econometrics ISSN

Transcription

1 Economerics 2013, 1, ; doi: /economerics Aricle economerics ISSN Ten Things You Should Know abou he Dynamic Condiional Correlaion Represenaion Massimiliano Caporin 1 and Michael McAleer 2, 3, 4, * OPEN ACCESS Deparmen of Economics and Managemen Marco Fanno, Universiy of Padova, Via del Sano 33, Padova, Ialy; massimiliano.caporin@unipd.i Economeric Insiue, Erasmus School of Economics, Erasmus Universiy Roerdam and Tinbergen Insiue, 3000 DR Roerdam, Neherlands Deparmen of Quaniaive Economics, Compluense Universiy of Madrid, Pozuelo de Alarcón, Madrid, Spain Insiue of Economic Research, Kyoo Universiy, Kyoo , Japan * Auhor o whom correspondence should be addressed; michael.mcaleer@gmail.com Received: 13 May 2013; in revised form: 7 June 2013 / Acceped: 14 June 2013 / Published: 21 June 2013 Absrac: The purpose of he paper is o discuss en hings poenial users should know abou he limis of he Dynamic Condiional Correlaion (DCC) represenaion for esimaing and forecasing ime-varying condiional correlaions. The reasons given for cauion abou he use of DCC include he following: DCC represens he dynamic condiional covariances of he sandardized residuals, and hence does no yield dynamic condiional correlaions; DCC is saed raher han derived; DCC has no momens; DCC does no have esable regulariy condiions; DCC yields inconsisen wo sep esimaors; DCC has no asympoic properies; DCC is no a special case of Generalized Auoregressive Condiional Correlaion (GARCC), which has esable regulariy condiions and sandard asympoic properies; DCC is no dynamic empirically as he effec of news is ypically exremely small; DCC canno be disinguished empirically from diagonal Baba, Engle, Kraf and Kroner (BEKK) in small sysems; and DCC may be a useful filer or a diagnosic check, bu i is no a model. Keywords: DCC represenaion; BEKK; GARCC; saed represenaion; derived model; condiional correlaions; wo sep esimaors; assumed asympoic properies; filer JEL Classificaions: C18; C32; C58; G17

2 Economerics 2013, Inroducion The 21s cenury has seen subsanial and growing ineres in he analysis of dynamic covariances and correlaions across invesmen insrumens. In paricular, here has been grea emphasis paid o he analysis of financial asses (see [1] and [2], among ohers, and he references cied in he surveys by [3] and [4]). More recenly, here has been growing ineres in energy finance, paricularly oil (see [5], [6], [7] and [8] among ohers). In his research sream, he mos widely-used represenaion is a variaion of Mulivariae Generalized AuoRegressive Condiional Heeroskedasiciy (GARCH), namely Dynamic Condiional Correlaion (DCC), as inroduced by [1]. The baseline represenaion has been exended in several direcions, dealing wih he parameerizaion (see [2], [6], and [9], among ohers), he inroducion of addiional elemens, such as asymmery (see [2] and [10], among ohers), and he proposal of alernaive esimaion mehods (see [11] and [12], among ohers). Despie he growing ineres in DCC and is cenral role in he esimaion of dynamic correlaions, several imporan issues relaing o his represenaion seem o have been ignored in he financial economerics lieraure. These imporan issues include he absence of any derivaion of DCC and is mahemaical properies, and a lack of any demonsraion of he asympoic properies of he esimaed parameers (for a summary of hese issues, see [13]). In his respec, a useful conribuion is [14], who demonsraes he inconsisency of he wo sep esimaor of he parameers of DCC. In fac, mos published papers dealing wih dynamic correlaions simply do no discuss saionariy of he model, he regulariy condiions, or he asympoic properies of he esimaors. Anoher criical elemen of DCC is associaed wih he consrucion of he dynamic condiional correlaions. In fac, he represenaion seems o provide esimaed dynamic correlaions as a bi-produc of sandardizaion, and no as a direc resul of he equaion governing he mulivariae dynamics. This will be clarified below. An alernaive represenaion which avoids his las criicism, bu neverheless has