Spatial GARCH: A spatial approach to multivariate volatility modelling

Transcription

1 Spaal GARCH: A spaal approach o mulvarae volaly modellng S.A. Borovkova Vrje Unverse Amserdam H.P. Lopuhaä Delf Unversy of Technology Absrac Ths paper nroduces a new approach o modellng he condonal varance n a mulvarae seng. I s essenally a combnaon of he popular GARCH model class wh a spaal componen, nspred by generalzed space-me models. The resulng spaal GARCH model akes no accoun boh emporal and spaal dependences n he condonal varance equaon, by allowng he condonal varance of a parcular marke o depend on pas volaly shocks n oher markes. The ner-marke dependences are summarzed n a wegh marx, whch s specfed beforehand usng exogenous nformaon such as marke capalzaon or counres GDPs. The man aracons of he spaal GARCH model are s smplcy, economc relevance, racably and a low number of parameers, compared o oher mulvarae GARCH models. We specfy he spaal GARCH model class, dscuss he choce of he wegh marx, saonary condons and show how he model parameers can be esmaed by he maxmum lkelhood mehod usng an erave GARCH maxmum lkelhood procedure. We apply he spaal GARCH (,) model wh varous wegh marces o he reurns from major sock markes. We fnd ha he spaal GARCH (,) model s very well sued for modellng volales of an ensemble of sock markes, as he spaal parameers are hghly sgnfcan. Moreover, spaal GARCH model excellenly capures he hgh kuross presen n squared reurns, whle regular GARCH fals o do so. Exendng he model o ncorporae leverage effecs leads o furher mprovemen n he volaly f. We compare wegh marces based on he nverse ravel dsance, GDP and marke capalzaon and examne whch wegh marx provdes a more accurae descrpon of spaal dependence. Keywords: condonal varance, space-me models, GARCH, wegh marx, maxmum lkelhood esmaon.

2 . Inroducon.. Overvew Volaly s one of he mos mporan conceps n fnance. Esmang and modellng volaly s essenal n he areas such as porfolo selecon, rsk managemen and opon prcng. The semnal work by Engle (98) nroduced he so-called auoregressve condonal heeroscedascy model (ARCH), whch, ogeher wh s many exensons, s now commonly used o model and forecas volaly of fnancal me seres. The mos mporan exenson s he so-called generalzed ARCH, or GARCH model by Bollerslev (986). The low order GARCH model (GARCH (,)) ganed a parcular populary, havng jus hree parameers, whle exhbng a good performance for mos fnancal seres. Over me, a broad range of exensons of he orgnal GARCH model have been developed o capure emprcal feaures of fnancal me seres. Mos wdely used ones are he socalled exponenal GARCH (EGARCH) and GJR-GARCH (also called asymmerc GARCH), whch ake no accoun leverage effecs,.e., he phenomenon of ncreasng volaly ha accompanes large and negave reurns. Exensons of a unvarae GARCH model o he mulvarae framework have he general name MGARCH (mulvarae GARCH). Generally, hese models aemp o descrbe he condonal dynamcs of he enre varance-covarance marx; he examples of such models are he VEC (vecor auoregresson-lke specfcaon of Bollerslev e al. (988)) and BEKK model by Baba, Engle, Kraf and Kroner (99) (see also Engle and Kroner (995)). There are wo man dffcules arsng n he mulvarae case: a large (and rapdly growng wh he dmenson) number of parameers and ensurng posve defneness of he condonal covarance marx. I s recognzed ha unresrced mulvarae GARCH models are characerzed by he curse of dmensonaly, n ha hey exhb oo many parameers. Ths makes he esmaon of such models oo complex and, n case of lmed amoun of daa, compleely non-feasble. The curse of dmensonaly has lmed he applcaon of unresrced GARCH models o sysems of hgher dmensons. Engle and Kroner (995) have shown ha, n general mulvarae GARCH specfcaons, he number of parameers s proporonal o he fourh power of he number of asses. So many resrced versons of mulvarae GARCH models have been nroduced. The Consan Condonal Correlaon model by Bollerslev e al. (988) (abbrevaed CCC) suggess he smples soluon o he dynamc modellng of he condonal covarance marx: he varances are assumed o have unvarae GARCH specfcaons, whle he (condonal) correlaon marx s

3 assumed consan. The Dynamc Condonal Correlaon (DCC) model by Engle () removes he (raher resrcve) condon of consan correlaons, by also allowng hose o have an auoregressve srucure. The CCC and DCC models parally overcome he dmensonaly problem; hese models are perhaps he wo mos popular condonal covarance models up o dae and have been also suded n Chrsodoulaks and Sachell () and Tse and Tsu (). Orano () explored ways of denfyng groups of me seres wh smlar correlaon srucures, o furher reduce he number of parameers. Hafner and Reznkova () suggesed a new esmaon mehod for he DCC model, amed a reducng he esmaon bas n hgher dmensons. A comparave analyss of some of hese alernave MGARCH specfcaons s performed by Ross and Spazzn () by means of Mone Carlo smulaons. A generalzed verson of he dagonal CCC, called Exended Consan Condonal Correlaon (ECCC), was defned by Jeanheau (998) and furher suded by He and Terasvra (4) and Conrad and Karanasos (). In hs model, he condonal correlaon marx s sll consan, bu he unvarae GARCH specfcaon for each of he condonal varances s exended by ncludng oher lagged condonal varances and squared reurns. Our model s a specal case of he ECCC model, whereby we mpose he dependence srucure on he condonal varances beforehand, usng exernal characerscs raher han smulaneously esmae hese dependences from he daa. Fnancal markes have become more nerconneced over he pas decade, as he recen fnancal crss has shown. Already Bauwens e al. (6) observe ha volales move ogeher across asses and markes. Shocks o volaly n one regon ofen spll over o oher regons and lead o ncreased volaly elsewhere. Ths s parcularly pronounced for European counres, whch are members of he same moneary unon (he Eurozone ), bu s also observed ousde of he EU. Hanhard and Ansoegu (9) have found ha equy markes are no negraed across he enre Eurozone, bu ha he sock markes of Europe s bgges economes (Germany, France and UK) are negraed. Also, hey repor ha he majory of European equy markes are negraed wh he German sock marke and ha here s nerdependence among he sock markes n Benelux. The relaonshp beween globalzaon and sock reurns has been nvesgaed also by Lam and Ang (6). They fnd ha sgnfcan relaonshps exs beween sock marke reurns, he global marke rsk facor and macroeconomc facors. Ths nerconnecon of world sock markes s also reaffrmed by Knf e al. (5), who sae ha correlaons beween naonal markes have been ncreasng. In he pas hree years, many arcles appeared ha documen he phenomenon of ncreased nerdependence, conagon The orgnal Eurozone counres () nclude: Ausra, Belgum, Fnland, France, Germany, Greece, Ireland, Ialy, Luxembourg, Neherlands, Porugal and Span. 3

