Thresholds, News Impact Surfaces and Dynamic Asymmetric Multivariate GARCH *

Transcription

1 Thresholds, News Impac Surfaces and Dynamic Asymmeric Mulivariae GARCH * Massimiliano Caporin Deparmen of Economic Sciences Universiy of Padova Michael McAleer Deparmen of Quaniaive Economics Compluense Universiy of Madrid March 009 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 codes: C3, C5, C5 * We would like o hank he Co-edior, wo referees, Monica Billio, Gabriele Fiorenini, Paolo Paruolo, Domenico Sarore and he paricipans of he Second Ialian Congress of Economerics and Empirical Economics 007 (Rimini, Ialy) and of 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. Corresponding auhor: Massimiliano Caporin, Diparimeno di Scienze Economiche Marco Fanno, Universià degli Sudi di Padova, Via Del Sano, Padova Ialy el fax massimiliano.caporin@unipd.i

2 . 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, Come and Lieberman (003), and Ling and McAleer (003) 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 wellknown 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, 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. 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)).

3 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 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). 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)) 3

4 Despie he lack of heoreical conribuions dealing wih asymmery and variance spillovers in he CCC and DCC models, 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)). We do no follow his srand of lieraure as he effecs of he variance misspecificaion on he correlaion dynamics are no known, so ha we prefer o generalise he variance dynamics. The model o be presened here could be exended following he GARCC model 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. 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 4

5 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. For he momen, GARCH-in-mean effecs are no considered, so i is assumed for simpliciy ha any mean componen has been modeled adequaely. The primary focus is on he mean residuals under he following equaions: Y = E Y I ε, E ε I 0, E ε ε I + = = Σ = D R D () in which Y, and, I is he informaion se available a ime E Y I is he condiional mean of ε is he n-dimensional mean residual vecor a ime. The mean residuals have a condiionally ime-dependen covariance marix ha can be decomposed ino he conribuions of he condiional variances and he condiional correlaions 3. Finally, condiional volailiies, given by: D diag ( σ,, σ,,..., σ n, ) D is a diagonal marix of =, and R is a (possibly ime-dependen) correlaion marix. 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, which is a full symmeric marix, is no obained by a Cholesky decomposiion of he correlaion Γ, 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. 5

6 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 Γ = U U. Finally, le ½ z = D ε denoe he sandardized innovaions, wih R as he correlaion marix. Define he vecors of condiional variances and squared innovaions as H ( σ ) ( ),, σ,,... σ, dg D D = = and = ( ) e ε ε ε n,,, n,,... define he Dynamic Asymmeric MGARCH (hereafer DAMGARCH) model:, respecively. The following equaions s H = W + B H + G i i m i= m= r, () { ( )} l where G m = A, m + Ψ, mg m I ( ε m ) ( ε m dɶ ) ( ε m dɶ ) l = {,, } and G = A + Ψ G I ( ε ) m m m m m = (3), (4) where, Bi, i =,,..., s, A, m, =,,..., 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 ɶ = 0 or d ɶ = d, where d is an n-dimensional vecor ha defines he upper (or lower) bounds of subse 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, I ( ε ), and assumes he value if he elemen ε i, belongs o is specific subse, and 0 oherwise. 6

7 (we will address below he srucure of he subses, he srucure of d and is usefulness 6 ). We highligh here ha he vecor d ɶ = 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 ɶ = 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 G m and G m define 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 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 L G I m m m = q Ψ L = Ψ L + Ψ L Ψ L,, m, m,, m,, m, q, (5) 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. 7

8 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 + B H + + G e l { ( )} G = A + Ψ G I ε =,, (6) so ha we can rewrie he ARCH erm as 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 8

9 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 dɶ dɶ, = ( ε ) ( ε ) ( ε ) and a second erm associaed o he asymmery persisence: l Ψ G I dɶ dɶ. = ( 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 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, marginal densiies of he univariae mean innovaions, ε i,, respecively. ε, and over he Consider he use of he mulivariae densiy. In his case, define mulivariae innovaion densiy, so ha: n S R as he suppor of he I ( ε ), ε S = 0, oherwise, (8) 9

