and wo, respecively. Sae hree is he regime ha spo and fuures reurn series have a nearly consan correlaion. Sae wo is he sae wih a highes volailiy of correlaion and he volailiy of correlaion in sae one is somewhere in beween. The regime probabiliies of being in each regime are ploed in figures 3 o 5. As for he corn daa, he hedge raios esimaed by using OLS, DCC, and IS DCC(5) are ploed in figure 6 and he sae-dependen ime-varying correlaions in each regime are ploed in figure 7 and 8. IS DCC(5) decomposes correlaions ino five differen saes wih differen volailiies in correlaions. The volailiies of correlaions are equal o 6 To save space, only hose figures for hree-sae case of whea and five-sae case of corn are illusraed here and o make he correlaion figures more clearly, only pos-000 period is ploed for he five-sae case of corn. 5

26 0, 0.038, 0.75, 0.35 and 0.58 in sae wo, one, hree, five and four, respecively. Sae wo is he regime ha spo and fuures reurn series have a nearly consan correlaion. Sae one and hree have relaively smaller volailiies in correlaions. Insead, as illusraed in figure 8, sae four and five have relaively larger volailiies in correlaions and volailiy in correlaion is larger in sae four han in sae five. The regime probabiliies of being in each regime for he five-sae case of corn are ploed in figures 9 o 3. VI. CONCLUSIONS The focus of his aricle has been invesigaing he effecs of muli-regime swiching in correlaion on fuures hedging via an independen swiching dynamic condiional correlaion GARCH ( IS DCC ) model. IS DCC avoids he pahdependency and recombining problems inheren in he MS DCC which possess problems of compuaional inensive and analyical inracabiliy. To auhor s knowledge, no exising paper invesigaes muli-regime correlaion fuures hedging. This migh be he fac ha previous regime swiching hedging models allow a fully model swiching and he poenial overparameer and convergence problems limi he discussion of he possible number of sae o wo. Empirical resuls from commodiy fuures hedging exercise show ha ouperforms sae-independen DCC and hree-sae IS DCC exhibis superior hedging effeciveness when full sample period is invesigaed. Sub-sample periods IS DCC hedging resuls show ha wo-sae IS DCC has beer performance for majoriy of he commodiies in he pos-000 period and IS DCC wih more han wo saes have beer 6

27 performances for majoriy of he commodiies in he pre-000 period. Overall, he conribuion of his paper is wofold. The proposed IS DCC provides a general framework for modeling muli-sae regime swiching ime-varying correlaion and resuls of hedging exercises illusrae he imporance of modeling his feaure for opimal dynamic fuures hedging. 7

28 Table I Summary Saisics for Spo and Fuures Reurns (In Percenage) of Full Sample and Two Sub-Sample Periods Sample Period: Sample Period: Sample Period: Spo Fuures Spo Fuures Spo Fuures Spo Fuures Spo Fuures Spo Fuures WHEAT CORN WHEAT CORN WHEAT CORN Mean Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera *** 60.36*** 38.40*** 79.33*** 07.79*** 88.95*** 07.4*** 39.3*** 93.0*** 7.0*** 93.96*** 43.09*** COCOA COFFEE COCOA COFFEE COCOA COFFEE Mean Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera *** 9.99*** 80.80*** 999.0*** *** 33.83*** 359.0*** 480.7*** 6.46*** 3.09*** 67.*** 96.6*** CRUDE OIL NATURAL GAS CRUDE OIL NATURAL GAS CRUDE OIL NATURAL GAS Mean Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera 49.89*** 93.66*** *** 8.07*** 36.0*** 86.0*** ***.90*** 85.00*** 99.75*** 59.65*** 8.40*** HEATING OIL PLATINUM HEATING OIL PLATINUM HEATING OIL PLATINUM Mean Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera 90.8*** 98.76*** 0.53*** 933.8*** *** *** 5.*** 90.89*** 94.6***.43*** *** 37.64*** Noe: *** indicaes significance a he % level and reurns are calculaed as he differences in he logarihm of price muliplied by 00. 8