4 and volaly spllovers durng he laes fnancal and Eurozone soveregn deb crss (e.g., Polson and Sco (), Chang and Wang (), Ogum (), Peng and Ng (), o name jus a few). All hs research movaes us o ncorporae some spaal srucure no a mulvarae GARCH model. The space-me model class (known as STAR Space-Tme Auoregressve modes, wdely appled n geology and ecology) s perfecly sued for hs ask, as hs model class explcly ncludes boh emporal and spaal dependences. Our spaal GARCH model s characerzed by a wegh marx, whch ncorporaes he spaal srucure apror, meanng ha does no have o be esmaed from he daa. We sugges several possble wegh marces: based on he geographc dsances beween counres, marke capalzaons or GDPs. The spaal coeffcens measure he aggregae effec of oher counres on a parcular sock marke, whch dramacally decreases he number of parameers. We develop an erave esmaon procedure based on he unvarae maxmum lkelhood esmaon of GARCH models. We ncorporae he leverage effec no he model and apply o he daly log-reurns of major sock exchange ndces: of he counres whn he orgnal Eurozone (excep Luxembourg), UK and US sock markes. The mos mporan concluson ha can be drawn from our emprcal sudy s ha he spaal GARCH (,) model s excellenly sued o descrbe emporal and spaal dependences beween he sock markes whn he Eurozone, bu also ncludng US and UK sock markes. The spaal parameers are sgnfcan for nearly all sock markes and spaal GARCH (,) models provde a sgnfcanly beer f for he volales of he major sock marke ndces han unvarae GARCH models. The paper s srucured as follows. The nex wo subsecons brefly revew he relevan GARCH and STAR models. In Secon we nroduce he spaal GARCH model, dscuss saonary condons and descrbe he erave MLE procedure. Secon 3 s devoed o he emprcal sudy and Secon 4 concludes... GARCH processes Le r be he logreurn on day and assume ha he mean reurn s zero. Le ( ) be a dscreeme Whe Nose process wh mean zero and un varance, and le F be he nformaon se of all nformaon up o (bu no ncludng) me. The ARCH (q) model s defned as: r, h 4

5 h q q q r r... r r. () Here h s he condonal varance of r : h E r F ] he coeffcens:,,,,..., q. [. A non-negavy condon s mposed on ARCH models ypcally requre a hgh order q o adequaely descrbe he changng volaly. The GARCH (p,q) model overcomes hese dffcules, by allowng he condonal varance o depend also on s own prevous values: h q p r h,,,. () I s ofen assumed ha are..d. Normal (,) random varables. In hs case, r F ~ N(, h ). Oher dsrbuons also are used, such as he Suden- or he generalzed error dsrbuon, whch allow for faer als han he normal dsrbuon. The low-order GARCH (,) model s ofen suffcen o capure volaly cluserng n he daa and s he mos wdely used model n quanave fnancal research (Engle and Kroner (995), Vogelvang (5)). He we also resrc ourselves o he GARCH model of lag (,); he exenson o hgher GARCH order wll be evden. The uncondonal varance n GARCH(,) model s gven by var( r ) E[ r ] (3) ( ) and s posve and fne as long as. Ths s he necessary and suffcen condon for he varance saonary of a GARCH (,) process. If, he uncondonal varance s no defned (non-saonary n varance) and, f, a un roo n he varance process exss (he process s srcly saonary wh nfne varance), whch s also ermed Inegraed GARCH or IGARCH. The asymmerc GARCH model (A-GARCH, also called GJR model, for Glosen, Jagannahan and Runkle (993)) s an exenson of GARCH, whch akes no accoun possble asymmerc responses of volaly o posve and negave reurns. Glosen e al. (993) argue ha posve unancpaed reurns resul n a downward revson and negave unancpaed reurns n an upward revson of he condonal volaly. In he case of equy reurns, hs asymmery s largely arbued o leverage effecs: a fall n he sock value of a frm causes a rse n frm s deb o equy rao. Shareholders reac 5

6 by percevng he fuure cashflow sream as more rsky, snce hey hold he resdual rsk of he frm. The GJR-GARCH(,) model deals wh hs feaure and s specfed as follows: h (4), r h r I where I = f r and oherwse. The las erm s added o accoun for leverage effecs and s characerzed by >. The nverse leverage effec ( < ) s observed n commody markes, where downward prce movemens ncrease he volaly, raher han upward ones. The non-negavy condon for he model coeffcens n GJR-GARCH model becomes:,, and. Hence, he erm can be negave, as long as. The saonary condon of GJR model (4) becomes, for a symmerc reurns dsrbuon (Lng and McAleer ()). For an asymmerc dsrbuon, he / n he above formula s replaced by he expeced number of negave reurns. The parameers of GARCH models are esmaed by he condonal maxmum lkelhood (ML) or quaz-maxmum lkelhood (QML) mehod, where he maxmum of he log-lkelhood funcon s found numercally. For a good revew on praccal ssues of applyng GARCH models, see e.g., Zvo (9)..3. Space-me models Space-me models have been appled exensvely o geologcal, ecologcal and socal me seres, as hey focus on lnkages beween me seres recorded a dfferen locaons. Typcal examples are daly ol producons a several ses or crme raes a varous cy locaons. Space-me models address he phenomenon of spaal correlaon : a collecon of seres a N fxed locaons exhbs sysemac dependence beween he seres a each locaon and s neghbourng locaons. Space-me auoregressve models (abbrevaed as STAR) have been nroduced by Pfefer and Deusch (98). The man feaure of hese models s he presence of a spaal componen n he auoregressve specfcaon. Ths spaal componen s represened by he wegh marx, whch allows for herarchcal orderng of neghbours of each se and characerzes he dependence of a se on s neghbours. In geology, ecology and socal scences one ofen uses unform, bnary or nverse dsance wegh marces, where he weghs depend on eher he number of drec neghbours of a se or he (nverse) dsance beween a se and s neghbours (he argumen beng ha ses ha are close are more nerdependen ha hose furher away from each oher). The exac specfcaon of he wegh 6