no discussion of he mahemaical properies or demonsraion of he asympoic properies of he esimaors, has been proposed by [15]. However, his represenaion seems o have araced considerably less ineres in he lieraure. I should be menioned ha many empirical applicaions involving DCC and relaed represenaions show ha he impac of news can be raher limied, hereby making he esimaed condiional correlaions similar o hose implied by simple BEKK models (see [16] and [17], a leas in small cross-secional problems (for furher deails, see [9] and [18]). This paper highlighs some criical issues associaed wih he use of he DCC and relaed represenaions o make poenial users aware of he inheren problems hey migh encouner. The main message is no agains he use of DCC, which is he mos popular represenaion of dynamic condiional correlaions, bu is inended o be cauionary, so ha users can undersand and appreciae he limis of DCC. In fac, we sugges ha DCC be regarded as a filer or as a diagnosic check, as in he Exponenially Weighed Moving Average approach adoped in he firs versions of he [19] mehodology. When an equaion has no been derived in a rigorous way, and for which we do no have any explici deails regarding he exisence of momens, derivaion and esabiliy of he saionariy condiions, and demonsraed asympoic properies of he esimaors, i should no be considered as a model, bu raher as a filer or a diagnosic check for esimaing and forecasing dynamic condiional

3 Economerics 2013, correlaions. We will elaborae on his issue in he remainder of he paper afer highlighing he criical aspecs of he DCC framework. The plan of he paper is o discuss en hings you should know abou he DCC represenaion. These caveas are discussed in Secion 2. Some concluding remarks are given in Secion Ten Caveas abou DCC The DCC represenaion was inroduced by [1] o capure he empirically observed dynamic conemporaneous correlaions of asse reurns. The represenaion can be given as follows. Denoe by r he vecor conaining he log-reurns of k asses. The densiy of he reurns is characerized by he absence of serial correlaion in he mean reurns, and by he presence of ime-varying second-order momens: r I 1 D ( µ,σ ) (1) where I denoes he informaion se o ime -1, 1 µ is he uncondiional mean, which is generally equal, or very close, o zero, Σ is he dynamic condiional covariance marix, and D is a generic mulivariae densiy funcion depending on he mean vecor and dynamic condiional covariance marix. Following [12], he covariance marix can be decomposed ino he produc of dynamic condiional sandard deviaions and dynamic condiional correlaions: where D = diag( σ, σ,..., σ ), ( ) 1 2 k Σ = DRD (2) diag a is a marix operaor creaing a diagonal marix wih he vecor a along he main diagonal, and R is a dynamic correlaion marix. From Equaions (1) and (2), he marginal densiy of each elemen of r has a ime-varying condiional variance, and can be modeled, for example, as a univariae GARCH process. The DCC represenaion focuses on he dynamic evoluion of R in Equaion (2), and recovers ha quaniy by considering he dynamics of he condiional variance of he sandardized residuals, which are defined as follows: η = D µ (3) 1 ( r ) By consrucion, he sandardized residuals have second-order uncondiional momen equal o E ηηʹ = R (4) wih R being he uncondiional correlaion, hereby moivaing he focus on sandardized residuals o recover he dynamics for he condiional correlaions. In pracice, he sandardized residuals can be used o verify empirically he exisence of dynamics in he condiional correlaions, for insance, by means of a rolling regression approach. Moreover, if he daa generaing process of he reurns is given in Equaions (1) and (2), he dynamic condiional covariance of he sandardized residuals is given as: E ηηʹ I 1 = R (5)

