Taking Advanage of Global Diversificaion: A Muivariae-Garch Approach Elena Kaloychou *, Soiris K. Saikouras, Gang Zhao Cass Business School, Ciy Universiy London, 106 Bunhill Row, London EC1Y 8TZ Firs draf: June 2008. This version: May 2009. Absrac We gauge he economic value of mulivariae covariance esimaors by assessing he risk-reurn performance of he resuling mean-variance efficien porfolios. A dynamic asse allocaion framework is deployed, where he mulivariae covariance forecass compee agains simpler nonparameric rivals widely used in he indusry. A wo-layer porfolio consrucion process for global asse allocaion is developed o overcome he problem of handling large dimensional covariance srucures. Based on an ou-of-sample volailiy iming seup he empirical resuls sugges ha he mulivariae covariance forecass ouperform heir nonparameric counerpars and he proposed wo-layer global asse allocaion is favored over convenional porfolio selecion approaches. JEL Classificaion: C32, C52, C53, F21, G11, G15 Keywords: Dynamic Condiional Correlaion, Asymmery, Covariance forecas, Dynamic Asse Allocaion, Porfolio performance. The auhors would like o hank paricipans a he 28h Inernaional Symposium on Forecasing, June 2008, Nice, he Briish Accouning Associaion Annual Conference, April 2009, Dundee, he FMA European Meeing, June 2009, Turin, he Mulinaional Finance Sociey Conference, July 2009, Cree, and seminar paricipans a Cass Business School for useful commens. The usual disclaimer applies. *Corresponding auhor: Tel. +44(0) 207 040 5259; E-mail: e.kaloychou@ciy.ac.uk (E. Kaloychou) 1
2.1 INTRODUCTION There has been a voluminous lieraure on ime series models for he esimaion of asse covariances, and many differen specificaions have been proposed for his purpose. Mos previous sudies eiher focus on he heoreical properies of novel esimaors or compare hem from a saisical poin of view. As inferences in regards o he relaive performance of alernaive covariance esimaors have been largely confined o he use of saisical evaluaion merics, he debae regarding he economic gains of differen covariance models is sill open-ended. The pracical usefulness of mulivariae ime-varying covariance esimaors o asse allocaion has been hindered by he lack of an appropriae way o handle a large number of asses. Esimaing variance-covariance marices using he enire asse se ypically enails huge compuaional burden of maximizing he likelihood, whereas wo-sep quasi likelihood approaches become biased due o he incidenal parameers problem. The quesion of fiing large dimensional ime-varying covariance models has also caused an upsurge of recen research (Engle, Shephard and Sheppard, 2008). In his sudy, we confron alernaive mulivariae covariance esimaors from a acical asse allocaion perspecive. The conribuion o he exan lieraure is hreefold. Firs, he covariance esimaors are assessed on he basis of on economic value relaive forecas accuracy is gauged by he abiliy o yield superior risk-reurn porfolio performance. Second, while previous sudies mainly focused on in-sample comparisons an ou-of-sample evaluaion is conduced, where wo forecasing seings for he mulivariae covariance models are deployed. In addiion, he RiskMerics exponenial smooher ha has been widely used in he indusry as a simple and viable way of esimaing condiional covariances when faced wih a large number of asses is used as a rival o assess he poenial gains from mulivariae covariance models. Hisorical moving average and random walk forecass are also used as naïve benchmarks. Third, we propose a novel way o ackle he problem encounered in he esimaion of ime-varying covariances for high dimensional porfolios of hundreds of asses. We sidesep esimaing he covariance marix for a huge asse se, by dividing he esimaion process in wo-layers, namely, he naional marke and he domesic marke layer. The approach is demonsraed in he conex of global secor allocaion. The empirical evidence suggess ha here is subsanial economic value in ime-varying mulivariae covariance forecass. Incorporaing mulivariae condiional covariance models in he asse allocaion sraegy generaes a beer risk-reurn profile han 2
he porfolio based on he simple covariance forecass. Furhermore, he proposed wo-layer global asse allocaion sraegy ouperforms he convenional approach by providing invesmen superior performance irrespecive of he porfolio consrucion echnique employed. The remainder of he paper is organized as follows. Secion 2.2 reviews he relaed lieraure. Secion 2.3 describes he daa used and provides some summary saisics. Secion 2.4 delineaes he ime-varying covariance models and develops he forecasing framework and he novel global asse allocaion sraegy. Secion 2.5 repors he empirical resuls and Secion 2.6 concludes. 2.2 BACKGROUND LITERATURE A burgeoning area in financial economics has been he volailiy dynamics of asse reurns and heir comovemen. The inegraion of modern financial markes and he booming of srucured financial insrumens (such as derivaives) have boh conribued o his research rend. In wha follows, his secion explores he lieraure relaed o he covariance dynamics of financial asses and heir impac on global diversificaion. 2.2.1 Sylized facs of Volailiy and Correlaion There is an exensive lieraure on he relaionship beween sock reurns and heir volailiy, and he consensus is ha volailiy is ime-varying and asymmeric in naure. Black (1976) conduced he firs empirical sudy on he relaion beween sock reurns and volailiy. He found ha volailiy was ypically higher afer he sock marke falls han afer i rises, so fuure condiional sock volailiy was negaively linked wih he curren sock reurn. Black argued ha he phenomenon may be due o he increase in leverage surfacing when he marke value of a firm declines. In a similar fashion, Chrisie (1982) documened parallel resuls on a larger sample size (379 firms) over a longer sample period (1962-78). The above findings are furher suppored by Schwer (1989), who analyzed he relaion of sock volailiy a economic aciviy, financial leverage and sock rading aciviy using monhly marke indices from 1857 o 1987. The leverage and volailiy feedback inerpreaions of asymmeric volailiy differ in regards of causaliies: leverage hypohesis ress on he conjecure ha reurn shocks lead o changes in condiional volailiy; whereas he volailiy feedback heory conends ha reurn shocks are caused by changes in condiional volailiy. Bekaer & Wu (2000) provided a new framework o examine he wo poenial explanaions of he asymmery, which is a 3
Condiional Capial Asse Pricing Model (CAPM) model wih a GARCH-M parameerizaion. They observed he peaks in he porfolio volailiy ypically correspond o large marke declines and suggesed ha volailiy feedback was he dominan cause of volailiy asymmery. In addiion, hey showed he main mechanism behind asymmery for high and medium leverage porfolios was covariance asymmery i.e. negaive shocks subsanially increase beas (condiional covariances beween porfolio reurns and marke porfolio), while posiive shocks had a mixed impac on beas. There is also evidence suggesing ha price flucuaions in one asse can ransmi o oher asses. This phenomenon is usually known as volailiy spillovers, and has araced he aenion of academics and praciioners alike. A an inernaional level, he consensus view is ha he linkage among global markes has been srenghened by he inegraion of financial markes and he growing rading aciviy among counries. The silen culpris behind such a rend have been he free flow of goods and capial, as well as he revoluion in informaion echnology. Eun & Shim (1989) invesigaed he inernaional ransmission mechanism of sock marke movemens, and documened subsanial amoun of inerdependence in inernaional equiy markes. The observed innovaions occurring in he US marke (1979-1985) rapidly ransmied o oher markes simulaneiy and in he same direcion. In conras, he informaion ransmissions from small markes o big markes were found o be weak and no significan, while he reverse roue was far more pronounced. Koumos & Booh (1995) invesigaed he ransmission mechanism of price reurns and volailiy across he US, UK and Japanese markes. They found srong evidence ha volailiy spillovers in a given marke were much more pronounced when he news arriving from he las marke was bad, i.e. negaive shocks increase volailiy ransmission considerably more han posiive shocks. Baele (2003) examined he magniude and ime-varying naure of volailiy spillovers from he aggregae indices of European (EU) and US markes o 13 local European equiy marke indices and documened an asymmeric effec. The US, as a proxy for he world marke, coninued o be he dominae influence in European equiy markes. Conagion increases during periods high equiy marke volailiy. Worhingon & Higgs (2004) analyzed he ransmission of equiy reurns and volailiy beween developed and Asian equiy markes. The evidence suggesed ha all Asian markes are highly inegraed, and heir co-movemen escalaed significanly when domesic volailiy increased. The price and volailiy spillover phenomenon is also well documened a a company level wihin naional equiy markes. Conrad, Gulekin & Kaul (1991) showed ha in he 4
US equiy marke (all firms recorded on he CRSP ape) here was a size asymmery in he predicabiliy of he volailiies ransmission of large versus small firms over 1962-1988. Harris & Pisedasalasai (2006) invesigaed he reurn and volailiy ransmission mechanisms beween large and small sock in he US sock and showed ha reurns and volailiies of large socks were imporan in predicing he fuure dynamics of smaller socks, bu no vice versa. More recenly, he research focus has recenly shifed owards modeling he dynamics of correlaions and covariance. Undersanding he dynamics of condiional covariance is crucial for inernaional porfolio diversificaion. Bollerslev, Engle & Wooldridge (1988) showed ha he condiional covariances beween financial asses were ime-varying. Longin & Solnik (1995) argued ha correlaions beween inernaional equiy markes increased over ime. Similarly, Ang & Bekaer (1999) found, via a regime swiching framework, evidence for he exisence of a high volailiy regime, in which monhly reurns (MSCI for US, UK and Germany 1970-97) were more correlaed and had lower means. In line wih he previous empirical evidence (Longin & Solnik, 1995) on inegraed equiy markes, hey also documened he presence of a high volailiy-high correlaion regime which ended o coincide wih a bear marke. A he same ime, i is rue hose financial markes are procyclical, and ha business cycles affec he inensiy of price movemens and he associaed correlaions. Along hese lines, Erb, Harvey & Viskana (1994) linked he ime variaion in correlaions o he phases of business cycles. They invesigaed he economic changes of he G-7 indusrial counries and found ha correlaions are higher in recessions relaive o expansions, and low when wo counries business cycles are ou of phase. Longin & Solnik (2001) invesigae he ime-varying naure of correlaion beween financial asses by deploying exreme value heory o he condiional correlaion of exreme observaions. Using monhly equiy index reurns for he five larges equiy markes across he world from 1959 o 1996 hey corroborae previous findings, ha correlaions rise in bear markes, bu no in bull markes. Bekaer, Harvey & Ng (2005) aemped o explain he cause of asymmeric correlaion a marke level, especially in exreme cases i.e. during financial crises. They proposed an asse pricing model, which posied ha changes in condiional correlaion were caused by global, regional and counry-specific fundamenals and found ha for some crises (e.g. Mexican crisis, 1994), he sudden jump in cross-correlaion beween indices was a resul of boh marke s excess reurns being exposed o a common facor and no of volailiy spillovers. However, hey did find economically meaningful increases in he residual 5
correlaion 1 in Asia, during he crisis. Alhough mos of he empirical sudies in his area have acceped he fac ha correlaion and covariance do vary over ime, i is only unil recenly, ha researchers have sared o observe and analyze he srucural breaks in he ime-varying correlaions. Billio & Pelizzon (2003) documened an increase in he condiional correlaion of European equiy markes in he afermah of he EMU inroducion, and ha he effec was no only regional, bu also had a fundamenal impac on global markes. Cappiello, Engle & Sheppard (2006) proposed an Asymmeric Generalized DCC mulivariae model (AG-DCC) o capure correlaion dynamics. They examined daily daa on FTSE All-World sock marke indices of 21 counries, and he 5-year average mauriy bond indices for he 13 counries over 1987-01. They found srong evidence of asymmeries in condiional covariance of boh equiy and bond reurns. More imporanly, hey documened significan evidence of a srucural break. Tha is, when he EMU was inroduced in January 1999, he level of condiional correlaion significanly increased, bu his sor of increase had no been found in he level of condiional volailiy. Hyde, Bredin & Nguyen (2007) invesigaed he weekly correlaion dynamics across 13 Asia-Pacific equiy markes, a European index and he US marke index 2 over 1991-06. Similar o previous sudies, hey esablished significan asymmeries in correlaions, a srucural break for he Asian crisis and an increase in correlaions over ime reinforcing he greaer marke inegraion. 2.2.5 Inernaional Diversificaion Diversificaion is he answer o he enhancemen of he risk-reurn profile in any invesmen decision, and he fund managemen profession has relied on his principle for a number of decades o manage a vas amoun of wealh. The increased rend of financial globalizaion and marke inegraion, however, has promped he quesion of wheher inernaional diversificaion is sill an effecive risk reducion approach. The quesion becomes even more relevan when financial crises and heir associaed effecs come under he spoligh. A few decades laer, Markowiz (1952) produced one of he mos valuable breakhroughs in he modern invesmen managemen arena. He formalized he inuiion of porfolio heory and proposed a porfolio selecion echnique on he basis of is risk-reurn profile, as opposed o merely compiling porfolios wih asses ha individually have 1 The residual correlaion refers o he correlaion of he exceed reurns, or, he residual errors from he corresponding reurn equaions. 2 The 13 Asia-Pacific equiy markes are presened by heir domesic sock exchange indices. The index for US marke is S&P500 and he index for European marke is a value weighed index of reurns from France, Germany, Ialy and he UK. 6