7 marx s a maer lef o he model bulder who may choose weghs o reflec he confguraon of a specfc applcaon. Space-me models ofen have sgnfcan advanages over vecor auoregressve models: he dependence srucure can be mposed beforehand, from some known locaon characerscs of he problem (e.g., permeably of sol n geologcal applcaons) and hence, he number of parameers o be esmaed s grealy reduced. In many applcaons, space-me auoregressve models have proven o ouperform VAR models, n erms of boh goodness of f and forecasng ably (see e.g., Sarors (5), Gacno (6)). The Generalzed Space-Tme Auoregressve model (GSTAR, Borovkova e al. ()) s a generalzaon of he STAR model class. The GSTAR model of auoregressve order p and spaal order s gven by he followng vecor represenaon: p Z( ) [ sz( s) s W Z( s)] e( ), s where e () are random error dsurbances wh mean zero (usually assumed..d.), he dagonal () ( N ) marces sk, k, conan he unknown parameers on he dagonal: ( sk,..., sk ), k, and he gven wegh marx W mees he followng condons: N w j j for all and w. (6) The GSTAR model represens he vecor me seres a me : Z () as a lnear combnaon of s own pas observaons and he weghed sum of he pas observaons n neghbourng ses. The scalar represenaon of he GSTAR model for any se (=,...,N) s p ( ) ( ) Z ( ) [ s Z ( s) s wj Z j ( s)] e ( ). s j N Noe ha he number of parameers o esmae s only Np, so grows lnearly wh N (and no as N, as s he case for a vecor auoregressve model). (5) A STAR model s a specal case of GSTAR, where all he auoregressve and spaal regressve () ( N ) parameers are he same for all locaon:..., k,. GSTAR s much less resrcve, sk sk allowng for locaon-specfc parameers and hence, dealng wh spaal heerogeney. GSTAR models of a hgher spaal order can be defned, by defnng a herarchal orderng of neghbours and specfyng dsnc wegh marces for dfferen spaal orders. However, n mos 7

8 applcaons, he spaal order of one (where all ses are consdered as poenal neghbours of each oher) s suffcen o adequaely model spaal dependences. I s shown n Borovkova e al. () ha he GSTAR model of spaal and auoregressve orders s ( saonary f ) ( ) ( ) ( ),. For a hgher auoregressve order, he saonary condon can be gven n a marx equaon form, bu no n he form of smple resrcons on he parameers. Borovkova e al. (8) has suded OLS esmaon of STAR models and showed conssency and asympoc normaly of he esmaed parameers. In he nex secon, we shall exend a GARCH model by addng a spaal componen n he spr of a GSTAR model and descrbe how he spaal GARCH model can be esmaed by he erave maxmum lkelhood mehod.. Spaal GARCH models.. The model A spaal GARCH model s essenally a combnaon of a GARCH model and a space-me model: s obaned by addng a spaal componen no he GARCH condonal varance equaon. We shall frs specfy he model n a unvarae form (for a sngle marke varance) and hen provde he marx noaon. Le r, be he logreurn of sock marke a me wh mean zero. Le random varables Z, be ndependen n and dencally dsrbued wh zero mean, un varance and consan covarance marx C. Then we le r h, Z,,, where h, s he condonal varance of he logreurns, whch we wre as, N h, a, a, r b, h a, w r b, w h, (7), j j, j N j j, j where wj are he componens of he spaal wegh marx, specfed beforehand and sasfyng he N j same condons as above: w and w for all. j The equaon (7) defnes he spaal GARCH model of order (,) for he locaon (sock marke). The spaal componens of he condonal varance equaon are he las wo erms of he rgh hand sde of (7). All lagged squared logreurns and lagged condonal varances of he N- oher sock markes are mulpled by he spaal weghs w j. Ths resuls no wo spaal exogenous varables n he condonal varance equaon and wo addonal parameers a, b, and, whch measure he 8

9 nfluence of he aggregaed lagged varances and squared reurns a all oher locaons on he varance a a parcular locaon. The specfcaon of a spaal GARCH model of a hgher order can be acheved by ncludng he pas squared logreurns and varances a hgher lags no (7), usng he same wegh marx. In he marx noaon, he spaal GARCH (,) model for an ensemble of all N sock markes condonal varances can be wren as follows: H A A A W R B B W H, (8) ( ) ( ) where he vecors of condonal varances and squared logreurns are respecvely H ) T ( h,,..., h, N and R ( r,..., r ) he N x N dagonal parameer marces T,, N, Ak and T A s he parameer vecor A ( a,,..., a, N ) and Bk are A k ak, a k,... a k, N b k, bk,, B k,... k,. bk, N The enre condonal varance-covarance marx of he vecor T ( r,,..., r, N s gven R ) by DCD, where D s he dagonal marx conanng he condonal volales modelled by he spaal GARCH model (7): h, h, D.... h, N A smple way o assure posvy of condonal varances s o requre ha all he weghs and coeffcens are non-negave: wj for all and j; a,, a,, a,, b, and b, for all. However, as Nelson and Cao (99) have shown for unvarae GARCH processes and Conrad and Karanasos () for exended CCC models, he posvy condons on all GARCH coeffcens s no necessary for he posvy of varance and n many emprcal cases hese may be oo resrcve, rulng ou possble negave volaly feedback. Conrad and Karanasos () suded he exended CCC models and saed necessary and suffcen condons (n erms of he process parameers) for he posvy of varance; hese condons are summarzed n Theorem of her paper. I can be seen easly ha our spaal GARCH (,) model s equvalen o he ECCC model of order one, wh he parcular form of he parameer marces A A A W and B B W. So for he condonal B 9