29 c f c f Table II. Esimaes of Unknown Parameers of Alernaive Models Daa period is from January 99 o December 007 WHEAT CORN DCC IS-DCC() IS-DCC(3) DCC IS-DCC() IS-DCC(3) IS-DCC(4) IS-DCC(5) Mean Equaion Mean Equaion (0.090) (0.3) (0.7) (0.0)*** (0.098)*** (0.087)*** (0.03)*** (0.08)*** (0.09) (0.090) (0.4) (0.06)*** (0.)** (0.090)*** (0.034)*** (0.034)*** Volailiy Equaion Volailiy Equaion (0.590)*** (4.005 (0.757)*** (0.95)*** (0.94)*** (0.07)*** (0.6)*** (0.3)*** (.8)*** (.87) (.43)*** (0.44)*** (0.37)*** (0.88)*** (0.9)*** (0.83)*** c (0.037)*** (0.043)*** (0.035)*** (0.0)*** (0.0)*** (0.04)*** (0.009)*** (0.006)*** f (0.04)*** (0.09)* (0.040)*** (0.08)*** (0.09)*** (0.04)*** (0.009)*** (0.007)*** c (0.066)*** (0.388)* (0.08)*** (0.09)*** (0.06)*** (0.06)*** (0.09)*** (0.06)*** f (0.6)** (0.905) (0.94)** (0.03)*** (0.030)*** (0.033)*** (0.035)*** (0.00)*** Correlaion Equaion Correlaion Equaion (0.037)*** (0.5) (0.063)** (0.040)*** (0.074)*** (0.066)*** (0.08) (0.054)* (0.056)*** (0.058)*** (0.030)* (0.006)*** (0.008)*** (0.006) (0.04) (0.04)** (0.046)*** (0.048)*** (0.070)* (0.00)*** (0.030)*** (0.7)*** (3.974) (0.090)*** (0.06)** (0.080)*** (0.075)*** (.34) (0.36)* (0.06)*** (0.060)*** (0.4)* (0.007) (0.008)** (.6) (.686) (0.9) )* (0.03)*** (0.056)*** (0.337) (0.00)*** (0.007) LRT Noe:. IS DCC i sands for he i -sae independen swiching DCC GARCH model.. Figures in parenheses are sandard errors and *, ** and *** indicae significance a he 0% level, 5% level and % level, respecively. 3. LRT sands for he likelihood raio es. The likelihood raio es saisics is given by LRT lnl( i ) lnl( i), where L(i) is he likelihood value of IS DCC i. The criical values a % for i,3,4, 5 are 3.8, 6.8, 0.09 and 3., respecively. 9

30 Table II. Coninue Esimaes of Unknown Parameers of Alernaive Models Daa period is from January 99 o December 007 c f c f COCOA COFFEE DCC IS-DCC() IS-DCC(3) DCC IS-DCC() IS-DCC(3) IS-DCC(4) Mean Equaion Mean Equaion (0.099) (0.088) (0.) (0.057) (0.097) (0.005)* (0.03)* (0.07) (0.094) (0.3) (0.055) (0.) (0.043) (0.037)* Volailiy Equaion Volailiy Equaion (0.40)*** (0.95)*** (0.)*** (0.654)*** (0.883)*** (0.85)** (0.733)*** (0.03)** (0.077)** (0.098)** (0.794)*** (.88)*** (0.357)* (0.646)*** c (0.04)*** (0.0)*** (0.0)*** (0.07)*** (0.08)*** (0.05)*** (0.08)*** f (0.00)*** (0.008)*** (0.009)*** (0.07)*** (0.08)*** (0.05)*** (0.08)*** c (0.05)*** (0.0)*** (0.04)*** (0.038)*** (0.049)*** (0.03)*** (0.04)*** f (0.03)*** (0.00)*** (0.0)*** (0.037)*** (0.053)*** (0.04)*** (0.033)*** Correlaion Equaion Correlaion Equaion (0.05)*** (0.4)*** (0.7)*** (0.009)*** (0.06)*** (0.000)*** (0.033)*** (0.009)*** (0.6)** (0.08)*** (0.03)*** (0.05) (0.03)*** (0.09) (0.00)*** (0.09)** (0.09)*** (0.7)*** (0.30)*** (0.00)*** (0.09)*** (0.000)*** (0.03)*** (0.0)*** (0.04) (0.00)*** (0.033)*** (.84) (0.03)*** (0.7)*** (0.00)*** (0.09)*** LRT Noe:. IS DCC i sands for he i -sae independen swiching DCC GARCH model.. Figures in parenheses are sandard errors and *, ** and *** indicae significance a he 0% level, 5% level and % level, respecively. 3. LRT sands for he likelihood raio es. The likelihood raio es saisics is given by LRT lnl( i ) lnl( i), where L(i) is he likelihood value of IS DCC i. The criical values a % for i,3,4, 5 are 3.8, 6.8, 0.09 and 3., respecively. 30