4 Economerics 2013, Wihou disinguishing beween he dynamic condiional covariance and dynamic condiional correlaion marices, [1] presens he following equaion based on he ouer cross-producs of he sandardized residuals: ( 1 α β) αη 1η ʹ 1 β 1 Q = Q+ + Q (6) where Equaion (6) has scalar parameers, as in he mos common DCC represenaion, Q is assumed o be a posiive definie marix wih uni elemens along he main diagonal (which is alleged o be a condiional correlaion marix), he wo scalar parameers saisfy a sabiliy consrain of he form α + β < 1, and he sequence Q purporedly drives he dynamics of he condiional correlaions. However, as he marix Q in Equaion (6) does no saisfy he definiion of a (dynamic) condiional correlaion marix, as in Equaion (2), [1] inroduces he following sandardizaion: ( ( )) ( ) ( ) R = diag dg Q Qdiag dg Q where dg ( A ) is a marix operaor reurning a vecor equal o he main diagonal of marix A. (In discussing he equivalen of Equaion (7) above, namely Equaion (25) in [1], a ypographical error is presen as he exponen is repored as 1 insead of 0.5.). I is clear ha Equaion (7) is a simple sandardizaion, suggesing ha he primary saisic of ineres, namely he dynamic condiional correlaion marix, can be compued from (6). However, o sae he obvious, a dynamic condiional correlaion marix is a sandardizaion of a dynamic condiional covariance marix, bu no every sandardizaion, such as ha in Equaion (7), is consisen wih a dynamic condiional correlaion marix. This lack of equivalence is even more obvious if i canno be demonsraed (as disinc from being saed) ha Equaion (6) is a dynamic condiional correlaion marix. (A simple illusraion would be o divide 10 elephans by 20 elephans, which is no a correlaion despie being a fracion.) I should be clear ha, as he second erm on he righ-hand side of Equaion (6) is no a dynamic updae of a condiional correlaion marix, he represenaion in Equaion (6) canno be a dynamic condiional correlaion marix. Bearing hese poins in mind, he following caveas should be seriously considered before using he DCC represenaion DCC is Based on he Condiional Second-Order Momen of he Sandardized Residuals, and Hence does no Direcly Yield Condiional Correlaions The simple observaion of Equaion (6) recognizes he srucure of he scalar BEKK model of dynamic condiional correlaions (see [16] and [17]), poining o an inheren conradicion in he DCC model. Combining Equaions (1), (2), and (3) leads o: η = D 1 r µ Moreover, using Equaion (7), his is equivalen o η = D 1 r µ " # ( ) D( 0,R ) ( ) D$ 0,diag dg ( Q ) ( ) 0.5 Q diag ( dg ( Q )) 0.5 % ' & (7)

5 Economerics 2013, However, if we consider he Q dynamic recurrence in Equaion (6) as a BEKK model, hen migh be inerpreed as he dynamic condiional covariance marix of he innovaion erm, which is η, and i hereby suggess ha he following holds: η = D 1 r µ ( ) D( 0,Q ) (8) which is inconsisen wih wha is implied in Equaions (1) (3) and (5). Therefore, we noe an implici conradicion in he way he DCC correlaion dynamics are derived. In addiion, a dynamic condiional correlaion marix may be obained only hrough he sandardizaion in Equaion (7). However, we can also noe an inconsisency beween he dynamic condiional expecaion repored in Equaion (5) and he way in which he dynamic condiional correlaion marix is obained in Equaion (7). Such inconsisency causes furher problems as Q is no he condiional covariance of η, as shown in Equaion (5), and is no he condiional correlaion of η as i is jus posiive definie, bu need no correspond o a dynamic condiional correlaion marix. The las remark can easily be verified by visual inspecion of he esimaes of Q, which are ypically no considered in empirical analysis. However, by using several daases, i is sraighforward o show ha he elemens of Q can be greaer han 1 (see, for example, [20]). As a consequence, i migh be saed ha he sequence Q is a convenien device for obaining dynamic condiional correlaions bu, as i sands, has no proper inerpreaion as eiher a dynamic condiional covariance or dynamic condiional correlaion marix. This leads o anoher cavea abou DCC DCC Is Saed Raher Than Derived From he previous commens, i clearly emerges ha DCC is a saed represenaion, bu i is no a derived model ha is based on he relaionship beween he innovaions o reurns and he sandardized residuals. Moreover, he DCC represenaion