DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND

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1 DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH NEW ZEALAND Thresholds News Impac Surfaces and Dynamic Asymmeric Mulivariae GARCH* Massimiliano Caporin and Michael McAleer WORKING PAPER No. 3/00 Deparmen of Economics and Finance College of Business and Economics Universiy of Canerbury Privae Bag 4800 Chrischurch New Zealand

2 WORKING PAPER No. 3/00 Thresholds News Impac Surfaces and Dynamic Asymmeric Mulivariae GARCH* Massimiliano Caporin and Michael McAleer April 00 Absrac: DAMGARCH is a new model ha exends he VARMA-GARCH model of Ling and McAleer (003) by inroducing muliple hresholds and ime-dependen srucure in he asymmery of he condiional variances. Analyical expressions for he news impac surface implied by he new model are also presened. DAMGARCH models he shocks affecing he condiional variances on he basis of an underlying mulivariae disribuion. I is possible o model explicily asse-specific shocks and common innovaions by pariioning he mulivariae densiy suppor. This paper presens he model srucure describes he implemenaion issues and provides he condiions for he exisence of a unique saionary soluion and for consisency and asympoic normaliy of he quasi-maximum likelihood esimaors. The paper also presens an empirical example o highligh he usefulness of he new model. Keywords: mulivariae asymmery condiional variance saionariy condiions asympoic heory mulivariae news impac curve. JEL Classificaions: C3 C5 C5 Acknowledgemens: The auhors would like o hank Monica Billio Gabriele Fiorenini Paolo Paruolo Domenico Sarore and he paricipans a he Second Ialian Congress of Economerics and Empirical Economics 007 (Rimini Ialy) and he Workshop in Compuaional and Financial Economerics 007 (Geneva Swizerland) for helpful commens and suggesions. The firs auhor acknowledges financial suppor from he Ialian MUR proec Cofin Economeric analysis of inerdependence sabilisaion and conagion in real and financial markes. The second auhor wishes o acknowledge he financial suppor of he Ausralian Research Council Naional Science Council Taiwan Cener for Research on he Inernaional Economy (CIRJE) Faculy of Economics Universiy of Tokyo and a Visiing Erskine Fellowship College of Business and Economics Universiy of Canerbury New Zealand. Deparmen of Economic Sciences Universiy of Padova Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands *Corresponding Auhor: Massimiliano Caporin massimiliano.caporin@unipd.i phone: fax:

3 WORKING PAPER No. 3/00 Thresholds News Impac Surfaces and Dynamic Asymmeric Mulivariae GARCH*. Inroducion Saring wih he seminal work of Engle (98) and Bollerslev (986) for univariae models and Bollerslev (990) and Engle and Kroner (995) for mulivariae models he modeling of condiional variances covariances and correlaions has araced considerable ineres in he risk and financial volailiy lieraure. Several exensions and generalizaions have been suggesed for boh he univariae and mulivariae represenaions (see for example Bollerslev Chou and Kroner (99) Bollerslev Engle and Nelson (994) Li Ling and McAleer (00) McAleer (005) and Bauwens Lauren and Rombous (006)). The numerous proposed models have been applied o vasly differen daa ses including exchange rae forecasing sock price volailiy predicion and marke risk measuremen hrough Value-a-Risk forecass. In comparison wih he developmen of model specificaions he heoreical conribuions have been limied. In fac he condiions for he exisence of a unique saionary and ergodic soluion and for he asympoic heory of he parameer esimaes have become available only for a subse of he proposed models (among ohers see Bougerol and Picard (99) and Ling and McAleer (00a) (00b) for univariae GARCH models Jeanheau (998) Come and Lieberman (003) Ling and McAleer (003) McAleer e al. (008) and Hafner and Preminger (009) for mulivariae GARCH MGARCH henceforh models). Furhermore in he mulivariae model case he diagnosic checking of model adequacy is poorly covered in he lieraure being resriced o some recen papers considering mulivariae exensions of he well-known Lung-Box es saisic (see Ling and Li (997) and Tsay (998)). One of he mos imporan opics in he financial economerics lieraure is he asymmeric behavior of condiional variances. The basic idea is ha negaive shocks have a differen impac on he condiional variance evoluion han do posiive shocks of a similar magniude. This issue was raised by Nelson (990) in inroducing he EGARCH model and was also considered by Glosen Noe ha he VARMA-GARCH model proposed by Ling and McAleer (003) ness some oher mulivariae GARCH represenaions including he CCC model of Bollerslev (990). However his class of models is non-nesed wih respec o he BEKK and Vech GARCH represenaions developed in Engle and Kroner (995) (see also Caporin and McAleer (008)).

4 Jagannahan and Runkle (99) Rabemananara and Zakoian (993) and Zakoian (994) for he univariae case. For hese models some general resuls apply including he condiions for saionariy and asympoic heory for he quasi-maximum likelihood esimaes (see Ling and McAleer (00a) and (00b)). However resricing aenion o only a single asse may be oo sringen paricularly if he primary goal is he measuremen of he risk of an invesmen or a porfolio. In such cases we could be ineresed in analyzing he effecs of a shock on a se of asses wih a possible disincion beween asse-specific shocks and marke shocks. In addiion o he possible mean effecs his paper focuses on he variance and covariance effecs monioring an asse s condiional variance reacion o anoher asse s specific shock. For insance he model we propose may be used o sudy he effecs of an oil price shock on oil price volailiy and on he volailiies of he socks belonging o he auo secor or he effecs of a marke shock (ha could be represened by an unexpeced macroeconomic shock) on he condiional variances of a se of socks. Furhermore in our modeling approach we will disinguish he effecs of a shock s sign from hese coming from he shock s size. Noe ha he possible combinaion of sign and size may depend on he oher asse s sign and size wih increased complexiy according o he chosen mulivariae framework. A relaed issue he so-called leverage effec (negaive shocks should increase condiional variances while posiive shocks should induce a reducion in he condiional variances) will no be addressed given ha our main focus is on condiional variance asymmery. Informaion on variance asymmery could be useful for asse pricing porfolio consrucion (given he relaionship of such a shock-propagaion mechanism wih he asse correlaions and heir beas) and for marke risk measuremen (see among ohers Hafner and Herwarz (998) Hansson and Hordahl (998) and he references in Bauwens e al. (006)). The srucures needed o monior esimae and use he condiional variance asymmeries should be included in an appropriae mulivariae model. A recen conribuion in his direcion was McAleer Hoi and Chan (009) who provided a mulivariae generalizaion of he GJR model of Glosen e al. (99). However heir approach is limied o a specific disincion beween posiive and negaive shocks and is based on an exension of he univariae analysis. The MGARCH lieraure includes several models wih asymmery wih ineresing examples given in Kroner and Ng (998) and De Goei and Marquering (004). In he cied papers he asymmery erm eners eiher a Vech or BEKK represenaion (for definiions see Engle and Kroner (995)). However McAleer Hoi and Chan (009) seem o be Noe ha he erm leverage is used by some auhors o idenify wha we call asymmery ha is a differen effec of negaive and posiive shocks of he same magniude on he condiional variance (for an example see Bauwens e al. (006))

5 he only auhors o have deal wih asymmery in he VARMA-GARCH model of Ling and McAleer (003). Noe ha he VARMA-GARCH model ness he CCC model of Bollerslev (990) so ha he inroducion of asymmery in he VARMA-GARCH model of Ling and McAleer (003) can be exended direcly o he variance dynamics in he CCC model and hence also he variance and covariance dynamics in he DCC model of Engle (00) and he GARCC model of McAleer e al (008). Despie he lack of heoreical conribuions dealing wih asymmery and variance spillovers in he DCC model he economeric lieraure includes several papers dealing wih simple GARCH() specificaions and asymmeric effecs in he correlaions (see among ohers Cappiello Engle and Sheppard (006)). The model o be presened here does no allow he correlaion marix o follow a dynamic evoluion such as in he DCC model. This choice is moivaed by he fac ha we also aim a providing heoreical resuls which could have no been derived by he inclusion of dynamic correlaions. Our modelling approach could be exended following he GARCC specificaion of McAleer e al. (008) ha allow for correlaion dynamics and possibly also asymmery in he correlaions. The purpose of his paper is o provide a general framework in which boh mulivariae variance asymmery and spillover effecs are considered o derive he condiions o ensure he exisence of a unique saionary and ergodic soluion and o prove he consisency and asympoic normaliy of he Quasi-Maximum Likelihood Esimaor (QMLE) for he parameers of ineres. In addiion o he radiional asymmeric effec we include ime dependence in he asymmeric componen of he variances hereby exending he ideas of Caporin and McAleer (006). We propose he Dynamic Asymmeric MGARCH (DAMGARCH) model ha allows for ime-varying asymmery wih spillover effecs. The ineracions beween variances may depend boh on a direc relaion beween he condiional variances (as in sandard MGARCH models) and on spillover effecs from he asymmeric componen of he GARCH model. As DAMGARCH is a generalizaion of he DAGARCH model of Caporin and McAleer (006) i inheris many of he properies of DAGARCH namely he possibiliy of explaining asymmery as well as persisence in asymmery. DAMGARCH also represens a generalizaion of he VARMA-GARCH model of Ling and McAleer (003). Therefore i is non-nesed wih respec o he Vech class of models of Engle and Kroner (995). Noe ha he DAMGARCH model generalizes he exising MGARCH models wih asymmery for he inclusion of spillovers in he asymmery erm for he definiion of asymmery over a se of hresholds and for he generalizaion of he indicaor funcions ha drive he asymmeric effec. Clearly he price o pay for such a generalizaion is he increase in he number of parameers and he subsequen complexiy of model esimaion. However following he sandard

