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1 econsor Make Your Publicaions Visible. A Service of Wirschaf Cenre zbwleibniz-informaionszenrum Economics Allen, David E.; McAleer, Michael; Powell, Rober; Singh, Abhay K. Working Paper Volailiy Spillover and Mulivariae Volailiy Impulse Response Analysis of GFC News Evens Tinbergen Insiue Discussion Paper, No /III Provided in Cooperaion wih: Tinbergen Insiue, Amserdam and Roerdam Suggesed Ciaion: Allen, David E.; McAleer, Michael; Powell, Rober; Singh, Abhay K. (2016) : Volailiy Spillover and Mulivariae Volailiy Impulse Response Analysis of GFC News Evens, Tinbergen Insiue Discussion Paper, No /III This Version is available a: hp://hdl.handle.ne/10419/ Sandard-Nuzungsbedingungen: Die Dokumene auf EconSor dürfen zu eigenen wissenschaflichen Zwecken und zum Privagebrauch gespeicher und kopier werden. Sie dürfen die Dokumene nich für öffenliche oder kommerzielle Zwecke vervielfäligen, öffenlich aussellen, öffenlich zugänglich machen, verreiben oder anderweiig nuzen. Sofern die Verfasser die Dokumene uner Open-Conen-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gesell haben sollen, gelen abweichend von diesen Nuzungsbedingungen die in der dor genannen Lizenz gewähren Nuzungsreche. Terms of use: Documens in EconSor may be saved and copied for your personal and scholarly purposes. You are no o copy documens for public or commercial purposes, o exhibi he documens publicly, o make hem publicly available on he inerne, or o disribue or oherwise use he documens in public. If he documens have been made available under an Open Conen Licence (especially Creaive Commons Licences), you may exercise furher usage righs as specified in he indicaed licence.

2 Volailiy Spillover and Mulivariae Volailiy Impulse Response Analysis of GFC News Evens David E. Allen a, Michael McAleer b, Rober Powell c, and Abhay K. Singh c a School of Mahemaics and Saisics, Universiy of Sydney, School of Business, Universiy of Souh Ausralia b Deparmen of Quaniaive Finance, Naional Tsing Hua Universiy, Taiwan, Economeric Insiue, Erasmus School of Economics, Erasmus Universiy, Roerdam, The Neherlands, Deparmen of Quaniaive Economics, Compluense Universiy of Madrid, Spain, Insiue of Advanced Sciences, Yokohama Naional Universiy, Japan c School of Business and Law, Edih Cowan Universiy, Perh, Ausralia Absrac: This paper applies wo measures o assess spillovers across markes: he Diebold Yilmaz (2012) Spillover Index and he Hafner and Herwarz (2006) analysis of mulivariae GARCH models using volailiy impulse response analysis. We use wo ses of daa, daily realized volailiy esimaes aken from he Oxford Man RV library, running from he beginning of 2000 o Ocober 2016, for he S&P500 and he FTSE, plus en years of daily reurns series for he New York Sock Exchange Index and he FTSE 100 index, from 3 January 2005 o 31 January Boh daa ses capure boh he Global Financial Crisis (GFC) and he subsequen European Sovereign Deb Crisis (ESDC). The spillover index capures he ransmission of volailiy o and from markes, plus ne spillovers. The key difference beween he measures is ha he spillover index capures an average of spillovers over a period, whils volailiy impulse responses (VIRF) have o be calibraed o condiional volailiy esimaed a a paricular poin in ime. The VIRF provide informaion abou he impac of independen shocks on volailiy. In he laer analysis, we explore he impac of hree differen shocks, he onse of he GFC, which we dae as 9 Augus 2007 (GFC1). I ook a year for he financial crisis o come o a head, bu i did so on 15 Sepember 2008, (GFC2). The hird shock is 9 May Our modelling includes leverage and asymmeric effecs underaken in he conex of a mulivariae GARCH model, which are hen analysed using boh BEKK and diagonal BEKK (DBEKK) models. A key resul is ha he impac of negaive shocks is larger, in erms of he effecs on variances and covariances, bu shorer in duraion, in his case a difference beween hree and six monhs. Keywords: Spillover Index, Volailiy Impulse Response Funcions (VIRF), BEKK, DBEKK, Asymmery, GFC, ESDC. JEL: C22, C32, C58, G32.