does no saisfy he definiion ha relaes dynamic condiional correlaions o dynamic condiional covariances, as given in Equaion (2). As such, he inerpreaion of DCC as a represenaion ha may yield dynamic condiional correlaions is inherenly flawed. This quandary also begs he quesion as o wheher DCC is acually a model, namely a se of assumpions, or alernaively as a represenaion wih explici and esable mahemaical properies and derivable saisical properies. A furher moivaion for he previous claim is inherenly relaed o he consrucion of he condiional correlaions wihin he DCC represenaion. Generally speaking, condiional correlaions can be derived from a condiional covariance model by sandardizaion of he covariances, namely 1 1 ρ = σ σ σ. However, such a procedure canno be applied o he DCC represenaion because ij, ij, i, j, he covariance is obained as σ = σ σ q q q. Consequenly, he radiional way of deriving ij, i, j, ij, i, j, correlaions applies o he condiional marix Q and no o he full condiional covariance. We conclude ha we canno derive condiional correlaions in he usual way due o he presence of wo sandardizaions raher han jus one. The above discussion also affecs he many represenaions which are obained as generalizaions of he DCC represenaion including, among ohers, [2], [6] and [14]. Moreover, his has furher consequences for he model srucure and he associaed saisical properies. Q

6 Economerics 2013, DCC Has No Momens This follows from he saed raher han derived properies of he represenaion (see [20] for furher deails). Therefore, here is no connecion beween univariae models of condiional variance, such as ARCH ([21]) and GARCH ([22]), and mulivariae models of condiional correlaions. This is in marked conras o he direc connecion beween he alernaive univariae condiional volailiy models and he BEKK mulivariae model of dynamic condiional covariances (see [16] and [17]), and he direc connecion beween univariae condiional volailiy models and he GARCC mulivariae model of dynamic condiional correlaions (see [20]). Neverheless, we observe ha he financial economerics lieraure includes several oher models and approaches where he underlying sochasic componen is characerized, for insance, by he known exisence of lower-order momens while higher-order momens, or even he variance, migh no exis. In any even, such approaches have been used exensively, wih useful and ineresing empirical resuls, wihin a risk managemen framework DCC Does No Have Tesable Regulariy Condiions This follows from poin (3) above. In paricular, [1] (p. 342) refers o reasonable regulariy condiions and sandard regulariy condiions, wihou saing hem explicily. The auhor of [14] (pp ) assumes ha he unsaed regulariy condiions, whaever hey migh be, are saisfied. An exension of he DCC represenaion o incorporae asymmeries is developed in [2], bu explici regulariy condiions are no provided. Wih no esable regulariy condiions, such as log-momen or second momen condiions, he inernal consisency of he model canno be checked. There is, herefore, no evidence as o wheher he purpored esimaes of dynamic condiional correlaions have any connecion o he definiion of dynamic condiional correlaions. The absence of explici regulariy condiions and of explici momen affecs also he derivaion of asympoic properies of he parameer esimaes. The auhor of [1] suggess he following wo sep approach for esimaing DCC parameers. Wihin a Quasi Maximum Likelihood framework, we have he following Gaussian log-likelihood for one observaion of he reurns r : Following he decomposiion in (2), we have: 1 2 ln Σ 1 2 r # Σ 1 r # 1 2 ln D R D 1 2 r " D 1 R 1 D 1 r " = 1 2 ln D # = % 1 2 ln D $ ln R ± 1 2 r" D 1 D 1 r" 1 2 r" D 1 R 1 D 1 r" r " D C ( ) l (, ) = l Θ + Θ Θ V V V C & # 1 D 1 r" (+ 1 ' 2 ln R η η" 1 2 η R & 1 % η" ( $ ' where i is shown ha he single observaion likelihood can be decomposed ino wo erms, namely a funcion of he variance parameers only, Θ, and a funcion of boh he variance and correlaions V