aracive characerisics. Tobin (1958) expanded Markowiz s (1952) work by adding a risk-free asse o he analysis. This made i possible o leverage or deleverage porfolios on he efficien fronier. Through leverage, porfolios on he capial marke line were able o ouperform porfolios on he efficien fronier. Solnik (1976) produced he firs formal sudy on inernaional diversificaion. Using weekly US and European daa, over 1966-71, he showed ha hrough inernaional diversificaion, invesors can ge a porfolio wih much lower aggregae volailiy compare o domesic diversificaion. His findings indicaed ha in domesic markes, he number of securiies wihin a porfolio was negaively relaed o porfolio risk (in line wih Markowiz, 1952), while even greaer reducion of risk can be aained by diversifying inernaionally. De Sanis & Gerard (1997) argued ha alhough severe marke declines were conagious, gains from inernaional diversificaion were sill achievable. Using monhly MSCI dollar-denominaed reurns on sock indices for he G7 plus Swizerland 3 and a condiional CAPM model wih mulivariae GARCH-M hey showed ha spillover effecs on price and volailiy are shor-lived (several days in mos cases), and will no have significan impacs on a porfolio diversified globally, rebalanced on a low frequency (e.g. monhly or quarerly). The findings suggesed ha alhough holding an inernaionally diversified porfolio provided lile proecion agains severe US marke declines, he long-erm benefis from he diversificaion sill remained economically aracive. Recenly, Ang & Bekaer (1999, 2002) showed ha, a high volailiy-high correlaion regime did exis, which ended o coincide wih a bear marke. However, by examining he relaionship beween he US, UK and German markes from 1970s o 1990s, hey found he evidence on higher volailiy was much sronger han he evidence on higher correlaion and lower means. They suggesed ha he benefis of inernaional diversificaion sill exised even in he high correlaion regime for long invesmen horizons. Goezmann, Li & Rouwenhors (2005) used a long span of hisorical daa o invesigae wheher global diversificaion sraegies served invesors well over he las cenury, and he poenial of inernaional diversificaion in he fuure. In order o explain he benefis from global diversificaion, hey decomposed hem ino a) he variaion in he average correlaion in equiy markes hrough ime and b) he variaion in he invesmen opporuniy se. From 1850 ill 2000, all he available daa on over 80 inernaional equiy markes had been colleced. The analysis suggesed ha he srucure of global correlaions shifed considerably over ime, and i was currenly near is hisorical high. They found 3 The larges European marke no included in he G7. 7
ha approximaely fify percen of he diversificaion benefis ha achieved by invesors oday were due o he increasing number of new invesmen opporuniies in he world markes; and he oher half due o a lower average correlaion beween he recenly available markes and exising markes, and among he new markes hemselves. 2.2.6 Evaluaion of Covariance Esimaors The upsho of he exan lieraure is ha alhough, under exreme condiions, invesors canno gain exra proecion from global diversificaion, he long-erm benefis are sill aainable. Since global asse allocaion is sill an aracive invesmen roue, he efficien consrucion of such a global porfolio comes o he forefron. The key inpu is he asse covariance srucure bu he quesion emanaing is wheher i pays off o ry and accuraely capure all he sylized facs of covariance srucure of asse reurns. De Sanis & Gerard (1997) noed ha he co-movemens beween financial asses were shor-lived, which negaes he value of accouning for correlaion dynamics in a long-erm invesmen sraegy. Fleming, Kirby & Osdiek (2001) examined he economic value of volailiy iming for shor-horizon invesors in equiy, bond and gold fuures. They found ha he predicabiliy capured by condiional volailiy models is economically significan and he dynamically formed porfolio ouperforms he uncondiionally efficien saic porfolio wih he same arge expeced reurn and ransacion coss. Balasubramanyan (2004) compared he performance of porfolios consruced on he basis of differen variance-covariance marix esimaion approaches. Porfolios consising of US, UK and Japan socks were rebalanced daily and six mulivariae GARCH models had been employed. The porfolio incorporaing ime-varying correlaion wih asymmeric volailiy and spillovers ouperformed is simple counerpar ha ignored such informaion. Engle & Colacio (2006) invesigaed he variance minimizaion problem subjec o a required reurn. They derived an imporan resul, ha he efficiency loss of he porfolio was minimized when he esimaed correlaion was equal o he rue value. Using a wo-asse framework, hey calculaed he loss of efficiency 4 while keeping he asses expeced reurns fixed. They demonsraed ha he asymmeric DCC specificaion was he bes performer in a simple sock and bond porfolio. Using a consan correlaion model during volaile correlaion phases can be very cosly: as much as 40% of reurn can be dismissed, if he wrong condiional correlaion model were employed. 4 The loss of efficiency refers o he loss in he reurn-risk raio when use he incorrec correlaion informaion o consruc he porfolio. 8
2.3 THE DATA The paper is based on daily reurns for en indusry indices and one broad marke index for he Japanese, UK and US equiy markes obained from Thomson DaaSream Inernaional. The daa perain o he en broad economic secors: Energy (ENG), Basic Maerial (BML), Consumer Goods (CGS), Consumer Service (CSV), Financial (FIN), Healh Care (HCR), Indusrial (IND), Technology (TEC), Telecommunicaion (TEL), and Uiliy (UTL). The sample spans he period from July 1 1996 o May 31 2007, which resuls in a oal of 2,849 logarihmic daily reurns for each index. Summary saisics for he seleced indices are presened in Table 1. [Inser Table 1: Panel A-D] The mean reurns over he period are posiive for mos secors and naional markes in he sample, wih he excepions of he Japanese HCR and TEC secors and he UK TEC secor. All daily reurns are non-normally disribued, paricularly in he form of lepokurosis. The exen of skewness differs across markes. The UK and US marke indices and mos of he secors herein are significanly negaively skewed, whereas in Japan half of he secors show posiive skewness and no significan skewness is observed a he naional marke level. The lack of symmery in he reurn disribuion is consisen wih previously repored findings (Bekaer & Wu, 2000; Glosen, Jagannahan & Runkle, 1993). The ADF es rejecs he hypohesis of a uni roo for all reurns series a he 1% significance level. Table 2 repors he uncondiional indusry wide correlaions wihin each marke. The [Inser Table 2: Panel A-D] secor correlaions appear o be high over he sample period in all markes. For Japan, UK and US he mean uncondiional correlaions beween secors are 54.9%, 44.3% and 52.5%, respecively 5. A he counry level, he UTL (67.2%, Japan), IND (55.9%, UK) and TEL (63.3%, US) secors have he highes average uncondiional correlaion wih he res of he secors in he same counry, while he ENG (33.4%, Japan), TEC (37.1%, UK) and he CGS (37.2%, US) are on average, over he sample period, he leas correlaed. Overall, as borne ou by he uncondiional correlaions averaged across secors and markes IND and HCR are he secors exhibiing he highes cross-correlaions, a 57.1% and 56.9%, respecively. The ENG secor has he lowes average uncondiional correlaion wih he oher secors (45.2%) albei sill high. A he naional level, he UK and US equiy markes exhibi uncondiional correlaion of approximaely 40%. The associaion beween he US and Japanese equiy 5 The hree mean correlaions are significan a he 95% level wih -saisics a 38.8, 35.3 and 27.26, respecively. The -saisic is compued as (-2)/(1-2 ) and follows a Suden- disribuion wih degrees of freedom (-2). The criical value a he 95%significance level is 0.01253. 9
markes is much weaker, only 11.6%, whereas UK and Japanese equiy markes is 27.6%. Volailiy spillovers have been well documened in previous sudies, which implies ha he increase in he volailiy of one asse(s) is ransmied o anoher asse. In order o gauge he exen of volailiy ransmission across secors and naional markes, we look a he correlaion beween he curren volailiy of index i, proxied by daily squared 2 2 reurns ( R ), wih he lagged volailiy of index j ( R ). Table 3 repors he correlaion i beween he day- volailiy of index i and he day -1 volailiy of index j. [Inser Table 3: Panel A-D] In general, he Japanese equiy marke has he lowes level of volailiy spillovers among secors (8.9% on average), while he US equiy marke experienced he highes exen of cross-secor volailiy ransmission (15.6% on average). Across all hree equiy markes, he U s volailiy change has he larges impac on he fuure volailiy of he oher secors. The average cross-secor correlaion beween he lagged daily volailiy of he UTL secor and he curren daily volailiy of all oher secors is 15.1% (when averaged across he hree counries), followed by he FIN secor (14.5%) and CSV secor (13.5%). The TEC secor has he smalles influence (9.9%) on he oher secor daily volailiy flucuaions. Overall, he volailiy ransmission effec averaged across all secors and over he hree counries is 12.6%. The highes degree of volailiy spillover among naional equiy markes is documened from he UK o Japan. The correlaion beween he day -1 volailiy of he UK equiy marke and he day- volailiy of he Japanese equiy marke and is approximaely 30%. The spillover measure from he US o he Japanese and UK equiy marke is around 21%. The spillover effec from Japan o he res of he world markes is relaively weak, in line wih he previous empirical findings. j 1 2.4 METHODOLOGY The ensuing empirical analysis is based on he consrucion of global indusry porfolios and heir comparison on he basis of risk-reurn profile. The porfolios differ in heir way of (i) esimaing asse correlaions and variances, and (ii) incorporaing his informaion in asse allocaion. To his end, a novel approach o inernaional diversificaion is proposed and esed using naional sock marke and secor indices from Japan, UK and US. Two compeing dynamic covariance esimaion frameworks are deployed. The firs one is he mulivariae GARCH (MGARCH) family of esimaors, while he second one is he simpler 10
approaches (e.g. RiskMerics 6 ) widely used in he indusry. In order o gauge he qualiy of he differen MGARCH specificaions over he in-sample period, realized variances and correlaions are obained. Monhly covariance forecass are generaed and used for dynamic porfolio rebalancing hroughou he in- and ou-of-sample periods 7. Finally, an ou-of-sample evaluaion of he volailiy iming sraegies is conduced in order o appraise he economic differences of asse allocaion based on rival covariance forecass. In wha follows, Secion 2.4.1 delineaes he economeric framework, Secion 2.4.2 presens he porfolio consrucion process and he economic evaluaion framework, while Secion 2.4.3 ses ou he forecasing echniques. In Secion 2.4.4, we show how he MGARCH framework can be embedded in our proposed porfolio consrucion process. 2.4.1 Mulivariae GARCH Esimaors A naural economeric framework for esimaing he covariance marix of asse reurns is he MGARCH class of models. Four differen ypes of MGARCH specificaions are considered in his sudy and heir key characerisics are described below. The firs model we use is he Baba-Engle-Kraf-Kroner (BEKK) model which was inroduced by Engle & Kroner (1995). The BEKK model draws upon he more general VEC-GARCH specificaion of Bollerslev, Engle & Wooldridge (1988). Le H be he symmeric [n x n] asse variance-covariance marix and vec(.) he operaor ha sacks he columns of he lower riangular par of is marix argumen. The firs-order VEC parameerizaion can be expressed as vec(h ) = vec(ω) + A vec(r -1 r -1) + B vec(h -1) (1) where he Ω is an [n x n] consan marix, r is he column vecor of he cross-secion of n asse reurns a ime and A, B are boh [n x n] parameer marices. Albei very flexible, he VEC model is bedeviled by wo issues. Firs, i does no guaranee he posiive definieness of H wihou addiional resricions. Second, even afer imposing resricions o ensure he symmery of H, he number of parameers o be esimaed in a firs order VEC is very large, n(n+1)/2+2(n(n+1)/2) 2, unless n is small. The BEKK formulaion can be viewed as a resriced version of (1), which guaranees he posiive definieness of H. The firs-order form of he BEKK model specifies he covariance marix H as follows H = C C + A (r -1 r -1) A + B H -1 B (2) where C is an [n x n] upper riangular marix. The full BEKK specificaion requires 6 RiskMerics was iniially a VaR mehodology proposed by JP Morgan (1996). Now i has become a sandard model for porfolio risk assessmen. 7 Rebalancing akes place on he firs rading day of every monh. 11
n 2 +n(n+1)/2 parameers o be esimaed. This sudy considers wo special cases of he full BEKK esimaor, where A and B are scalar and diagonal [n x n] marices. The diagonal BEKK esimaes n+n(n+1)/2 parameers. The second class of MGARCH models used is correlaion models. Correlaion models rely on decomposing he condiional covariance ino condiional sandard deviaions and correlaions. They have he advanage of esimaing fewer parameers han he BEKK. The simples is he Consan Condiional Correlaion (CCC) GARCH model inroduced by Bollerslev (1990) which imposes ime invarian correlaions. The CCC model is esimaed in wo seps. Firs, a univariae GARCH(p,q) model is fied o each reurn series o generae he condiional variance h i, i = 1,, n. Then, he covariance is specified as where D diag h,..., h 1 n H = D R D (3) and R is a posiive definie [n x n] marix wih [R] ii = 1 and -1< [R] ij < 1. The uncondiional correlaion marix is ypically used as an esimaor for R. The consan correlaion assumpion has been found o be oo resricive in several empirical sudies (Kroner & Ng, 1998; Ang & Bekaer, 1999; Tse & Tsui, 2002). The covariance decomposiion in (3) can be exended o allow for non consan condiional correlaions by inroducing dynamics in he correlaion marix R. Among he many specificaions proposed for he evoluion of R he Dynamic Condiional Correlaion (DCC) GARCH model of Engle (2002) is he mos popular. The DCC esimaor has he same firs sep as he CCC approach, bu for each series he sandardized errors, ε i = u i / h i, are also produced alongside he condiional variance. In he second sep, he ime-varying correlaion marix is formalized via he following dynamic process Q = (1-a-b)Q + a ε -1 ε -1 + b Q -1 (4) where Q is an [n x n] symmeric marix;q = E[ε ε ] is he uncondiional covariance marix of sandardized innovaions, esimaed by is sample counerpar 1 Q T T i1 ', a and b are scalars. The ime varying correlaion is specified as R = (Q *) -1 Q (Q *) -1 (5) where Q * = diag( q i,, q n ) and ensures ha R has he srucure of a correlaion marix as long as Q is posiive definie 8. The scalar formulaion in (4) poses idenical dynamics for all asse correlaions and permis no ransmission of pas shocks beween asses. Appendix A1 ses ou in deail he DCC specificaion in he wo-asse case. 8 Q will be posiive definie wih probabiliy one if (Q A QA B QB) is posiive definie. 12