10 where S is a subse of S. Furhermore, we have l S = S, Si S = Ø, i, =,,..., l, i. (9) = As an example, we may define he following hree subses of S : { ε : ε i L,,...n }, { ε εi U } S = < d i =, S = : > d, i =,...n, 3, S = S S S. 3 (0) 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 ɶ = 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 i, ) indicaor funcion is he univariae counerpar of equaion (8), namely: 0

11 ɶ if d = d ( εi, ) i, i, I =, =,,..., k ( εi, ), ε < d 0, oherwise, ε > d i, i, I =, = k +,..., l 0, oherwise (.a) ɶ,,, ( ) 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 (a,b) is εi, di,, while for = l (ha is, he las subse), he condiion becomes εi, > di, l wih di, <... < di, k < 0 < di, k + <... < di, l, 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 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 he variables and are defined accordingly o he srucure of ( i, ) d may be differen over I ε, namely d { d,,..., d, l} = for =,..., k, d = { d,,..., d, l} for = k +,..., l, and d = 0n for = k, k +. Under (8), he definiion of d depends on he relaions used o define he S subses. In he example in (0), we have d = d Lin, 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

12 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 = + A I ( ε ) A e, 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 7 See Table for parameer number of several MGARCH models.

13 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. 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 3

14 R correlaion marix saisfying Γ Γ = 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, = D Γ 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 In he following, i is assumed ha he hresholds are fixed over he probabiliy densiy funcion of he η. The srucural hresholds d can be fixed a priori or deermined by a quanile relaion, d = F ( α) 8. Furhermore, he erm hresholds will be used only wih respec o he marginal i 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 d, i < εi, DΓ d, 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 ( ε n, )). Finally, for =, he condiion is εi, DΓ d, i, while for = l, he condiion is εi, > DΓ dl, i. Equaion (3) refers o he indicaor funcions as defined in (.b). The previous commens and he discussion on hreshold 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. 4

15 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:, d < ε d I ( ε ) =, =,,..., l 0, oherwise (4) 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: {,,, } S = ε : d < ε d, i =,,..., n, =,,..., l. (5) i i i Noe ha equaion (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. 5

16 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: 4 ɶ ɶ ɶ ( ). (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: Aɶ = Aɶ 4 Aɶ, Ψ ɶ = Ψɶ 4 Ψɶ.,,.,.,., Aɶ 3 = Aɶ Aɶ 4, Ψ ɶ 3 = Ψɶ Ψɶ., 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 6

17 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 rivariae case, assuming mulivariae normaliy and noing ha innovaions, we may define hree hreshold vecors: η. As an example, for he η is a vecor of independen 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). Furhermore, 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), and refer o he DAMGARCH model as defined in equaions from () o (4), wih one of he following resricions: i) he model has no dynamic asymmery effec (ha is, he parameer marices Ψ,... Ψ l, r are all zero and hus he model collapses on a muliple hreshold asymmery specificaion which is a direc generalizaion of McAleer e al. (009)); or, ii) he parameer marices A,... Al, r, Ψ,... Ψ l, r, B... Bs are all diagonal. The resricions are needed o obain he model srucural properies as generalizaions of he proofs 7

18 in Ling and McAleer (003) and McAleer e al. (009). The model parameer vecor θ is defined as follows: vec ( W ) : vec ( A, )...: vec ( A l, r ) : vec ( Ψ, )...: vec ( Ψ l, r ) 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. 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 zɶ are non-negaive i=,, s, =,, r; 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 zɶ L B L = 0 are ouside he uni circle. Assumpion 4: r I E G z L i= i ɶ i and s i Bi L are lef coprime, and saisfy oher idenifiabiliy i= condiions given in Jeanheau (998) (he condiions are given in he proof o Theorem 3). 8