10 varances o be posve, he condons (C)-(C3) of Theorem of Conrad and Karanasos () mus apply. Checkng hese condons for a parcular se of esmaed parameers s raher nvolved; Conrad and Karanasos exensvely suded he bvarae GARCH (,) case, for whch hey derved a se of nequales ha he parameers mus sasfy. For example, hey showed ha, n bvarae case, boh dagonal elemens of he parameer marx B canno be negave smulaneously and ha negave volaly feedback n boh drecon s ruled ou. The varance specfcaon (7) models spllovers of volales across markes. Economcally nerpreable spaal dependences beween markes should be specfed beforehand by means of he wegh marx W. The wegh marx should reflec nerrelaons beween he markes or asses under consderaon. If sock marke volales of several counres are modelled, he weghs should reflec lnkages beween he counres economes; hence, should be based on nerpreable economc facors. The frs, raher obvous choce s he wegh marx based on he nverse of he ravel dsances beween sock marke s ces. Ths choce can creae he frs mpresson of he spaal dependences beween he sock markes. Arguably, nearby counres have more economc and fnancal nerconnecons (due o hsorcal and geographcal lnkages) and herefore her sock markes can have a greaer nfluence on each oher. Ths s especally he case n an economc and moneary unon such as he Eurozone. Flavn e al. () sae ha geographcal varables maer when examnng equy marke lnkages. They menon ha, n parcular, he number of overlappng openng hours and a common border end o ncrease cross-counry sock marke correlaons. However, because of he ncreasng globalsaon, s no obvous wheher dsance s a major economc facor ha conrbues o spaal correlaon. Globalsaon can make oher economc facors more relevan, for example, he marke capalzaon of he sock markes or he gross domesc produc of he counry n whch he sock marke s suaed. I s lkely ha a larger marke capalzaon of a sock marke leads o a greaer nfluence on oher sock markes; he same nerpreaon holds for GDP: a larger GDP s expeced o lead o a more sgnfcan sock marke whch has more nfluence on oher sock markes. The evdence for ha s provded by e.g., Hanhard and Ansoegu (9), who documen a sgnfcan nfluence of Germany (large GDP and large sock marke capalzaon) on all Eurozone sock markes. Alernavely, a wegh marx can be based on some measures of ner-connecedness of economes, e.g., volumes of expors beween counres or he amoun of bank holdngs of anoher counry s asses. However, hese daa are hard o oban and hey may poorly reflec he way he correspondng sock markes are ner-relaed. If volales of several socks are modelled smulaneously, such as n Tsay () or Nakaan and Terasvra (8), he bnary weghs can be used: hese weghs are equal eher one or zero, dependng on, e.g., wheher wo companes belong o he same ndusry classfcaon or no. If we wsh o lnk varances of dfferend knds of markes (sock, fxed ncome, commody, foregn

11 exchange), we can use an emprcal way o choose he wegh marx: we can esmae he long-erm hsorcal correlaons beween markes daly squared logreurns or realzed varances and base he wegh marx on hese correlaons. Such a wegh marx s used n several spaal modellng applcaons n geology, ecology and genecs n he absence of oher, more nerpreable weghs. The advanage of hs mehod s ha s always avalable and ha here s pleny of hsorcal daa o esmae he long-erm hsorcal correlaons. In our emprcal sudy, we wll also consder he spaal GARCH (,) model wh an added componen for leverage effec. We defne he spaal GJR-GARCH (,) model as follows:, N h, a, a, r b, h a, w r b w h r I, (), j j, j N, j j, j,, where = f r, and oherwse. Agan, we expec ha > n mos sock marke I, applcaons. If < (as he case for commody markes), he followng posvy condon s ofen mposed: a for all., The condon for he weak saonary of he spaal GARCH model follows from he correspondng saonary condon for ECCC models, derved by Jeanheau (998) and furher suded n He and Terasvra (4). Ths resul s, n urn, a specal case of a more general resul of Bollerslev and Engle (993) and Engle and Kroner (995) for vecor GARCH (p,q) models whou he assumpon of he consan condonal correlaon. The spaal GARCH (,) model, defned above, s weakly and srcly saonary f he larges (n modulus) egenvalue of he marx A A W ) ( B B ) s less han one. In ha case, he uncondonal varances are gven by S ( I ( A B ) ( A B ) W ) A, where S s he vecor of he uncondonal locaon-specfc varances. ( W For a specfc locaon, he uncondonal varance s gven by s a ( a b ( a ) b N j ) w j s j, where s j, j,..., N are he uncondonal varances a oher locaons. So for posvy of he uncondonal varances s suffcen o requre ha a, a b and a b for all locaons. Noe ha he saonary condon, specfed above, s expressed n erms of he larges egenvalue of he marx A A W ) ( B B ) and no as some smple nequales for parameer values, as s ( W

12 he case for GARCH (,). However, due o he specfc dagonal form of he parameer marces and he resrcon on he wegh marx, we can formulae a smple suffcen saonary condon on he parameers of he spaal GARCH (,) model. Ths condon, saed n he Proposon below, s que src and ceranly s no necessary, bu can serve as an easy frs check of he model s saonary. If, for some parcular applcaon, hs condon fals, does no mean ha he model s no saonary; n ha case he larges (n modulus) egenvalue of he marx ( A AW ) ( B B W ) mus be compued and verfed ha s less han one. Proposon. The spaal GARCH (,) model (8) s weakly and srcly saonary f max a b max a b. Proof: Noe ha, for any marx norm, we have: A B A B, AB A B and A for any egenvalue of A. Take he maxmum A norm: A N j max a and denoe M A A W ) ( B B ). Then, usng he properes j ( W of he norm and he fac ha W, we have ha M A B A B W max a b max a b M and he maxmum (n absolue value) egenvalue of A A W ) ( B B ) s less han one ( W (whch mples he weak and srong saonary of he model, accordng o Jeanheau (998)) f he rgh hand sde of he above nequaly s less han one... ML esmaon The spaal GARCH model (7), (8) can be esmaed by means of he condonal MLE. Some modfcaons wll be requred o deal wh he spaal dependence on oher locaons varances. Le R ( r,,... rr, N ) be he vecor of log-reurns a me (=,...,n), = ( a, a, a, b, b, ) he vecor of parameers a locaon and he vecor of all unknown parameers. The oal (condonal) lkelhood s N,..., Rn ) f ( Rn Fn ) f ( Rn Fn )... f ( R F ) f ( r,,..., rn, ) LF( ) f ( R N LF ( ), () where f r,..., r ) f ( r F ) f ( r F )... f ( r ) s he lkelhood funcon for (, n, n, n n, n, F locaon and s he locaon-specfc se of he parameers. The above expresson follows from he