This resricion is relaxed in he Asymmeric Generalized DCC (AG-DCC) version of Sheppard (2002), which exends (4) by subsiuing he scalar parameers (a, b) wih marices, and also accommodaing for asymmeries in he condiional variance-covariance Q = (Q - A Q A B Q B G NG) + A ε -1 ε -1 A + B Q -1 B + G -1-1 G (6) where A, B and G are [n x n] parameer marices, = I[ε <0] ε ( indicaes he elemen-by-elemen Hadamard produc), and N = E[ ] where expecaion is replaced by is sample analogue, N 1 T T i1 '. Model (6) posulaes ha covariance dynamics are asse-specific and news can be passed on o oher asses. In addiion, he las erm allows negaive shocks o have a sronger impac on he evoluion of variances and covariances. For esimaion racabiliy we follow Sheppard (2002) and Cappiello, Engle & Sheppard (2006) and consider he diagonal version of (6) hroughou he analysis (ADCC model). Inerdependence of asse volailiies and correlaions is no allowed for in his seing. A more resriced scalar version is also esimaed where A= [a], B= [b ] and C= [c]. Appendix A2 gives deails of he ADCC specificaion in he wo-asse case. We consider also an exended version of (6) which accommodaes srucural breaks in boh he long-run mean and he dynamics of correlaions (ADCC-break) as in Cappiello, Engle & Sheppard (2006). The ADCC-break model accouns for wo regimes as follows Q d( Q d( A 1 AQ A BQ B GN G ) (1 d)( Q 1 1 1 1 1 1 1 A BQ 1 1 1 1 1 1 1 1 1 B G G ) (1 d)( A 1 1 1 1 2 AQ A BQ B G N G ) 2 2 2 2 1 1 2 2 2 2 A BQ 2 1 2 2 B G 2 2 G 2 1 1 2 ) (7) where d is a break indicaor defined as d=1 for <, and 0 else; Q 1 = E[ε ε ] for < τ, Q 2 = E[ε ε ] for τ, wih N1, N2 similarly defined. The News Impac Surface (NIS) for MGARCH models is he analogue o a news impac curve for univariae models. The NIS funcion ƒ(ε 1,ε 2) porrays how he condiional correlaion of wo relaed asses reacs o heir join pas informaion (Kroner & Ng, 1998). For he ADCC models considered, he NIS for he correlaion is given by 9 f, ) c ( a a g g ) (8) ( i j ij i j i j i j i j where c ij is he ij h elemen of he consan marix in (6). In he presence of asymmery, g i and g j boh significan, i is expeced ha join bad news has a greaer impac on fuure correlaion han join good news or a combinaion of good/bad news, ceeris paribus. Pu differenly, for given shocks correlaion increases more when boh asses suffer (ε i < 0, ε j <0). Model esimaion is by quasi maximum likelihood (QML). Appendix B gives deails of 9 This is a simplified form of he NIS funcion under he assumpion of lineariy. The exac NIS funcion is given in Appendix C. 13
he esimaion mehod and log-likelihood funcions. Inferences are based on Bollerslev-Wooldridge robus sandard errors (Bollerslev & Wooldridge, 1992). Individual significance and join hypohesis ess are based on -saisics. 2.4.2 Covariance Esimaion and Porfolio Consrucion Sraegies The sample is divided ino an in-sample esimaion period 01/07/96 o 28/06/02 of fixed lengh 1565 days (T-T₁=72 monhs), and a holdou evaluaion period 01/07/02 o 31/05/07 of 1284 days (T₁=59 monhs). The in-sample period serves as a esing plaform for selecing he bes performing covariance esimaor among he various MGARCH specificaions. The seleced MGARCH specificaions in erms of porfolio performance are subsequenly o be scruinized in an ou-of-sample forecasing exercise. The condiional ~ covariance (Ω ) is aken as he populaion covariance measure and is proxy ) for he ( in-sample porfolio performance evaluaion is he monhly realized covariance, ~ M r r, 1 where M is he number of rading days wihin he corresponding monh and r τ are he daily reurns. For comparison purposes benchmark porfolios based on he ex-pos realized covariance marix are also conrased agains he mulivariae approaches. Once he daily esimaes of he condiional variance-covariance marix are generaed from each model, hey are aggregaed o monhly using M H H 1, where he Hτ is he MGARCH covariance marix for day τ, and M is he number of rading days wihin monh. A dynamic mean-variance framework is deployed o consruc porfolios based on he monhly variance-covariance esimaes from he differen MGARCH models. We consider an invesor wih a monhly invesmen horizon who allocaes funds across n risky asses plus a riskless securiy according o he following sraegies: maximize he expeced porfolio reurn subjec o a arge condiional volailiy σ p* (max-r), minimize he condiional porfolio variance subjec o a arge expeced reurn p* (min-v), or maximize he expeced uiliy of he invesmen (max-u). The arge volailiy and expeced reurn are proxied by he corresponding in-sample averages for he marke index. The hree-monh Japanese inerbank loan, LIBOR, and Treasury bill middle raes are employed as risk free asses for Japan, UK and US, respecively. Le = E -1[R ] and H = E -1[Ω ] denoe, respecively, he [n x 1] vecor of he risky asse condiional expeced reurns, and he [n x n] marix of he condiional covariance marix for 14
monh. For he in-sample analysis, he ex-pos monhly asse reurn and he esimaed condiional covariance marix is used o proxy and H. According o he max-r and min-v and sraegies he invesor solves he following opimizaion problems, and max p w 1 w I R w * s.. p w Hw f (9) min w Hw w * s.. p w 1 wi R f (10) where w is an [n x 1] vecor of porfolio weighs on he risky asses, R f is he reurn on he risk free asse, I is an [n x 1] vecor of 1s. No shor selling consrains are imposed in order o guaranee feasibiliy of he soluion. The risky asse opimal weigh vecors for he wo sraegies are as follows. 1 H * 2 ( p ) For he max- R sraegy, w 1 R f I ' H R f I * 1 p R f H R f I For he min-v sraegy, w, 1 R I ' H R I f f R f I. while he weigh on he risk free asse is (1 w I). For he max-u sraegy, we assume he invesor is risk averse, wih consan absolue risk aversion. Thus, he uiliy maximizaion problem for monh is formulaed as max U( R w p ) w ' v w ' H w (11) where U(R p) is he invesor s uiliy for a given end of period porfolio reurn R p and is he risk aversion of he invesor se a 2.5. No riskless asse is involved in his sraegy as wih a riskless asse in he invesmen opporuniy se, he opimal uiliy-maximizing allocaion under U(.) would assign all he wealh o he riskless asse. A shor selling resricion is, however, imposed in he max-u sraegy. We employ a 130/30 shor selling resricion, which is ypically adaped in he fund managemen indusry. The 130/30 shor-sell resricion implies ha he fund manager can hold no more han 30% of he oal wealh shor, and no more han 130% of he iniial wealh in long posiions. The porfolios consruced are rebalanced monhly under each covariance esimaor o produce a sequence of porfolios spanning he in-sample period. The adequacy of volailiy iming sraegies based on alernaive MGARCH specificaions is judged on he basis of heir 15
porfolio risk-reurn profile. The porfolio evaluaion crieria used are he in-sample average monhly porfolio reurn (Re), Sharpe raio (SR) and invesor uiliy (U) for he max-r sraegy, porfolio volailiy (Sd) for he min-v sraegy, and invesor uiliy (U) for he max-u sraegy. The evaluaion framework described above is used for he in-sample model selecion. The seleced MGARCH specificaion under each economic crierion is carried forward o he ou-of-sample analysis. 2.4.3 Forecas Framework We use he in-sample daily daa in order o esimae he MGARCH model parameer se {Θ}, and generae muli-sep-ahead (1 o 30 day) ou-of-sample covariance forecass. The winner MGARCH models from he in-sample model selecion are esimaed over an iniial window of fixed lengh 1565 days, denoed [1,τ*], and a se of covariance marix forecass, { Hˆ } * M * 1, is generaed for each day in he firs forecas monh *+1 (July 2002). The daily covariance forecass are based on simulaed reurn series for days {τ*+n, n=1,,m} and ~ MGARCH parameer esimaes {} using informaion up o day τ*. These are hen aggregaed ino a monhly forecas ha will be used as inpu for porfolio rebalancing. The window is hen rolled forward M days o [1+M, τ*+m] o obain he second se of daily covariance forecass, for each day in monh *+2 (Augus 2002), and so forh unil 59 ieraions for all he monhs in he ou-of-sample period. Evenually, one-monh-ahead covariance forecass are produced for all ou-of-sample monhs. Albei only ineresed in monhly covariance forecass he MGARCH models are fied direcly o daily reurns. Using aggregae monhly daa o esimae asse reurn volailiy and heir comovemen is likely o resul in less accurae forecass (Andresen, Bollerslev & Lange, 1999) and downward biased esimaes because he naure of serial correlaion observed in financial reurns is affeced by he sampling frequency. Auocorrelaion is ypically posiive in he shor-horizon (daily/weekly) reurns, bu i could be negaive in longer-horizon (monhly/annually) reurn series (Fama & French, 1988). In oher words, mean-reversion is documened in low frequency daa, while persisence is eviden a higher frequencies. As a resul, for daily daa he deviaion from he mean (volailiy) is generally larger han for monhly, hereby using aggregae monhly reurns would lead o underesimaion of volailiies. Covariance Forecass based on Simulaed Reurns The objecive is o simulae he asse reurn process M-days ahead and use he 16
simulaed reurns as inpus for generaing muli-sep-ahead covariance forecass. The Mone Carlo Reurn Generaing Process (RGP) used in his paper posis ha fuure reurns are a weighed average of heir recen pas and a random normally disribued innovaion wih mean and variance parameers ha are updaed monhly. The simulaion reurn DGP for asse i on he n h ou of sample day is as follows rˆ k i, ~ iid N ˆ, vˆ ), n=1,, M (12) i, * n (1 k) ri, *nm T i ( i i j 1 2 ˆ i (1 ) ri, * j, ˆ (1 ) j 1 T v i r (13) j 1 j 1 2 i, * j The innovaions i are assumed o come from he same disribuion wih mean and variance ˆ, vˆ ) esimaed each monh using exponenial smoohing (decay facor = 0.97) over a ( i i window of T=100 days. Four scenarios are examined wih degree of freedom parameer k = 0, 5%, 10%, and 15%. The simulaion RGP draws upon he fac ha here is persisence in reurns even a monhly horizons. The basic scenario of k=0 implies ha we simulae he one-monh-ahead reurn by replicaing he exac reurns paern observed in he previous monh. I follows by definiion ha E τ-m[r τ-m+1]=e τ-m[r τ-m+2]= = E τ-m[r τ]=r τ-m. In pracice, however, we do observe mean reversion in monhly reurns and he inroducion of he ghos feaure in he RGP conforms o his conenion. In his spiri, we allow he reurn o flucuae around he paern observed over he las monh, while conrolling for he magniude of his flucuaion hrough he degree of freedom parameer k. We consruc R=1 simulaed samples R { j} j MC 1, each of hem conaining he daily reurn pah { r * 1, r * 2,..., r * M } for he following M-rading-day monh. The covariance forecas { Hˆ, ˆ,..., ˆ * 1 H * 2 H * M } for every rading day in he monh is generaed using he esimaed ~ MGARCH parameer se {}, and he simulaed daily reurn series. The one monh-ahead covariance forecas can hen be obained. We also consruc a simpler covariance forecas based on a weighed average of ex-pos ~ * L monhly MGARCH covariance esimaes over a L-lengh window { H }. The forecas * for monh *+1 is generaed using he las six monhs covariance esimaes, Hˆ ~ H where L = 6 monhs and he decay parameer = 0.97 as * 1 * L ( L* 1) * 1 L advocaed by RiskMerics for monhly daa. 17
Simple Benchmark Covariance Models The ou-of-sample performance of he porfolios based on he above forecass of he seleced MGARCH models is assessed agains hose of simple rival covariance esimaors. The hree naïve compeiors for esimaing variance-covariance marices are he Random Walk (RW), Moving Average (MA) and he RiskMerics Exponenial Weighed Moving Average (EWMA). The RW variance-covariance forecas for monh *+1 is he realized variance-covariance of he previous monh, ˆ RW ~ H * 1 *. The hisorical MA model formulaes nex monh s volailiy as an equally-weighed average of he pas realized volailiies over a rolling window of lengh L=6 monhs. So he one-monh-ahead forecas of he variance-covariance marix of he reurns series is * L MA * 1 (1 ) L * 1 Hˆ L ~. The EWMA forecas is a weighed average pas observaions which gives more imporance o recen reurns informaion over he rolling L-monh window, ˆ ~, EWMA H * 1 * L ( L* 1) * 1 L and = 0.97. All approaches are also compared o porfolios formed on he basis of he ex pos realized covariance esimaor, ~. 2.4.3 Two-layer Global Secor Allocaion Sraegy In his secion, we propose an asse allocaion sraegy designed o ackle high dimensional porfolios. The raionale behind his approach, as well as is implemenaion in he conex of inernaional diversificaion is ouined below. Global porfolio selecion has radiionally focused on naional equiy marke indices. However, since secor correlaions are generally lower han marke correlaions we sugges ha diversificaion based on secor indices across differen naional markes can be poenially more fruiful. Inernaional secor diversificaion involves he esimaion of a high dimensional covariance marix 10, which is compuaionally expensive and can hamper esimaion accuracy. I is a fac ha he efficiency of he esimaed covariance marix decreases wih he number of asses (Engle & Sheppard, 2001; Silvennoinen & Teräsvira, 2008), while some of he MGARCH esimaors (e.g. BEKK) are no well-suied o a muliasse seing. In order o exploi global diversificaion, and conquer he esimaion difficulies of he increasing number of parameers we propose a wo-layer global secor 10 For insance, he Diagonal-DCC MGARCH for a porfolio wih aggregae marke indices requires esimaion of 3 x 3 + 3 x 2=15 parameers, while furher diversifying among he en secors he number of parameer o be esimaed rises o 3(10 x 3 + 10 x 2) = 150. 18