3 1. INTRODUCTION The similariies beween GARCH and VARMA-ype models provide a foundaion for he approach o generalize impulse response analysis, as inroduced by Sims (1980), o he analysis of shocks in financial volailiy. Previous alernaive approaches in he lieraure have been made owards racing he impac of various ypes of shocks hrough ime (see, for example, Koop e al. (1996), Engle and Ng, (1993), Gallan e al. (1993), and Lin (1997)). Koop e al. (1996) defined generalized impulse response funcions for he condiional expecaion using he mean of he response vecor condiional on hisory and a curren shock, as compared wih a baseline ha condiions only on hisorical innovaions. The wo merics we use o capure spillovers in his paper build upon hese approaches. Diebold and Yilmaz (2009, 2012) develop measures of reurn and volailiy spillovers based on vecor auoregressive (VAR) models. The Diebold and Yilmaz (2012) varian of heir measure is based on generalized impulse responses and capures boh direcional and ne spillovers beween markes. The firs se of analyses in his paper is based on he applicaion of hese merics o analyse daily realized volailiy (RV) merics aken from he Oxford-Man Insiue of Quaniaive Finance Realized Library for he S&P500 and he FTSE index (See Gerd e al., (2009)). The Diebold and Yilmaz Spillover Index mehod has araced some aenion in he lieraure. Kloessner and Wagner (2012) presen an algorihm, o explore he rue range of he (2009) spillover index, in which he impulse response funcions depended on he ordering of variables in he VAR. However, his issue has been avoided in he Diebold and Yilmaz (2012) measure. Aler and Beyer (2013) explore he dynamics of he European Sovereign Deb Crisis using a meric based on he Spillover Index. Diebold and Yilmaz (2014, 2016) have expanded heir mehod o measure financial firm inerconnecedness. The second se of analyses feaure Hafner and Herwarz s (2006) Volailiy Impulse Response Funcions (VIRFs) which also exend he generalized impulse response funcions framework provided by Koop e al. (1996). Their approach is novel in ha VIRF explores he condiional variance raher han he condiional mean. Given ha GARCH models can be viewed as being linear in he squared innovaions, and ha mulivariae GARCH models are known o have a VARMA represenaion wih non-gaussian errors, Hafner and Hewarz (2006) adop his paricular srucure o calculae condiional expecaions of volailiy analyically in heir VIRF analysis. 2

4 Panopoulou and Panelidis (2009) examine volailiy ransmissions beween he U.S. and he res of he G-7 counries using daily sock marke reurn daa and repor ha he linkages beween he markes had changed subsanially wih naional markes becoming more inerdependen. They provide evidence of direc volailiy spillovers, running mainly from he US and poin o more rapid informaion ransmission during he laer years of heir sudy. Their analysis is he closes in spiri o he curren sudy, bu hey do no examine he impac of asymmeric shocks in heir GARCH framework or employ he Diebold and Yilmaz (2012) Spillover Index analysis. Jin e al., (2012), use VIRF o analyse he ransmission of shocks in crude oil markes, whils Le Pen and Sevi (2010), underake a similar analysis of elecriciy markes. Ohlsen e al., (2014) employ VIRF analysis o explore he relaionship beween energy and equiy markes. More generally, in he GARCH lieraure here has been a longsanding concern wih modelling volailiy ransmission. An early sudy by Koumos and Booh (1995) examined price volailiy spillovers for he US, he UK and Japan in he conex of a mulivariae EGARCH model which permied he capure of possible asymmeries in he volailiy ransmission mechanism. These auhors found evidence of price spillovers, and exensive and reciprocal second momen ineracions, which were asymmeric, i.e. negaive innovaions in a given marke increased volailiy in he nex marke o rade more han posiive innovaions. We furher explore his issue in he curren paper. Furhermore, Ross (1989), suggesed ha under appropriae condiions, he variance of price change equals he rae of informaion flow, and hus provided a direc link beween he second momen and he flow of informaion, in an arbirage free economy. In a coninuaion of his logic, Engle e al., (1990) noed ha a possible explanaion for ARCH effecs and an explanaion of he phenomenon of volailiy clusering, mus lie eiher in he arrival process of news, or in marke dynamics in response o he news. If informaion comes in clusers, hen he asse reurns or prices may exhibi ARCH behaviour, even if he marke perfecly and insananeously adjuss o he news. The curren paper follows in his radiion and uses impulse response analysis o analyse he ransmission of shocks across markes. In our Generalized VIRF (GVIRF), we consider hree major news evens which ac as shocks o he volailiy of our wo series. The onse of he GFC, which we dae as 9 Augus 2007 (GFC1), began wih he seizure in he banking sysem precipiaed by BNP Paribas announcing ha i was ceasing aciviy in hree hedge funds ha specialised in US morgage deb. I ook one year for he financial crisis o come o a head, bu i did so on 15 Sepember 2008 when he US governmen allowed he invesmen bank Lehman Brohers o go bankrup (GFC2). The dae 9 May 2010 marked he poin a 3