13 fac ha we assumed ha he nnovaons Z, are normally dsrbued, hen he reurns Z, are ndependen n boh and. If we also assume ha r, are also condonally normal, ndependen n and and he condonal densy ( r, F ) s normal wh mean zero and varance h,. f The above represenaon of he lkelhood shows ha he oal lkelhood funcon s he produc of he lkelhood funcons for each locaon, meanng ha he enre mulvarae spaal GARCH model can be esmaed by applyng he (normal) maxmum lkelhood esmaon procedure separaely o each ndvdual locaon. So for each locaon, he followng model has o be esmaed: h a a r b h a X b Y, (3),,,,,,,,,, where X N, wjr, j and Y, j N w h j, j. j The model (3) can be seen as a GARCH (,) model wh wo exogenous varables X, and Y, (GARCH-X). Maxmum lkelhood esmaon of such models s sraghforward and s ncorporaed n mos packages, such as Malab, Splus or Evews. However, here one essenal dffculy arses: o calculae he values of Y,, he esmaed varances h, j a all oher locaons mus be known. These are no known beforehand, as hey resul from applyng he very same spaal GARCH model ha we are ryng o esmae. To overcome hs problem, we sugges he followng erave soluon. On he frs eraon sep, we esmae (for each locaon) he regular unvarae GARCH (,) model and oban he nal se of parameers ( a, a, b ) and he seres of esmaed varances ( h,..., ). Ths nal varance esmaes are used as varance proxes, o calculae he realzaons, h n, of he exogenous varables, Y,, and o subsequenly esmae he spaal GARCH model. On he nex eraon sep, we esmae he model (3) for each locaon, usng he values of Y, calculaed wh he varances ( h,..., ) esmaed on he prevous sep, and oban he new vecors of parameer, h n, esmaes ( a, a, b,, a b ) and he new seres of esmaed varances: ( h,..., ).. We erae,,,, h n, hs esmaon procedure, re-calculang he values of Y, wh he prevous sep varances and updang he parameer esmaes, unl convergence s acheved,.e., unl he esmaed parameers do no change much from one sep o he nex. The applcaon of hs erave esmaon procedure o he hsorcal daa from major sock markes (descrbed n he nex secon) shows ha he parameer convergence s acheved whn 3 o 6 eraon seps. 3

14 In he nex secon we wll apply he spaal GARCH (,) model (7) and he spaal GJR-GARCH (,) model () o he major sock markes of Europe, US and UK and compare hem o he collecon of unvarae (and unrelaed) GARCH (,) models. We wll use he nverse dsance, GDP-based and marke capalzaon-based wegh marces and assess whch one leads o a beer volaly f. We compare he performance of he spaal GARCH models wh unvarae GARCH models, so we wll no perform any comparson wh mulvarae GARCH models. Ths s because we do no aemp o model he enre varance-covarance marx, bu concenrae on modellng a collecon of condonal varances. Also, our spaal model has only wo more parameers han GARCH (,) model, so comparng wh unvarae GARCH more far han wh over-paramerzed MGARCH models. 3. Emprcal sudy 3.. Daa The daa consss of daly log-reurns on blue-chp sock marke ndces of he nal Eurozone counres (excludng Luxembourg), UK and US, rangng from January unl he end of. Fgure n he Appendx shows he graphs of daly squared log-reurns (an obvous proxy for daly volaly) for The Neherlands (NED), Uned Saes (US) and Uned Kngdom (UK). On all graphs, volaly cluserng s clearly vsble. I s also clear ha volaly clusers occur more or less smulaneously n dfferen counres. However, here are also counry-specfc volaly peaks, such as ha of he NYSE Compose ndex on Ocober 3, 8. Tha day, he NYSE Compose experenced huge gans as nvesors be ha he wors of he cred crss was over, followng a seres of global naves o resolve he fnancal crss. We can also observe peaks ha occur n wo ou of hree counres. For example, he AEX and FTSE exhbed excessve gans of respecvely 9.98 % and 6.8% on March 3, 3. These gans were due o he favorable cour rulng on Corus merger of 999, n whch boh he Neherlands (Hoogovens) and he UK (Brsh Seel) large companes were represened. We wll nclude a dummy varable no he GARCH equaons o accoun for hese abnormal reurns. Table shows he hsorcal correlaon marx of he squared log reurns. These correlaons are on average.5, bu for some pars of counres can reach as hgh as.9 (e.g., beween France and Neherlands, France and Ialy or UK and Neherlands). The frs wegh marx s based on he nverse ravel dsances beween he sock marke ces. The weghs are normalzed, so ha hey add up o n each row. Because of he normalzaon, he resulng wegh marx (Table n he Appendx) s no symmerc: for example, w NED, BEL. 3, For more nformaon: hp://money.cnn.com/8//3/markes/markes_newyork/?posverson=835 4

15 whle w BEL, NED. 8. I can be observed ha weghs w, US (ha should measure he nfluence of he US sock marke on oher markes) are que low, whch conradcs our nuon. Also, noe ha, for example, he larges wegh w, corresponds o Belgum, whch s agan no nuve. The NED j second wegh marx (Table 4) s based on GDPs of counres as of, provded by he IMF (and shown n Table 3). Ths marx seems more realsc, as US, UK and Germany ge he hghes weghs, as we mgh expec. Fnally, he marke capalzaon wegh marx (shown n Table 5) s obaned from marke capalzaons as of 3 (also repored n Table 3); as expeced, s very smlar o he GDP wegh marx. 3.. Esmaon resuls We esmae he spaal GARCH (,) model (7) on he bass of he hreen sock markes reurns, for all hree wegh marces. We compare he performance of he spaal GARCH (,) models o unvarae GARCH (,) models. We also esmae he spaal GJR-GARCH (,) model (), o esablsh wheher ncludng he leverage effec leads o a beer f. Frs, we performed he Engle (98) es for heeroscedascy for all 3 seres. The null hypohess of homoscedascy s rejeced for all sock markes. Then, unvarae GARCH and spaal GARCH models were esmaed by maxmzng he condonal quas-maxmum lkelhood (for spaal GARCH, he erave MLE mehod descrbed above was used, maxmzng he lkelhood funcon () a each eraon). To accoun for possble condonal non-normaly, we used he heeroscedascy conssen covarance of Bollerslev and Wooldrdge (99). For spaal GARCH, he parameer convergence was acheved whn 3 o 6 eraons; he convergence creron was se a 5% of he parameer values. Fgures, 3 and 4 show he parameer esmaes for he spaal GARCH (,) model wh respecvely he nverse dsance, GDP and marke capalzaon wegh marces (he parameers a are greaer han zero for all locaons, bu also all que small and so are no shown). These fgures show ha nearly all spaal parameers are hghly sgnfcan for GDP and marke capalzaon wegh marces and spaal ARCH parameers are sgnfcan for he nverse dsance wegh marx (n ha case, spaal GARCH parameers are que close o zero excep for Ausra and USA). Ths ndcaes ha he spaal volaly spllovers are ndeed observed and sgnfcan among he consdered sock markes. Spaal ARCH coeffcens are generally larger han spaal GARCH ones, ndcang ha he laes squared reurns from oher markes maer more for he fuure volaly levels han he prevous values of oher markes varances. 3 Through Daasream, annual ndex repors and relevan webses such as 5