allocaion sraegy, which builds upon wo sub-porfolios, he Domesic Secor (DS) and he Naional Marke (NM). The invesmen opporuniy se is he en secors of each marke and he naional marke indices. The proposed Global Secor (GS) allocaion sraegy decomposes he ineracion among markes ino comovemen wihin a naional marke (secors), and beween he markes. Wealh allocaion is underaaken in wo layers. The firs layer consiss of a mean-variance efficien NM porfolio, where informaion on marke comovemen is he inpu. Le NM W porfolio of marke indices, denoe he K-dimensional column vecor of opimal weighs for he NM NM R NM and D diag h,..., h 1 K be he [K x K] correlaion marix of he naional indices, be he corresponding sandard deviaion marix. The NM porfolio variance can be compued as follows NM NM NM NM NM NM h W D R D W (15) In he second layer, he wealh invesed in each equiy marke is furher divided across secors based on he opimal mean-variance scheme, where informaion on DS comovemen is he inpu. The hree porfolios comprising S domesic secor indices are generaed, based on he esimaed [S x S] secor covariance marix. The DS porfolio variance is h DS wih associaed expeced reurn r DS DS DS DS DS DS D R D W W (16) 19. Finally, he GS porfolio is consruced by combining he NM and DS porfolios. In order o resric he number of MGARCH esimaed parameers he correlaions of he K domesic secor porfolios ( R are assumed o equal o he marke wide correlaion ( R h GS W W NM NM NM DS DS DS NM D R D W DS NM DS NM D R D W GS DS NM wih expeced porfolio reurn r r W. DS ) beween wo markes ). The GS porfolio variance is The invesmen sraegy provides a rade-off beween esimaion racabiliy and global diversificaion gains. More imporanly, by using wo separae layers of asse allocaion, a he naional and secor levels, he sraegy offers a dual proecion mechanism. When global markes become more urbulen, he weighing scheme of he NM is geared owards he marke ha is relaively less affeced by he bad global economic condiions. Insead of simply increasing invesmen in his relaively less vulnerable marke, he DS scheme (17)
provides a way of opimizing he invesmen wihin markes. This approach implies ha he wide marke index may no necessarily be he mos efficien porfolio in a paricular marke. We empirically show, using he realized covariance informaion over he ou-of-sample period, ha he secor porfolio (DS) can yield a superior risk-reurn profile o he marke index porfolio (NM). However, inernaional invesors can sill rely on naional marke indices o choose how and when o diversify across markes. Wheher he GS allocaion sraegy proposed ouperforms he NM and/or DS porfolio is ye anoher quesion pursued in he ensuing empirical analysis. 2.5 EMPIRICAL RESULTS The firs secion examines he oupus of differen MGARCH specificaions over he in-sample period, whereas he following secion deals wih he in-sample performance evaluaion and model selecion. The final secion invesigaes he ou-of-sample porfolio performance of he proposed wo-layer global allocaion sraegy and addresses he quesion of wheher our laer combined wih he MGARCH esimaed covariances could be fruiful for fund managers. 2.5.1 Saisical Inferences on MGARCH Esimaors The MGARCH models are esimaed over he in-sample period for he naional indices and he domesic secors in Japan, UK and US. 11 The models are: scalar and diagonal BEKK, scalar CCC, scalar and diagonal DCC, ADCC, DCC-break and ADCC-break, a oal of eleven specificaions. For he wo-sep DCC-ype esimaors he univariae GARCH mus firs be defined. Thus, we fi a GARCH (1,1) and E-GARCH (1,1,1) o daily reurns and he Akaike (AIC) and Schwarz (SIC) informaion crieria are deployed for selecing he mos appropriae model. Appendix Table A3 ses ou he resuls. The asymmeric E-GARCH is largely significan, bu he simple GARCH specificaion is favored by he AIC/SBC for all secor and marke index reurns. Given he specificaion of he firs-sep univariae condiional volailiy, he MGARCH models are hen esimaed over he 6-year in-sample period for each of he four sub-porfolios. The resuls are repored in Table 4. [Inser Table 4: Panel A - D] Table 4, Panels A-C show he MGARCH esimaion resuls for he DS porfolios in he hree 11 The BEKK MGARCH esimaor does no converge for he covariance esimaion of he DS porfolio for Japan and UK, as well as he NM porfolio. 20
markes. Panels A1-C1 repor he model parameers when here is no srucural break in he specificaion. Boh he scalar and diagonal ADCC sugges ha he asymmeric effec in correlaions albei significan is negligible. Table 4, Panels A2-C2 se ou he esimaion resuls when a srucural break is inroduced in he correlaion dynamics. Following he perinen lieraure (Baele, 2003; Billio & Pelizzon, 2003 among ohers), he breakpoin is he onse of he European Moneary Union (EMU) on 01/01/1999, when all he EMU members irrevocably fix heir exchange rae and he Euro is inroduced o replace he single naional currency. The radical ransform of he European money marke influenced he economy of he EMU member counries and ha of he closely inegraed UK marke. This has also affeced he US dollar cash flows, hrough is impac on ineres raes aached o he Euro-dollar. The change in he erm srucure influences he value of he US dollar and consequenly impacs he US economy. The fundamenal changes in he European and US money markes will affec he domesic equiy markes, and furher ransmi o he Japanese marke due o he srong degree of globalizaion (Hamao, Masulis, & Ng, 1990; Koumos & Booh, 1995). Alhough he ime aken for he effecs of money marke ransformaion in Europe o ransmi o he US and hen furher influence Japan is less clear, seing he srucural breakpoin on he inroducion of EMU seems reasonable. The findings in Table 4, Panels A2-C2 indicae a dramaic change in he dynamic srucure of condiional correlaions following he inroducion of EMU. Ineresingly, for Japan he diagonal ADCC-break model poins owards a subsanial degree of correlaion asymmery for mos secors in he pre-emu period, bu he effec became significanly less marked pos-emu. The asymmery parameer has decreased dramaically for 7 ou of 10 secors (ENG, BML, IND, CGS, CSV, FIN and TEC). The TEC indusry suffers he bigges drop wih he correlaion asymmery parameer slumping from 0.137 o 0.006, followed by BML whose asymmery parameer drops from 0.05 o 0.003. Similar inferences hold for he UK and US markes. The exen of asymmery weakens significanly in 7 secors. In he UK, he CSV secor experiences he bigges drop in he correlaion asymmery parameer, from 0.144 o 0.012, whereas in he US, i is he FIN secor for which he asymmeric effec drops from 0.064 o 0.003. In all markes, he average asymmery effec loses significance in he pos-emu period as indicaed by he scalar ADCC-break. Overall, i appears ha he asymmeric impac of shocks on he covariance srucure of equiy reurns is smoohed ou pos-emu. A possible inerpreaion could be he quick recovery of global markes from he Asian crisis, which increased invesor confidence abou unfavorable price movemens. The degree of persisence in condiional correlaion, measured by (a 2 + b 2 + g 2 ), also 21
undergoes a srucural break. Condiional correlaion for mos secors becomes significanly more persisen afer he inroducion of he EMU. For insance, for he ENG secor in Japan he degree of persisence in condiional correlaion is 0.843 (diagonal DCC-break) and 0.840 (diagonal ADCC-break) in he pre-break period, while hese rise o 0.990 and 0.997, respecively, in he pos-break period. Appendix D repors he resuls of saisical ess on differences beween he parameers in he pre- and pos-emu periods. Appendix E presens echnical deails on how he ess are formulaed. Table 4, Panel D illusraes he MGARCH esimaion resuls for he naional marke indices. Correlaion asymmery is less noable for marke indices han secors. In line wih he secor evidence, correlaion persisence a he marke level increased pos-emu, bu i is sill much smaller in magniude. The effec of he srucural break in correlaion dynamics, is furher illusraed by means of he NISs in Figure 1. [Inser Figure 1] The NISs refer o he pre- and pos-emu periods for wo secors (IND and FIN), which behave significanly differen in he wo periods. In summary, i seems ha accouning for a srucural break in correlaion dynamics is imporan as asymmeric effecs dampen afer he inroducion of he EMU and secor/marke correlaions become more sluggish. Increased persisence can resul in higher uncondiional correlaions. Finally, he relaive ranking of alernaive MGARCH specificaions in erms of in-sample fi is presened in Table 4, Panels A3-D3 for each sub-porfolio. AIC and SIC boh favor he mos parsimonious scalar DCC model, whereas he likelihood raio es (LR es) favors he mos flexible diagonal ADCC-break for he domesic secors and he diagonal ADCC for he naional marke indices. 2.5.2 Performance Comparison of MGARCH Esimaors I is well known ha he bes-fi model does no necessarily lead when i comes o economic value. In his secion, we invesigae he economic ranking of alernaive covariance esimaors wih a view o selec hose ha yield superior porfolio performance. The NM and DS porfolios are buil based on hree porfolio selecion sraegies, min-variance, max-reurn and max-uiliy. The invesor allocaes funds beween he marke/secors and he risk-free asse, and dynamically rebalances he porfolio based on he monhly updaed covariance. The bes esimaor is chosen for each sraegy using various porfolio 22
performance crieria, and is o be scruinized in he ou-of-sample volailiy iming exercise. Table 5 ses ou he in-sample evaluaion of he esimaors in he conex of he hree porfolio sraegies considered. [Inser Table 5: Panel A - D] Table 5, Panel A shows he performance evaluaion of he max-reurn sraegy porfolios. The bes performing model is seleced using hree crieria, average monhly Sharpe Raio (SR), Reurn (Re), and invesor uiliy (U) - noe ha in his sraegy we fix he volailiy a a arge level, which is approximaed as close as feasible by each esimaor. The esimaors seleced by he SR are he diagonal DCC-break model for he domesic secor Japanese (DS-JP) porfolio, he scalar DCC-break for boh he domesic secor porfolios of UK (DS-UK) and US (DS-US), and he scalar DCC model for he NM porfolio 12. In erms of maximum Re he CCC esimaor is unanimously he bes in-sample performer. Under he U measure he DCC-break (diagonal or scalar) is favored for he DS-JP and DS-US and he NM porfolios, whereas he scalar ADCC-break is favored for he DS-UK. Table 5, Panel B presens he in-sample performance of he porfolios based on he min-variance invesmen sraegy. The bes model is assessed in erms of average volailiy Sd. The min-variance sraegy has a predeermined arge reurn level, so he Re is fixed across esimaors and volailiy is he only differeniaing aspec 13. The esimaors seleced under Sd are he diagonal DCC-break for DS-JP, and he scalar DCC-break for DS-UK, DS-US, and NM. In Panel C, he performance of he max-uiliy porfolios is illusraed. I emerges ha he uiliy maximizing esimaors are he diagonal ADCC for boh he DS-JP and DS-UK, and he scalar DCC-break for DS-US, whereas he scalar ADCC-break for he NM porfolio. Because, unlike in he oher wo sraegies, we impose a 130/30 shor-sell resricion and disallow access o risk-free asse on he max-u, he resuling porfolios are relaively more volaile (low SR, high Sd), and score worse in erms of average uiliy. Table 5, Panel D summarizes he winner models across sraegies and equiy markes. For compleeness, all crieria are considered, bu he model seleced for he ou-of-sample porfolio rading will be based on he mos relevan crierion for each sraegy. The resuls sugges ha ha for he max-r sraegy a simple CCC provides he highes average reurn. This is no surprising as when concerned mosly abou generaing reurn, risk becomes largely irrelevan. All oher crieria poin owards ime varying covariance models. This 12 In case of a ie beween models, he simples specificaion wins. 13 In line wih our monhly rebalancing (volailiy iming) sraegy, our SR and U measures are compued by averaging he monhly Sharpe raios and uiliies over he in-sample period raher han dividing he average monhly Re and Sd (a measure which reflecs long-run performance). In he laer case, given he same reurn level, he model ranking in erms of Sd, SR and U are idenical, whereas i is no he case here. 23
resul suggess ha even when he invesmen sraegy is geared owards porfolio reurn invesors do need o adequaely capure covariance dynamics in order o achieve high risk-reurn efficiency. The mos dominan model in erms of maximum SR and U is he (scalar/diagonal) DCC-break. As invesors become more concerned abou risk, accouning for he dynamics of correlaion becomes increasingly imporan. For insance, in he min-v sraegy he (scalar/diagonal) DCC-break is he one ha achieves he minimum risk and maximum SR. In he max-u sraegy, he ADCC model ouperforms is peers in erms of monhly uiliy. Given he risk-aversion implied by he ypical uiliy funcion, he models ha enail he highes uiliy are ofen he ones providing he highes risk-reurn rade-off (SR). In general, he empirical resuls in Table 5, Panel D bear ou some ineresing general findings. When invesors are ineresed in risk-reurn radeoff he accurae esimaion of covariance dynamics is more imporan. Across sraegies in half of he insances, he model seleced by he SR also provides he lowes Sd and highes U. A he secor level, he mos successful model ends o be he scalar/diagonal DCC-break, while he max-u sraegy also endorses asymmeries in he DCC specificaion. For he naional marke index porfolio he CCC appears o perform raher well, which may indicae ha ime variaion in correlaion is less noable a a marke raher han secor level. For he naional marke indices he scalar ADCC-break is favored under he max-reurn sraegy, whereas he CCC under he min-variance and max-u. In he ligh of he saisical and economic inferences so far, hree findings are worh emphasizing. Firs, in-sample fi and porfolio performance do no go in andem. The model ha provides he highes LLF is he diagonal ADCC-break, bu i is no he mos desirable when i comes o invesmen performance. According o he AIC/SBC, he bes in-sample fi model is he scalar DCC, however, superior invesmen performance poins oward somewha more elaborae specificaions. Second, accouning for srucural breaks in he dynamics of he covariance evoluion can enhance boh he saisical properies and economic value of covariance models. No only do srucural break models generally provide beer fi, bu also low porfolio volailiy (under he min-v sraegy) and high SR (under boh he min-v and max-r). Third, i seems ha in some cases ignoring he effec of asymmeric correlaion can be cosly, even hough he effec is weak. The empirical evidence suggess ha he asymmeric effec on reurns correlaion is significan bu raher low. However, in he maximum uiliy sraegy where he invesor faces shor-selling consrains and no risk free asse, allowing for asymmery improves he porfolio risk-reurn 24