6 pracice in his srand of he lieraure resriced parameerizaions ha do no affec oo much he model flexibiliy could be considered. Differenly o miigae he compuaional complexiy of full model esimaion we propose a muli-sage esimaion approach. We presen a simple empirical analysis o compare a bivariae DAMGARCH model wih basic CCC specificaions where he condiional variances follow a sandard GARCH() model he asymmeric GJR-GARCH of Glosen e al. (99) or he DAGARCH model of Caporin and McAleer (006). The model proposed provides a higher likelihood and relevan insighs ino he asymmeric dynamics in he DAX and FTSE sock marke indices. Throughou he paper we use he following noaion: : denoes horizonal marix concaenaion; ρ ( A) idenifies he eigenvalue of marix A wih larges absolue value; vec(a) sacks he columns of marix A; vecu(a) sacks he columns of he lower riangular par of A below he main diagonal; diag(a) is a diagonal marix wih he vecor a on he main diagonal; dg(a) is he vecor conaining he elemens on he main diagonal of A; denoes he Hadamard marix muliplicaion. The remainder of he paper has he following srucure. Secion defines he DAMGARCH model and considers hree specific issues namely he definiion of hresholds (subsecion.) asympoic properies of he model and of he QMLE (subsecion.) and esimaion of DAMGARCH (subsecion.3). In Secion 3 we inroduce he News Impac Surface and presen a simulaed example of he possible forms of he funcion depending on he relaions beween he condiional variances. Secion 4 presens an empirical analysis of wo of sock marke indices comparing DAMGARCH wih a se of CCC models. Secion 5 gives some concluding commens.. DAMGARCH: Mulivariae GARCH wih Dynamic Asymmery In wha follows Y represens an n-dimensional vecor of observable variables. The primary focus is on he mean residuals under he following equaions: Y = E Y I ε E ε I 0 E εε I + = =Σ = D RD () in which I is he informaion se available a ime E Y I is he condiional mean of Y and ε is he n-dimensional mean residual vecor a ime. The mean residuals have a

7 condiionally ime-dependen covariance marix ha can be decomposed ino he conribuions of he condiional variances and he condiional correlaions 3. Finally is a diagonal marix of condiional volailiies given by: D diag ( σ σ... σn ) D = and R is he correlaion marix. Noe ha as saed in he inroducion we use a consan correlaion marix oherwise heoreical resuls could have no been derived. We also sress ha heoreical resuls for he DCC model of Engle (00) and for all specificaions derived from he DCC have no been rigorously proved; see he discussion in Caporin and McAleer (009). I is also assumed ha he sandardized and uncorrelaed innovaions η = Γ D ε are independen wih ΓΓ = R. Noe ha as disinc from sandard pracice Γ which is a full symmeric marix is no obained by a Cholesky decomposiion of he correlaion marix. Differenly Γ comes from he eigendecomposiion of he correlaion marix. In fac R U U = where U is he marix of eigenvecors and is he diagonal marix of eigenvalues. Using he fac ha UU = I he ideniy marix Γ can be se o le z = D ε denoe he sandardized innovaions wih R as he correlaion marix. ½ Γ= U U. Finally Define he vecors of condiional variances and squared innovaions as H = σ σ... σ = dg D D and = ( ) e ε ε... ε respecively. The following equaions ( ) ( ) n n define he Dynamic Asymmeric MGARCH (hereafer DAMGARCH) model: s H = W+ BH + G i i m i= m= () r { ( )} l where G m= A m+ψ mg m I ( ε m) ( ε m d ) ( ε m d ) (3) l = { } and G = A +Ψ G I ( ε ) m m m m m = (4) 3 I is implicily assumed ha he covariance dynamics are a by-produc of he condiional variances and dynamic condiional correlaions. Therefore he Vech and BEKK represenaions (see Engle and Kroner (995)) are no direcly comparable wih he model developed in his paper.

8 where Bi i =... s Am =... l m=... r Ψ m =... l m=... r and G are n-dimensional square marices W and G are n-dimensional vecors l is he number of subses in which he suppor of he mulivariae probabiliy densiy funcion of ε has been pariioned (ha is here may be l hreshold vecors 4 ). In addiion I ( ε ) is a scalar (or a diagonal marix) indicaor funcion 5 (is srucure will be furher specified below) ha verifies if he vecor ε (each i componen of ε ) belongs o subse of he oin suppor (of he marginal suppor) d = 0 or d = d where d is an n-dimensional vecor ha defines he upper (or lower) bounds of subse (we will address below he srucure of he subses he srucure of d and is usefulness 6 ). We highligh here ha he vecor d = 0 is a selecion vecor allowing a direc inclusion of he hresholds or bounds d in he GARCH equaion. Noe also ha he vecors d characerize he hresholds used in he definiion of he asymmeric componens. These hresholds are no necessarily explicily included in he GARCH equaion (when d = 0 ) while hey define he pariions of ε in all cases (and hus always ener in he funcions I ( ε ) as will be shown below). We call d he vecors of observed hresholds and he definiion will be moivaed below. In he derivaion of he asympoic properies we will use an alernaive represenaion of he DAMGARCH model which is given in Appendix A.. The wo erms and G m G mdefine he ARCH componen of he model: he firs is a funcion dependen on he hresholds and he second erm which in urn drives he dynamic asymmery. Finally we noe ha G in (3) is measurable wih respec o he informaion se a ime bu i eners equaion () wih a lag of a leas. Addiional commens on he inerpreaion of hese wo elemens are given below. The indicaor funcion and herefore he number of hresholds (or number of subses) can be defined no only on he innovaion vecors bu also on a larger number of erms. In fac i is 4 The erm hresholds is no appropriae when considering a mulivariae densiy for which he hreshold may be a vecor wih differen componens as he marginal densiy may have differen hresholds. In dealing wih mulivariae densiies reference will insead be made o a pariion of he densiy suppor ha defines some subses. 5 If a scalar he funcion I ( ε ) assumes he value if he vecor ε belongs o subse and 0 oherwise. If a diagonal marix each elemen in he diagonal of I ( ε ) is an indicaor funcion based on he marginal of he mean residuals I ( ε i ) and assumes he value if he elemen ε i belongs o is specific subse and 0 oherwise. 6 Noe ha he inclusion of d induces a coninuous news impac surface as will be shown below. The represenaion adoped here generalises he coninuous news impac curve of Engle and Ng (993) and Caporin and McAleer (006) o he mulivariae case.

9 possible o generalize I ( ε ) o I ( ε ε... ε m). However in his case he number of hresholds (or subses) may increase appreciably. Considering only he sign of he innovaion a single lag leads o l = while he use of wo lags leads o l = 4 wih an exponenial increase in he number of pariions. Pu differenly we can generalize equaions (3) and (4) by increasing he number of lags for he erms G and G. In his case we can wrie: l ( ) { ( ) ( ε )} G = A +Ψ LG I m m m = q Ψ L =Ψ L+Ψ L Ψ L m m m mq (5) wih an obvious increase in he number of parameer marices (similarly for G ). Secion.3 considers he esimaion problem and includes a discussion of he role of he number of parameers and feasible represenaions. Noe ha equaion () defines he dynamics of he condiional variances on he basis of (i) pas condiional variances and (ii) pas squared innovaions. While he firs erm represens he sandard GARCH componen he second erm does no explicily include he sandard ARCH componen. In fac he represenaion we choose can be recas wih a slighly differen srucure showing he asymmeric variance dynamic as an addiion o he VARMA-GARCH srucure of Ling and McAleer (003). Acually in a simple case assuming d =0 = r=s= (ha is where he hresholds do no direcly ener in he ARCH componen definiion which allows he omission of G ) we can wrie he ARCH coefficiens of he model as follows: ( A ) { ( )} H = W+ BH + + G e l G = A +Ψ G I ε = (6) so ha we can rewrie he ARCH erm as

10 l { ( )} A+ G = A + A +Ψ G I ε = (7). Equaion (7) includes he radiional ARCH erm A and addiional ARCH marices he A =... l marices which are modifying he ARCH coefficiens depending on he hresholds and a dynamic asymmeric componen G. Equaion (7) also highlighs ha a sufficien condiion for he idenificaion of boh he A (wih =... l) and he A marices requires ha a las one of he A =... l marices mus be se o zero. In his case he marices A =... l will define he differenial effecs on he condiional variances (ARCH componen) of each subse wih respec o a baseline subse. I may be considered a mulivariae generalizaion of he GJR model where we have a sandard ARCH coefficien and an addiional coefficien ha is added o he ARCH componen only for negaive shocks. In he DAMGARCH model we have a baseline ARCH componen referred as an example o small posiive innovaions and addiional effecs for negaive small innovaions negaive large innovaions and posiive large innovaions. Furhermore in equaion (3) we allowed for he direc dependence of he ARCH par from he hresholds in order o induce a coninuous news impac curve as in Engle and Ng (993). Finally he ARCH par of DAMGARCH can also be inerpreed as he sum of wo componens: leing r= for simpliciy we may idenify a sandard ARCH par (hreshold dependen and hence asymmeric) ha is given by l AI d d = ( ε ) ( ε ) ( ε ) and a second erm associaed o he asymmery persisence: l Ψ G I d d. = ( ε ) ( ε ) ( ε ) This second erm includes a componen ( ) G ha carries he asymmeric effec up o ime - (depending on he informaion se a ime -). The asymmeric behavior of ime condiional variances depends hus on pas shocks (hrough he firs erm) bu also on he asymmery behavior

11 in he previous period (hrough he second erm). The paern of asymmery effecs over he condiional variances. G will capure he ime-varying Pu differenly we can define DAMGARCH as a MGARCH model in which he ime-varying ARCH coefficiens A +Ψ G = l depend on he pariion o which ime - shock vecor belongs namely he A marices and on an auoregressive componen ha drives he persisence in he ARCH coefficiens as parameerized by he Ψ marices. Caporin and McAleer (006) provide a deailed discussion of he inerpreaion of he DAGARCH coefficiens which can be generalized direcly o he DAMGARCH model. A deeper discussion of he indicaor funcion is required. We propose wo alernaive srucures which are defined over he mulivariae densiy of he mean innovaion vecor ε and over he marginal densiies of he univariae mean innovaions ε i respecively. Consider he use of he mulivariae densiy. In his case define mulivariae innovaion densiy so ha: n S as he suppor of he I ( ε ) ε S = 0 oherwise (8) where S is a subse of S. Furhermore we have l = S = S S S = Ø i =... l i. i (9) As an example we may define he following hree subses of S : { ε: εi L...n } { ε εi U } S = < d i = S = : > d i =...n 3 S = S S S. 3 (0)