31 c f c f Table II. Coninue Esimaes of Unknown Parameers of Alernaive Models Daa period is from January 99 o December 007 CROUD OIL NATURAL GAS DCC IS-DCC() IS-DCC(3) IS-DCC(4) IS-DCC(5) DCC IS-DCC() IS-DCC(3) IS-DCC(4) Mean Equaion Mean Equaion (0.78) (0.6) (0.677) (0.05)*** (0.9)* (0.354)** (0.48)* (0.555)* (0.7)* (0.73) (0.4) (0.677) (0.05)*** (0.)* (0.35)** (0.306)** (0.367)* (0.86)** Volailiy Equaion Volailiy Equaion (0.490)*** (0.5)*** (0.868)*** (0.65)*** (0.768)*** (.5)*** (.05)*** (.95)*** (.38)*** (0.48)*** (0.56)*** (0.95)*** (0.9)*** (0.78)*** (.485)*** (.93)*** (.74)*** (.39)*** c (0.030)*** (0.05)*** (0.07)*** (0.007)*** (0.05)*** (0.039)*** (0.04)*** (0.039)*** (0.038)*** f (0.09)*** (0.04)*** (0.06)*** (0.007)*** (0.06)*** (0.036)*** (0.035)*** (0.035)*** (0.036)*** c (0.035)*** (0.03)*** (0.046)*** (0.006)*** (0.039)*** (0.036)*** (0.037)*** (0.036)*** (0.037)*** f (0.033)*** (0.03)*** (0.047)*** (0.007)*** (0.039)*** (0.034)*** (0.039)*** (0.043)*** (0.048)*** Correlaion Equaion Correlaion Equaion (0.0)*** (0.05) (0.05)*** (0.03)*** (0.07)*** (0.03)*** (0.05)*** (0.039)*** (0.)*** (0.034)*** (0.006)*** (0.034)*** (0.00)*** (0.035)*** (0.73)*** (0.004) (0.03)*** (0.003) (0.038)*** (0.04) (0.)*** (0.00)*** (0.007) (0.037)*** (0.034)*** (0.045)*** (0.09) (0.06)*** (0.033)*** (0.06) (0.03)*** (0.046) (0.04) (0.06) (0.035)*** (.88) (0.039)*** (0.00) (0.037)*** (0.8)*** (.790) (0.033)*** (0.476) (0.038)*** (0.053) (0.7)** (0.00) (0.6) (0.040)*** (0.035)*** LRT Noe:. IS DCC i sands for he i -sae independen swiching DCC GARCH model.. Figures in parenheses are sandard errors and *, ** and *** indicae significance a he 0% level, 5% level and % level, respecively. 3. LRT sands for he likelihood raio es. The likelihood raio es saisics is given by LRT lnl( i ) lnl( i), where L(i) is he likelihood value of IS DCC i. The criical values a % for i,3,4, 5 are 3.8, 6.8, 0.09 and 3., respecively. 4. The LRT of crude oil is sill significan. However, we do no proceed o IS DCC6 since here are fify parameers o be esimaed and empirically, increasing he number of saes o five does no creae significan gains compared o four saes. 3