7 Economerics 2013, parameers, Θ and V Θ, respecively. Noe ha he firs likelihood componen is based on a C correlaion marix se o he ideniy marix, which is hen used o recover he variance parameers only. The second likelihood componen is used o esimae he correlaion parameers, condiionally on he firs sage likelihood esimaed parameers ( Θ ˆ V ). In [1], i is suggesed ha he firs likelihood componen can be furher decomposed ino he sum of univariae likelihoods represening he marginal conribuion of each reurn series, under he assumpion of independence. This is a firs simplificaion imposed o deal wih he curse of dimensionaliy ha generally affecs mulivariae GARCH models (see [23] for furher deails). In addiion, o simplify he compuaional burden associaed wih he maximizaion of he second sage ˆ T C Θ, Θ =Σ l Θˆ, Θ, [1] suggess replacing he marix Q wih he sample likelihood L ( V C) = 1 ( V C) correlaion marix of he sandardized residuals η, hereby inroducing an inermediae 1.5 sep. The previously oulined approach enails a number of assumpions which are generally no saisfied by empirical daa, as follows: - Marginal variances are assumed o be independen, which rules ou any form of spillovers or feedback across variances and shocks of he various asses. This is relaed o he general idea of having dependence across asses governed only by he correlaions. However, his is no always he case, and shocks of differen asses can affec he variance of a single asse. - The sample correlaion marix is assumed o be an appropriae esimaor for he marix Q, which is no necessarily a correlaion marix. - The approach is called wo sep, when in realiy i is a hree sep procedure when sample correlaions are used for Q, and is a proper wo sep procedure when he correlaion likelihood ( ) Σ Θ Θ is maximized wih respec o he full parameer se Θ, and condiionally on he C T C = 1 l V, C variance parameers Θ. V However, he possible incompaibiliy beween he assumpions leading o he esimaion approach described above do no preven is use, which can be moivaed and suppored by is compuaional simpliciy, an imporan issue of which users should be aware. Neverheless, he asympoic properies of he wo sep esimaor are no discussed in [1], apar from a reference o [24], which remains an unpublished manuscrip and does no, in fac, demonsrae any asympoic properies for he DCC parameers. We have he addiional following cavea: 2.5. DCC Yields Inconsisen Two Sep Esimaors The auhor of [1] (p. 342) saes ha he sandardized residuals in equaion (6) are a Maringale difference by consrucion in suggesing how o esimae he parameers of DCC by resoring o sandard ARMA mehods. Moreover, he fac ha he errors are a maringale difference sequence allows recovery of general resuls for mulivariae GARCH processes and, in paricular, hose in [17]. However, [14] poins ou ha he DCC represenaion canno be inerpreed as a linear mulivariae GARCH, and his leads o he inconsisency of he wo sep DCC esimaor discussed above. The inconsisency is governed by he fac ha in equaion (6) he marix Q is no he expecaion of he

8 Economerics 2013, sandardized residuals cross-producs. Therefore, i is no possible o obain a maringale difference by rewriing equaion (6) in a companion VARMA form. The primary meri of [14] is in highlighing he inconsisency problem, bu he proposed soluion sill suffers from he same roubles afflicing he DCC represenaion of [1]. In fac, [14] discusses argeing and a modificaion o DCC o enable consisen esimaion. However, he assumes ha he esimaors of he modified DCC represenaion are asympoically normal under sandard regulariy condiions, wihou saing wha he condiions migh be. I has been shown in [18] and [23] ha dynamic condiional correlaions can be esimaed consisenly by using an indirec DCC represenaion based on he BEKK model, bu asympoic normaliy canno be esablished DCC Has No Desirable Asympoic Properies In [14], [20] and [23] i has been shown ha he esimaed parameers of he DCC represenaion under he sandard wo sep approach have no asympoic properies. Moreover, he asympoic properies of he join maximum likelihood esimaor (for all parameers in one sep) are no known. In