12 In his example assuming ha d L is a small negaive number and d U is a large posiive number he pariion disinguishes exreme evens from he remaining elemens of S. The direc dependence of variances from hresholds (see equaion 3) was inroduced in order o induce coninuiy of he news impac o condiional variances. When he indicaor funcions are defined over he mulivariae densiy suppor coninuiy may no be simply achieved. In fac hresholds may have a complex represenaion (see he examples in Appendix A.). In hese cases we mus se d = 0 and coninuiy may be obained appropriaely defining he A coefficien marices associaed o he pariion. In he example of equaion (0) we should se A =A +A a and A 3 =A +A b where A a and A b are parameer marices. Noe his is similar o having a sandard ARCH coefficien and wo addiional componens associaed wih exreme evens of eiher posiive or negaive sign and a baseline ARCH componen A. Noe ha he hresholds in his case bracke vecors. Pu differenly we may define he hresholds over he marginal densiies of he innovaions. In his case we may define he I ( ε ) funcion as a diagonal marix of dimension n wih I ( ε i ) on he main diagonal. In urn I ( ε i ) is he indicaor funcion for he inclusion of ε i in he -h subse defined over he probabiliy densiy suppor of ε i. The I ( ε ) indicaor funcion is he univariae counerpar of equaion (8) namely: i if d = d ( εi ) i i I = =... k ( εi ) ε < d 0 oherwise ε > d i i I = = k+... l 0 oherwise (.a) ( ) if d = 0 di < εi di I εi = =... l 0 oherwise (.b) where he subse is expressed as a segmen on he suppor of he probabiliy densiy funcion of ε i. Furhermore for = (ha is he firs subse) he condiion in (ab) is εi di while for = l (ha is he las subse) he condiion becomes εi > dil wih d <... < d < 0 < d <... < d i ik ik+ il ha is he k-h hreshold is equal o zero for all variables. The las assumpion is imposed in order o simplify he model srucure. Finally he indicaor funcion disinguishes posiive and negaive values in order o induce coninuiy in he news impac surface which will be defined below. Noe

13 ha by defining he hresholds over he marginal suppor we skip possible problems associaed wih he brackeing of vecors given ha we bracke single elemens of he innovaion vecor. Furhermore a single vecor may have elemens saisfying differen funcions I ( ε ) wihou any consrain adding flexibiliy o he model. Noe ha he observed hreshold equivalence (apar from he zero hreshold) over differen componens of he innovaions was no imposed in () implying ha we can have differen observed hreshold values for each ε i. In he following secion we will show ha he indicaor funcion defined in (8) may be more general han ha defined in (). Furhermore we will show how o represen he indicaor funcion ha follows () in he form of (8) by defining he pariions of ε densiy funcion suppor appropriaely. If we follow () in defining he indicaor funcions hen he elemens of d may be differen over he variables and are defined accordingly o he srucure of ( ) = for =... k d = { d... d l} for = k+... l and d = 0n for = kk +. Under (8) he definiion of I ε i namely d { d... d l } d depends on he relaions used o define he S subses. In he example in (0) we have d = di Ln d = 0 n and d3 = duin where i n is an n-dimensional vecor of ones and 0 n is an n- dimensional vecor of zeros. However noe ha he definiion of he d elemens may be more complex under (8) han in () as will be shown below. Finally noe ha he indicaor funcion defined on he oin probabiliy suppor can be represened in marix form (insead of he scalar case previously used) by simply replacing one wih an ideniy marix. The developmen of he DAMGARCH model is similar in spiri o Ling and McAleer (003) McAleer e al. (007) and McAleer e al. (009). In fac assuming a consan correlaion marix and imposing he condiion ha G = Ae (an n-dimensional square parameer marix ha is no influenced by asymmeric behavior) yields he VARMA-GARCH model of Ling and McAleer (003). Moreover he GARCC model proposed in McAleer e al. (008) could be obained assuming a ime-dependen srucure for he condiional correlaion marix again under he resricion G = Ae. A relaed approach was used in McAleer e al. (009) for he inroducion of asymmeric condiional variances in he MGARCH framework. In his case he appropriae marix is given by: G = + AI ( ε ) A e

14 in addiion he marix indicaor funcion is defined as in () wih a single hreshold se o zero for all ε i and wih he explici inclusion of he ARCH parameer marix. This model is also he mulivariae counerpar of he GJR model of Glosen Jagannahan and Runkle (99) wih consan correlaions. Is represenaion using a pariion over he suppor of ε includes n subses associaed wih all he possible combinaions of posiive and negaive values in he elemens of ε. As he DAMGARCH model can nes all he previous cases i follows ha he CCC model of Bollerslev (990) he DCC model of Engle (00) and he varying condiional correlaion (VCC) model of Tse and Tsui (00) may be also considered as special cases of DAMGARCH. The CCC model is obained by seing G = Ae assuming ha he marices A and B are diagonal and ha he model has a consan condiional correlaion marix. The DAMGARCH model exends curren mulivariae represenaions of GARCH by inroducing muliple hresholds and ime-dependen asymmery. However DAMGARCH has a similar limiaion of he sandard mulivariae represenaion namely he problem of (high) dimensionaliy 7. In order o resolve his problem diagonal represenaions can be used such as a separae univariae DA-GARCH model for each innovaion variance. Diagonaliy implies ha all he parameer marices are diagonal while no resricions are imposed on he hresholds which could differ according o he variables involved. Furhermore block srucures could be considered as in Billio e al. (006). In ha case he parameer marices could be pariioned and resriced on he basis of a paricular asse classificaion... Defining Thresholds and Model Specificaions As given in Caporin and McAleer (006) he use of muliple hresholds wih ime-varying condiional variances may creae problems in he definiion of hresholds. In fac if he hresholds are designed o idenify he queues of he innovaion densiy hey mus be defined over he sandardized innovaions as he hresholds should adap o movemens in he condiional variances. Consider a simple example in which a ime series follows a GARCH() process bu wihou any mean dynamics. If we focus on he upper α -quanile of he mean disribuion his quanile is a funcion of he condiional variance and of he quanile of he sandardized innovaion densiy. 7 See Table for parameer number of several MGARCH models.

15 Thus in univariae represenaions hresholds have o be defined over he sandardized innovaion eiher by fixing a se of values or a se of perceniles a priori. Coninuing wih his example assume ha he lowes hreshold for mean innovaions ε is fixed a d so ha he indicaor funcion for his case is I( ε ) = ( ε < d ). The probabiliy associaed wih L his indicaor funcion gives: L ( ε < L) = ( σ < L) = ( < σ L) = z( σ L) P d P z d P z d F d () where F z (.) is he cumulaive densiy of he sandardized innovaions z. We noe ha he probabiliies are funcions of he condiional variance. Therefore fixing a value for d L is no equivalen o defining a quanile on he mean innovaion probabiliy densiy funcion. A more appropriae hresholds formulaion should consider fixing heir value over he sandardized innovaion densiy (see Caporin and McAleer (006) for furher deails). A similar srucure is needed for mulivariae represenaions as hresholds mus hen be defined over sandardized innovaions. However a furher difficuly arises wih regard o he definiion of hresholds according o he oin or marginal densiies. The wo approaches are equivalen if and only if here is zero correlaion among he variables. For his reason we believe ha hresholds have o be defined over he sandardized and uncorrelaed innovaions ha is on he innovaions defined as η = Γ D ε where Γ is a symmeric marix obained from he specral decomposiion of he R correlaion marix saisfying ΓΓ = R. Making a parallel wih simulaneous equaion sysems he shocks η may be compared wih he srucural shocks and depending on heir values hey affec he observed shocks ε and heir variance dynamics. Following his saemen we noe ha he observed hresholds d defined over he mean innovaions ε will be ime dependen: in fac we define hem as d = DΓ d where d represens he vecor of srucural hresholds defines wih respec o η. Observed hresholds may be ime dependen given ha hey hresholds are a funcion of condiional sandard deviaions and condiional correlaions boh of hem possibly ime dependen. Differenly he srucural hresholds are assumed o be ime independen

16 In he following i is assumed ha he hresholds are fixed over he probabiliy densiy funcion of he η. The srucural hresholds d = F ( α) i 8 d can be fixed a priori or deermined by a quanile relaion. Furhermore he erm hresholds will be used only wih respec o he marginal densiies while he erm suppor pariions will be used wih respec o he oin densiy. Noe ha he inroducion of a hreshold srucure on marginal or oin densiies will be equivalen only in special cases namely when he correlaions are all equal o zero. Thresholds and pariions can be defined as follows. Consider firs he definiion of hresholds over marginal densiies. Assume ha he hresholds are fixed over he componens of η. Finally define F (.) as he oin cumulaive densiy and F(.) i =... n as he marginal cumulaive densiies of he η. I follows ha: i DΓ εd id< di Γ i I ( εi ) = =... l 0 oherwise (3) where d is he vecor of srucural hresholds defined over he η innovaions. Noe ha he condiion in equaion (3) is based on he elemens of a ime-dependen hreshold vecor so ha he indicaor marix funcion is given by I ( ε) = diag( I ( ε ) I ( ε )... I ( εn )). Finally for = he condiion is εi D Γ d i while for = l he condiion is εi > D Γ dl i. Equaion (3) refers o he indicaor funcions as defined in (.b). The previous commens and he discussion on hreshold definiion are valid also for he indicaor funcions of (.a). The correlaion marix decomposiion we used differs from he sandard Cholesky one. In fac we preferred o consider a symmeric decomposiion based on eigenvalues and eigenvecors in order o exclude he ordering effecs induced by he Cholesky decomposiion. Pu differenly he pariion over he oin densiy of ε is defined as: dε < d I ( ε) = =... l 0 oherwise (4) 8 Noe ha he sandardised innovaions are also uncorrelaed so ha he hresholds and he quaniles may be defined over eiher he marginal or he oin disribuion funcion.