16 Anoher observaon s ha boh posve and negave values of parameers occur, ndcang eher negave volaly spllovers or a compensaon for relavely hgh values of anoher (G)ARCH parameer (spaal vs non-spaal). Generally, seems ha he (G)ARCH and spaal (G)ARCH parameers compensae each oher: he greaer s one, he lower s he oher. Also, boh spaal parameers for nearly all locaons ake over from he regular (G)ARCH parameers, whch sgnfcanly dmnsh n values compared o unvarae GARCH model parameers. For all wegh marces, he condons ha assure posvy of he uncondonal varances: a, a b and a b hold for all locaons. The condon of he Proposon : max a b max a b does no hold for any of he wegh marx (ndcang ha hs condon s ndeed oo srong), bu he calculaed maxmal (n absolue value) egenvalues of he esmaed marces A ˆ Bˆ ) ( Aˆ Bˆ ) W are less han one for all hree cases (.9 for he nverse ( dsance wegh marx,.97 for he GDP wegh marx and.98 for he marke capalzaon wegh marx), ponng ou owards he exsence of a saonary soluon o he spaal GARCH equaon (8). The spaal GARCH model leads o a dramac mprovemen n he model f, compared o he collecon of unvarae GARCH models. Ths s llusraed n Fgure 5 on he example of FTSE. The unvarae GARCH volales are smooher and dsplay more perssence han he spaal GARCH volales; however, he spaal GARCH model does a much beer job a capurng he observed volaly cluserng, exreme squared reurns and perods of hgh volaly: an mporan aspec where he unvarae GARCH fals compleely. The same paern s observed for all oher markes (n a lesser degree for he US sock marke). Ths s confrmed by observng he sandardzed resduals, whch, for all locaons, are approxmaely normal (resdual skewness s nearly zero and resdual kuross s approxmaely 3 for all locaons) for he spaal GARCH bu no for unvarae GARCH models, where he null-hypohess of normaly of he sandardzed resduals s srongly rejeced for all locaons. The rade-off for hs effec s he fac ha he unvarae GARCH leaves no resdual ARCH effecs (esed by Engle s LM sascs), whle for he spaal GARCH resduals, he same LM sascs ndcae some remanng ARCH effecs. Ths s possbly due o he fac ha he regular GARCH parameers are overaken by he spaal GARCH parameers, whch can be solved by addng a few more lagged ARCH and GARCH erms. Table 6 shows he overall measures of model f for AEX, FTSE and NYSE Compose ndces (servng as examples), for all models and wegh marces. Observaons on Table 6 exend o all oher sock markes: he RMSE, log-lkelhood and BIC all mprove sgnfcanly by gong from unvarae o he spaal GARCH model. The average mprovemen n log-lkelhood s 6%; smlar resuls hold 6

17 for RMSE and BIC. Ths s observed for all markes (agan, n a lesser degree for he US sock marke, where he mprovemen n erms of he lkelhood s raher margnal - hs s as expeced, snce he effec of oher sock markes on he US sock marke s no ha hgh). Includng he leverage effec no he spaal GARCH model furher mproved he measures of f (however, he effec of ncludng leverage was no as dramac as gong from unvarae o spaal GARCH). For all seres, he leverage coeffcens of he spaal GJR-GARCH model () were posve and hghly sgnfcan and he log-lkelhood furher ncreased by.5% on average. Regardng he relave performance of varous wegh marces, we can observe he followng. Frs, he spaal GARCH models wh hree dfferen wegh marces perform comparable o each oher. For European counres, however, he model wh he nverse dsance wegh marx performs slghly beer, as he example of AEX shows n Table 6. The US sock marke volaly s modelled equally well by all hree wegh marces, whle he case of UK les somewhere n he mddle. Ths ndcaes ha, for he Eurozone, he geographc proxmy s an mporan deermnan of he sock markes neracons, as reflecs hsorcally esablshed economc lnks. Fnally, Table 7 shows he correlaons beween unvarae and spaal GARCH volales and he daly volaly proxy for AEX, FTSE and NYSE. Agan, he resuls shown n hs able hold for all 3 markes: hese correlaons are much hgher for he spaal han for he unvarae GARCH models. In all, we fnd ha he spaal GARCH model (wh or whou leverage effecs) sgnfcanly ouperforms he unvarae GARCH ones on nearly all crera, whle requres only margnal addonal complexy and compuaonal effor. 4. Conclusons We presened he spaal GARCH model class, where we capure he marke nerdependences and volaly spllovers by means of a pre-defned and economcally relevan wegh marx. The model has a very low number of parameers: hs number grows lnearly wh he dmenson and no as he second (or even fourh) power of he dmenson, as n oher mulvarae volaly models. The applcaon of he spaal GARCH model o an ensemble of major sock markes has shown ha he spaal parameers are hghly sgnfcan, ndcang he presence of volaly spllovers beween markes. The spaal GARCH model exhbs much beer f han he regular GARCH model; he f s furher mproved by addng he componen for he leverage effec. The model s capable o capure 7

18 skewness and hgh kuross presen n he daa, n suaons where he regular GARCH model fals o do so. The spaal GARCH model was appled here o naonal sock markes wh he wegh marces based on he dsances beween sock marke s ces, counres GDPs or sock marke capalzaons. For he Eurozone, he dsance-based wegh marx performed he bes, ndcang he mporance of geographc proxmy as he measure of esablshed economc lnks whn he regon. The spaal GARCH s also excellenly sued o modellng an ensemble of ndvdual socks, wh he wegh marx consruced accordng o he well-known Indusry Classfcaon Benchmark. I can also be appled o varous oher mulvarae cases, such as o a porfolo of commodes, wh he wegh marx reflecng he commody classes (meals, agrculural, energy). The applcaon of he spaal GARCH model o hese suaons s he work n progress. Our furher research focuses on esablshng he asympoc properes of he ML esmaors, such as conssency and asympoc normaly. Moreover, gven here s ample evdence for sgnfcan volaly spllovers, he causaly relaonshps beween sock marke volales deserve furher nvesgaon. References Bauwens L. S., Lauren S. and Rombous J.V.K. (6) Mulvarae GARCH Models: A Survey. Journal of Appled Economercs, Vol, pp Baba, Y., R. F. Engle, D. Kraf, and K. Kroner (99) Mulvarae Smulaneous Generalzed ARCH, UCSD, Deparmen of Economcs, unpublshed manuscrp. Bollerslev T. (986) Generalzed Auoregressve Condonal Heeroskedascy. Journal of Economercs, Vol 3, pp Bollerslev, T., R.F. Engle and J.M. Wooldrdge (988) A capal asse prcng model wh me varyng covarances. Journal of Polcal Economy, 96, pp Bollerslev, T. and J. M. Wooldrdge (99) Quas maxmum lkelhood esmaon and nference n dynamc models wh me varyng covarances. Economerc Revews, Vol., pp Bollerslev, T. And R.F. Engle (993) Perssence n condonal varances. Economerca, Vol. 6, pp

