Comparing Multivariate GARCH Models by Problem Dimension

Transcription

1 Comparing Mulivariae GARCH Models by Problem Dimension Massimiliano Caporin and Michael McAleer Absrac In he las 5 years, several Mulivariae GARCH (MGARCH) models have appeared in he lieraure. The wo mos widely known and used are he Scalar BEKK model and he DCC model. Some recen research has begun o examine MGARCH specificaions in erms of heir ou-of-sample forecasing performance. In his paper, we provide an empirical comparison of a se of MGARCH models, namely BEKK, DCC, Correced DCC (cdcc), CCC of Bollerslev, Exponenially Weighed Moving Average, and covariance shrinking, using he hisorical daa of 89 US equiies. Our mehods conribue o he lieraure in several direcions. Firs, we consider a wide range of models, including he recen cdcc model and covariance shrinking. Second, we use a range of ess and approaches for direc and indirec model comparison. Third, we examine how he model rankings are influenced by he cross-secional dimension of he problem. Keywords: covariance forecasing, model confidence se, model ranking, model comparison, MGARCH, ime series analysis. Inroducion Mulivariae Volailiy Models (MVM) have araced a considerable ineres over he las decade. This may be associaed wih he increased availabiliy of financial daa, he increased compuaional powers of compuers, and he fac ha he financial indusry has begun o realize he possible advanages of hese models. The recen lieraure on he opic has moved from he inroducion of new models o he efficien esimaion of exising models. Among he mos highly cied opics are he Massimiliano Caporin, Universiy of Padova, Deparmen of Economics and Managemen Marco Fanno ; massimiliano.caporin@unipd.i. Michael McAleer, Erasmus Universiy Roerdam; michael.mcaleer@gmail.com

2 PAGE 4 Name of Firs Auhor and Name of Second Auhor curse of dimensionaliy and feasible model esimaion. In fac, he feasibiliy of model esimaion is now of cenral ineres, wih many sudies proposing appropriae parameerizaions of known models (Billio e al, 2006, Franses and Hafner, 2009, Caporin and Paruolo, 2009, Bonao e al., 2009, Asai e al., 2009), or focusing on special esimaion mehods (Engle and Kelly, 2008, Engle e al., 2008, Fan e al., 2007). Despie he heoreical properies ypically being assumed under unsaed and unesable regulariy condiions, many proposed models have been used widely in empirical financial sudies. Wihin his framework, a differen problem arises: How can we compare and rank models characerized by a differen srucure? Some research has recenly appeared in he lieraure o ackle he problem of evaluaing alernaive covariance models (see Engle and Sheppard (2008), Clemens e al. (2009), and Paon and Sheppard (2009)). These papers presen limied comparisons across a small range of models. The mehods of comparison used in he previous conribuions could be viewed as wo large classes (see Paon and Sheppard (2009)), namely he direc and indirec evaluaion of volailiy forecass. The firs group includes he Mincer-Zarnowiz regression (Mincer and Zarnowiz, 969), Diebold-Mariano es (Diebold and Mariano, 996, and Wes, 996, 2006), Realiy Check of Whie (2000), Superior Predicive Abiliy (SPA) es of Hansen (2005), and he Model Confidence Se (MCS) approach of Hansen e al. (2005). The second group includes approaches based on he comparison of loss funcions adaped o he needs of covariance forecass. This is he case, for insance, of asse allocaion and risk managemen. The ess direcly comparing he covariance forecass fi he general framework of loss-funcion comparison, as discussed in Clemens e al. (2009) and Paon and Sheppard (2009). The Diebold-Mariano and Wes approaches are valid for pairwise comparisons of he models, while Realiy check and SPA require he idenificaion of a benchmark model, whereas MCS does no require a benchmark specificaion. Overall, he MCS approach seems o be he preferred one and he mos appropriae as i provides a saisical es and a mehod for deermining which models are saisically equivalen wih respec o a given loss funcion. In his paper we conribue o he lieraure on covariance forecas evaluaion in several ways. Firs, our selecion of models o be compared differs from hose of previous sudies. Similarly o he lieraure we consider he CCC model of Bollerslev (990), DCC model of Engle (2002), Scalar BEKK model wih argeing of Ding and Engle (2002), and he naïve Exponenially Weighed Moving Average approach. We complemen his se by including he cdcc model of Aielli (2008), and he covariance shrinking approach of Ledoi and Wolf (2003, 2004). Second, we use he weighed likelihood raio es of Amisano and Giacomini (2007), which is close o a loss-funcion based comparison of equal predicive abiliy based on a likelihood loss funcion. The es will be applied boh in he direc evaluaion of covariance forecass and as an alernaive o he Diebold-Mariano es. Third, we will evaluae and rank he alernaive models over differen crosssecional dimensions, saring from wo asses, and up o 89 asses, which we selec from he S&P00 consiuens (a similar daase has been used in Engle e al., 2008). By comparing models over an increasing number of variables, we will examine if esimaion error and model error play a role in he forecass of condiional covariance models.