17 where he condiion is saisfied if and only if he vecor ε is included in he pariion of he oin probabiliy suppor. Specifically equaion (4) is equivalen o equaion (7) as we can wrie he subse as: { } Sε = d : ε < d i =... n =... l. i i i (5) Noe ha equaions (3) and (4) are no equivalen represenaions and is possible o move from one o he oher when he correlaions are all equal o zero and only in very special cases. Consider a bivariae example o illusrae he poin. Assume ha he correlaion beween he wo variables is equal o zero. Then he following figure represens a pariion ha can be obained using eiher he marginal or he oin hreshold definiion (specifically a single hreshold ha is se o zero): [Inser Figure here] For he marginal hreshold case we have l = and a single hreshold ha is se equal o zero. For he oin pariion we have l = 4 wih each subse idenifying a quadran of he Caresian plane. However Figure represens a suppor pariion which is defined under he oin probabiliy bu which canno be obained using he marginal hreshold definiion. [Inser Figure here] This pariion disinguishes beween he cases where boh variables are negaive and he remaining combinaions. The fac ha equaions (3) and (4) are equivalen does no mean ha he models defined over he oin or he marginal hresholds are also equivalen. In fac he represenaion (4) over he oin suppor is associaed wih a more flexible model. In he case of he marginal hresholds i follows ha: [ ] ( ε ) [ ] ( ε ) G = A +Ψ G I + A +Ψ G I (6) whereas over he oin suppor i follows ha:

18 4 ( ). (7) G = A +Ψ G I ε = The wo represenaions are based on he same oin suppor pariion. However he second represenaion is more flexible since i allows differen variance reacions for each of he four subses of he Caresian plane. The wo equaions are equivalen under he following parameric resricions: le x be he firs componen and y he second such ha = idenifies he subse wih boh componens negaive = 4 idenifies he subse wih boh componens posiive = defines he subse wih posiive x and negaive y and = 3 defines he subse wih negaive x and posiive y. I follows ha (6) and (7) are equivalen if: A = A 4 A. Ψ. = Ψ 4 Ψ.. A 3 = A A. 4 Ψ. 3 = Ψ Ψ. 4. (8) where. denoes he firs column of a marix. Appendix A. includes wo addiional examples on pariions defined over he oin suppor. When he correlaions are no zero he ransformaion of srucural hresholds ino observed hresholds may creae non-coniguous (or no dense) ses in he suppor of ε ha makes he brackeing of vecors as in equaion (4) almos impossible. Differenly when hresholds are defined over marginal densiies he only effec of he correlaions on he condiional variances is hrough he hresholds hemselves. In fac in he limiing case of diagonal specificaions he condiional variances are close o be driven by univariae DAGARCH models given ha he only link is in he hresholds (diagonal specificaions exclude any spillover effec in he GARCH coefficiens as well as in he ARCH and asymmery erms). Finally noe ha if he diagonal specificaion is coupled wih uncorrelaed sandardized residuals η hen he DAMGARCH model collapses exacly on a collecion of univariae DAGARCH models (now here are no links beween he condiional variances and he sandardized residuals). Wihin he DAMGARCH model he hresholds are no endogenous bu mus be fixed a priori on he basis of a disribuional assumpion for he srucural residuals η. As an example for he

19 rivariae case assuming mulivariae normaliy and noing ha η is a vecor of independen innovaions we may define hree hreshold vecors: zα 0 z α d = z α d 0 d 3 z = = α z 0 z α α where z α idenifies he α quanile of he univariae normal densiy. When he model has been esimaed he researcher may es he disribuional assumpions possibly updae he beliefs and re-esimae he DAMGARCH model. In addiion he hresholds may be defined on using he empirical densiies of he η. In his case an ieraive esimaion procedure should be use as we evidence in he esimaion secion... Saionariy and Asympoic Theory In his paper we focus on he variance model srucure. The inclusion of ARMA mean componens can be obained using he resuls in McAleer e al. (009). We sress ha we assume a consan correlaion marix R so ha he exension o ime-dependen correlaions can be obained as an exension of he resuls in McAleer e al. (008). In he following we provide he assumpions and he heorems saing he saionariy and he asympoic properies of he DAMGARCH model. All he proofs are repored in Appendix A.4. The assumpions and he heorems are a direc exension of he resuls in Ling and McAleer (003) refer o he DAMGARCH model as defined in equaions from () o (4). The model parameer vecor θ is defined as follows: W : vec ( A )...: vec( A lr ) : vec ( Ψ )...: vec ( Ψ lr ) θ =. : vec ( B )...: vec ( Bs ) : vecu ( R) In addiion following McAleer e al. (009) and McAleer e al. (008) we assume ha he parameer space Θ is a compac subspace of Euclidean space such ha θ is an inerior poin of Θ. Noe ha he hresholds are no included in he parameer vecor. Assumpion : E Y I = 0.

20 As a direc consequence of Assumpion he mean residuals are observable. Assumpion : The innovaions η = Γ D ε are independenly and idenically disribued. The srucural hresholds are defined over he η. The srucural hresholds are known. As saed in Assumpion we assume he knowledge of srucural hresholds which are fixed a quaniles of he underlying srucural innovaions following he descripion in he previous secion. The following addiional assumpions are needed o derive he condiions o ensure he exisence of a unique ergodic and saionary soluion o he DAMGARCH model. Assumpion 3: R is a finie and posiive definie symmeric marix wih ones on he main diagonal and ρ ( R) having a posiive lower bound over he parameer space Θ; all elemens of B i and E G z are non-negaive i= s = r (where G and z are defined in Theorem below); W has elemens wih posiive lower and upper bound over Θ; and all he roos of r s i i i i= i= i I E G z L BL = 0 are ouside he uni circle. Assumpion 4: r i i i= I E G z L and s i BL i are lef coprime and saisfy oher idenifiabiliy i= condiions given in Jeanheau (998) (he condiions are given in he proof o Theorem 3). Assumpion 5: a leas one of he following se of resricions is saisfied: i) he model has no dynamic asymmery effec (ha is he parameer marices Ψ... Ψ lr are all zero and hus he model collapses on a muliple hreshold asymmery specificaion which is a direc generalizaion of McAleer e al. (009)); ii) he parameer marices A... Alr Ψ... Ψ lr B... Bs are all diagonal. The resricions in Assumpion 5 are needed o obain he model srucural properies as generalizaions of he proofs in Ling and McAleer (003) and McAleer e al. (009). Theorem : Under assumpions ()-(5) he DAMGARCH model of equaions ()-(4) admis a unique second-order saionary soluion H measurable wih respec o he informaion se I -

21 where I - is a σ-field generaed by he innovaions z. The soluion expansion: H has he following causal H = W + M A ξ + i = i= (9) M = 0 : I : 0 n n (3 nlr+ ns) n 3 nlr n ( s ) (0) zg : zg : zg... zg : zg : zg zb... zb 0(3 nl n) 3 nlr 0(3 nl n) ns A = I r -r -r s 3 nl( r-) 3 nl( r-) 3nl 3 nl( r-) ns G- : G- : G-... Gr - : Gr - : Gr - B... Bs () = 0 I 0 ns ( -) 3 nlr ns ( -) ns ( -) n ( ) ( ) b c zw i e G G 0 ( ) W 0 ( ) ξ l 3nl r n s () ( εε ) ( ) ( ε ) e = dg e = dg d d e = dg d G m ( n nl) (3) ( A m+ψ mg m I ( ε m) ) A m+ψ mg m I ( ε m) ( ) ( Alm +Ψlm G m Il( ε m) ) : :... =...: (4) ( ( )) / [ ] ε and ( ) z = diag dg z z E z = I z = D D = diag H n (5) where he quaniy G comes from he alernaive represenaion of he DAMGARCH model described in Appendix A.. Hence { } Y H are sricly saionary and ergodic. The following heorem saes he condiions o ensure he exisence of momens for he DAMGARCH model.

22 E A < b Theorem : Under assumpions () o (5) if ρ ( ) hen he bh momens of Y are finie. Where b is a sricly posiive ineger and b denoes he Kroneker produc of b marices A defined in Theorem. We assume he coefficiens are esimaed by means of Quasi-Maximum Likelihood following Bollerslev and Wooldridge (99). A deeper discussion of he DAMGARCH esimaion and he relevan implemenaion issues is included in secion.3. In order o prove he consisency of he QML esimaes we inroduce he following assumpion on logarihmic momens as in Jeanheau (998). Assumpion 6: For any θ Θ we have E θ log ( Σ ) < where ( ½ ) ( ½ Σ ) = diag H R diag H and H is defined in (9). The following wo heorems define consisency and asympoic normaliy of he quasi-maximum likelihood esimaor for he DAMGARCH model parameers. Theorem 3: Define ˆ θ as he quasi-maximum likelihood esimaes of DAMGARCH. Under he condiions given by Jeanheau (998) repored in Appendix A.4 and he heorems in Ling and McAleer (003) we have ˆ p θ θ. Theorem 4: Suppose ha is Y generaed by equaions ()-(4) saisfying assumpions ()-(6). Given he consisency of he QMLE for DAMGARCH under condiions 4.i) 4.ii) and 4.iii) we have ( ˆ L θ θ) ( 0 ) n N Σ θ Ω θσ θ : L 4.i) θθ ' exiss and is coninuous in an open and convex neighbor of θ ; 4.ii) n L θθ ' ˆ θ converges in probabiliy o a finie non-singular covariance marix L θθ ' Σ θ = E n for any sequence ˆ θ such ha ˆ p θ θ ;