32 c f c f Table II. Coninue Esimaes of Unknown Parameers of Alernaive Models Daa period is from January 99 o December 007 HEATING OIL PLATINUM DCC IS-DCC() IS-DCC(3) IS-DCC(4) IS-DCC(5) DCC IS-DCC() IS-DCC(3) IS-DCC(4) Mean Equaion Mean Equaion (0.0) (0.069) (0.038) (0.063 )** (0.064)** (0.069)* (0.080) (0.069)* (0.90) (0.037) (0.069) (0.039) (0.060)* (0.06)* (0.07)* (0.084) (0.069)* (0.94) Volailiy Equaion Volailiy Equaion (0.343)*** (0.48)*** (0.50)*** (0.40)*** (0.44)*** (0.05)*** (0.05)*** (0.049)*** (0.086)*** (0.3)*** (0.438)*** (0.568)*** (0.368)*** (0.384)*** (0.049)*** (0.049)*** (0.049)*** (0.075)*** c (0.0)*** (0.06)*** (0.06)*** (0.09)*** (0.048)*** (0.03)*** (0.03)*** (0.0)*** (0.0)*** f (0.04)*** (0.06)*** (0.07)*** (0.09)*** (0.048)*** (0.0)*** (0.0)*** (0.00)*** (0.0)*** c (0.0)*** (0.05)*** (0.07)*** (0.0)*** (0.0)*** (0.06)*** (0.06)*** (0.04)*** (0.08)*** f (0.04)*** (0.09)*** (0.03)*** (0.03)*** (0.07)*** (0.03)*** (0.03)*** (0.03)*** (0.04)*** Correlaion Equaion Correlaion Equaion (0.00)*** (0.009) (0.050)*** (0.00)*** (0.05)*** (0.06)*** (0.053)*** (0.049)*** (0.080)*** (0.045)*** (0.009) (0.009) (0.057) (0.05)*** (0.076)*** (0.040)* (0.07)*** (0.06)*** (0.05)*** (0.04)*** (0.087)*** (0.047)*** (0.003)*** (0.33) (0.73) (0.034)*** (0.06) (0.05)*** (0.003)** (0.05)*** (0.00)*** (0.056)*** (0.07) (0.080)*** (0.046)*** (0.034) (0.05) (7.39) (0.09) (0.075)*** (0.044)*** (0.09)*** (0.07)*** (0.06)*** (0.06)*** (0.)*** (0.047)*** (0.003)** (0.) (0.3) LRT Noe:. IS DCC i sands for he i -sae independen swiching DCC GARCH model.. Figures in parenheses are sandard errors and *, ** and *** indicae significance a he 0% level, 5% level and % level, respecively. 3. LRT sands for he likelihood raio es. The likelihood raio es saisics is given by LRT lnl( i ) lnl( i), where L(i) is he likelihood value of IS DCC i. The criical values a % for i,3,4, 5 are 3.8, 6.8, 0.09 and 3., respecively. 3

33 Table III Ou-of-Sample Hedging Effeciveness. Hedging period is from January 008 o December 008 Variance of Hedged Porfolio Reurn Percenage Variance Reducion Improvemen of IS-DCC(3) over Oher model Expeced Weekly Uiliy 3 Uiliy Gain of Dynamic Hedging Models over OLS 4 Variance of Hedged Porfolio Reurn Percenage Variance Reducion Improvemen of IS-DCC(3) over Oher model Expeced Weekly Uiliy 3 Uiliy Gain of Dynamic Hedging Models over OLS 4 WHEAT (RC=0.074*) 5 CORN (RC=0.000***) Unhedged OLS % 6.% %.37% DCC %.30% % 0.58% IS-DCC() %.% % 0.0% IS-DCC(3) % % IS-DCC(4) % -0.6% IS-DCC(5) % -0.5% COCOA (RC=0.43) COFFEE (RC=0.085*) Unhedged OLS %.63% %.66% DCC %.88% %.65% IS-DCC() %.55% %.83% IS-DCC(3) % % IS-DCC(4) % 0.83% CRUDE OIL (RC=0.00***) NATURAL GAS (RC=0.67) Unhedged OLS %.50% % 0.33% DCC % 0.30% % 0.65% IS-DCC() % 0.4% % 0.03% IS-DCC(3) % % IS-DCC(4) % 0.09% % 0.50% IS-DCC(5) % 0.07% HEATING OIL (RC=0.097*) PLATINUM (RC=0.039**) Unhedged OLS % 0.74% % 3.50% DCC %.0% % 0.33% IS-DCC() % 0.6% % 0.9% IS-DCC(3) % % IS-DCC(4) % % % 0.0% IS-DCC(5) % 0.0% Noe:. Percenage variance reducions are calculaed as he differences of variance of unhedged posiion and esimaed variance of aleraive models over variance of unhedged posiion muliplied by 00.. Improvemen of IS DCC3 over oher hedging sraegies is defined as he difference of he percenage variance reducion of IS DCC3 and he percenage variance reducion of alernaive hedging sraegies 3. Expeced weekly uiliy is calculaed based on equaion (3) 4. Uiliy gains of dynamic hedging models over OLS are defined as he differences of he expeced uiliies of alernaive dynamic models and he expeced uiliy of OLS. 5. RC sands for he Whie s realiy check p-value esing he null ha no improvemen of he bes IS DCC over OLS. *, ** and *** indicae significance a he 0% level, 5% level and % level, respecively. 33