and uiliy profiles. Correlaion asymmeries become relevan form an economic viewpoin when invesors are no allowed o pu (large amouns of) funds in he risk-free asse. 2.5.3 Payoffs from he Two-Layer Global Asse Allocaion Sraegy In his secion we assess he poenial of our proposed wo-layer global asse allocaion sraegy based on he ex pos realized covariance process in he ou-of-sample period. To his end, we sar by verifying wheher he domesic porfolio consising of secor indices delivers a beer risk-reurn profile han he corresponding naional marke index. If rue, he implicaion is ha marke porfolio, as replicaed by he naional marke index, may no be he opimal invesmen opporuniy in a given equiy marke. Second, we es wheher our proposed wo-layer global asse allocaion sraegy is superior o invesing in eiher he naional marke or he domesic secor indices of an individual marke. Table 6 illusraes he performance of he differen approaches o asse allocaion based on he ex-pos ( rue ) variance-covariance informaion over he ou-of-sample period. [Inser Table 6: Panel A - C] The approaches differ in (i) he asses in he invesmen opporuniy se (domesic secor indices, naional marke indices, global secor indices and he benchmark marke index), and (ii) he crierion for allocaing wealh across asses (max-r, min-v and max-u). The same shor-selling and risk-free asse condiions hold for he passive marke index racking. We firs invesigae wheher he DS porfolios ouperform heir benchmark marke index rivals under he hree invesmen sraegies. In he conex of he max-r sraegy (Panel A), i is obvious ha he SR (average and sandardized), U and Re of he DS porfolios are much higher han heir marke index counerpars are. For he min-v sraegy (Panel B), again he DS porfolios remarkably ouperform heir domesic marke index benchmark in erms of SR (average and sandardized) and Sd. Turning o he max-u sraegy (Panel C), he DS porfolios clearly ouperform heir corresponding domesic marke index on he basis of all performance measures 14. The upsho is ha he invesor could obain a beer risk-reurn profile by diversifying asses across secors insead of racking he naional marke index. Figures 2 4 graphically se ou he comparison beween he DS porfolios and domesic marke index porfolios under he hree sraegies. There is consensus ha despie he increasing inegraion of global financial markes, 14 The uiliy of he max-u porfolios is negaive, reflecing heir high volaile naure due o resricions on invesing in he risk-free asse and shor-selling. 25
inernaional diversificaion is sill aracive (De Sanis & Gerard, 1997; Ang & Bekaer 1999, 2002; Goezman, Li & Rouwenhors, 2005). Our empirical resuls poin o anoher direcion; he ype of asses used is much more imporan han he number of markes o diversify in. Table 6, Panel A suggess he all hree DS porfolios considerably ouperform he NM porfolio despie he laer being diversified across hree equiy markes. This corroboraes ha domesic secor diversificaion is more beneficial han inernaional diversificaion. Noneheless, boh he NM and DS porfolios are beaen by he GS porfolio under he min-v and max-u sraegies. Similar o he NM porfolio bu insead of using naional indices he GS porfolio uses secor level indices from differen equiy markes o disribue funds. The resuls under he min-v sraegy (Panel B) sugges ha doing so significanly improves he SR and Sd of he porfolio. The performance of he GS is, however, raher unsable as indicaed by he relaively low sandardized SR. In he max-u sraegy (Panel C), he average U and Sd of he GS porfolio are also much beer hen hose for he NM counerpar. The resuls show srong suppor o he conjecure ha he novel wo-layer global asse allocaion is superior o eiher naional marke or domesic secor diversificaion. However, he GS porfolio does no fare well under he max-r sraegy, where i shows a relaively low reurn, low SR and high volailiy (Panel A). Addiionally, he average monhly GS volailiy σ(re) is 22.84%, much higher han he NM (2.416%) and DS (2.21% - 2.55%) porfolios. The explanaion behind he poor performance of he GS porfolio may lie in he assumpions behind he wo-layer allocaion: (i) he naional marke indices correlaions are he inpus for compuing he weighs o be invesed in he domesic secor (DS) porfolios of each marke. Tha is, he correlaion among naional markes serves as a proxy of he correlaion among he corresponding DS porfolios when we compue he variance of he GS porfolio. In order o guaranee feasibiliy in he consrained opimizaion problems for he max-r and min-v sraegies we do no impose any resricions on shor-selling and risk-free asses. However, he porfolio as implied by he marke index does no involve shor-selling or risk-free asse. Thus, he naional marke index may no longer capure he characerisics of he corresponding DS porfolio, and his may cas some doub on he validiy of he firs layer GS asse allocaion. In he min-v sraegy shor-selling is only a small proporion of he whole porfolio and so he naional marke index correlaions can proxy quie well he correlaions beween he secor porfolios. The average shor-sell percenage in he min-v GS porfolio is only 1.25% wih a maximum shor-sell a 9.7%. On he oher hand, he average shor-sell proporion in he max-r GS porfolio is 33% wih a maximum of 278%. Wheher he naional marke 26
index can sill represen he general characerisics of a DS porfolio wih such excessive shor-selling becomes quesionable. Thus, he relaively poor performance of he max-r sraegy could sem from he fac ha he naional marke correlaions are no longer a good proxy for he domesic secor porfolio correlaions. 2.5.4 MGARCH covariance forecass: Economic gains or rouble? The nex quesion concerns he performance of our proposed approach o global asse allocaion when based on forecased covariance. For each performance crierion he NM, DS and GS porfolios are consruced based on he MGARCH models seleced in Secion 2.5.2. An addiional se of GS porfolios is formed based on simpler covariance forecass, reaed as benchmarks for gauging he economic value added of MGARCH models. Table 9 shows he ou-of-sample performance of he GS, he NM and DS porfolios according o he differen crieria. In order o isolae he economic value of he various covariance marix forecass, we fix he inpu of expeced reurn o be eiher he observed reurn over he previous monh = R -1, or he average reurn of he las 12 monhs 1 12 R i 15. 12 i1 [Inser Table 9: Panel A - C] Panel A ses ou he performance of he porfolios based on he max-r sraegy and he MGARCH model seleced based on he SR crierion. The empirical resuls sugges ha when he expeced reurn is esimaed as he reurn of previous monh, GS porfolio does no provide a sound invesmen efficiency no maer which forecas echnique is deployed. 1 However, when swich he reurn seing o 12 12 R i i1, he GS porfolio based on he RW covariance forecas enjoys boh he highes SR and Re. The performance saisics in Panel B are based on he MGARCH models seleced by he (Sd) crierion under he min-variance sraegy. I is clear ha, he GS porfolio yields he lowes ou-of-sample Sd among differen asse allocaion sraegies no maer which expeced reurn seing or forecas mehod is deployed. Besides, under boh expeced reurn seings, he porfolios based on he MG-EWMA covariance forecas have he lowes monhly volailiy. Moreover, he average volailiy of he porfolios under he = R -1 seing is generally lower han heir counerpars derived wih he alernaive expeced reurn seing. 15 Excep for he Min.Sd porfolio consrucion sraegy, since under his sraegy, he weighing scheme is only affeced by he informaion of expeced variance-covariance marix. 27
Panel C shows he performance of he max-u porfolios based on he MGARCH esimaors seleced by he U measure. The GS porfolio enjoys he bes ou-of-sample performance in erms of boh he U and Sd measure for all he forecas mehods. Furhermore, he resul shows ha he porfolio consruced based on 1 he 12 12 R i i1 reurn seing generally enjoy a higher (U) and lower (Sd) han is counerpars based on he alernaive reurn seing. Under his average reurn seing, he GS porfolio based on he MG-EWMA covariance forecas echnique delivers he bes ou-of-sample performance in erms of boh (U) and (Sd). In general, he empirical findings reveal ha he GS porfolio ouperforms he domesic secor and naional marke porfolios in erms of he relevan performance measure for each sraegy. In erms of forecas qualiy, he MG-EWMA ranks bes among he rival forecas mehods. Based on he argumen of Engle & Colacio (2006), he porfolio based on he bes variance-covariance forecas should provide he lowes porfolio volailiy. From he empirical resul, i is clear ha he porfolios based on he MG-EWMA forecass mehod enjoy he lowes Sd figures across all he hree porfolio consrucion sraegies. In erms of economic value, he MG-EWMA forecas mehod ouperforms is counerpars. The porfolios based on he MG-EWMA covariance forecas provide he bes ou-of-sample invesmen performance in wo ou of hree porfolio consrucion sraegies (see Sd under min-variance, and, U under max-u) and bea he random walk and RiskMerics benchmarks. The only excepion is he max-reurn sraegy, where he porfolio based on he RW forecas mehod enjoys he bes ou-of-sample performance. However, he max-reurn sraegy focuses more on expeced reurn insead of volailiy and so he expeced covariance inpu becomes relaively less imporan. 2.6 CONCLUSIONS The esimaion and forecas of large dimensional variance-covariance marix is crucial for he implemenaion of modern porfolio heory. Various esimaion echniques wih differen model specificaions have been developed for his purpose. However, a comprehensive economic comparison among he differen approaches has no been carried ou as ye. In his sudy, we assess he esimaion and forecas qualiy of various academic sae of he ar covariance models by gauging he risk-reurn profile of he porfolio consruced based on hem. In addiion, he pervasive problem of esimaing a high dimensional covariance marix is addressed. To his end, a dynamic mean-variance 28
framework wih hree porfolio consrucion sraegies is deployed and a wo-layer secor allocaion sraegy ha rades-off esimaion racabiliy and diversificaion is proposed. The analysis reveals a mismach beween model saisical fi and economic value. In general, he mos flexible esimaor (diagonal ADCC-break), in erms of model specificaion, has he bes saisical performance. However, when he porfolio consrucion sraegy and performance crierion are more geared owards expeced reurn he CCC is he bes performing MGARCH model in erms of economic value, while he he models from DCC-break family are favoured when risk comes ino play. Our wo-layer global asse allocaion sraegy developed o conquer he difficuly of esimaing high dimensional asse covariance marices provides a beer risk-reurn performance han is rival asse allocaion sraegies when consruced based on boh he rue variance-covariance informaion. The MGARCH covariance forecas mehod provides he bes ou-of-sample risk-reurn performance across he porfolio consrucion sraegies and beas simple benchmark covariance forecas approaches such as he random walk and he indusry sandard RiskMerics. REFERENCES Andersen T. G., Bollerslev, T. and Lange. S. (1999) Forecasing financial marke volailiy: sample frequency vis-à-vis forecas horizon. Journal of Empirical Finance 6:457-477. Ang, A. and Bekaer, G. (1999) Inernaional Asse Allocaion wih Time-Varying Correlaions. Working paper of Columbia Universiy. Ang, A. and Bekaer, G. (2002) Inernaional asse allocaion wih regime shifs. Review of Financial Sudies 15:1137-1187. Baele, L. (2003) Volailiy spillover effecs in European equiy markes. Working Papers of Faculy of Economics and Business Adminisraion, Ghen Universiy, Belgium: 03/198. Balasubramanyan, L. (2004) Do ime-varying covariances, volailiy comovemen and spillover maer? Working Paper Series of Pennsylvania Sae Universiy. Bekaer, G. and Wu, G. (2000) Asymmeric volailiy and risk in equiy markes. The Review of Financial Sudies 13:1-42. Bekaer, G., Harvey, C. and Ng, A. (2005) Marke inegraion and conagion. Journal of 29
Business 78:39-69. Billio, M. and Pelizzon, L. (2003) Volailiy and shocks spillover before and afer EMU in European sock markes. Journal of Mulinaional Financial Managemen 13:323-340. Black, F. (1976) Sudies of sock price volailiy changes. Proceedings of he 1976 meeings of he American Saisical Associaion, Business and Economics Saisics Secion: 177-181. Bollerslev, T. (1990) Modeling he coherence in shor-run nominal exchange rae: a mulivariae Generalized ARCH approach. Review of Economics and Saisic 72:498-50. Bollerslev, T., Engle, R. and Wooldridge, J. (1988) A capial asse pricing model wih ime-varying covariances. The Journal of Poliical Economy 96:116-131. Bollerslev, T. and Wooldridge, J. (1992) Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariance. Economic Review 11:143-172. Cappiello, L., Engle, R. and Sheppard, K. (2006) Asymmeric Dynamics in he correlaions of global equiy and bond reurns. Journal of Financial Economerics 4:537-572. Chrisie, A. (1982) The sochasic behaviour of common sock variances: Value, leverage, and ineres rae effecs. Journal of Financial Economics 10:407-432. Conrad, J., Gulekin, M. and Kaul, G. (1991) Asymmeric predicabiliy of condiional variances. The Review of Financial Sudies 4:597-622. De Sanis, G. and Gerard, B. (1997) Inernaional asse pricing and porfolio diversificaion wih ime-varying risk. The Journal of Finance 52:1881-1912. Engle, R. (2002) Dynamic condiional correlaion: a simple class of mulivariae generalized auoregressive condiional heeroskedasiciy models. Journal of Business and Economic Saisics 20:339-350. Engle, R. and Kroner, K. (1995) Mulivariae simulaneous GARCH. Economeric Theory 11:122-150. Engle, R. and Colacio, R. (2006) Tesing and valuing dynamic correlaions for asse allocaion. Journal of Business & Economic Saisics 24:238-253. Engle, R. and Sheppard, K. (2001) Theoreical and empirical properies of Dynamic Condiional Correlaion MVGARCH. Working paper No. 2001 15, Universiy of California, San Diego. Eun, C. and Shim, S. (1989) Inernaional ransmission of sock marke movemens. The Journal of Financial and Quaniaive Analysis 24:241-256. Erb, C., Harvey, C. and Viskana, T. (1994) Forecasing inernaional equiy correlaions. Financial Analyss Journal November/December-1994:32-45. Fama, E. F. and French, K. R. (1988) Permanen and emporary componens of sock prices. The Journal of Poliical Economy 96:246-273. 30