3 Conribuion Tile PAGE 3 2 Feasible covariance and correlaion models for large problem dimensions The condiional densiy of a k-dimensional vecor of financial variables (reurns) in deviaion from heir mean follows: ( ) x = I D Σ () µ ε ~ 0, where I - is he informaion se a ime -, D(.) denoes a mulivariae densiy wih ime-varying covariance marix. We do no consider he effecs of differen mean specificaions and assume he mean is fixed a he sample mean for a range of alernaive covariance models. As a resul, all forecas discrepancies are due o differences in he expeced covariances, while all in-sample differences are due o differences in he esimaed covariance models. The firs model we esimae is he Scalar BEKK wih argeing consrain (see Engle and Kroner, 995; Ding and Engle, 200; Caporin and McAleer, 2008, 2009, 200): ( ) ( ) Σ = Σ + α ε ε Σ + β Σ Σ. (2) where he inercep is equal o he uncondiional covariance marix E[ ε ε '] Σ =. Scalar BEKK in (2) is feasible even for very large cross-secional dimensions as i conains only wo parameers ha mus be esimaed by maximum likelihood, namely he parameers driving he model dynamics. We hen consider hree models based on a decomposiion of he covariance marices ino variances and correlaions. The firs is he CCC model of Bollerslev (990) which, saring from (), assumes ha he covariance marix saisfies Σ = D RD where D is a diagonal marix of condiional sandard deviaions, and R is an uncondiional correlaion marix. We assume ha all he condiional variances follow a simple GARCH(,) process wihou asymmery in order o make he model direcly comparable wih Scalar BEKK. In he CCC model, he correlaion marix R is deermined using a radiional sample esimaor. The model is esimaed using a wo-sep approach, namely he condiional variances on each specific series, and hen esimae he uncondiional correlaion marix using he sandardized residuals. This approach makes he model feasible, even wih a large number of asses. The DCC model of Engle (2002) was proposed as a generalizaion of he CCC model, and assumes a ime-varying condiional correlaion in Σ = D R D following, ( ) ½ R = Q Q Q Q = Q I (3) ( ) ( ) Q = S + α D ε ε D S + β Q S (4) where D is he same as for he CCC model, S is he uncondiional correlaion marix, and α and ß are he scalar parameers driving he model dynamics. Following Engle (2002), he model is esimaed wih a muli-sep approach ha clearly reduces he

4 PAGE 4 Name of Firs Auhor and Name of Second Auhor efficiency bu makes he model feasible wih large cross-secional dimensions. Noe ha he model in (3)-(4) includes argeing, as defined by Caporin and McAleer (200). Aielli (2008) shows ha he esimaor of S used in he muli-sep esimaion of he DCC is inconsisen. In order o resolve his serious issue, Aielli (2008) inroduces he cdcc model, which replaces (4) wih ( ) ( ) Q = S + α Q D ε ε D Q S + β Q S (5) where he parameer marix S is symmeric, has uni elemens over he main diagonal, and is now he covariance marix of he innovaions Q D ε, which are no observable. The modificaion resores consisency bu again exposes he model o he curse of dimensionaliy as he marix S in (5) has o be esimaed. Aielli (2008) also suggess a feasible esimaion mehod for large problem dimensions. The las wo models considered are he Exponenially Weighed Moving Average model and he Covariance Shrinking approach of Ledoi and Wold (2003, 2004). The EWMA model provides a recursion for he evaluaion of he condiional covariance marix, which is based on a single parameer λ which we fix a 0.94: ( λ) ε ε λ Σ = + Σ (6) By consrucion, he EWMA is feasible even for very large cross-secional dimensions. Finally, we consider he covariance