heir exension of DCC, [2] do no esablish any asympoic properies. In a recen conribuion, [11] claim o prove consisency of he esimaes of he DCC represenaion in Theorem 1, bu he proof refers o pseudo-rue parameers raher han he parameers of ineres. In Theorem 2, he auhors assume consisency of he esimaed parameers of ineres (such ha he pseudo-rue parameers are idenical o he parameers of ineres) in claiming a proof of asympoic normaliy (see [23] for furher deails). I is clear ha he availabiliy of asympoic properies is sill an open quesion. As a consequence, he reliabiliy of sandard inferenial procedures, such as saisical significance, or likelihood raio esing across nesed DCC represenaions, remains unknown, and should be considered only on he basis of appropriae simulaion experimens DCC Is No a Special Case of GARCC, Which Has Tesable Regulariy Condiions and Sandard Asympoic Properies In [20] he Generalized Auoregressive Condiional Correlaion (GARCC) model is derived based on he relaionship beween he innovaions o reurns and he sandardized residuals, using a vecor random coefficien auoregressive process. The scalar and diagonal versions of BEKK are also shown o be special cases of a vecor random coefficien auoregressive process, hough no he Hadamard and full BEKK models. The GARCC model provides a moivaion for dynamic condiional correlaions ha saisfy he definiion of a condiional correlaion marix, and hence can be shown o produce dynamic condiional correlaions. As an applicaion of a vecor random coefficien auoregressive process, he GARCC model also has esable regulariy condiions, and he esimaed parameers can be shown o be consisen and asympoically normal.

9 Economerics 2013, DCC Is No Dynamic Empirically and Variance Misspecificaion Impac Is No Known Are he purpored dynamic condiional correlaions real or apparen, and do hey arise solely from he sandardizaion of he dynamic condiional covariances? Wha is he impac of variance misspecificaion? Wih respec o he firs quesion, we refer o he parameer esimaes which are generally observed in empirical sudies, whereby β is large and close o 1, while α is ypically small and less han As a resul, he condiionally dynamic marix Q may appear o be dynamic as boh parameer esimaes are saisically significan. However, he limied impac of he shocks, driven by he small esimae of α, makes Q almos consan, wih limied oscillaions around he uncondiional Q value of Q. In addiion, such a limied dynamic effec migh be amplified in he dynamic correlaion marix R, as defined in (7), due o he presence of he sandardizaion. This may well confuse he user of he DCC represenaion, who migh no be aware ha ha he purpored dynamics are spurious. Moving o he second quesion, we give he underlying inuiion saring from a classical example. In he conex of he Box-Jenkins procedure, if an ARMA(1,1) is esimaed when an ARMA(2,1) represenaion is correc, hen he residuals migh sill show some AR dynamics. For insance, in he limiing case of real roos for he ARMA(2,1) model, if he roos of he esimaed model correspond o hose of he rue model (such ha he AR componen capures precisely one of he wo roos of he ARMA(2,1) model), hen he residuals would sill be an AR(1) process. Therefore, esimaing he residuals wih an AR filer could possibly capure he remaining dynamics. Transposing he same argumen ino he GARCH framework, he condiional variance migh be esimaed as GARCH(1,1), bu he correc model migh have asymmery, leverage, jumps, hresholds and/or higher ime-varying momens. As a resul, he parameer esimaes migh be biased. The sandardized residuals, which are ypically no checked for furher condiional heeroskedasiciy (as he common wisdom is ha GARCH(1,1) should be sufficien), may have remaining heeroskedasiciy, however mild. Fiing sandardized residuals using a GARCH(1,1) model, which is he diagonal erm of he DCC represenaion, will capure some dynamics. Even if he condiional correlaions happen o be consan, he condiional covariances across he sandardized residuals may appear o be dynamic because of he