23 4.iii) n L θ ˆ θ converges in law o a mulivariae normal disribuion N ( 0 Ω ) θ wih covariance marix equal o L Ω θ = lim E n θ ˆ θ L θ' ˆ θ..3 Esimaion We already menioned ha he esimaion of DAMGARCH could be considered hrough a quasimaximum likelihood approach following Bollerslev and Wooldridge (99). This means ha we can define an approximae likelihood funcion L( θ ) ha depends on he condiional covariance T ( ). Tradiionally he approximae likelihood funcion is marix L( θ) = l ( θ) = l Σ ( θ) T = = derived from a mulivariae normal disribuion. In he MGARCH lieraure here also exiss a wosep esimaion approach ha considers univariae esimaion of he condiional variances and mulivariae esimaion of he correlaion parameers following Bollerslev (990) and Engle (00). I should be noed ha he wo-sep approach canno be used wih DAMGARCH for wo reasons: firs given he dependence of he condiional variance dynamics from he observed hresholds ha in urn are defined over he condiional variances and correlaions; second by he inclusion of possible spillover effecs across condiional variances in he radiional GARCH marix. The wosep approach could be used only under he srong assumpion of independence beween he mean residuals ε and absence of spillovers in he GARCH componen of he model. Noe ha even when all he parameer marices are diagonal (ha is when here are no spillovers beween variables) bu he correlaions are no zero he wo-sep approach canno be used given ha he observed hresholds sill depend on he correlaion marix (he compuaion of hresholds requires knowledge of he correlaion marix). Alhough all he parameers could be esimaed a leas in principle by maximizing he likelihood funcion he model complexiy creaes several implemenaion and numerical opimizaion problems. These are presen unless very resricive parameerizaions or limied dimension sysems are considered. By fiing he full model or wih even a moderae number of variables he full esimaion induces a sensible increase in he compuaional ime. Numerical opimizaion problems could be reduced by implemening firs-order derivaives which will be considered in fuure exensions and applicaions of he curren paper. Here in order o reduce he compuaional burden we sugges he following approximaed esimaion procedure. Recall ha he number of variables is denoed by n. Thus we sugges he following seps:

24 ) assume ha he sandardized and uncorrelaed residuals are disribued according o a sandard normal variables fix he srucural hresholds d for =...l a heoreical quaniles of he normal disribuion; ) esimae a sandard GARCH model on a univariae basis and save he condiional variances GARCH σ he sandardized residuals ( ε ) σ d = d =... l; GARCH i GARCH i η = ε σ for i= n and he hresholds GARCH i i GARCH i 3) esimae a univariae DAGARCH model (see Caporin and McAleer (006)) using he GARCHσ i d residuals hresholds and save he condiional variances η = ε σ for i= n; DAGARCH i i DAGARCH i DAGARCH σ and he sandardized 4) compue he uncondiional correlaion marix (using he sample esimaor) on he η = η : η :... η series and save he correlaion marix DAGARCH DAGARCH DAGARCH DAGARCH n R n he uncorrelaed residuals Rη = Rη : Rη :... Rη n and he hresholds d =Γ Dd =... l (as defined in equaion ()); DAMGARCH 5) es using sandard approaches he disribuion assumpion of sep ) and if necessary updae he d hresholds; noe ha he hreshold may be updaed eiher modifying he disribuional assumpion or by compuing hem using he empirical model residuals. i If we assume ha he model follows a diagonal specificaion in he GARCH condiional variance dynamics and all correlaions esimaed in sep 4) are zero hen he previous seps allow he complee model esimaion. The user us needs o validae he disribuional assumpions in sep 5) and if needed updae he esimaes of sep 3) and 4). Some ieraions of seps 3)-5) are needed if he hresholds are derived from empirical model residuals. Alernaively he algorihm should proceed wih he following seps. Noe ha Seps )-5) in his case are used o derive a reasonable vecor of coefficien saring values hereby reducing he compuaional ime. 6) esimae he condiional variance parameers by fixing he correlaion marix and using he hresholds defined in seps 4) and 5); hen save he condiional variances DAMGARCH σ and i he sandardized residuals η = ε σ for i= n; DAMGARCH i i DAMGARCH i

25 7) compue he uncondiional correlaion marix (using he sample esimaor) on he η = η : η :... η series and save he correlaion DAMGARCH DAMGARCH DAMGARCH DAMGARCH n marix R n he uncorrelaed residuals Rη = Rη : Rη :... Rη n and he hresholds ( )... d ε =Γ Dd = l (as defined in equaion ()); DAMGARCH 8) es he disribuion assumpion of sep ) using sandard approaches and if necessary updae he d hresholds; noe ha he hreshold may be updaed eiher modifying he disribuional assumpion or by compuing hem using he empirical model residuals; 9) ierae seps 6) o 8) unil convergence of he full model likelihood funcion (ieraions are needed because we separae he esimaion of condiional variance parameers from he esimaion of correlaions). Given he parameer esimaes sandard errors could be compued by numerical mehods on he full sysem likelihood (ha is by he oin use of numerical gradien and Hessian compuaion in a Quasi-Maximum Likelihood approach following Bollerslev and Wooldridge (99)). Clearly he proposed approach is subopimal bu full sysems esimaion is likely o be viable only for smalldimensional sysems. Noably when full sysem esimaion is considered ieraive esimaion approaches should be in any case used if he iniial assumpion of innovaion densiy is no suppored by he daa or if empirical hresholds are used. The asympoic properies of DAMGARCH are derived under an assumpion of model idenifiabiliy and using a normal likelihood (ha is he QML sandard approach). In he DAMGARCH model wo differen deviaions from normaliy may be considered: lepokurosis and asymmery in he underlying shock densiy (differen from he asymmery in he condiional variances capured by he model). Given ha he srucural hresholds are based on he underlying innovaions model performance could be improved if he densiy used o deermine hresholds is closer o he rue densiy. For his purpose seps 5) and 8) of he ieraive esimaion procedure allow he disribuional assumpion o be checked. Noe ha he asympoic resuls are no affeced by he misspecificaion of he likelihood funcion as hey are derived wihin a QML framework. A his poin a discussion of a feasible model srucure and on he number of parameers is needed. The general model has a very high number of parameers: recall ha n is he number of asses s is he GARCH order and r he ARCH order. Furhermore l is he number of hresholds (ha is we have l componens in he asymmeric GARCH srucure) and q is he order of he hreshold funcion G. Therefore he oal number of parameers is: n for he condiional variance consans

26 n s for he GARCH componen n ( q) l + for he hreshold componen and n ( n ) for he correlaion marix namely n n ( s l l q) n ( n ) /. Clearly his is an inracable number of parameers even for small dimensional sysems. However several resricions could be considered: he use of diagonal parameer marices (a he cos of excluding any spillover effecs among he condiional variances bu allowing for an easier muli-sep esimaion procedure); inroducing resricions on he asymmery dynamic (acing on he erm G ); fixing he number of hresholds a a small value such as one (l = ) for posiive-negaive or as an example o hree (l = 4) for disinguishing among large and small posiive (negaive) values; or a combinaion of all of he above resricions. Furhermore we can expec ha he sandard GARCH orders should be small possibly equal o ; similarly we may expec he hreshold dynamics order o be small. Finally noe ha if he model follows a pure ARCH dynamic (resricing s o zero) wo-sep esimaion procedures are direcly available. Table repors some examples resricing o hree he hreshold number imposing he sandard GARCH orders and fixing he asymmery dynamics order o one. The number of DAMGARCH parameers is also compared wih several alernaive models. We show ha he number of parameers in DAMGARCH is of order O(n ) namely he same order as he sandard BEKK model bu lower han he order of he general Vech model which is O(n 4 ). Furhermore he diagonal specificaion of DAMGARCH wih common dynamics in he asymmery has a parameric dimension ha is comparable o ha of he CCC model bu wih addiional ineresing properies. / [Inser Table here] 3. The News Impac Surface implied by DAMGARCH Engle and Ng (993) inroduced he news impac curve which is a useful ool for evaluaing he effecs of news on he condiional variances. The differen reacions of he condiional variances o posiive and negaive shocks moivaed he GJR and EGARCH represenaions of Glosen e al. (99) and Nelson (990) respecively. Boh models permi a richer parameerizaion of he news impac curve as compared wih he sandard GARCH model. As an exension Caporin and McAleer (006) provided he news impac curve in he presence of muliple hresholds and dynamic asymmery in condiional volailiy.

27 This secion provides a mulivariae exension of he news impac curve for he DAMGARCH model. Wihou loss of generaliy consider a simple model wih wo variables srucural shocks normally disribued hree hresholds se o zero o he 5% and 95% quaniles and all oher orders resriced o one. These values lead o he following DAMGARCH represenaion: H = W+ BH + G (6) { ( ε ) ( ε ) ( ε ) } 4 G = A +Ψ G I d d = 4 (7) { ( )} G = A +Ψ G I ε. = (8) The parameer marices have been se o he following specificaions: A = A A3 A = = = (9) W = B R 0.0 = = Ψ= Noe ha he firs (second) asse s condiional variance depends on he second (firs) asse s large posiive (negaive) shocks (see marices A and A 4 ). Furhermore he wo asses condiional variances are linked by a spillover effec (see marix B ). Finally Ψ is consan over. Tradiionally he news impac curve represens he variance movemens in response o an idiosyncraic shock assuming ha all pas variances are evaluaed a he uncondiional variance implied by he model. For he simple GARCH() model his implies: NIC z = ω + βσ + ασ (30)

28 where z represens he idiosyncraic componen. In he DAMGARCH model assuming ha he correlaions are consan he news impac surface is given by: NIS = W + BE [ H ] + ( A E[ G] ) diag dg I ( z)( z d )( z d +Ψ Γ Γ ) E[ H ] i= (3) where he derivaion of he formula and he expecaions are defined Appendix A.. Noe ha he News Impac Surface is coninuous by consrucion when hresholds are defined over he marginal densiy suppor as in (.a). Differenly when hresholds are defined over he mulivariae densiy suppor of η or when d = 0 he coninuiy of he News Impac Surface is no always guaraneed. As we have already argued i may be induced by an appropriae definiion of DAMGARCH parameer marices (see secion ). As an example we repor he News Impac Surfaces for he wo asse example for wo differen cases: he firs wih he coefficiens repored in (3) while he second is wihou any relaion beween he variances (i.e. all marices are diagonal excluding he correlaion marix). [Inser Figures 3 o 6 here] Noe ha when he asses are no correlaed and here is no spillover or asymmeric behavior he NIS collapses o he radiional News Impac Curve as shown for he firs asse in Figure 3 and for he second in Figure 4. When we inroduce spillovers he asymmeric componen comes ino play modifying he NIS for he moniored asses depending on he shocks affecing he oher asse (see Figures 5 and 6). If here is no spillover he NIS for each asse collapses on he News Impac Curve of a univariae DAGARCH model. 4. Dynamic Asymmeric Effec: an empirical example