shrinking approach of Ledoi and Wolf (2003, 2004). The auhors proposed a mehod ha is designed o find a compromise beween he large esimaion errors in he sample covariance and he misspecificaion error in he esimaors of he covariance. Following he covariance shrinking approach, we define he expeced covariance for ime as follows: ( λ) S λf Σ = + (7) where S is he sample covariance marix deermined up o ime -, and F is a srucured esimaor deermined using he informaion se o ime -, and is called shrinkage arge. The coefficien λ, he shrinkage consan, has o be esimaed, and depends on he form of he shrinkage arge (for furher deails, see Ledoi and Wolf (2003, 2004)). In he following, we will consider as he shrinkage arge he covariance wih consan correlaion, as described in Ledoi and Wolf (2004). 3 Comparing compeing covariance and correlaion models I is assumed ha he models are o be compared using ou-of-sample forecass, where forecass are made one period ahead and for an evaluaion period from T+ o T+h. Models are esimaed wih a rolling approach and, in order o avoid any dependence on he mean dynamics, we fi he mean using is sample esimaor across all models (he sample mean is esimaed wih he same rolling approach). The one-sep-ahead

5 Conribuion Tile PAGE 3 covariance forecass for ime T+i are denoed by Σ ˆ m, where m is he model index T + i (m=,2, M). We follow Paon and Sheppard (2009) and consider separaely he direc and indirec evaluaion mehods. Wihin he firs group, we include wo approaches based on common loss funcions, namely he Diebold-Mariano es, and he MCS approach of Hansen e al. (2005), and he es proposed by Amisano and Giacomini (2007). For he Diebold-Mariano es, we consider he MSE loss funcion: m m ( ) ( ) lf ˆ ˆ m, T + i = 2 2 vec Σ k T + i et ie + T + i vec ΣT + i et + ie (8) T + i k where e = x ˆ µ (noe ha he observed ime T+i reurn is used), and he ime T + i T + i T + i T+i rue volailiy is approximaed by et ie. + T + i The es of equal predicive abiliy corresponds o checking he null hypohesis of zero loss funcion differenials, H : 0 0 E lf j lfl E LF = jl =, where i and j are wo differen model indices, h lf j = lf, and LF jl = lf m, T + i j lf. The es saisic is l h i= jl, h = jlσ jl 0, where ( h LF jl ) given as h LF ( h LF ) N ( ) σ is he heeroskedasiciy and auocorrelaion (HAC) consisen esimae of he asympoic variance of hlf jl. The Amisano-Giacomini (2007) is based on he logarihmic scores of wo compeing models over he forecas evaluaion period. We firs denoe he model m log-scores as: ( ˆ ) log f = log Σ ˆ m + e Σ m e. (9) m, T + i T + i T + i T + i T + i Amisano and Giacomini (2007) hen consider he quaniy L jl, T + i = w( et i ) + log f j, T + i log f where j and l represen wo differen models, l, T + i is a weighing funcion. The null hypohesis of equal predicive abiliy of and w( e T + i ) he wo models is : = 0, where H 0 E L jl, T, h L h jl, T, h L jl, T + i h i= =, while he alernaive hypohesis refers o differen predicive abiliy. The es saisic is given as: = σ ( ) ( ) where ( L jl, T, h ) L h L N jl, h jl, T, h jl, T, h 0, σ is he heeroskedasiciy and auocorrelaion consisen esimae of he asympoic variance Var hl jl, T, h. If he null hypohesis is rejeced, he es saisic sign could be used o deermine he model preference (posiive values suggess a preference for model j). Wihin he direc model evaluaion framework, we consider equal weighs for all poins over he forecas horizon. Posiive values of he es saisic are associaed wih a preference for he firs model. In considering he Diebold-Mariano es, posiive values of he es saisic show evidence for a preference for he second model (ha is, wih smaller losses).