19 Borovkova S., Lopuhaä H.P., and Nuran B. () Generalzed STAR model wh expermenal weghs. Proceedngs of he 7h Inernaonal Workshop on Sascal Modelng. Borovkova S., Lopuhaä H.P., and Nuran B. (8) Conssency and asympoc normaly of leas squares esmaors n generalzed STAR models. Sasca Neerlandca, Vol 6, pp Chang M. H. and Wang L.M. () Volaly conagon: A range-based volaly approach. Journal of Economercs, Vol 65, pp Chrsodoulaks, G A., Sachell, S E. () Correlaed ARCH (CorrARCH): Modellng he mevaryng condonal correlaon beween fnancal asse reurns. European Journal of Operaonal Research, Vol. 39, pp Conrad C. And Karanasos M. () Negave volaly spllovers n he unresrced ECCC-GARCH mnodel. Economerc Theory, Vol. 6, pp Engle, R. F. (98) Auoregressve Condonal Heeroskedascy wh Esmaes of he Varance of Uned Kngdom Inflaon, Economerca, Vol 5 (4), pp Engle R. F. and Kroner K.F. (995) Mulvarae Smulaneous Generalzed ARCH. Economerc Theory, Vol, pp. -5. Engle, R.F. () Dynamc condonal correlaon: A smple class of mulvarae GARCHmodels. Journal of Busness and Economc Sascs, Vol., pp Flavn, T.J., M. J. Hurley and F. Rousseau () Explanng Sock Marke Corrlaon: a Gravy Model Approach, The Mancheser School Supplemen, pp Gacno V.D. (6) A generalzed Space-Tme ARMA model wh an applcaon o regonal unemploymen analyss n Ialy. Inernaonal Regonal Scence Revew, Vol 9, pp Glosen, L. R., Jagannahan, R. and Runkle, D. E. (993) On he Relaon Beween he Expeced Value and he Volaly of he Nomnal Excess Reurn on Socks. The Journal of Fnance, Vol 48 (5), pp

20 Hafner, C.M. and O. Reznkova (). On he esmaon of dynamc condonal correlaon models. Compuaonal Sascs and Daa Analyss, Vol. 56, pp Hanhard, A. and Ansoegu, C. (9) Measurng European Sock Marke Inegraon Va a Sochasc Dscoun Facor Approach. ESADE Busness School, Barcelona, Span. He, C. and T. Terasvra (4) An exended consan condonal correlaon GARCH model and s fourh-momen srucure. Economerc Theory, Vol., pp Jeanheau, T. (998) Srong conssency of esmaons of mulvarae ARCH model. Economerc Theory, Vol 4, pp Knf, J., Kolar, J. and Pynnonen, S. (5) Wha drves correlaon beween sock marke reurns? Inernaonal evdence. Unversy of Vaasa, Workng paper. Lam, S.S. and Ang, W.W. L. (6) Globalzaon and Sock Marke Reurns. Global Economy Journal, Vol. 6. Lng, S. And McAleer, M. () Saonary and he exsence of momens of a famly of GARCH processes. Journal of Economercs, Vol 6, pp Nakaan, T. And T. Terasvra (8) Posvy consrans on he condonal varances n he famly of condonal correlaon GARCH models. Fnance Research Leers, Vol. 5, pp Nelson, D.B. and Cao, C.Q. (99) Inequaly consrans n he unvarae GARCH model. Journal of Busness and Economc Sascs, Vol, pp Ogum, G. () Equy Volaly Transmsson and Conagon beween he US and Emergng Sock Markes: The Role of he US Subprme Crss. Workng paper, La Serra Unversy. Orano, E. () Idenfyng fnancal me seres wh smlar dynamc condonal correlaon. Compuaonal Sascs and Daa Analyss, Vol. 54, pp. -5. Polson, N.G. and J.G. Sco () An asse prcng model for he conagon age. Bloomberg Vew. Avalable a: hp://

21 Peng, Y. And Ng, W.L. () Analysng fnancal conagon and asymmerc marke dependence wh volaly ndces va copulas. Annals of Fnance, Vol. 8, pp Pfefer P.E and Deusch S.J. (98) Idenfcaon and Inerpreaon of Frs order Space Tme ARMA. Technomercs, Vol, p Ross, E. and F. Spazzn () Model and dsrbuon uncerany n mulvarae GARCH esmaon: A Mone Carlo analyss. Compuaonal Sascs and Daa Analyss, Vol. 54, pp Sarors A. (5) STARMA Models for Crme n he Cy of Sao Paolo. Workng Paper. Tse, Y.K. and A.K.C. Tsu () A mulvarae GARCH model wh me-varyng correlaons. Journal of Busness and Economc Sascs, Vol., Vogelvang B. (5) Economercs: Theory and Applcaons wh Evews. Essex: Pearson Educaon Lmed. Zvo E. (9) Praccal Issues n he Analyss of Unvarae GARCH Models. Handbook of Fnancal Tme Seres, Sprnger-Verlag.