misspecificaion. Therefore, sandardizaion does no filer ou he dynamics in he covariances due o he biases in he iniial GARCH(1,1) esimaes. As a resul, he condiional correlaions may appear o be dynamic (wih significan parameer esimaes) due o misspecificaion in he firs sep. However, no research seems o have followed his line of research, and so i is no clear wha he poenial impac of he condiional variance misspecificaion migh be on he condiional correlaion dynamics DCC Canno Be Disinguished Empirically from Diagonal BEKK in Small Sysems In [18], i has been shown ha he esimaes of he dynamic condiional correlaions from a scalar BEKK model in effec an indirec DCC represenaion are very similar o hose from he DCC represenaion. This suppors he argumen ha he DCC represenaion can mimic dynamic condiional correlaions, a leas for small financial porfolios. Theoreical argumens o suppor his claim are presened in [9]. However, here is no empirical evidence of he similariies/dissimilariies

10 Economerics 2013, beween he dynamic correlaions obained from he DCC represenaion and hose obained, for insance, from a diagonal BEKK model. As a consequence, we canno verify if he empirical fi provided by he DCC represenaion migh be beer han he fi obained from a more general BEKK model in which he dependence across he condiional variance and covariances is aken ino accoun. Clearly, he advanage of he DCC represenaion is in he ease of esimaion, bu mainly in large sysems DCC May Be a Useful Filer or a Diagnosic Check, bu I Is No a Model A significan problem in empirical pracice is ha many users seem o be under he misapprehension ha DCC is a model when i is no. DCC has no obvious or desirable mahemaical or saisical properies. Neverheless, DCC may be a useful filer or a diagnosic check ha can capure he dynamics in wha are purpored o be condiional correlaions, even if hey arise hrough possible model misspecificaion. In his conex, he DCC filer may perform well empirically. In fac, he populariy of he DCC represenaion is moivaed by wo main elemens, namely he ease in esimaion, and he abiliy of he filer o capure he possible presence of dynamic correlaions of condiional variance misspecificaion. Consequenly, he DCC filer may play a useful role in forecasing ou-of-sample dynamic condiional covariances and correlaions. 3. Conclusions The paper discussed en hings poenial users should know abou he Dynamic Condiional Correlaion (DCC) represenaion for esimaing and forecasing ime-varying condiional correlaions. The reasons given for being cauious abou he use of DCC included he following: DCC represens he dynamic condiional covariances of he sandardized residuals, and hence does no yield dynamic condiional correlaions; DCC is saed raher han derived; DCC has no momens; DCC does no have esable regulariy condiions; DCC yields inconsisen wo sep esimaors; DCC has no asympoic properies; DCC is no a special case of GARCC, which has esable regulariy condiions and sandard asympoic properies; DCC is no dynamic empirically as he effec of news is ypically exremely small; DCC canno be disinguished empirically from diagonal BEKK in small sysems; and DCC may be a useful filer or a diagnosic check, bu i is no a model. The compuaional advanages of he DCC represenaion migh become relevan when focusing on large sysems. However, here is no empirical evidence on he comparison of condiional correlaions obained direcly from he DCC represenaion and indirecly from he BEKK model in a large cross secion of asses. As a resul, we canno verify if he use of he DCC filer provides condiional pahs ha are similar o hose obained from a viable alernaive model. On he oher hand, he BEKK model is more general as i allows for direc spillovers and feedback effecs across condiional variance and covariances, as well as indirec spillovers and feedback effecs across condiional correlaions. The GARCC model is also a viable alernaive as i saisfies he definiion of a dynamic condiional correlaion marix, and also has demonsrable, as disinc from assumed, regulariy condiions and asympoic properies.