6 PAGE 4 Name of Firs Auhor and Name of Second Auhor The Diebold-Mariano and Amisano-Giacomini ess permi pairwise comparisons of models. However, he es oucomes do no ensure ha an opimal model is clearly idenified. For his reason, we consider he Model Confidence Se approach, which performs a join forecas comparison across all models. The MCS performs an ieraive selecion procedure, esing a sep j he null hypohesis of equal predicing abiliy of all models included in a se M (he saring se l M conains all models) under a given loss 0 funcion. The null hypohesis has he form H : 0,,, 0 E lf jl lf jl E LF = jl = j > l j l M. Noe ha he same procedure, l as well as all he subsequen saisics and ess, could also be used wih he Amisano- Giacomini log-scores, wih he cauion of changing heir sign (hereby ranslaing hem from gains o losses). Hansen e al. (2005) propose wo saisics o es he null hypohesis: ( ) R, h = max j, l M LF jls LF jl l where ( jl ) and ( ( )) 2 SQ, h = LF jls LF jl j, l M l, j> l s LF is a boosrap esimae of he variance of LF jl, and he p-values of he es saisics are deermined using a boosrap approach. If he null hypohesis is rejeced a a given confidence level, he wors performing model is excluded from he se. Such a model is idenified as j = arg max j M LF jl Var LF jl l l Ml l Ml where he variance is compued again using a boosrap mehod. In he empirical analysis given below, we will use boh he Diebold-Mariano loss funcion, as well as of he Amisano-Giacomini log-scores (minus he log-scores o ransform hem ino losses) (see Hansen e al. (2005) for furher deails on MCS). For he indirec evaluaion of he mulivariae models, we consider an asse allocaion framework and compare he impac of model choice by conrasing he performances of specific porfolios: (i) equally weighed porfolio, denoed as EW, which is no exposed o he asse reurn mean esimaion error, and is superior o many oher porfolios (see De Miguel e al. (2009)); and (ii) global minimum variance porfolio wih and wihou shor selling consrains, denoed as GMV and GMVB, respecively. Given he porfolio mean and variance forecass (he las are model dependen), and he realized porfolio reurns, we compare models using he Amisano-Giacomini es for he EW sraegy (GMV and GMVB have been excluded o avoid disorions in he es saisic due o he esimaion error of porfolio weighs). Furhermore, we compare models using boh he MSE loss funcion as well as he QLIKE loss funcion defined in Paon and Sheppard (2009) wihin a Diebold-Mariano framework as well as for he Model Confidence Se. / 2 4 Daa descripion and seleced models In order o compare he models presened in he previous secions, we have seleced a daase similar o ha of Engle e al. (2009). We downloaded from Daasream he S&P00 consiuens a he end of March Then we seleced only hose asses wih

7 Conribuion Tile PAGE 3 oal reurn indices available from he beginning of 997 o he end of March The seleced period conains 394 daily reurns. On hese daa we fi he models defined in Secion 2, using boh Mulivariae Normal and Mulivariae Suden densiies for BEKK, DCC and cdcc. The 9 specificaions considered are evaluaed for differen problem dimensions, such ha each model is esimaed for 2, 3, 4, 5, 0, 5, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80 and 89 asses. In hese empirical applicaions, asses are ordered alphabeically, and we progressively enlarge he number of variables used in he model esimaion and evaluaion seps. We esimae he models wih a rolling approach, and hen compare he ou-of-sample forecass for one year of daily observaions. All models for all problem dimensions are esimaed daily and are used o produce one-sep-ahead forecass. We consider wo differen ou-of-sample evaluaion periods. In he firs, we focus on exreme marke condiions and compare models for he period April 2008 March This could be considered as a model sress-es o deermine if more highly parameerized models are preferred o simpler or naïve specificaions as hey are no exposed o parameer uncerainy and insabiliy. For his forecas range, we also perform a robusness check by invering he asse order o verify he absence of disorions due o asse order. The second forecas evaluaion period is for 2006, when he marke was in a low volailiy sae and was rending upward. This second comparison allows verificaion of wheher he model ranking migh be affeced by overall marke condiions. All he empirical resuls are repored in a se of ables and graphs in he following secions. Tables are summarized and repored in he paper, while he enire se of empirical resuls is available from he auhors upon reques. 