Fleming, J., Kirby, C., and Osdiek, B. (2001) The economic value of volailiy iming. Journal of Finance 56:329 352. Glosen, L. Jagannahan, R. and Runkle, D. (1993) On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance 48:1779-1802. Goezmann, W., Li, L. and Rouwenhors, K. (2005) Long-erm global marke correlaions. Journal of Business 78:1-38. Hamao, Y. Masulis, R. and Ng, V. (1990) Correlaions in price changes and volailiy across inernaional sock markes. The Review of Financial Sudies 3:281-307. Harris, R. and Pisedasalasai, A. (2006) Reurns and volailiy spillovers beween large and small socks in he UK. Journal of Business Finance & Accouning 33:1556-1571. Hyde, S., Bredin, D. and Nguyen, N. (2007) Correlaion dynamic beween Asia-Pacific, EU and US sock reurns. Working Paper of Universiy of Mancheser. Kroner, K. and Ng, V. (1998) Modeling asymmeric comovemens of asses reurns. Review of Financial Sudies 11:817-844. Koumos, G. and Booh, G. (1995) Asymmeric volailiy ransmission in inernaional sock markes. Journal of Inernaional Money and Finance 14:747-762. Longin, F. and Solnik, B. (1995) Is he correlaion in inernaional equiy reurns consan: 1960-1990? Journal of Inernaional Money and Finance 14:3-26. Longin, F. and Solnik, B. (2001) Exreme correlaion of inernaional equiy markes. Journal of Finance 56:649-676. Markowiz, H. (1952) Porfolio selecion. The Journal of Finance 7:77-91. Schwer, G. (1989) Why does sock marke volailiy change over ime? Journal of Finance 44:1115-1153. Solnik, B. (1976) Why no diversify inernaionally raher han domesically? Financial Analys Journal 30:48-54. Tobin, J. (1958) Liquidiy preference as behavior owards risk. The Review of Economic Sudies 25:65-86. Tse, Y. and Tsui, A. (2002) A mulivariae generalized auoregressive condiional heeroskedasiciy model wih ime-varying correlaions. Journal of Business and Economic Saisics 20:351-362. Welch, B. L. (1947) The generalizaion of suden s problem when several differen populaion variances are involved. Biomerika 34:28-35. 31
Table 1: Daily Sock Reurn Disribuion (%) Panel A: Japanese Equiy Secor Indices Name BML CGS CSV ENG FIN HCR IND TEC TEL UTL Mean 0.008 0.004 0.008 0.014 0.017-0.016% 0.002-0.011 0.021 0.006 Sd. Dev. 0.949 1.835 1.841 1.384 1.051 1.790 1.834 1.158 1.402 1.408 Skewness 0.093** 0.048 0.009-0.108*** 0.141*** 0.301*** 0.089* 0.093** -0.146*** 0.055 Kurosis 6.123*** 6.306*** 4.812*** 4.568*** 5.453*** 6.097*** 5.428*** 5.187*** 5.540*** 5.742*** Jarque-Bera es 1165.5 1302.7 391.2 298.4 725.9 1185.3 705.8 573.6 778.4 896.9 p-value 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 ADF es -50.17-52.51-53.98-53.88-46.86-59.01-51.49-46.69-51.37-54.50 p-value. 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 Noe: The number of observaions is 2849 for all secors. ADF is he Augmened Dickey-Fuller es for he null of a uni roo. The lag is chosen based on a max lag of T 0.5 /2 27, and a downward selecion procedure based on he Schwarz Informaion Crierion (SIC) so as no series correlaion is presen. *, ** and ***, denoe significance a he 10%, 5% and 1% level, respecively. Panel B: UK Equiy Secor Indices Name BML CGS CSV ENG FIN HCR IND TEC TEL UTL Mean 0.027 0.010 0.008 0.031 0.031 0.020 0.009-0.058 0.019 0.042 Sd. Dev. 1.271 1.855 1.099 1.458 1.382 1.200 1.478 2.699 1.758 0.980 Skewness -0.129*** 0.092** -0.092** -0.015-0.061-0.120*** -0.717*** -0.593*** 0.148*** -0.016 Kurosis 5.645*** 7.557*** 6.376*** 5.451*** 6.603*** 6.145*** 11.550*** 10.708*** 5.118*** 5.364*** Jarque-Bera 841.2 2477.0 1361.5 715.3 1547.6 1185.0 8950.1 7242.0 544.8 665.4 p-value. 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ADF saisic -49.85-53.89-50.18-34.10-50.26-39.49-47.83-50.71-34.60-54.31 p-value. 0.01% 0.01% 0.01% 0.00% 0.01% 0.00% 0.01% 0.01% 0.00% 0.01% 32
Table 1: Daily Sock Reurn Disribuion (CONT D) Panel C: US Equiy Secor Indices Name BML CGS CSV ENG FIN HCR IND TEC TEL UTL Mean 0.027 0.010 0.027 0.046 0.041 0.027 0.033 0.030 0.014 0.019 Sd. Dev. 1.369 1.267 1.256 1.415 1.353 1.050 1.278 2.015 1.325 1.110 Skewness 0.083* -0.251*** -0.186*** -0.023 0.114*** -0.154*** -0.136*** 0.223*** -0.119*** -0.405*** Kurosis 6.318*** 9.006*** 8.554*** 4.860*** 6.446*** 8.347*** 7.415*** 6.847*** 6.039*** 10.214*** Jarque-Bera 1314.4 4325.5 3690.2 412.1 1420.2 3416.3 2330.5 1785.9 1106.3 6275.0 p-value. 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% ADF saisic -52.51-54.88-38.94-40.38-51.60-34.40-39.68-54.04-54.61-51.13 p-value. 0.01% 0.01% 0.00% 0.00% 0.01% 0.00% 0.00% 0.01% 0.01% 0.01% Panel D: Naional Equiy Marke Indices Name Japan UK US Mean 0.002 0.020 0.028 Sd. Dev. 1.224 0.990 1.094 Skewness -0.075-0.285*** -0.105*** Kurosis 4.998*** 5.975*** 6.474*** Jarque-Bera 483.8 1105.8 1459.3 p-value 0.00% 0.00% 0.00% ADF Saisic -52.5-52.9-54.4 p-value 0.00% 0.00% 0.00% See noe in Panel A 33
Table 2: Uncondiional Daily Reurn Correlaion Panel A: Japan Secor Wide BML CGS CSV ENG FIN HCR IND TEC TEL UTL BML 1 0.219 0.144 0.303 0.482 0.319 0.349 0.464 0.311 0.416 CGS 0.219 1 0.668 0.653 0.442 0.541 0.286 0.615 0.575 0.512 CSV 0.144 0.668 1 0.833 0.420 0.582 0.272 0.638 0.683 0.577 ENG 0.303 0.653 0.833 1 0.592 0.702 0.440 0.788 0.800 0.783 FIN 0.482 0.442 0.420 0.592 1 0.559 0.458 0.706 0.577 0.655 HCR 0.319 0.541 0.582 0.702 0.559 1 0.502 0.762 0.607 0.766 IND 0.349 0.286 0.272 0.440 0.458 0.502 1 0.541 0.378 0.614 TEC 0.464 0.615 0.638 0.788 0.706 0.762 0.541 1 0.718 0.817 TEL 0.311 0.575 0.683 0.800 0.577 0.607 0.378 0.718 1 0.655 UTL 0.416 0.512 0.577 0.783 0.655 0.766 0.614 0.817 0.655 1 Panel B: UK Secor Wide BML CGS CSV ENG FIN HCR IND TEC TEL UTL BML 1 0.447 0.597 0.479 0.571 0.390 0.588 0.391 0.390 0.372 CGS 0.447 1 0.485 0.335 0.482 0.345 0.492 0.297 0.304 0.281 CSV 0.597 0.485 1 0.442 0.723 0.494 0.581 0.605 0.636 0.468 ENG 0.479 0.335 0.442 1 0.522 0.428 0.409 0.264 0.346 0.377 FIN 0.571 0.482 0.723 0.522 1 0.597 0.558 0.471 0.574 0.490 HCR 0.390 0.345 0.494 0.428 0.597 1 0.374 0.236 0.389 0.459 IND 0.588 0.492 0.581 0.409 0.558 0.374 1 0.414 0.405 0.327 TEC 0.391 0.297 0.605 0.264 0.471 0.236 0.414 1 0.514 0.221 TEL 0.390 0.304 0.636 0.346 0.574 0.389 0.405 0.514 1 0.353 UTL 0.372 0.281 0.468 0.377 0.490 0.459 0.327 0.221 0.353 1 Panel C: US Secor Wide BML CGS CSV ENG FIN HCR IND TEC TEL UTL BML 1 0.691 0.655 0.467 0.639 0.521 0.716 0.436 0.467 0.396 CGS 0.691 1 0.735 0.354 0.687 0.535 0.725 0.554 0.526 0.368 CSV 0.655 0.735 1 0.349 0.764 0.593 0.785 0.679 0.631 0.373 ENG 0.467 0.354 0.349 1 0.370 0.396 0.409 0.235 0.321 0.451 FIN 0.639 0.687 0.764 0.370 1 0.635 0.767 0.580 0.604 0.447 HCR 0.521 0.535 0.593 0.396 0.635 1 0.635 0.390 0.514 0.425 IND 0.716 0.725 0.785 0.409 0.767 0.635 1 0.647 0.588 0.424 TEC 0.436 0.554 0.679 0.235 0.580 0.390 0.647 1 0.551 0.237 TEL 0.467 0.526 0.631 0.321 0.604 0.514 0.588 0.551 1 0.377 UTL 0.396 0.368 0.373 0.451 0.447 0.425 0.424 0.237 0.377 1 D: Naional Equiy Marke Indices Japan UK US Japan 1 0.276 0.116 UK 0.276 1 0.399 US 0.116 0.399 1 34
Table 3: Spillover Effecs of he Daily Realized Volailiy - Correlaions Beween RV i, and RV j,-1 Panel A: Japan Secor Wide ENG BML IND CGS HCR CSV TEL UTL FIN TEC ENG 0.255 0.116 0.063 0.074 0.056 0.071 0.083 0.041 0.060 0.047 BML 0.095 0.139 0.057 0.037 0.081 0.078 0.030 0.053 0.069 0.025 IND 0.045 0.059 0.103 0.073 0.047 0.071 0.104 0.038 0.072 0.093 CGS 0.066 0.092 0.153 0.176 0.065 0.100 0.110 0.065 0.122 0.136 HCR 0.093 0.108 0.126 0.105 0.171 0.105 0.096 0.083 0.081 0.137 CSV 0.059 0.080 0.065 0.053 0.068 0.106 0.055 0.073 0.074 0.047 TEL 0.053 0.058 0.119 0.091 0.066 0.072 0.248 0.101 0.049 0.171 UTL 0.122 0.131 0.196 0.182 0.163 0.156 0.131 0.228 0.109 0.161 FIN 0.046 0.085 0.061 0.054 0.045 0.076 0.031 0.044 0.169 0.035 TEC 0.019-0.001 0.071 0.044 0.045 0.012 0.125 0.045 0.025 0.136 Panel B: UK Secor Wide ENG BML IND CGS HCR CSV TEL UTL FIN TEC ENG 0.182 0.106 0.115 0.154 0.163 0.201 0.144 0.164 0.164 0.059 BML 0.118 0.229 0.126 0.096 0.137 0.110 0.093 0.099 0.133 0.027 IND 0.103 0.100 0.154 0.109 0.111 0.113 0.097 0.111 0.108 0.032 CGS 0.126 0.170 0.170 0.216 0.138 0.165 0.117 0.156 0.176 0.058 HCR 0.174 0.118 0.100 0.135 0.211 0.283 0.169 0.205 0.254 0.073 CSV 0.155 0.105 0.172 0.160 0.175 0.184 0.154 0.194 0.153 0.081 TEL 0.125 0.065 0.122 0.145 0.147 0.231 0.192 0.182 0.203 0.068 UTL 0.154 0.099 0.106 0.126 0.131 0.178 0.126 0.217 0.209 0.044 FIN 0.131 0.093 0.141 0.149 0.179 0.163 0.128 0.161 0.183 0.075 TEC 0.079 0.044 0.039 0.074 0.062 0.063 0.083 0.030 0.111 0.141 Panel C: US Secor Wide ENG BML IND CGS HCR CSV TEL UTL FIN TEC ENG 0.126 0.137 0.110 0.083 0.153 0.128 0.131 0.232 0.130 0.081 BML 0.110 0.158 0.123 0.118 0.152 0.123 0.132 0.221 0.164 0.093 IND 0.083 0.129 0.128 0.098 0.133 0.105 0.137 0.190 0.149 0.117 CGS 0.103 0.115 0.118 0.112 0.138 0.136 0.139 0.182 0.172 0.127 HCR 0.143 0.171 0.173 0.139 0.281 0.193 0.128 0.182 0.219 0.164 CSV 0.100 0.136 0.171 0.110 0.185 0.132 0.145 0.184 0.166 0.133 TEL 0.115 0.166 0.197 0.143 0.111 0.153 0.158 0.233 0.142 0.102 UTL 0.162 0.180 0.231 0.162 0.142 0.227 0.268 0.426 0.274 0.215 FIN 0.099 0.162 0.152 0.129 0.181 0.134 0.135 0.202 0.213 0.096 TEC 0.099 0.186 0.243 0.152 0.163 0.185 0.163 0.180 0.207 0.183 Panel D: Naional Equiy Marke Indices Japan UK US Japan 0.075 0.297 0.210 UK 0.162 0.202 0.206 US 0.068 0.144 0.156 Noe: The figures in he above able ij correspond o he correlaion beween he realized volailiy (RV) on day of he secor in row i wih he realized volailiy on day -1 of he secor in column j. 35
Table 4: In-sample MGARCH Esimaion The Table below shows he parameers esimaed from he MGARCH models. The sample period is from July/1/1996 o June/28/2002. The a is he parameer of he ARCH facor in he second sep of he DCC-ype MGARCH model; he b is he parameer of he GARCH facor in he second sep of he DCC-ype MGARCH model; he g is he parameer of he asymmery effec in he second sep of he DCC-ype MGARCH model esimaion. The srucure break is on he dae: Jan/1/1999. Panel A-1: Domesic Secor Porfolio of he Japanese Equiy Marke Wihou Srucure Break Series Diagonal Models a^2 b^2 g^2 ENG DCC 0.011 *** 0.986 *** ADCC 0.012 *** 0.984 *** 0.000 *** BML DCC 0.021 *** 0.972 *** ADCC 0.020 *** 0.971 *** 0.006 *** IND DCC 0.025 *** 0.957 *** ADCC 0.024 *** 0.956 *** 0.005 *** CGS DCC 0.017 *** 0.969 *** ADCC 0.020 *** 0.966 *** 0.000 *** HCR DCC 0.016 *** 0.982 *** ADCC 0.017 *** 0.982 *** 0.000 *** CSV DCC 0.019 *** 0.967 *** ADCC 0.020 *** 0.965 *** 0.001 *** TEL DCC 0.019 *** 0.954 *** ADCC 0.017 *** 0.954 *** 0.005 *** UTL DCC 0.005 *** 0.987 *** ADCC 0.006 *** 0.986 *** 0.000 *** FIN DCC 0.031 *** 0.951 *** ADCC 0.030 *** 0.947 *** 0.010 *** TEC DCC 0.025 *** 0.954 *** ADCC 0.023 *** 0.954 *** 0.004 *** Scalar Models a b g DCC 0.016 *** 0.976 *** ADCC 0.016 *** 0.976 *** 0.000 *** The "*"s indicae he parameer significance level: *** -- 99%, ** -- 95%, * -- 90%. The "-" means he parameer is no significan a 90% confidence level. 36
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel A-2: Domesic Secor Porfolio of he Japanese Equiy Marke (CONT D) Wih Srucure Break Period 1 Period 2 Series Diagonal Models a^2 b^2 g^2 a^2 b^2 g^2 ENG DCC 0.014 *** 0.829 *** 0.009 *** 0.981 *** ADCC 0.013 *** 0.818 *** 0.009 *** 0.009 *** 0.988 *** 0.000 *** BML DCC 0.057 *** 0.739 *** 0.033 *** 0.903 *** ADCC 0.041 *** 0.734 *** 0.050 *** 0.031 *** 0.910 *** 0.003 *** IND DCC 0.027 *** 0.760 *** 0.035 *** 0.907 *** ADCC 0.021 *** 0.675 *** 0.025 *** 0.034 *** 0.909 *** 0.006 *** CGS DCC 0.015 *** 0.797 *** 0.017 *** 0.924 *** ADCC 0.008 *** 0.698 *** 0.037 *** 0.020 *** 0.914 *** 0.000 *** HCR DCC 0.010 *** 0.775 *** 0.015 *** 0.973 *** ADCC 0.007 *** 0.722 *** 0.012 *** 0.015 *** 0.965 *** 0.020 *** CSV DCC 0.030 *** 0.697 *** 0.025 *** 0.921 *** ADCC 0.028 *** 0.721 *** 0.011 *** 0.028 *** 0.914 *** 0.000 *** TEL DCC 0.012 *** 0.857 *** 0.042 *** 0.868 *** ADCC 0.012 *** 0.849 *** 0.007 *** 0.040 *** 0.879 *** 0.004 *** UTL DCC 0.016 *** 0.984 *** 0.006 *** 0.973 *** ADCC 0.023 *** 0.977 *** 0.000 *** 0.014 *** 0.753-0.029 - FIN DCC 0.083 *** 0.779 *** 0.074 *** 0.797 *** ADCC 0.064 *** 0.767 *** 0.050 *** 0.064 *** 0.818 *** 0.006 ** TEC DCC 0.048 *** 0.721 *** 0.042 *** 0.891 *** ADCC 0.037 *** 0.439 *** 0.137 *** 0.041 *** 0.891 *** 0.006 *** Scalar Models a b g a b g DCC 0.017 *** 0.924 *** 0.022 *** 0.932 *** ADCC 0.017 *** 0.924 *** 0.000-0.022 *** 0.932 *** 0.000 - The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. Panel A-3: Summary Table for Domesic Secor Porfolio of he Japanese Equiy Marke Scalar Models Diagonal Models LLF Num. of Para AIC SIC LLF Num. of Para AIC SIC CCC 50134*** 75 128.36 530.03 DCC 50657*** 50 78.33 346.12 DCC 50614*** 32 42.34 213.72 ADCC 50663*** 60 98.33 419.67 ADCC 50616*** 33 44.34 221.07 DCC-break 50911*** 70 118.32 493.22 DCC-break 50847*** 34 46.33 228.42 ADCC-break 50930 90 158.32 640.33 ADCC-break 50847*** 36 50.33 243.13 Noe: The LLF is he log-likelihood value; he AIC is he Akaike Informaion Crierion, AIC = 2 x K 2 x ln(llf), where K is he number of parameer; he SIC is he Schwarz Informaion Crierion, SIC = K x ln(llf) - 2 x ln(llf). The "*"s indicae he significance level of he LR es, wih he highes LLF esimaor as unresriced conrol model: *** -- 99%, ** -- 95%, * -- 90%. BEKK models are no included in he LR es as hey are no nesed wih he condiional correlaion specificaions. The es saisics is 2 x [LLF unresriciced LLF resriced], he criical value is drawn from he chi-square disribuion wih degree of freedom equals o he number of resriced parameers. 37
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel B-1: Domesic Secor Porfolio of he UK Equiy Marke Wihou Srucure Break Series Diagonal Models a^2 b^2 g^2 ENG DCC 0.044 *** 0.000 *** ADCC 0.004 *** 0.985 *** 0.011 *** BML DCC 0.008 *** 0.980 *** ADCC 0.009 *** 0.983 *** 0.001 *** IND DCC 0.008 *** 0.990 *** ADCC 0.010 *** 0.986 *** 0.001 *** CGS DCC 0.005 *** 0.976 *** ADCC 0.006 *** 0.970 *** 0.002 *** HCR DCC 0.023 *** 0.960 *** ADCC 0.027 *** 0.957 *** 0.001 *** CSV DCC 0.015 *** 0.974 *** ADCC 0.016 *** 0.972 *** 0.004 *** TEL DCC 0.026 *** 0.956 *** ADCC 0.024 *** 0.962 *** 0.000 *** UTL DCC 0.003 *** 0.995 *** ADCC 0.003 *** 0.996 *** 0.000 *** FIN DCC 0.023 *** 0.952 *** ADCC 0.025 *** 0.951 *** 0.003 *** TEC DCC 0.012 *** 0.978 *** ADCC 0.012 *** 0.978 *** 0.001 *** Scalar Models a b g DCC 0.012 *** 0.977 *** ADCC 0.011 *** 0.976 *** 0.002 *** The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. 38