29 This secion focuses on he esimaion of he DAMGARCH model and is comparison wih he simpler CCC-GARCH() CCC-GJR and CCC-DAGARCH models 9. We consider he daily closing levels of he DAX and FTSE 00 indices. The sample considered covers he period from 998 o 004 (734 daily observaions) and he daa were downloaded from Daasream. The wo markes are highly correlaed and may show srong dependence in he exreme reurns. Therefore we may expec a NIS ha is similar o ha repored in he previous secions. Table 5 repors he esimaed coefficiens of he DAMGARCH model while Tables 3 and 4 repor he CCC-GARCH() he CCC-GJR and he CCC-DAGARCH esimaes respecively. Furhermore Figures 7 and 8 repor he condiional variances esimaed by he hree models for he FTSE index and he percenage differences beween he CCC and DAMGARCH models. Figures 9 and 0 repor he NISs for he DAX and he FTSE implied by he DAMGARCH model. In his bivariae case we esimae a full DAMGARCH model wih hree hresholds and all oher orders se o. The hresholds were iniially fixed a zero a he upper and lower 0% ails under a sandardized normal disribuion for he uncorrelaed and variance sandardized residuals. Following he esimaion approach oulined in secion we verified he disribuion of he empirical model residuals which showed some deviaion from normaliy (asymmery for he FTSE reurns and mild lepokurosis for he DAX reurns). Given his observaion we decided o fix he hresholds using empirical srucural residuals and o ierae model esimaion and definiion of he hresholds unil convergence of he likelihood funcion (he sopping rule was se o a change in he likelihood value lower han -4 ). Convergence required only 4 ieraions. Some descripive saisics and he hreshold values used in he esimaion of DAMGARCH are included in Table 6. In addiion Figures and repor he frequency hisograms of he residuals. Graphical and descripive analyses show he differences beween he index innovaions and he impac of he deviaions from normaliy o he srucural hresholds (and is more eviden in FTSE). Noe ha in he following and in he Tables he parameer marices of DAMGARCH are mached wih a subscrip corresponding o he following pariion of marginal innovaions densiy: - large negaive values (below he lower hreshold 0% quanile); 9 The condiional variances follows in hese cases have he following specificaions: CCC-GARCH() σ ω αε βσ σ = ω + αε + γε I ε < 0 + βσ ; CCC-DAGARCH = + + ; CCC-GJR ( ) i i i i i i i i i 4 i = i+ i i + ( i + i i ) Ii ( zi )( i di ) di i i = σ ω βσ γ φ γ ε where = d σ z d are defined as he 5% 50% and 95% quaniles of a sandardized normal densiy i = ε σ he hresholds i i i 4 ( ) I ( z ) Ii ( zi ) = ( zi < di ) = and i ( i ) ( i i ) γ = γ + φ γ i i i i i i = I z = z > d = 3 4.

30 - negaive values; 3 - posiive values; 4 - large posiive values (above he upper hreshold 90% quanile). Three hresholds were also used for he DAGARCH specificaions and he coefficiens subscrip can be inerpreed as for he DAMGARCH model. The CCC models provide persisen condiional variance dynamics a finding ha is confirmed by he elevaed values of he B marix in he DAMGARCH model. The DAMGARCH model provides significan coefficiens in many parameer marices wih he excepion of he A marix associaed wih posiive innovaions. Similar findings are observed in he CCC-DAGARCH esimaes. There is an eviden inerrelaion beween he wo markes in paricular for large posiive shocks and he correlaion esimaed by DAMGARCH is similar o ha given by he CCC models. Comparing he fied condiional variances we noe some discrepancies in paricular during periods of high volailiy: he DAMGARCH peaks in he condiional variance seem in some cases o anicipae hose produced by CCC models an effec ha may be due o an improved forecasing abiliy. Furhermore he relaive percen changes of CCC models wih respec o he DAMGARCH ones reveals ha even if he paerns are very close (as we can observe in Figure 7) here are relevan differences in some cases he variances are doubles while in oher are halved. Noably he differences are very high even comparing DAGARCH and DAMGARCH condiional variances. Finally he log-likelihood provided by he DAMGARCH model is much higher han ha of he CCC models. Given here exiss a nesing relaionship beween he models sandard likelihood raio ess can be used (in he following p-values are derived presuming ha he asympoic densiy is he radiional one). These ess are in favor of he DAMGARCH model. The CCC-GJR model ness CCC-GARCH; in his case he es saisic has a value of 9.08 degrees of freedom and a p-value of less han -0. Differenly he likelihood comparison of CCC-GJR and DAMGARCH provides a es saisic equal o 8.5. The es has now 30 degrees of freedom: in he B marix (off diagonal coefficiens are zero) 6 in he Ψ i marices (all coefficiens resriced o zero) and in he A i marices (8 for zero resricions on he off-diagonal parameers plus 4 zero resricions for diagonal elemens in A and A 4 diagonal elemens in A and A 3 can be inerpreed as he ARCH effec for negaive and posiive shocks). In his case he P-value is less han -6 hereby supporing he inclusion of hreshold asymmery beween variances. Furhermore a comparison of he CCC- GJR and CCC-DAGARCH specificaions is in favor of he former. I seems ha he inclusion of muliple hresholds in he condiional variances is no useful a leas when we exclude any spillover across variances. Noe ha in he CCC-DAGARCH model we posulae ha he hresholds do no depend on he correlaion marix. More ineresingly he comparison of CCC-GJR and

31 DAMGARCH shows ha he mulivariae model wih dynamic asymmery variance spillovers and hresholds correlaion dependen provide relevan improvemens wih respec o radiional MGARCH models. To conclude he News Impac Surfaces repored in Figures 9 and 0 are similar o he example discussed in Secion 3. They show ha boh he DAX and FTSE condiional variances depend on he oher asse shocks wih a more pronounced effec in he case of DAX. [Inser Figures 7 o and Tables o 6 here] 5. Concluding Remarks This paper inroduced a new MGARCH model DAMGARCH which generalized he VARMA- GARCH model of Ling and McAleer (003) by inroducing mulivariae hresholds and imedependen asymmery in he ARCH componen of he model. As a resul he proposed parameerizaion is able o explain variance asymmery and hreshold effecs simulaneously wih variance spillovers. Furhermore we provided he condiions for he exisence of a unique saionary soluion and by generalizing he asympoic heory in Ling and McAleer (003) showed ha he quasi-maximum likelihood esimaors were consisen and asympoically disribued as mulivariae normal. In addiion we presened he analyic form of he mulivariae news impac curve which was labelled he news impac surface whose final purpose was a deailed graphical analysis of asymmery and leverage effecs. In an illusraive empirical applicaion i was shown ha he DAMGARCH model ouperformed he sandard CCC model in erms of he maximized log-likelihood values. An exended comparison of DAMGARCH and oher more radiional MGARCH models is lef for fuure research.

32 References Bauwens L. Lauren S. and Rombous J.K.V. (006) Mulivariae GARCH models: a survey Journal of Applied Economerics Billio M. Caporin M. and Gobbo M. (006) Flexible dynamic condiional correlaion mulivariae GARCH for asse allocaion Applied Financial Economics Leers Bollerslev T. (986) Generalized auoregressive condiional heeroskedasiciy Journal of Economerics Bollerslev T. (990) Modelling he coherence in shor-run nominal exchange raes: a mulivariae generalized ARCH approach Review of Economic and Saisics Bollerslev T. Chou R.Y. and Kroner K. F. (99) ARCH modeling in finance: a review of he heory and empirical evidence Journal of Economerics Bollerslev T. Engle R. F. and Nelson D. B. (994) ARCH models. In R.F. Engle and D. McFadden Handbook of Economerics Vol. 4 Elsevier Amserdam pp Bollerslev T. and Wooldridge J.M. (99) Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariances Economeric Reviews Bougerol P. and Picard N. (99) Saionariy of GARCH processes Journal of Economerics Caporin M. and McAleer M. (006) Dynamic asymmeric GARCH Journal of Financial Economerics Caporin M. and McAleer M. (008) Scalar BEKK and Indirec DCC Journal of Forecasing Caporin M. and McAleer M. (009) Do we really need boh BEKK and DCC? A ale of wo mulivariae GARCH models working paper available a Social Science Research Nework hp://ssrn.com/absrac=54967 Cappiello L. Engle R.F. And Sheppard K. (006) Asymmeric dynamics in he correlaions of global equiy and bond reurns Journal of Financial Economerics Come F. and O. Lieberman (003) Asympoic heory for mulivariae GARCH processes Journal of Mulivariae Analysis De Goei P. And Marquering W. (004) Modeling he condiional covariance beween sock and bond reurns: a mulivariae GARCH approach Journal of Financial Economerics Engle R.F. (98) Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion Economerica

33 Engle R.F. (00) Dynamic condiional correlaion: a simple class of mulivariae generalized auoregressive condiional heeroskedasiciy models Journal of Business and Economic Saisics Engle R.F. and Kroner K.F. (995) Mulivariae simulaneous generalized ARCH Economeric Theory -50. Engle R.F. and Ng V. (993) Measuring and esing he impac of news on volailiy Journal of Finance Glosen L.R. Jagannahan R. and Runkle D.E. (99) On he relaion beween he expeced value and volailiy of he nominal excess reurn on socks Journal of Finance Hafner C. and Herwarz H. (998) Time-varying marke price of risk in he CAPM approaches empirical evidence and implicaions Finance Hafner C. and Preminger A. (009) On asympoic heory for mulivariae GARCH models Journal of Mulivariae Analysis Hansson B. and Hordahl P. (998) Tesing he condiional CAPM using mulivariae GARCH- M Applied Financial Economics Kroner F. And Ng V. (998) Modelling asymmeric comovemens of asse reurns Review of Financial Sudies Jeanheau T. (998) Srong consisency of esimaors for mulivariae ARCH models Economeric Theory Li W.K. Ling S. and McAleer M. (00) Recen heoreical resuls for ime series models wih GARCH errors Journal of Economic Surveys Ling S. and Li W.K. (997) Diagnosic checking of nonlinear mulivariae ime series wih mulivariae ARCH errors Journal of Time Series Analysis 8(5) Ling S. and McAleer M. (00a) Saionariy and he exisence of momens of a family of GARCH processes Journal of Economerics Ling S. and McAleer M. (00b) Necessary and sufficien momen condiions for he GARCH(rs) and asymmeric power GARCH(rs) models Economeric Theory Ling S. and McAleer M. (003) Asympoic heory for a vecor ARMA-GARCH model Economeric Theory McAleer M. (005) Auomaed inference and learning in modeling financial volailiy Economeric Theory 3-6. McAleer M. Chan F. Hoi S. and Lieberman O. (008) Generalized auoregressive condiional correlaion Economeric Theory