5 Resuls for direc model comparison A summary of he resuls is repored in Table. The Diebold-Mariano and Amisano- Giacomini ess poin ou he negaive performance of he covariance shrinking approach over boh evaluaion periods. For boh ess and all problem dimensions, SHR always provides higher losses (lower log-scores) compared wih he oher models, apar from EWMA. In his las case, he ess are discordan. These findings are onfirmed by MCS. Wih respec o EWMA, we noe a sriking difference by comparing he resuls of he Diebold-Mariano es wih hose of he Amisano-Giacomini procedure. Wih he laer, EWMA provides larger losses han all he oher models when he number of asses exceeds 0, and his resul holds for boh evaluaion periods. If we consider he MSE loss funcion, EWMA is always equivalen o all he oher models, apar from SHR, which is ouperformed for more han 5 asses, bu only during he crisis period. Considering he non-naïve models, he Diebold-Mariano and Amisano-Giacomini es resuls are subsanially similar, which leads o he following observaions: i) All models are equivalen if we consider up o 0 asses when he models are compared over exreme marke condiions, while up o 5 if we compare models over he year 2006; if we consider he 2006 evaluaion period, many more models are saisically equivalen, even for medium problem dimensions; ii) Moving from he Normal o he Suden densiy does no improve he forecas accuracy of he BEKK, DCC and cdcc specificaions; iii) Scalar BEKK is always inferior o he CCC, DCC, and cdcc specificaions, irrespecive of he densiy; DCC and cdcc ouperform CCC, even for smaller

8 PAGE 4 Name of Firs Auhor and Name of Second Auhor dimensions (from he 4 asses case); cdcc ouperforms DCC under boh Normal and Suden densiies over he exreme marke condiion case, while he reverse holds for he 2006 forecas evaluaion period. These resuls are confirmed by he MCS approach, bu only for he Amisano-Giacomini loss funcion. For he MSE loss funcion under he crisis forecas evaluaion range, and a he % level, all models are subsanially equivalen for all problem sizes. If we compare models during he year 2006, resuls for he MSE and Amisano- Giacomini loss funcions are much closer. The main message we exrapolae from our resuls is he equivalence across many models when he marke is in a low volailiy sae, and he preference for dynamic condiional correlaion models. Moreover, when he marke is experiencing large and sudden changes in volailiy, dynamic condiional correlaion models may be preferred, bu he resuls are no consisen across all model comparison mehods. 6 Resuls for indirec model comparison If we consider he Diebold-Mariano ess, he resuls are somewha similar o hose obained from he direc model comparisons. Firs, during boh he crisis and he 2006 ou-of-sample periods, he SHR model underperforms all oher specificaions, even for small cross-secional dimensions. The crisis makes he esimaion of porfolio weighs exremely noisy, paricularly affecing he resuls based on he MSE loss funcion. Wih he exclusion of SHR and EWMA for he QLIKE loss funcions, all models are equivalen when compared for April 2008-March 2009, while some saisically significan differences appear when he comparison is based on These resuls raise some doubs abou he usefulness of dynamic covariance and correlaion models of a relaively complex naure when hey are used o deermine porfolio weighs. The join effec of he esimaion error on he model coefficiens and of he esimaion error of porfolio weighs would seem o make complex models virually equivalen o simple models, such ha increasing he complexiy of a model does no improve he efficiency of an allocaion sraegy. On he conrary, when one of he wo sources of error is serilized, by means of an EW sraegy, some discrepancies seem o appear: CCC and BEKK underperform DCC and cdcc; cdcc ouperforms DCC for large problem dimensions during he crisis, while he opposie holds during 2006 for he MSE loss funcion; he Mulivariae