22 ,,9,6,3 Daly volaly AEX-ndex,,8,4 Tme - Daly volaly NYSE Compose Index,6,,8,4 Squared log reurns Squared log reurns Appendx Fgure : Graphs wh daly volaly approxmaons for he AEX. FTSE and NYSE Compose ndex Daly volaly FTSE Squared log reurns Tme - Tme -

23 Table : Correlaon marx of he squared log reurns of 3 blue-chp sock marke ndces NED BEL GER FIN IRE AUT SPA POR ITA FRA GRE UK US NED. BEL.848. GER FIN IRE AUT SPA POR ITA FRA GRE UK US

24 Table : Inverse ravel dsance wegh marx NED BEL GER FIN IRE AUT SPA POR ITA FRA GRE UK US NED,3,85,359,7,578,364,9,653,6,5,58,9 BEL,8,67,95,63,53,37,85,699,863,33,58,83 GER,37,97,85,6,8,45,366,,9,496,86,34 FIN,79,98,53,8,5,55,48,837,849,656,89,45 IRE,68,4,688,474,57,66,585,679,35,336,74,88 AUT,988,8,96,64,549,5,4,48,89,7,748,36 SPA,776,874,599,39,79,635,84,967,9,485,99,99 POR,733,798,577,46,833,594,74,8,94,479,86,5 ITA,5,46,,449,64,393,73,56,354,594,96,34 FRA,393,8,63,3,767,576,566,4,99,84,749, GRE,837,866,35,733,634,4,764,635,38,863,757,8 UK,7,97,63,334,34,493,48,384,635,785,54,9 US,86,859,78,764,988,744,876,93,78,866,638,97 4

25 Table 3: Marke capalzaon and GDP Marke cap x Counry Code Name Index bl. Counry Code GDP x $ bl. NED AEX 43.4 BEL BEL 53. GER DAX FIN OMX HELSINKI 5.6 IRE ISEQ 59.3 AUT ATX 9.4 SPA IBEX POR PSI ITA FTSE MIB INDEX 6.8 FRA EURONEXT CAC 77. GRE FTSE/ATHEX.4 UK FTSE US NYSE COMPOSITE INDEX 4.9 NED 7 BEL 44 GER 34 FIN 94 IRE 87 AUT 35 SPA 46 POR 49 ITA 847 FRA 4 GRE 94 UK 88 US 576 5

26 Table 4: GDP wegh marx NED BEL GER FIN IRE AUT SPA POR ITA FRA GRE UK US NED,5,7,7,68,7,59,9,668,8,6,88,5456 BEL,5,5,69,67,5,54,89,66,793,5,89,54 GER,78,64,77,74,39,557,99,73,878,7,97,5978 FIN,49,47,7,66,4,5,88,656,787,4,83,5357 IRE,49,47,6,69,4,5,88,656,787,4,83,5356 AUT,5,48,3,69,67,5,89,66,79,5,88,5387 SPA,6,54,56,7,69,3,9,686,8,9,85,5599 POR,5,47,9,69,67,5,5,658,788,5,85,5368 ITA,65,56,76,73,7,3,53,94,836,,864,569 FRA,68,58,9,74,7,34,538,95,77,3,876,577 GRE,5,48,,69,67,5,5,89,659,79,86,5377 UK,69,59,96,74,7,34,54,96,79,85,3,5788 US,59,3,349,46,4,64,6,88,393,67,,76 6

27 Table 5: MarkeCap wegh marx NED BEL GER FIN IRE AUT SPA POR ITA FRA GRE UK US NED,84,653,56,3,6,55,,33,393,55,463,6699 BEL,3,649,56,3,59,54,,39,39,55,456,6666 GER,4,88,59,34,63,63,,349,44,58,543,766 FIN,3,83,648,3,59,53,,38,39,55,45,6647 IRE,3,83,646,55,59,53,,37,389,55,448,663 AUT,3,83,648,55,3,53,,38,39,55,45,665 SPA,33,84,654,56,3,6,,33,394,55,466,673 POR,3,83,645,55,3,59,53,37,389,55,447,664 ITA,36,86,666,57,33,6,58,,4,56,49,6833 FRA,37,86,67,57,33,6,59,,339,57,5,6877 GRE,3,83,648,55,3,59,53,,38,39,45,6647 UK,54,97,753,64,37,69,78,4,38,453,64,776 US,39,46,95,3,95,563,453,6,97,53,6,3679 7

28 Fgure : The esmaed parameers wh 95% confdence nervals for he spaal GARCH (,) model wh he nverse ravel dsance wegh marx. 8

29 Fgure 3: The esmaed parameers wh 95% confdence nervals for he spaal GARCH (,) model wh he GDP wegh marx. 9

30 Fgure 4: The esmaed parameers wh 95% confdence nervals for he spaal GARCH (,) model wh he marke capalzaon wegh marx. 3

31 Fgure 5. Squared log-reurns, unvarae GARCH (,) and spaal GARCH (,) volaly forecass for FTSE. DAILY_VOLATILITY_UK GARCH UK.6 SGARCH UK Condonal varance Condonal varance

32 Table 6. Resdual mean squared error (RMSE) and log-lkelhood values for he AEX, FTSE and NYSE Compose and all wegh marces. Inverse ravel dsance wegh marx AEX FTSE NYSE Compose RMSE Loglkelhood BIC RMSE Loglkelhood BIC RMSE Loglkelhood BIC GARCH (,) E E E Spaal GARCH (,) E E E SGJR-GARCH (,) E E E Marke Capalzaon wegh marx AEX FTSE NYSE Compose RMSE Loglkelhood BIC RMSE Loglkelhood BIC RMSE Loglkelhood BIC GARCH (,) E E E Spaal GARCH (,) 3.53E E E SGJR-GARCH (,) 3.899E E E GDP wegh marx AEX FTSE NYSE Compose RMSE Loglkelhood BIC RMSE Loglkelhood BIC RMSE Loglkelhood BIC GARCH (,) E E E Spaal GARCH (,) 3.588E E E SGJR-GARCH (,) 3.884E E E

33 Table 7. Correlaons of he condonal varances of varous models wh each oher and wh he daly volaly proxy. FTSE Daly Volaly GARCH SGARCH (D) SGJR-GARCH (D) SGARCH (M) SGJR-GARCH (M) SGARCH (G) SGJR-GARCH (G) Daly Volaly. GARCH SGARCH (D) SGJR-GARCH (D) SGARCH (M) SGJR-GARCH (M) SGARCH (G) SGJR-GARCH (G) NYSE Compose Daly Volaly GARCH SGARCH (D) SGJR-GARCH (D) SGARCH (M) SGJR-GARCH (M) SGARCH (G) SGJR-GARCH (G) Daly Volaly. GARCH.57. SGARCH (D) SGJR-GARCH (D) SGARCH (M) SGJR-GARCH (M) SGARCH (G) SGJR-GARCH (G) AEX Daly SGARCH SGARCH SGJR-GARCH SGJR-GARCH Volaly GARCH (D) SGJR-GARCH (D) (M) (M) SGARCH (G) (G) Daly Volaly. GARCH.488. SGARCH (D) SGJR-GARCH (D) SGARCH (M) SGJR-GARCH (M) SGARCH (G) SGJR-GARCH (G)