34 McAleer M. F. Chan and D. Marinova (007) An economeric analysis of asymmeric volailiy: heory and applicaion o paens Journal of Economerics 39 () McAleer M. Hoi S. and Chan F. (009) Srucure and asympoic heory for mulivariae asymmeric condiional volailiy Economeric Reviews 8 (5) Nelson D.B. (990) Condiional heeroskedasiciy in asse pricing: a new approach Economerica Rabemananara R. and Zakoian J.M. (993) Threshold ARCH models and asymmeries in volailiy Journal of Applied Economerics Tsay R.S. (998) Tesing and modelling mulivariae hreshold models Journal of he American Saisical Associaion 93 (443) Tse Y.K. and Tsui A.K.C. (00) A mulivariae generalized auoregressive condiional heeroscedasiciy model wih ime-varying correlaions Journal of Business and Economic Saisics Zakoian J.M. (994) Threshold heeroskedasic funcions Journal of Economic Dynamics and Conrol

35 Appendix A. Alernaive represenaion of he DAMGARCH model Equaions () - (4) can be represened in an alernaive way as follows: s r = + i i+ m m i= m= H W BH G G G m ( n nl) G m ( nl ) ( ) ( Alm +Ψlm G m Il( ε m) ) ( A m+ψ mg m I ( ε m) ) A m+ψ mg m I ( ε m) : :... =...: dg ( ε m d )( ε m d ) : dg ( ε m d )( ε m d ) :... =...: dg ( ε m d l)( ε m d l) ( ) G = G i I. m m l n where we used he equivalence ( ε d ) ( ε d ) = dg ( ε d )( ε d ) he following relaions. Furhermore using ( ε )( ε ) = ( εε + ε ε ) = ( εε ) + ( ) ( ε ) ( ε ) ( εε ) ( ) ( ε ) dg d d dg d d d d dg dg d d dg d dg d = dg + dg d d dg d = e + e + e we may rewrie G mas {( ) : ( ) :...: ( )} G = e + e + e e + e + e e + e + e m m m m m m l m l highlighing he fac ha he componen including he innovaion conains an elemen which is ime varying bu consan over all pariions a second elemen ime invarian bu changing across pariions and a cross erm which is ime varying and varying across pariions. Using his las resul he following represenaion can be derived:

36 s r a b c ( ) H = W + BH + G G + G G + G G i i m m m m m i= m= { } { } G = i e G = e : e :...: e G = e : e :...: e. a b c m l m l m m m m l Appendix A. Possible pariions defined over he oin suppor The flexibiliy of he pariions and of he models defined direcly over he oin suppor accommodaes paricular represenaions such as ha depiced in Figure 3 which focuses on very exreme evens. A naural quesion ha may arise is he idenificaion of common shocks or common componens. [Inser Figure A. here] Finally he pariions defined over he oin probabiliy suppor may also accommodae non-linear relaions beween asses. A simple example is he disincion beween exreme evens of an ellipical mulivariae disribuion as depiced in Figure A.. This may be ineresing for cases wih consan correlaions and hresholds defined over he sandardised bu correlaed innovaions. [Inser Figure A. here] Appendix A.3: Derivaion of he News Impac Surfaces and of he uncondiional esimaes Assume s= and r= and ha he indicaor funcion is defined over he marginal densiies so ha ( ) I ε is a diagonal marix. The model represenaion is given by H = W+ BH + G ( ε ) ( ε )( ε ) = l ( ε ) = l G = A +Ψ G I dg d d G = A +Ψ G I.

37 Lemma A. The focus of he indicaor funcion refers o subse : ( ) I ε. The following equaliy holds: ( ε ) ( ( ε ) ( ε )... ( ε )) ( ( ) ( )... ( )) ( ) I = diag I I I = diag I z I z I z = I z. n n Proof: Assume ha he indicaor funcions are defined as in (.a). Then for a given in l and a given i in k ε = ( ε < ε < ε ) = ( Γ < ε ) i Γ i ( [ η] ) ( [ η] i i ) i i i i ( i ) ( i ). ( ) ( ) ( ) I I d d I D d D d i i i i i i i = I DΓ d < DΓ DΓ d = I Γ d < Γ Γ d = I Γ d < z Γ d = I z i i A similar proof can be derived when indicaor funcions follows (.b). If he model has a unique saionary soluion we can wrie E[ H] = W + BE [ H ] + E G. As we are ineresed in he uncondiional values he expecaions con be rewrien as follows: E[ H] = W + BE [ H] + E G by exploiing he srucure of G and using Lemma A.:

38 l E G = E A +Ψ G I dg d d ( ) ( ε ) ( ε )( ε ) = = +Ψ = l ( A E[ G ] ) E I ( ε) dg ( ε d )( ε d ) = +Ψ = l ( A E[ G] ) E I ( ε) dg ( ε d )( ε d ). The uncondiional value of he asymmeric erm has o be numerically compued. The following equaliies hold: l [ ] = ( ε ) +Ψ ( ε ) = ( ) +Ψ [ ] ( ) E G A E I E G I A E I z E G E I z l = = ( ) [ ] ( ) [ ] = AE I z EG E I z AM EG M +Ψ = +Ψ = = l l where M E I ( z ) = (which is a diagonal marix). Noe ha his expecaion can be numerically evaluaed if he correlaions are consan over ime. Alernaively we sugges using approximaions and evaluae he quaniy using he uncondiional correlaions implied by he correlaion model. Given he values of linear sysem M we deermine he uncondiional value [ ] EG by solving he following l l vec ( E [ G] ) = vec AM + vec Ψ E [ G] M. = = Nex consider hen he following equaliy: ( ε )( ε ) = ( Γ )( Γ ) dg d d dg D z d z d D while if d = 0 we should consider if d = d ( ε )( ε ) = ( ) dg d d dg D z z D.

39 By focusing on he following expecaion (when d = d ): E I ( ε) dg ( ε d )( ε d ) E I ( z) dg D( z d )( z d = Γ Γ ) D and using he fac ha ( ) I z is diagonal we can wrie: E dg I z D z d z d D E = dg DI z d d Γ Γ = Γ Γ D ( ) ( )( ) ( )( η )( η ). This expression arises from he fac ha he diagonal of he produc wihin he inernal parenheses is equivalen o he produc of ( ) I η wih he dg(.) resul given above. Furhermore again using he fac ha our ineres is on he diagonal elemens we derive he following equaliies: E dg DI ( z)( η d )( η d Γ Γ ) D = = E dg ( DD) dg I ( z)( η d )( η d Γ Γ ) = = E dg ( DD) E dg I ( z)( η d )( η d Γ Γ ) = = E H dg E I z Γd Γd [ ] ( )( η )( η ). Where we use he diagonaliy of values namely dg( E[ D] E[ D] ) dg( E[ DD ] ) E[ H] D and we replace he expecaions wih heir uncondiional = = and Γ comes from he decomposiion of he uncondiional correlaion marix. Noe also ha he expecaion of E dg ( D D ) independen from he expecaion of E dg I ( z)( η Γd)( η Γd) is by he law of ieraed expecaions: he condiional sandard deviaions are a funcion of he informaion se a ime - while he innovaions are referred o ime. Thus defining

40 N E = I z z Γd z Γd ( )( )( ) i follows ha: E dg D I z z d z d D E H dg N E H dg N ( )( Γ )( Γ ) = [ ] ( ) = [ ] ( ) ( ) [ ]. ( ) [ ] ( ) = dg N E H = diag dg N E H Noe ha when d = 0 he previous resul is sill valid bu we mus redefine N as follows: ( ) N = E I z zz. I should be emphasized ha he uncondiional expecaion of he correlaion marix equals he correlaion marix if he marix is consan oherwise i has o be compued on he basis of a specified dynamic srucure. Collecing he various resuls we can hen wrie E G = A +Ψ E G E I z dg d d = l ( [ ]) ( ) ( ε )( ε ) = ( ) [ ] l ( [ ]) ( ) = A +Ψ E G diag dg N E H l ( ) ( ) ( ) [ ] E[ H ] = W + B E [ H ] + A +Ψ E [ G ] diag dg N E H. = Solving wih respec o he uncondiional variances gives ( ) l E[ H] = In B ( A +ΨE[ G] ) diag dg ( N ) W. = Noe ha he uncondiional value of he correlaion marix should be derived under he appropriae model ha is used o define he dynamic condiional correlaions unless he condiional correlaions are assumed o be consan. The uncondiional variance of DAMGARCH is equivalen o Σ= DRD

41 and correlaion argeing is imposed when he following equaliies hold: R * = R ( ) ( ( )) W = I B A +Ψ E G diag dg N H l * * * n = where R * and H * refer o he corresponding sample esimaors * N is evaluae using he decomposiion of R * * and E G depends on * M which is also evaluaed using he decomposiion of R *. Noe also ha he dynamic correlaion model has also o be re-cas in a way ensuring correlaion argeing. Noe ha his resul includes as special cases he CCC VARMA-GARCH VARMA-AGARCH and DCC models. Appendix A.4: Proofs of Theorems Proof of Theorem. Following Ling and McAleer (003) we firs rewrie DAMGARCH in he following form: X = AX + ξ where X = G G G G G G... G G G H H... H a b c a b c a b c r+ r+ s+ which has dimension (3nlr+ns)x and where he elemens here included are defined in Appendix A.. Noe ha he vecor X conains hreshold-dependen elemens a ime hreshold dependen componens as well as he innovaion (mean residuals) a ime. Using again he noaion inroduced in Appendix A. we consider he following represenaion of he variance dynamic:

42 s r a b c ( ). H = W + BH + G G + G G + G G i i m m m m m i= m= ( ) he equaion for H yields: Muliplying by z = diag dg ( zz ) s r a b c ( ) e = zw + zbh + zg G + zg G + zg G i i m m m m m i= m= given ha zh = e. The previous equaion implies a marix A wih he following srucure: zg : zg : zg... zg : zg : zg zb... zb 0(3 nl n) 3 nlr 0(3 nl n) ns A = I r -r -r s 3 nl( r-) 3 nl( r-) 3nl 3 nl( r-) ns G- : G- : G-... Gr - : Gr - : Gr - B... Bs 0 I 0 ns ( -) 3 nlr ns ( -) ns ( -) n. Furhermore = ( ) ( ) b c zw i e G G 0 ( ) W 0 ( ) ξ l 3nl r n s Given hese quaniies and following Ling and McAleer (003) we define he quaniy S m = ξ + A ξ m + i = i= where m =. Denoe by s m he elemen of order k in he summaion included in S m ha such Esm = Ee kξ+ e k A + i ξ i=

43 where e k is a vecor conformable wih S m comprising zeros and wih in posiion k. Given ha he marices A are no independen we should modify he proof of Ling and McAleer (003). We firs noe ha he following decomposiion holds for each marix A : A A Z = ZA G : G : G... G : G : G B... B 0(3 nl n) 3 nlr 0(3 nl n) ns = I r - r - r - s 3 nl( r-) 3 nl( r-) 3nl 3 nl( r-) ns G- : G- : G-... Gr - : Gr - : Gr - B... Bs z 0n (3 nlr+ ns n) = 0(3 nlr+ ns n) n I 3nlr+ ns n 0 I 0 ns ( -) 3 nlr ns ( -) ns ( -) n where A depends on he informaion se a ime - and Z depends on he informaion se a ime. Furhermore consider a simple DAMGARCH model where he lags of he condiional variance dynamics are all resriced o be equal o one which implies ha A depends only on he informaion a ime -.and i is independen from he model he following proof mus be adaped. Thus we have Z.When increasing any lag lengh or order of m = k + i i ξ = k [ ] i i ξ = k i= i= E s E e Z A e E Z E A Z E A e A where and are wo marices and A E G z... E G B... B 03 nl ( n+ 3 nl) r 03nl ns r s = I( 3 )( -) 0( 3 )( -) ( 3 ) 0 n+ nl r n+ nl r n+ nl ( n+ 3 nl)( r-) ns E G z... E G r B... B s 0 I 0 ns ( -) ( n+ 3 nlr ) ns ( -) ns ( -) n. (A.3.) To obain (A.3.) we have used he previously inroduced decomposiion of A and he equaliy

44 ( + i i) ξ = [ ] i i ξ i= i= E Z A E Z E A Z E A. (A.3.) If he model follows he general represenaion in equaion ()-(4) in he main ex (A.3.) is no valid due o he inclusion in A of he G... G erms. In fac he erms - r - G... G depend on - r - he pas values of he innovaions and are hus dependen on he pas values of Z. However under he resricion ha he model has no dynamic asymmery he erms G... G simplifies removing - r - he dependence on pas values of Z. In his las case he expecaions can be spli given he dependence of on ime - quaniies only (when his is no he case he expecaion wihin he A parenheses will involve addiional erms). A similar resul applies when he model parameer marices are diagonal: by expanding he erm G in (A.3.) and by using he diagonaliy of he parameer marices and of ( ) I ε we can show ha he expecaion in (A.3.) is sill valid. Unforunaely such an approach canno be used when he model has full parameer marices. When (A.3.) is valid i can be shown ha Assumpion 3 ensures he roos of he characerisic polynomial of A lie inside he uni circle hereby proving he convergence of A and hence of he whole erm. The remainder of he proof follows closely ha in Ling and McAleer (003) and in McAleer e al. (009) also wih respec o he proof of sric saionariy and ergodiciy. Proof of Theorem. Using Theorem and he resuls repored in Appendix A.4 he proof follows by direc exension of he resuls in McAleer e al. (009). Proof of Theorem 3. Consisency is obained by verifying he condiions given in Jeanheau (998) namely i) he parameer space Θ is compac; ii) for any θ Θ he model admis a unique sricly saionary and ergodic soluion; iii) here exiss a deerminisic consan k such ha and θ Θ Σ > k ;

45 iv) model idenifiabiliy; v) Σ is a coninuous funcion of he parameer vecor θ ; vi) E θ log Σ < 0 θ Θ. Noe ha he deerminan of he condiional covariance marix can be decomposed using equaion () ino Σ = D R D = D R where we have also used he assumpion of a consan condiional correlaion marix. Furhermore by Assumpion 3 D is sricly posiive and here exiss a consan k such ha D > k. In addiion again using Assumpion 3 here exiss a second consan k such ha R > k. Then we can define a hird consan k = k k such ha Σ > k and θ Θ where Θ is a compac subspace of an Euclidean space. This proves condiions i) and iii). Theorem ensures he exisence of a unique sricly saionary and ergodic soluion o DAMGARCH verifying condiion (ii). Assumpion 4 deals wih condiion (iv) ensuring idenifiabiliy while Assumpion 5 imposes he log-momen condiion (vi). Finally under Assumpion 4 i is eviden ha he condiional variances are a coninuous funcion of he parameer se proving condiion v). Condiion (ii) refers o a unique sricly saionary and ergodic soluion while Theorem provides condiions for second-order saionary soluion. However using he resuls in Ling and McAleer (003) Theorem 3. consisency can be proved under second-order saionary soluions. Proof of Theorem 4. Using he previous resuls he proof can be obained by direc exension of he Theorems and Lemmas in Ling and McAleer (003) and McAleer e al. (009).

46 ω α β DAX Coeff *S.dev FTSE Coeff *S.dev Correlaion 0.75 Log-Likelihood Table : CCC-GARCH esimaes bold values idenify significan coefficiens ω α γ β DAX Coeff *S.dev FTSE Coeff *S.dev Correlaion 0.73 Log-Likelihood Table 3: CCC-GJR-GARCH esimaes bold values idenify significan coefficiens DAX FTSE DAX FTSE ω Coeff Coeff Ψ (DAX) 00*S.dev *S.dev B (DAX) Coeff Coeff Ψ (FTSE) 00*S.dev *S.dev A (DAX) Coeff Coeff Ψ 3 (FTSE) 00*S.dev *S.dev A (DAX) Coeff Coeff Ψ 4 (FTSE) 00*S.dev *S.dev A 3 (FTSE) Coeff Correlaion *S.dev Log-Likelihood A 4 (DAX) Coeff *S.dev Table 4: CCC-DAGARCH esimaes bold values idenify significan coefficiens

47 DAX FTSE DAX FTSE ω Coeff Coeff Ψ (DAX) 00*S.dev *S.dev B (DAX) Coeff Coeff Ψ (FTSE) 00*S.dev *S.dev B (FTSE) Coeff Coeff Ψ (DAX) 00*S.dev *S.dev A (DAX) Coeff Coeff Ψ (FTSE) 00*S.dev *S.dev A (FTSE) Coeff Coeff Ψ 3 (DAX) 00*S.dev *S.dev A (DAX) Coeff Coeff Ψ 3 (FTSE) 00*S.dev *S.dev A (FTSE) Coeff Coeff Ψ 4 (DAX) 00*S.dev *S.dev A 3 (DAX) Coeff Coeff Ψ 4 (FTSE) 00*S.dev *S.dev A 3 (FTSE) Coeff Coeff Corr 00*S.dev *S.dev A 4 (DAX) Coeff Log-Likelihood *S.dev A 4 (FTSE) Coeff *S.dev Table 5: DAMGARCH esimaes bold values idenify significan coefficiens DAX FTSE Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Correlaion Jarque-Bera Probabiliy % quanile % quanile Table 6: descripive analysis of sandardized and uncorrelaed residuals (used for deermining empirical srucural innovaions) and empirical quaniles used o define model hresholds 0% (90%) quanile for he normal variable is -.8 (.8)

48 DAMGARCH (including correlaions) Asses number (number of correlaions) () 3 (3) DAMGARCH l=4 s=r=q= (6) (0) (45) (90) 00 (4950) n (n(n-)/) Full ( ) ( ) n n n + s + l + lq n + n + n( n ) s + l + lq n + n+ n( n ) s+ l+ n + n+ n( n ) s+ l+ n+ n+ n( n ) s+ r n+ n+ n( n ) s+ r n+ + sdcc + r n( n+ ) + ( s+ r) n + s+ r n( n+ ) n( n+ ) + ( s+ r) n + s+ r n( n+ ) n n n( n ) + ( s+ r) + Diagonal ( ) Common Dynamic ( ) Diagonal and Common Dynamic ( ) CCC (GARCH(sr) and correlaions) DCC (GARCH(sr) and correlaions) ( ) ( ) Diagonal BEKK(sr) Triangular BEKK(sr) ( ) BEKK(sr) Diagonal Vech(sr) ( ) Vech(sr) >5 0 6 ( ) Table : Model dimension ( s DCC and r DCC are he lag orders in he DCC model). DCC

49 x x y y Figure : Mulivariae GJR Represenaion Figure : Pariion Over he Join Suppor Asse Asse Asse Asse Figure 3: NIS of he firs asse wihou spillovers Figure 4: NIS of he second asse wihou spillovers

50 Asse Asse Asse Asse Figure 5: NIS of he firs asse wih spillovers Figure 6: NIS of he second asse wih spillovers FTSE DAX FTSE DAX Figure 9: DAX NIS Figure 0: FTSE NIS