Suden-based models underperform Mulivariae Normal-based models. Following he direc comparison, we deermine he model confidence ses over several problem dimensions. For he crisis period (April 2008 o March 2009) and he EW sraegy, he model confidence se includes all models a he % level for he MSE and QLIKE loss funcions. Minor differences appear a he 5% level: for MSE, wih only SHR marginally excluded; for QLIKE, he se of equivalen models includes only EWMA, cdcc(n), and DCC(N). Considering he MCS resuls for boh es saisics and for he GMV and GMVB sraegies, all models are equivalen, apar from he exclusion of EWMA a he 5% level under he GMV allocaion rule. This resul confirms our earlier resuls ha models are equivalen when porfolio weighs are esimaed. We hen analyze he resuls for he 2006 ou-of-sample period. Overall, he resuls confirm he previous findings ha many models are equivalen for esimaed porfolio

9 Conribuion Tile PAGE 3 weighs. However, we presume ha he greaer flexibiliy in separaely esimaing variances and correlaions provides some benefis over naïve and general covariance specificaions, in paricular, when he problem dimension is large. Furhermore, for small and medium problem dimensions, he esimaion error has a relevan role, making more complex covariance and correlaion models almos equivalen, if no worse, han EWMA. When he number of variables is increased, he flexibiliy of DCC and cdcc models becomes even more relevan han he esimaion error. In summary, he indirec comparisons sugges ha model performances are affeced by several sources of error. The esimaion error of model parameers is always presen. Esimaion error of porfolio weighs may play a relevan role during exreme marke condiions, and be so relevan as o make many models saisically equivalen in erms of forecass. 6 Conclusions From he empirical poin of view, Mulivariae GARCH models suffer seriously from he so-called curse of dimensionaliy. For his reason, several simple specificaions are ypically used, including he CCC, DCC and Scalar BEKK models. Alernaively, naïve mehods could be used, such as EWMA or he Covariance Shrinking approach. However, few sudies have considered a deailed ou-of-sample comparison of hese models. This paper has shed ligh on his opic, bu he oucome is far from conclusive. By using alernaive evaluaion mehods, including he direc and indirec approaches, pairwise and mulivariae mehodologies, and differen ou-of-sample evaluaion periods, he resuls are mixed. The only common finding is ha covariance shrinking mehods underperform he dynamic models, even for small cross-secional dimensions, a leas for he daase and periods considered in he paper. Less common oucomes sugges, for small problem dimensions, here is a higher probabiliy ha alernaive approaches will provide subsanially equivalen covariance forecass. This finding is less eviden for large problem dimensions, where simple dynamic specificaions, despie being highly resricive, may be superior o naïve specificaions based on calibraed coefficiens. In his case, models separaely capuring he variance and correlaion dynamics are marginally preferred o pure covariance models. Furhermore, he impac of several sources of error, such as esimaion error of he model parameers, esimaion error of porfolio weighs for indirec comparison, and errors associaed wih he choice of proxy, come ino play and can affec he oucomes, hereby suggesing he need for furher analysis. References Aielli, 2008, Consisen esimaion of large scale dynamic condiional correlaions, Working paper n. 47, Deparmen of Economics, Saisics, Mahemaics and Sociology, Universiy of Messina. Amisano, G., and Giacomini, R., 2007, Comparing densiy forecass via weighed likelihood raio ess, Journal of Business and Economic Saisics, 25, Asai, M., Caporin, M., and McAleer, M., 2009, Block srucure mulivariae sochasic volailiy, Available a SSRN: hp://ssrn.com/absrac=