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel B-2: Domesic Secor Porfolio of he UK Equiy Marke (CONT D) Wih Srucure Break Period 1 Period 2 Series Diagonal Models a^2 b^2 g^2 a^2 b^2 g^2 ENG DCC 0.002 *** 0.000 *** 0.044 *** 0.000 *** ADCC 0.000 *** 0.851 *** 0.048 *** 0.047 *** 0.000 *** 0.363 *** BML DCC 0.018 *** 0.501 *** 0.006 *** 0.981 *** ADCC 0.005 *** 0.901 *** 0.032 *** 0.005 *** 0.981 *** 0.004 *** IND DCC 0.023 *** 0.438 *** 0.010 *** 0.971 *** ADCC 0.006 *** 0.915 *** 0.058 *** 0.010 *** 0.968 *** 0.000 *** CGS DCC 0.041 *** 0.400 *** 0.004 *** 0.963 *** ADCC 0.016 *** 0.816 *** 0.038 *** 0.009 *** 0.883 *** 0.000 *** HCR DCC 0.023 *** 0.958 *** 0.009 *** 0.991 *** ADCC 0.015 *** 0.890 *** 0.041 *** 0.010 *** 0.990 *** 0.000 *** CSV DCC 0.065 *** 0.101-0.024 *** 0.952 *** ADCC 0.005 *** 0.725 *** 0.144 *** 0.025 *** 0.943 *** 0.012 *** TEL DCC 0.051 *** 0.766 *** 0.034 *** 0.942 *** ADCC 0.080 *** 0.875 *** 0.020 *** 0.044 *** 0.932 *** 0.000 *** UTL DCC 0.010 *** 0.000 *** 0.002 *** 0.960 *** ADCC 0.002 *** 0.000 *** 0.000 *** 0.002 *** 0.957 *** 0.000 *** FIN DCC 0.110 *** 0.072-0.021 *** 0.948 *** ADCC 0.020 *** 0.919 *** 0.034 *** 0.020 *** 0.946 *** 0.005 *** TEC DCC 0.006 *** 0.875 *** 0.014 *** 0.969 *** ADCC 0.003 *** 0.996 *** 0.001 *** 0.015 *** 0.959 *** 0.002 *** Scalar Models a b g a b g DCC 0.017 *** 0.911 *** 0.011 *** 0.972 *** ADCC 0.009 *** 0.922 *** 0.015 *** 0.011 *** 0.972 *** 0.000 - The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. Panel B-3: Summary Table for Domesic Secor Porfolio of he UK Equiy Marke Scalar Models Diagonal Models LLF Num. of Para AIC SIC LLF Num. of Para AIC SIC CCC 47311*** 75 128.47 530.144 DCC 47562*** 50 78.46 346.24 DCC 47547*** 32 42.46 213.84 ADCC 47595*** 60 98.46 419.80 ADCC 47574*** 33 44.46 221.20 DCC-break 47736*** 70 118.45 493.35 DCC-break 47704*** 34 46.45 228.55 ADCC-break 47766 90 158.45 640.46 ADCC-break 47708*** 36 50.45 243.26 Noe: The LLF is he log-likelihood value; he AIC is he Akaike Informaion Crierion, AIC = 2 x K 2 x ln(llf), where K is he number of parameer; he SIC is he Schwarz Informaion Crierion, SIC = K x ln(llf) - 2 x ln(llf). The "*"s indicae he significance level of he LR es, wih he highes LLF esimaor as unresriced conrol model: *** -- 99%, ** -- 95%, * -- 90%. BEKK models are no included in he LR es as hey are no nesed wih he condiional correlaion specificaions. The es saisics is 2 x [LLF unresriciced LLF resriced], he criical value is drawn from he chi-square disribuion wih degree of freedom equals o he number of resriced parameers. 39
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel C-1: Domesic Secor Porfolio of he US Equiy Marke Wihou Srucure Break Series Diagonal Models a^2 b^2 g^2 ENG BEKK 0.025 *** 0.980 *** DCC 0.006 *** 0.982 *** ADCC 0.006 *** 0.983 *** 0.000 *** BML BEKK 0.019 *** 0.980 *** DCC 0.026 *** 0.956 *** ADCC 0.021 *** 0.956 *** 0.018 *** IND BEKK 0.029 *** 0.969 *** DCC 0.021 *** 0.970 *** ADCC 0.019 *** 0.964 *** 0.013 *** CGS BEKK 0.021 *** 0.974 *** DCC 0.022 *** 0.960 *** ADCC 0.017 *** 0.958 *** 0.015 *** HCR BEKK 0.031 *** 0.960 *** DCC 0.013 *** 0.987 *** ADCC 0.013 *** 0.984 *** 0.004 *** CSV BEKK 0.029 *** 0.968 *** DCC 0.015 *** 0.979 *** ADCC 0.012 *** 0.980 *** 0.008 *** TEL BEKK 0.025 *** 0.966 *** DCC 0.007 *** 0.988 *** ADCC 0.007 *** 0.990 *** 0.000 *** UTL BEKK 0.036-0.960 *** DCC 0.007 *** 0.988 *** ADCC 0.009 *** 0.985 *** 0.000 *** FIN BEKK 0.031 *** 0.965 *** DCC 0.018 *** 0.971 *** ADCC 0.016 *** 0.969 *** 0.006 *** TEC BEKK 0.027 *** 0.966 *** DCC 0.016 *** 0.974 *** ADCC 0.015 *** 0.974 *** 0.004 *** Scalar Models a b g BEKK 0.146 *** 0.989 *** DCC 0.014 *** 0.979 *** ` ADCC 0.014 *** 0.979 *** 0.000 - The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. 40
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel C-2: Domesic Secor Porfolio of he US Equiy Marke (CONT D) Wih Srucure Break Period 1 Period 1 Series Diagonal Models a^2 b^2 g^2 a^2 b^2 g^2 ENG DCC 0.009 *** 0.839 *** 0.042 *** 0.479 *** ADCC 0.009 *** 0.990 *** 0.001 *** 0.048 *** 0.480 *** 0.000 *** BML DCC 0.017 *** 0.838 *** 0.025 *** 0.949 *** ADCC 0.003 *** 0.982 *** 0.015 *** 0.018 *** 0.955 *** 0.019 *** IND DCC 0.034 *** 0.826 *** 0.027 *** 0.941 *** ADCC 0.006 *** 0.951 *** 0.021 *** 0.024 *** 0.935 *** 0.012 *** CGS DCC 0.027 *** 0.758 *** 0.020 *** 0.956 *** ADCC 0.004 *** 0.945 *** 0.027 *** 0.015 *** 0.952 *** 0.018 *** HCR DCC 0.025 *** 0.851 *** 0.009 *** 0.991 *** ADCC 0.009 *** 0.943 *** 0.014 *** 0.010 *** 0.990 *** 0.000 *** CSV DCC 0.020 *** 0.838 *** 0.018 *** 0.969 *** ADCC 0.003 *** 0.955 *** 0.017 *** 0.014 *** 0.968 *** 0.017 *** TEL DCC 0.015 *** 0.931 *** 0.006 *** 0.964 *** ADCC 0.019 *** 0.859 *** 0.022 *** 0.013 *** 0.930 *** 0.000 *** UTL DCC 0.023 *** 0.977 *** 0.023 *** 0.000 *** ADCC 0.085 *** 0.708 *** 0.000 *** 0.035 *** 0.011 *** 0.000 *** FIN DCC 0.014 *** 0.899 *** 0.021 *** 0.953 *** ADCC 0.001 *** 0.931 *** 0.064 *** 0.021 *** 0.950 *** 0.003 *** TEC DCC 0.048 *** 0.800 *** 0.014 *** 0.967 *** ADCC 0.007 *** 0.915 *** 0.059 *** 0.014 *** 0.964 *** 0.006 *** Scalar Models a b g a b g DCC 0.018 *** 0.916 *** 0.015 *** 0.963 *** ADCC 0.008 *** 0.939 *** 0.013 *** 0.015 *** 0.963 *** 0.000 - The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. Panel C-3: Summary Table for Domesic Secor Porfolio of he US Equiy Marke Scalar Models Diagonal Models LLF Num. of Para AIC SIC LLF Num. of Para AIC SIC BEKK 49733 47 72.37 324.09 BEKK 49747 65 108.37 456.49 CCC 49472*** 75 128.38 530.06 DCC 49907*** 50 78.36 346.15 DCC 49867*** 32 42.37 213.75 ADCC 49917*** 60 98.36 419.70 ADCC 49868*** 33 44.37 221.10 DCC-break 50084*** 70 118.36 493.25 DCC-break 50050*** 34 46.36 228.45 ADCC-break 50131 90 158.36 640.36 ADCC-break 50055*** 36 50.36 243.16 Noe: The LLF is he log-likelihood value; he AIC is he Akaike Informaion Crierion, AIC = 2 x K 2 x ln(llf), where K is he number of parameer; he SIC is he Schwarz Informaion Crierion, SIC = K x ln(llf) - 2 x ln(llf). The "*"s indicae he significance level of he LR es, wih he highes LLF esimaor as unresriced conrol model: *** -- 99%, ** -- 95%, * -- 90%. BEKK models are no included in he LR es as hey are no nesed wih he condiional correlaion specificaions. The es saisics is 2 x [LLF unresriciced LLF resriced], he criical value is drawn from he chi-square disribuion wih degree of freedom equals o he number of resriced parameers 41
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel D-1: Naional Marke Index Porfolio Wihou Srucure Break Series Diagonal Models a^2 b^2 g^2 Japan DCC 0.000 *** 0.000 *** ADCC 0.000 *** 0.000-0.000 *** UK DCC 0.000 *** 0.000 - ADCC 0.000 *** 0.000-0.000 *** US DCC 0.042 *** 0.000 *** ADCC 0.122 *** 0.000 *** 0.000 *** Scalar Models A B G DCC 0.001 *** 0.000 - ADCC 0.001 *** 0.000-0.000 - The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. Panel D-2: Naional Marke Index Porfolio (CONT D) Wih Srucure Break Period 1 Period 1 Series Diagonal Models a^2 b^2 g^2 a^2 b^2 g^2 Japan DCC 0.000 *** 0.000-0.000 *** 0.000 - ADCC 0.000 *** 0.000-0.000 *** 0.000 *** 0.000 *** 0.000 *** UK DCC 0.000 *** 0.000-0.000 *** 0.000 - ADCC 0.000 *** 0.000-0.000 *** 0.115 *** 0.000 *** 0.000 *** US DCC 0.063 *** 0.000 *** 0.144 *** 0.000 *** ADCC 0.080 *** 0.000 *** 0.000 *** 0.000 *** 0.000 *** 0.000 *** Scalar Models a b g a b g DCC 0.000-0.000-0.016 *** 0.000 - ADCC 0.000-0.000-0.000-0.016 *** 0.000-0.000 - The "*"s indicaes he parameer's significance level: *** -- 99%, ** -- 95%, * -- 90%. 42
Table 4: In-sample MGARCH-Type Model Esimaion Resuls (CONT D) Panel D-3: Summary Table for he Naional Marke Index Porfolio Scalar Models Diagonal Models LLF Num. of Para AIC SBC LLF Num. of Para AIC SBC CCC 14709** 12 4.81 69.08 DCC 14715* 15 10.81 91.14 DCC 14709*** 11 2.81 61.72 ADCC 14720 18 16.81 113.21 ADCC 14709*** 12 4.81 69.08 DCC-break 14721 21 22.81 135.27 DCC-break 14711** 13 6.81 76.43 ADCC-break 14716*** 27 34.81 179.41 ADCC-break 14711*** 15 10.81 91.14 Noe: The LLF is he log-likelihood value; he AIC is he Akaike Informaion Crierion, AIC = 2 x K 2 x ln(llf), where K is he number of parameer; he SIC is he Schwarz Informaion Crierion, SIC = K x ln(llf) - 2 x ln(llf). The "*"s indicae he significance level of he LR es, wih he highes LLF esimaor as unresriced conrol model: *** -- 99%, ** -- 95%, * -- 90%. BEKK models are no included in he LR es as hey are no nesed wih he condiional correlaion specificaions. The es saisics is 2 x [LLF unresriciced LLF resriced], he criical value is drawn from he chi-square disribuion wih degree of freedom equals o he number of resriced parameers 43
Table 5: In-sample Porfolio Performance Comparison The Table below shows he performances of he porfolios based on esimaed variance-covariance marices of he differen MGARCH models. All he porfolios are rebalanced on a monhly basis over he period from July/1/1996 o June/28/2002. Panel A: Model Selecion under he max-reurn sraegy Layer JP Equiy Marke UK Equiy Marke US Equiy Marke Naional Marke Index MGARCH Models μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) BEKK - - - - - - - - 2.5126 14.2350% 5.7882% 0.0313% - - - - diagonal BEKK - - - - - - - - 2.5358 14.1780% 5.7190% 0.1365% - - - - scalar CCC 2.4289 16.7490% 7.2835% -0.7119% 2.5770 13.7740% 5.4748% 0.3104% 2.4238 14.5180% 6.2416% -0.6644% 1.4695 7.1788% 5.0042% -5.1236% scalar DCC 2.5381 16.2870% 6.5228% 0.2421% 2.6763 13.5490% 5.1490% 0.8418% 2.5148 14.3200% 5.7890% 0.0424% 1.4712 6.9242% 4.8533% -4.9949% scalar ADCC 2.5383 16.2900% 6.5236% 0.2428% 2.6761 13.5930% 5.1654% 0.8448% 2.5149 14.3220% 5.7895% 0.0427% 1.4712 6.9242% 4.8533% -4.9949% diagonal DCC 2.5480 16.2260% 6.4724% 0.2974% 2.6572 13.5370% 5.1881% 0.7310% 2.5251 14.3440% 5.7829% 0.0891% 1.4711 6.9249% 4.8542% -4.9960% diagonal ADCC 2.5512 16.2220% 6.4599% 0.3229% 2.6737 13.5350% 5.1460% 0.8236% 2.5262 14.4140% 5.8084% 0.0955% 1.4709 6.9249% 4.8547% -4.9979% scalar DCC-break 2.5490 16.3130% 6.4999% 0.3127% 2.6867 13.5590% 5.1290% 0.8928% 2.5369 14.3090% 5.7383% 0.1455% 1.4712 6.9377% 4.8549% -4.9890% scalar ADCC-break 2.5490 16.3130% 6.4999% 0.3127% 2.6855 13.6310% 5.1562% 0.9002% 2.5343 14.4020% 5.7776% 0.1407% 1.4712 6.9377% 4.8549% -4.9890% diagonal DCC-break 2.5575 16.3350% 6.4938% 0.3539% 2.6608 13.6540% 5.2267% 0.7552% 2.5265 14.3820% 5.8139% 0.0632% 1.4704 6.9368% 4.8578% -4.9966% diagonal ADCC-break 2.5555 16.3620% 6.5094% 0.3389% 2.6784 13.5980% 5.1662% 0.8401% 2.5271 14.4880% 5.8361% 0.0905% 1.4705 6.9389% 4.8585% -4.9961% Noe: The Sharpe Raio (SR), Uiliy (U) and average monhly porfolio reurn (Re) and volailiy (Sd) are calculaed from he porfolio consruced based on he maximum porfolio expeced reurn wih arge variance sraegy. The (SR) denoes he value of average monhly porfolio Sharpe-Raio over he in-sample period; he (Re) is he value of average monhly porfolio reurn; he (Sd) is he value of average monhly porfolio sandard deviaion over he in-sample period; he (U) is he value of average monhly invesor uiliy. - indicaes he paricular MGARCH model is no suiable for he mulivariae ime-series daa from he given porfolio. Bold indicaes he bes model under each model selecion crieria. 44
Table 5: In-sample Porfolio Performance Comparison (CONT D) The Table below shows he performances of he porfolios based on esimaed variance-covariance marices of he differen MGARCH models. All he porfolios are rebalanced on a monhly basis over he period from July/1/1996 o June/28/2002. Panel B: Model Selecion under he min-variance Invesmen Sraegy Layer JP Equiy Marke UK Equiy Marke US Equiy Marke Naional Marke Index MGARCH Models μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) BEKK - - - - - - - - 4.5644 0.6500% 0.1488% 0.2837% - - - - diagonal BEKK - - - - - - - - 4.6065 0.6500% 0.1474% 0.2872% - - - - scalar CCC 3.8789 0.0800% 0.0222% 0.0259% 29.3080 0.4700% 0.0170% 0.4285% 4.4049 0.6500% 0.1554% 0.2685% 3.5868 0.5015% 0.2177% 0.0552% scalar DCC 4.0533 0.0800% 0.0210% 0.0287% 30.4250 0.4700% 0.0163% 0.4301% 4.5701 0.6500% 0.1485% 0.2844% 3.5854 0.5015% 0.2176% 0.0553% scalar ADCC 4.0535 0.0800% 0.0210% 0.0287% 30.4260 0.4700% 0.0163% 0.4301% 4.5702 0.6500% 0.1485% 0.2844% 3.5854 0.5015% 0.2176% 0.0553% diagonal DCC 4.0690 0.0800% 0.0209% 0.0288% 30.2080 0.4700% 0.0164% 0.4298% 4.5889 0.6500% 0.1481% 0.2856% 3.5853 0.5015% 0.2176% 0.0553% diagonal ADCC 4.0741 0.0800% 0.0209% 0.0289% 30.3970 0.4700% 0.0163% 0.4301% 4.5913 0.6500% 0.1480% 0.2858% 3.5850 0.5015% 0.2176% 0.0553% scalar DCC-break 4.0706 0.0800% 0.0209% 0.0289% 30.5440 0.4700% 0.0162% 0.4302% 4.6104 0.6500% 0.1474% 0.2872% 3.5856 0.5015% 0.2170% 0.0558% scalar ADCC-break 4.0706 0.0800% 0.0209% 0.0289% 30.5350 0.4700% 0.0162% 0.4302% 4.6064 0.6500% 0.1475% 0.2869% 3.5856 0.5015% 0.2170% 0.0558% diagonal DCC-break 4.0842 0.0800% 0.0209% 0.0290% 30.2580 0.4700% 0.0164% 0.4298% 4.5919 0.6500% 0.1482% 0.2855% 3.5837 0.5015% 0.2172% 0.0554% diagonal ADCC-break 4.0811 0.0800% 0.0209% 0.0290% 30.4550 0.4700% 0.0163% 0.4300% 4.5938 0.6500% 0.1481% 0.2857% 3.5841 0.5015% 0.2173% 0.0554% See noe in Panel A. 45
Table 5: In-sample Porfolio Performance Comparison & Model Selecion (CONT D) The Table below shows he performances of he porfolios based on esimaed variance-covariance marices of he differen MGARCH models. All he porfolios are rebalanced on a monhly basis over he period from July/1/1996 o June/28/2002. Panel C: Model Selecion under he max-u Invesmen Sraegy Layer JP Equiy Marke UK Equiy Marke US Equiy Marke Naional Marke Index MGARCH Models μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(re) μ(sd) μ(u) BEKK - - - - - - - - 1.5010 7.6070% 5.3187% -4.7841% - - - - diagonal BEKK - - - - - - - - 1.5105 7.5908% 5.2342% -4.6631% - - - - scalar CCC 1.3220 7.5218% 5.5984% -5.3642% 1.7843 8.8502% 5.4563% -3.9739% 1.5115 7.8438% 5.4783% -4.9653% 0.5400 1.6100% 4.0304% -7.9725% scalar DCC 1.3575 7.3705% 5.3649% -4.9191% 1.7764 8.8319% 5.3957% -3.8102% 1.5558 7.8362% 5.2658% -4.5045% 0.5229 1.4907% 3.9877% -7.9704% scalar ADCC 1.3822 7.7709% 5.5679% -4.8768% 1.7882 8.5718% 5.2333% -3.8402% 1.5651 7.7897% 5.2048% -4.4964% 0.5264 1.5273% 4.0104% -7.9841% diagonal DCC 1.3788 7.7347% 5.5055% -4.9022% 1.7724 8.7067% 5.4295% -4.0674% 1.5546 7.8797% 5.2851% -4.5116% 0.5237 1.5017% 3.9935% -7.9738% diagonal ADCC 1.3883 7.6250% 5.3591% -4.8140% 1.8358 8.8349% 5.2510% -3.5574% 1.5613 7.6984% 5.1749% -4.5057% 0.5258 1.5196% 4.0004% -7.9725% scalar DCC-break 1.3777 7.6652% 5.4538% -4.8646% 1.8178 8.8437% 5.3386% -3.7103% 1.5528 7.7385% 5.1955% -4.4660% 0.5272 1.5409% 4.0065% -7.9736% scalar ADCC-break 1.3698 7.6119% 5.3927% -4.8829% 1.7952 8.5191% 5.2361% -3.7960% 1.5444 7.6959% 5.2075% -4.5146% 0.5339 1.5503% 4.0089% -7.9686% diagonal DCC-break 1.3840 7.7314% 5.5069% -4.8248% 1.8034 8.7065% 5.2941% -3.7638% 1.5538 7.7902% 5.2719% -4.5797% 0.5272 1.5353% 4.0095% -7.9816% diagonal ADCC-break 1.3425 7.4344% 5.4354% -5.0925% 1.8106 8.7715% 5.2339% -3.6434% 1.5612 7.9446% 5.3416% -4.5555% 0.5284 1.5286% 4.0017% -7.9693% See noe in Panel A. 46