11 Economerics 2013, As DCC is presenly he mos popular represenaion of dynamic condiional correlaions, poenial users are srongly encouraged o undersand and appreciae he limis of DCC in order o be able o use i as a sensible filer or as a diagnosic check for esimaing and forecasing dynamic condiional correlaions. Acknowledgmens The auhors mos are graeful o wo referees for very helpful commens and suggesions. For financial suppor, he second auhor wishes o acknowledge he Ausralian Research Council, Naional Science Council, Taiwan, and he Japan Sociey for he Promoion of Science. An earlier version of his paper was disribued as Ten Things You Should Know abou DCC. Conflic of Ineres The auhors declare no conflic of ineres. References 1. Engle, R. Dynamic condiional correlaion: A simple class of mulivariae generalized auoregressive condiional heeroskedasiciy models. J. Bus. Econ. Sa. 2002, 20, Cappiello L.; Engle, R.F.; Sheppard, K. Asymmeric dynamics in he correlaions of global equiy and bond reurns. J. Financ. Econ. 2006, 4, Bauwens, L.; Lauren, S.; Rombous, J.V.K. Mulivariae GARCH models: A survey. J. Appl. Econ. 2006, 21, Silvennoinen, A.; Terasvira, T. Mulivariae GARCH Models. In Handbook of Financial Time Series; Andersen, T.G., Davis, R.A., Kreiss, J.P., Mikosch, T., Eds.; Springer: Berlin, Germany, Lanza, A.; McAleer, M.; Manera, M. Modeling dynamic condiional correlaions in WTO oil forward and fuures reurns. Financ. Res. Le. 2006, 3, Billio, M.; Caporin, M.; Gobbo, M.; Flexible dynamic condiional correlaion mulivariae GARCH for asse allocaion. Appl. Financ. Econ. Le. 2006, 2, Hammoudeh, S.; Liu, T.; Chang, C.-L.; McAleer, M. Risk spillovers in oil-relaed CDS, sock and credi markes. Energy Econ. 2013, 36, Chang, C.-L.; McAleer, M.; Tansucha, R. Crude oil hedging sraegies using dynamic mulivariae GARCH. Energy Econ. 2011, 33, Franses, P.H.; Hafner, C.M. A generalized dynamic condiional correlaion model: Simulaion and applicaion o many asses. Econ. Rev. 2009, 28, Kasch, M.; Caporin, M. Volailiy hreshold dynamic condiional correlaions: An inernaional analysis. J. Financ. Econ. 2013, in press. 11. Engle, R.F.; Shephard, N.; Sheppard, K. Fiing Vas Dimensional Time-varying Covariance Models. Oxford Financial Research Cenre, Financial Economics Working Paper 30, Colacio, R.; Engle, R.F.; Ghysels, E. A componen model for dynamic correlaions. J. Econ. 2011, 164,

12 Economerics 2013, McAleer, M. Auomaed inference and learning in modeling financial volailiy. Econ. Theory 2005, 21, Aielli, G.P. Dynamic condiional correlaions: On properies and esimaion. J. Bus. Econ. Sa. 2013, in press. 15. Tse, Y.K.; Tsui, A.K.C. A mulivariae GARCH model wih ime-varying correlaions. J. Bus. Econ. Sa. 2002, 20, Baba, Y.; Engle, R.F.; Kraf, D.; Kroner, K.F. Mulivariae Simulaneous Generalized ARCH. Unpublished Manuscrip, Deparmen of Economics, Universiy of California, San Diego, CA, USA, Engle, R.F.; Kroner, K.F. Mulivariae simulaneous generalized ARCH. Econ. Theory 1995, 11, Caporin, M.; McAleer, M. Scalar BEKK and indirec DCC. J. Forecas. 2008, 27, Riskmerics TM. J.P. Morgan Technical Documen, 4h ed.; J.P. Morgan: New York, NY, USA, McAleer, M.; Chan, F.; Hoi, S.; Lieberman, O. Generalized auoregressive condiional correlaion. Econ. Theory 2008, 24, Engle, R.F. Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 1982, 50, Bollerslev, T. Generalised auoregressive condiional heeroscedasiciy. J. Econ. 1986, 31, Caporin, M.; McAleer, M. Do we really need boh BEKK and DCC? A ale of wo mulivariae GARCH models. J. Econ. Surv. 2012, 26, Engle, R.F.; Sheppard, K. Theoreical and Empirical Properies of Dynamic Condiional Correlaion Mulivariae GARCH. Naional Bureau of Economic Research working paper series 2001, no by he auhors; licensee MDPI, Basel, Swizerland. This aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion license (hp://creaivecommons.org/licenses/by/3.0/).