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11 Table : Summary of resuls for direc model comparisons Forecas sample January o December 2006 April 2008 o March 2009 DM es (MSE loss) Mos models are equivalen up o 5 asse dimension; SHR underperforms all models (EWMA excluded); EWMA ouperforms in some cases; Suden densiy does no improve over Normal; CCC ouperforms BEKK; DCC ouperforms BEKK and CCC; cdcc ouperforms BEKK, CCC and DCC; Mos models are equivalen up o 0 asse dimension; SHR underperforms all models (EWMA excluded); EWMA is equivalen o mos models; Suden densiy does no improve over Normal; CCC ouperforms BEKK; DCC ouperforms BEKK and CCC; cdcc ouperforms BEKK, CCC and DCC; AG MCS wih MSE loss Mos models are equivalen up o 5 asse dimension; SHR underperforms all models (EWMA excluded); EWMA underperforms wih more han 0 asses; Suden densiy does no improve over Normal; CCC ouperforms BEKK; DCC ouperforms BEKK and CCC; cdcc ouperforms BEKK, CCC and DCC; SHR is no included (a he % level); EWMA is always included; DCC is he bes model for larger problem dimensions; Many models equivalen for medium problem dimensions; Mos models are equivalen up o 0 asse dimension; SHR underperforms all models (EWMA excluded); EWMA underperforms wih more han 0 asses; Suden densiy does no improve over Normal; CCC ouperforms BEKK; DCC ouperforms BEKK and CCC; cdcc ouperforms BEKK, CCC and DCC; SHR and BEKK(T) are included only for small problem dimensions (a he 5% level); All models are equivalen a he % level; MCS wih AG loss SHR is included only for small problem dimensions; EWMA is included only for small problem dimensions; DCC and cdcc are he bes models for larger problem dimensions; Many models are equivalen for medium problem dimensions. All models are equivalen up o 5 asses; DCC(N) and cdcc(n) are he bes models for larger problem dimensions.; Noes: The firs column repors he quaniies used for he direc model comparisons. In he able, AG = Amisano-Giacomini, DM = Diebold- Mariano, MSC = Model Confidence Se, and MSE = Mean Squared Error. The second and hird columns repor a summary of resuls for he wo ou-of-sample periods. When he densiy is no repored for DCC, cdcc and BEKK, he commens refer o boh densiies. Commens in ialics idenify differen behavior across samples or loss funcions.

12 PAGE 4 Name of Firs Auhor and Name of Second Auhor Table 2: Summary of resuls of indirec model comparisons based on Diebold-Mariano and Model Confidence Se Forecas sample January o December 2006 April 2008 o March 2009 DM es (MSE loss) SHR underperforms all models; BEKK underperforms under GMV and GMVB; DCC and cdcc ouperform BEKK and CCC under EW; DCC ouperforms cdcc under EW; Suden densiy does no improve over Normal under EW; SHR underperforms all models under EW; Under GMV and GMVB all models are equivalen; DCC and cdcc ouperform BEKK and CCC under EW; cdcc ouperforms DCC under EW; Suden densiy does no improve over Normal under EW; DM es (QLIKE loss) MCS (MSE loss) SHR underperforms all models; EWMA underperforms under GMV; BEKK models underperform under GMVB; DCC and cdcc ouperform BEKK and CCC under EW; DCC ouperforms cdcc under EW; Suden densiy does no improve over Normal under EW; SHR is always excluded; EWMA and BEKK(T) are excluded for large problem dimensions under GMV and GMVB; BEKK and CCC are excluded under EW; EWMA, DCC and cdcc are included for large problem dimensions; SHR underperforms all models; EWMA underperforms under GMV; Mos models are equivalen under GMV and GMVB; DCC and cdcc ouperform BEKK and CCC under EW; cdcc ouperforms DCC under EW; Suden densiy does no improve over Normal under EW; All models are equivalen a he % level; A he 5% level under EW, SHR is excluded for large problem dimensions; MCS (QLIKE loss) SHR is always excluded; EWMA and BEKK(T) are excluded for large problem dimensions under GMV and GMVB; BEKK and CCC are excluded under EW; EWMA, DCC and cdcc are included for large problem dimensions. All models are equivalen a he % level; A he 5% level under EW, he se includes only EWMA, DCC(N) and cdcc(n). Noes: The firs column repors he quaniies used for he indirec model comparisons. In he able, DM = Diebold-Mariano, MSC = Model Confidence Se, MSE and QLIKE denoe he wo loss funcions, while EW, GMV and GMVB idenify he porfolio sraegies considered. The second and hird columns repor a summary of resuls for he wo ou-of-sample periods. When he densiy is no repored for DCC, cdcc and BEKK, he commens refer o boh densiies. If porfolio sraegies are no repored, he commens apply o all sraegies. Commens in ialics idenify differen behavior across samples or loss funcions.