Table 5: In-sample Porfolio Performance Comparison & Model Selecion (CONT D) The Table below shows he seleced models for each sub-porfolio under he differen model selecion crieria. The porfolios are derived based on esimaed variance-covariance marices of he differen MGARCH models. Panel D: Bes Performing Model across Selecion Crieria Invesmen Sraegy max-reurn min-variance max-u Layer μ(sr) μ(re) μ(sd) μ(u) μ(sr) μ(sd) μ(sr) μ(re) μ(sd) μ(u) JP Equiy Marke diagonal DCC-break CCC diagonal ADCC diagonal DCC-break diagonal DCC-break diagonal DCC-break diagonal ADCC scalar ADCC diagonal ADCC diagonal ADCC UK Equiy Marke scalar DCC-break CCC scalar DCC-break scalar ADCC-break scalar DCC-break scalar DCC-break diagonal ADCC CCC scalar ADCC diagonal ADCC US Equiy Marke scalar DCC-break CCC diagonal BEKK scalar DCC-break scalar DCC-break scalar DCC-break scalar ADCC diagonal ADCC-break diagonal ADCC scalar DCC-break Naional Marke Index scalar DCC CCC scalar DCC scalar DCC-break CCC scalar DCC-break CCC CCC scalar DCC scalar ADCC-break 47
Table 6: Porfolio Performance Based on he True Variance-Covariance Informaion. The Table below shows he performances of he porfolios based on rue variance-covariance marices esimaed using daa over he ou-of-sample period. All he porfolios are rebalanced on a monhly basis over he period from July/1/2002 o May/31/2007. Panel A: Max-reurn Porfolio Consrucion μ(re) μ(sd) μ(sr) μ(sr)/σ(sr) μ(u) μ(u) /σ(u) σ(re) Min(Re) Max(Re) JP Domesic Secor (DS-JP) 17.5708% 5.7446% 3.0587 6.8890 0.0321 1.2583 2.5506% 11.7940% 22.3080% JP Domesic Marke Index 4.0936% 5.7446% 0.7126 1.3726-0.1027-3.4427 2.9825% 0.2716% 12.7330% UK Domesic Secor (DS-UK) 13.8759% 4.5826% 3.0280 6.2912 0.0242 1.0970 2.2056% 8.5188% 18.8290% UK Domesic Marke Index 3.6629% 4.5826% 0.7993 1.7954-0.0779-3.8200 2.0402% 0.7096% 8.3889% US Domesic Secor (DS-US) 15.2814% 5.0990% 2.9969 6.1688 0.0253 1.0229 2.4772% 9.2348% 20.8350% US Domesic Marke Index 3.6088% 5.0990% 0.7078 1.5739-0.0914-3.9855 2.2930% 0.3152% 8.3640% Naional Marke Index (NM) 6.5953% 5.0299% 1.3112 2.7294-0.0598-2.4745 2.4164% 2.0893% 11.7900% Global Secor (GS) 10.2860% 8.3874% 0.8878 0.2936-0.0915-0.4404 22.8437% -38.3462% 55.7106% Panel B: Min-variance Porfolio Consrucion JP Domesic Secor (DS-JP) 0.0800% 0.0169% 4.8855 6.8773 0.0004 5.8357 0.0000% 0.0800% 0.0800% JP Domesic Marke Index 0.0800% 0.2765% 1.1318 1.3625-0.0034-0.6140 0.0000% 0.0800% 0.0800% UK Domesic Secor (DS-UK) 0.4700% 0.0142% 34.4761 6.0963 0.0043 69.8262 0.0000% 0.4700% 0.4700% UK Domesic Marke Index 0.4700% 0.1728% 8.2895 1.5846 0.0019 0.5671 0.0000% 0.4700% 0.4700% US Domesic Secor (DS-US) 0.6500% 0.1246% 5.4564 6.0477 0.0034 6.2277 0.0000% 0.6500% 0.6500% US Domesic Marke Index 0.6500% 17.7022% 1.2051 1.4430-0.1121-0.2607 0.0000% 0.6500% 0.6500% Naional Marke Index (NM) 0.5015% 0.2175% 3.1917 2.6032 0.0002 0.0879 0.0000% 0.5015% 0.5015% Global Secor (GS) 0.3106% 0.0051% 167.9009 1.0548 0.0030 19.6972 0.0152% 0.2772% 0.3763% Panel C: Max-U Porfolio Consrucion JP Domesic Secor (DS-JP) 5.0185% 3.5328% 1.4790 2.0003-0.0340-1.1796 3.1534% 0.1332% 14.3930% JP Domesic Marke Index 0.9360% 5.1887% 0.2837 0.3385-0.1142-1.7803 3.9874% -8.3845% 10.9090% UK Domesic Secor (DS-UK) 5.2298% 4.0055% 1.6662 1.6411-0.0374-0.6371 4.0174% -9.0268% 16.8760% UK Domesic Marke Index 0.7091% 4.5645% 0.4123 0.5017-0.0894-1.0553 3.6733% -12.7380% 8.5948% US Domesic Secor (DS-US) 5.0233% 3.8304% 1.6483 1.9026-0.0339-0.6324 3.7549% -4.8657% 19.2530% US Domesic Marke Index 0.7734% 4.4295% 0.3317 0.4314-0.0893-1.2664 3.3499% -11.8230% 8.3355% Naional Marke Index (NM) 1.7339% 3.1868% 0.7420 0.7782-0.0553-1.1921 2.9554% -7.0061% 7.5324% Global Secor (GS) 5.6847% 2.7060% 2.42306 2.3329-0.0064-0.2040 2.7828% -2.1448% 12.0028% Noe: The performance saisics in he able are calculaed from he porfolio consruced based on he maximize porfolio reurn subjec o porfolio variance (max-reurn), minimize porfolio variance subjec o arge reurn (min-variance), and maximize invesor uiliy (max-u) sraegies. The domesic marke index porfolio under each panel is consruced follow he same invesmen sraegy as he DS porfolios in he same panel. Insead of using various secor level indices, he domesic marke index porfolio only uses he corresponding marke index o consruc he porfolio. The (SR) means he value of average monhly porfolio Sharpe-raio over he in-sample period; he σ(sr) is he sandard deviaion of he monhly porfolio Sharpe-raio over he in-sample period; he (SR) / σ(sr) is he raio beween average monhly porfolio Sharpe-Raio and he sandard deviaion of he monhly Sharpe-Raio over he in-sample period; he (Re) means he value of average monhly porfolio reurn over he in-sample period; he (Sd) is he average monhly porfolio sandard deviaion over he in-sample period, he (U) means he value of average monhly uiliy funcion over he in-sample period; he σ(u) is he sandard deviaion of he monhly invesor uiliy over he in-sample period; he (U) / σ(u) is he raio beween average monhly invesor uiliy and he sandard deviaion of he monhly invesor uiliy over he in-sample period; he σ(re) is he volailiy of he monhly porfolio reurns; and, he Min(Re) and Max(Re) is he minimum and maximum monhly reurn figure during he sample period, respecively. 48
Table 7: Ou-of-Sample Porfolio Performance Comparison The Table below shows he performances of he porfolios based on esimaed variance-covariance marices of he seleced MGARCH models from he in-sample model selecion period. All he porfolios are rebalanced on a monhly basis over he period from July/1/2002 o May/31/2007. Panel A: Max-Reurn Sraegy using Models Seleced by Sharpe-Raio Crierion Model for Domesic Secor Porfolio: Japan diagonal DCC-break UK scalar DCC-break US scalar DCC-break Model for Naional Indices Porfolio: DCC Performance Measure Expeced Reurn = R -1 12 1 R i 12 i1 Forecas Mehod Porfolio μ(sr) μ(re) μ(sd) μ(sr) μ(re) μ(sd) Simple MA GS -0.07086-0.42874% 8.69739% 0.05557 0.50169% 8.82681% NM 0.19348 1.12520% 5.39475% 0.28821 1.31818% 5.50831% DS-JP 0.06524 0.56128% 6.89596% 0.23409 1.35879% 7.06520% DS-UK 0.08104 0.70164% 5.37628% 0.18589 0.90367% 5.27777% DS-US -0.11880-0.17996% 6.23481% -0.02008-0.32917% 5.89718% Simple EWMA GS -0.07229-0.44254% 8.65616% 0.05761 0.51275% 8.80290% NM 0.19382 1.12448% 5.38583% 0.28864 1.31593% 5.50125% DS-JP 0.06587 0.56185% 6.87456% 0.23414 1.35993% 7.04531% DS-UK 0.08009 0.69165% 5.35530% 0.18710 0.90611% 5.27001% DS-US -0.11906-0.18359% 6.21374% -0.01802-0.31996% 5.88543% RW GS -0.22424-2.85564% 13.73351% 0.32762 6.58554% 21.28043% NM 0.21216 1.28513% 5.94191% 0.27841 1.36949% 6.10942% DS-JP 0.19516 2.53840% 9.69024% 0.01005 0.61659% 12.26602% DS-UK 0.09117 0.96098% 7.79406% 0.28742 2.59851% 11.90933% DS-US -0.14891-0.62918% 8.27653% 0.08374-0.92903% 11.35809% MGARCH MG-EWMA GS -0.03945-0.16814% 7.04461% 0.07618 0.60364% 6.75207% NM 0.19340 1.06037% 5.02327% 0.30369 1.33308% 5.11859% DS-JP 0.05750 0.48371% 6.26387% 0.19644 0.96958% 6.39799% DS-UK 0.12526 0.71649% 4.84600% 0.19812 0.79602% 4.60046% DS-US -0.06156 0.11961% 5.43018% 0.01688-0.05841% 5.12203% MG-Simulaion df=0 GS -0.13108-2.43535% 13.42599% 0.26163 3.85532% 14.76338% NM 0.21223 1.23442% 5.61232% 0.30415 1.40599% 5.63635% DS-JP 0.10819 1.01223% 6.79582% 0.17142 1.00524% 6.79669% DS-UK 0.05190 0.52882% 8.53044% 0.27272 2.51772% 9.34049% DS-US -0.13364-0.47783% 9.73951% 0.09046-0.27369% 10.51403% df=0.05 GS -0.11542-2.12316% 14.33986% 0.26629 4.06978% 15.82231% NM 0.20981 1.26993% 5.82070% 0.30182 1.44778% 5.84112% DS-JP 0.10446 1.02829% 7.04701% 0.17198 1.05237% 7.05079% DS-UK 0.04720 0.53042% 8.86361% 0.25078 2.41903% 9.63433% DS-US -0.13279-0.44979% 9.82401% 0.09676-0.26566% 10.85535% Noe: The performance saisics in he able are calculaed from he porfolio consruced based on he maximize porfolio expeced reurn wih arge variance (max-reurn) porfolio consrucion sraegy. The (SR) means he value of average monhly porfolio Sharpe-raio over he ou-of-sample period; he (Re) means he value of average monhly porfolio reurn over he ou-of-sample period; he (Sd) is he average monhly porfolio sandard deviaion over he ou-of-sample period. Two independen simulaions wih he same degrees of freedom (d.f.) specificaion have been carried ou, he figure in he able is he average resul of he wo simulaions. 49
Table 7: Ou-of-Sample Porfolio Performance Comparison (CONT D) The Table below shows he performances of he porfolios based on esimaed variance-covariance marices of he seleced MGARCH models from he in-sample model selecion period. All he porfolios are rebalanced on a monhly basis over he period from July/1/2002 o May/31/2007. Panel B: Min-Variance Sraegy using Models Seleced by Variance Crierion Model for Domesic Secor Porfolio: Japan diagonal DCC-break UK scalar DCC-break US scalar DCC-break Model for Naional Indices Porfolio: scalar DCC-break Performance Measure 12 1 Expeced Reurn = R -1 R 12 Forecas Mehod Porfolio μ(re) μ(sd) μ(re) μ(sd) Simple MA GS 0.30592% 0.00609% 0.30692% 0.06640% NM 0.31557% 0.25061% 0.45021% 0.62418% DS-JP 0.03282% 0.02212% 0.04338% 0.06797% DS-UK 0.43099% 0.01757% 0.43535% 0.05726% DS-US 0.29322% 0.15789% 0.26727% 0.51574% i1 i Simple EWMA GS 0.30594% 0.00610% 0.30701% 0.06610% NM 0.31554% 0.25055% 0.45030% 0.62400% DS-JP 0.03284% 0.02213% 0.04336% 0.06787% DS-UK 0.43097% 0.01755% 0.43546% 0.05721% DS-US 0.29292% 0.15778% 0.26811% 0.51510% RW GS 0.30609% 0.00838% 0.31921% 0.09013% NM 0.31949% 0.26470% 0.43628% 0.66482% DS-JP 0.03796% 0.02894% 0.03177% 0.08369% DS-UK 0.43203% 0.02446% 0.44871% 0.07631% DS-US 0.27594% 0.20480% 0.26565% 0.66088% MGARCH MG-EWMA GS 0.30599% 0.00551% 0.30874% 0.06041% NM 0.31712% 0.24784% 0.45735% 0.62011% DS-JP 0.03272% 0.02200% 0.04119% 0.06677% DS-UK 0.43123% 0.01774% 0.43505% 0.05770% DS-US 0.29682% 0.15298% 0.28213% 0.50481% MG-Simulaion df=0 GS 0.30589% 0.00825% 0.58343% 0.08852% NM 0.32242% 0.26264% 0.45051% 0.65436% DS-JP 0.03433% 0.02276% 0.04057% 0.06705% DS-UK 0.43065% 0.02352% 0.44691% 0.07488% DS-US 0.27755% 0.18886% 0.28849% 0.65863% df=0.05 GS 0.30612% 0.00783% 0.58432% 0.08977% NM 0.32117% 0.26164% 0.44940% 0.65504% DS-JP 0.03426% 0.02276% 0.04054% 0.06708% DS-UK 0.43061% 0.02357% 0.44566% 0.07493% DS-US 0.27821% 0.18537% 0.29212% 0.65681% Noe: The performance saisics in he able are calculaed from he porfolio consruced based on he minimize porfolio expeced variance wih arge reurn (min-variance) porfolio consrucion sraegy. The (Re) means he value of average monhly porfolio reurn over he ou-of-sample period; he (Sd) is he average monhly porfolio sandard deviaion over he ou-of-sample period. Two independen simulaions wih he same degrees of freedom (d.f.) specificaion have been carried ou, he figure in he able is he average resul of he wo simulaions. 50
Table 7: Ou-of-Sample Porfolio Performance Comparison (CONT D) The Table below shows he performances of he porfolios based on esimaed variance-covariance marices of he seleced MGARCH models from he in-sample model selecion period. All he porfolios are rebalanced on a monhly basis over he period from July/1/2002 o May/31/2007. Panel C: Max-U Sraegy using Models Seleced by Uiliy Crierion Model for Domesic Secor Porfolio: Japan diagonal ADCC UK diagonal ADCC US scalar DCC-break Model for Naional Indices Porfolio: scalar ADCC-break Performance Measure 12 1 Expeced Reurn = R -1 R 12 Forecas Mehod Porfolio μ(u) μ(re) μ(sd) μ(u) μ(re) μ(sd) Simple MA GS -0.06792 0.92035% 3.38798% -0.05086 0.60760% 2.43832% NM -0.07177 0.87072% 3.58680% -0.06777 0.81524% 3.38185% DS-JP -0.09049 1.16561% 4.42593% -0.07663 0.86009% 3.59237% DS-UK -0.09900 0.57900% 4.68573% -0.08554 0.53547% 4.15653% DS-US -0.09460 0.32598% 4.43001% -0.07808 0.53063% 3.78322% i1 i Simple EWMA GS -0.06674 0.98648% 3.33943% -0.05325 0.36002% 2.43888% NM -0.07159 0.84881% 3.55615% -0.06797 0.79959% 3.38279% DS-JP -0.08835 1.24278% 4.33740% -0.07521 0.91023% 3.54123% DS-UK -0.10035 0.56377% 4.75757% -0.08645 0.48865% 4.17596% DS-US -0.09386 0.32159% 4.42842% -0.08202 0.14313% 3.80194% RW GS -0.06901 0.69257% 3.28385% -0.05772 0.58287% 2.77087% NM -0.07350 0.82335% 3.63029% -0.06960 0.75568% 3.44122% DS-JP -0.08506 1.26420% 4.10517% -0.07588 0.91175% 3.54935% DS-UK -0.09664 0.62947% 4.60936% -0.09079 0.56856% 4.38960% DS-US -0.09502 0.24755% 4.44654% -0.08192 0.43842% 3.99365% MGARCH MG-EWMA GS -0.06711 0.90000% 3.36502% -0.04965 0.67369% 2.41762% NM -0.07124 0.83005% 3.54098% -0.06728 0.79457% 3.32776% DS-JP -0.09212 1.13163% 4.49803% -0.07723 0.91756% 3.66023% DS-UK -0.09794 0.57931% 4.64720% -0.08380 0.60178% 4.10626% DS-US -0.09314 0.41850% 4.41177% -0.07936 0.50688% 3.89440% MG-Simulaion df=0 GS -0.07183 0.46881% 3.29508% -0.05917 0.29877% 2.65837% NM -0.07306 0.78297% 3.58784% -0.06881 0.74444% 3.37772% DS-JP -0.08752 1.32746% 4.26223% -0.07694 0.92202% 3.60956% DS-UK -0.09873 0.38320% 4.55455% -0.09142 0.38761% 4.31582% DS-US -0.09764 0.12137% 4.46089% -0.08495 0.15166% 3.94013% df=0.05 GS -0.07282 0.56329% 3.39918% -0.05650 0.56052% 2.65729% NM -0.07380 0.79670% 3.63529% -0.06902 0.72680% 3.38567% DS-JP -0.08802 1.37297% 4.31542% -0.07882 0.81108% 3.64382% DS-UK -0.10162 0.33021% 4.67853% -0.09163 0.39332% 4.33685% DS-US -0.09714 0.25236% 4.52038% -0.08211 0.46590% 3.96350% Noe: The performance saisics in he able are calculaed from he porfolio consruced based on he maximize invesor uiliy (max-u) porfolio consrucion sraegy wih 130/30 shor-sell consrain. The (U) means he value of average monhly porfolio uiliy over he ou-of-sample period; he (Re) is he average monhly porfolio reurn over he ou-of-sample period; he (Sd) is he average monhly porfolio sandard deviaion over he ou-of-sample period. Two independen simulaions wih he same degrees of freedom (d.f.) specificaion have been carried ou, he figure in he able is he average resul of he wo simulaions. 51
Figure 1: Pre- and Pos-Even Correlaion News Impac Surface The srucural break poin is se on he dae of 1 s, Jan, 1999. The NISs are obained from he esimaion of diagonal ADCC-break MGARCH model. UK-Indusrial and UK-Financial secors US-Indusrial and US-Financial secors.