19 Theorem : Under assumpions ()-(3), he DAMGARCH model of equaions ()-(4) admis a unique second-order saionary soluion, H ɶ, measurable wih respec o he informaion se I -, where I - is a σ-field generaed by he innovaions zɶ. The soluion expansion: H ɶ has he following causal Hɶ = W + M A ξ i+ = i= M = 0 : I : 0 n n (3 nlr+ ns) n 3 nlr n ( s ) zɶ G : zɶ G : zɶ G... zɶ G : zɶ G : zɶ G zɶ B... zɶ B 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 -... G -r : G -r : G -r B... Bs 0 I 0 n( s-) 3 nlr n( s-) n( s-) n (9) (0) () ( ) ( ), b, c = z W, i e, G, G,0 ( ), W,0 ( ) ɶ () ξ l 3nl r n s ( ε ε ) ( ), ( ε ) e = dg e = dg d d e = dg d (3) G m ( n nl) ( A, m + Ψ, mg m I ( ε m )) A, m + Ψ, mg m I ( ε m ) ( ) ( Al, m + Ψl, mg m Il ( ε m )) : :... =...: ( ( )), / [ ], ε, and ( ) zɶ = diag dg z z E zɶ = I z = D D = diag H (5) n (4) where he quaniy G is defined in he alernaive represenaion of he DAMGARCH model described in Appendix A.. Hence, {, } Y H ɶ are sricly saionary and ergodic. The following heorem saes he condiions o ensure he exisence of momens for he DAMGARCH model. 9

20 E A < b Theorem : Under assumpions () o (3), 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 5: For any θ Θ, we have E ( H ) θ log < 0, where 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 ()-(4). 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 Σθ ΩθΣ θ : L 4.i) θ θ ' exiss and is coninuous in an open and convex neighbor of θ ; 4.ii) n L θ θ ' ˆ θ converges in probabiliy o a finie non-singular covariance marix L θ θ ' Σ θ = E n for any sequence ˆ θ such ha ˆ p θ θ ; 0

21 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 En θ ˆ θ L θ ' ˆ θ..3 Esimaion I was menioned briefly in secion. ha he esimaion of DAMGARCH could be considered hrough a quasi-maximum 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 marix, L( θ ) = l ( θ ) = l Σ ( θ ) T T ( ). Tradiionally, he approximae likelihood = = funcion is derived from a mulivariae normal disribuion. In he MGARCH lieraure, here also exiss a wo-sep 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 wo-sep 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). The model complexiy creaes several implemenaion and numerical opimizaion problems, even wih simplified and limied dimension sysems. In order o reduce he compuaional burden, he following approximaed esimaion procedure is suggesed. Noe ha full esimaion can be considered a he cos of sensibly increasing compuaional ime. Furhermore, numerical opimizaion problems could be reduced by implemening firs-order derivaives, which will be considered in fuure exensions and applicaions of he curren paper. Recall ha he number of variables is denoed by n. Thus, we sugges he following seps:

22 ) assume ha he sandardized and uncorrelaed residuals are disribued according o a sandard normal variables, fix he srucural hresholds quaniles of he normal disribuion; d for =,...l a heoreical ) 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 ɶ 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 d GARCH σ i, hresholds and save he condiional variances DAGARCH σ and he sandardized i, residuals ɶ η = ε σ for i=, n; DAGARCH i, i, DAGARCH i, 4) compue he uncondiional correlaion marix (using he sample esimaor) on he ɶ η = η : η :... η ɶ ɶ ɶ 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 D d =,... 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. 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 ɶ η = ε σ for i=, n; DAMGARCH i, i, DAMGARCH i,

23 7) compue he uncondiional correlaion marix (using he sample esimaor) on he ɶ η = η : η :... η ɶ ɶ ɶ 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 D d = 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, 3

24 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. 4

25 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 + B H + G, (6) { ( ε ) ( ε ɶ ) ( ε ɶ ) } 4 G = A + Ψ G I d d, (7) = 4 { ( )} 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) where z represens he idiosyncraic componen. In he DAMGARCH model, assuming ha he correlaions are consan, he news impac surface is given by: 5