5 which he focus of concern swiched from he privae secor o he public secor. By he ime he IMF and he European Union announced hey would provide financial help o Greece, he issue was no longer he solvency of banks bu he solvency of governmens, and his marks he onse of he European Sovereign Deb Crisis (ESDC). The major difference beween he wo approaches is ha he firs uilises a VAR approach o joinly analyse a ime-series of he daily RV series for he wo markes, as represened by he S&P500 and he FTSE. The mehod feaures an analysis of he average of he RV series for he wo markes and he merics applied capure spillovers o and from he wo markes and he ne spillovers. The VIRF analysis is developed in he conex of a mulivariae GARCH approach, incorporaing assymeric effecs, and feauring analysis of he impulse responses of he condiional volailiy series. Given ha volailiy is condiional, i makes sense o condiion he model on volailiy a a given poin in ime, raher han an average. Hence, we use hree differen poins in ime, or subsamples, in our basic series o capure impacs a he onse of he GFC, he heigh of he GFC, and he beginning of subsequen European Sovereign deb crisis. The remainder of he paper is as follows. In Secion 2 he research mehods and daa are discussed, including he Spillover Index, volailiy impulse response funcions, mulivariae GARCH models, he regulariy condiions for BEKK and diagonal BEKK (DBEKK) models, he riangular, Hadamard and full BEKK models, and diagonal and scalar BEKK models. The empirical resuls are discussed in Secion 3, and some concluding remarks are given in Secion RESEARCH METHODS AND DATA We use wo differen parameric approaches o explore he ransmission of volailiy shocks across markes; he Diebold and Yilmaz (2009, 2012) Spillover Index and he Hafner and Herwarz (2006) mulivariae volailiy impulse response analysis. The nex sub-secions inroduce he mehods used. 2.1 Spillover index Diebold and Yilmaz (2009) develop a measure of reurn and volailiy spillovers based on vecor auoregressive (VAR) models in he broad radiion of Engle, Io and Lin (1990). They concenrae on variance decomposiions, and hey demonsrae how i is possible o aggregae spillover effecs across 4

6 markes, capuring a grea deal of informaion ino a single spillover measure. They consruc heir measure using variance decomposiions associaed wih an N variable VAR. They proceed by aking each asse i, and adding he shares of is forecas error variance coming from shocks o asse j, for all j i, and hen hey add across all i, i =1,..., N. The variance decomposiions allow permi hem o spli he forecas error variances of each variable ino pars aribuable o he various sysem shocks. They aggregae and conduc a spillover index. A drawback of he Diebold and Yilmaz (2009) spillover index is ha i relies on Cholesky-facor idenificaion of VARs, meaning ha he resuling variance decomposiions can be dependen on variable ordering. In addiion heir (2009) measure capures oal spillovers bu no direcional spillovers. Diebold and Yilmaz (2012) exend heir (2009) meric o make i invarian on ordering, by using generalised impulse response funcions, and consruc i in a manner ha capures direcional spillovers. They proceed in he following manner. They consider a covariance saionary N-variable VAR(p), Φ, where, ~0, is a vecor of i.i.d. disurbances. The moving average represenaion is, where he coefficien marices obey he recursion Φ Φ Φ, wih an ideniy marix and 0 for 0. The Diebold and Yilmaz (2009, 2012) spillover index measures use variance decomposiions, which permi hem o decompose he forecas error variances of each variable ino pars aribuable o he various sysem shocks. The innovaion in Diebold and Yilmaz (2012) is ha hey employ a generalized VAR framework in he manner of Koop, Pesaran and Poer (1996) and Pesaran and Shin (1998). The generalized framework permis correlaed shocks bu reas hem appropriaely using he hisorically observed disribuion of he errors. Diebold and Yilmaz (2012) define own variance shares as he fracion of he H-sep-ahead error variances in forecasing due o shocks o, for 1,2,,, and cross variance shares, or spillovers, as he fracions of he H-sep-ahead error variances in forecasing resuling from shocks o, for i, 1,2,,, such ha. Diebold and Yilmaz (2012) wrie he generalised H-sep-ahead forecas error variance decomposiions by, for H=1,2,., resuling in Where. 1 is he variance marix for for he error vecor, is he sandard deviaion of he error erm for he i h equaion and is he selecion vecor wih one as he i h elemen and zero oherwise. 5

7 Given ha hey have used generalised impulse response funcions he sum of he elemens of each row of he variance decomposiion able is no equal o 1: They use he informaion available in he variance decomposiion marix for he consrucion of he spillover index by normalizing each enry of he variance decomposiion marix by he row sum as: In his consrucion, 1and, Deibold and Yilmaz (2012) hen proceed o consruc a oal volailiy spillover index as:,,.100, They furher consruc a direcional spillover measure o illuminae how volailiy spills across from differen asses or asse classes. They do his by using he normalized elemens of he generalized variance decomposiion marix. Their measure of direcional volailiy spillover received by marke i from oher markes j as: By conras direcional volailiy spillovers ransmied by marke i o marke j as: 100. Finally, Diebold and Yilmaz (2012) compue ne spillovers from marke i o all oher markes j as:. The ne volailiy spillover is he difference beween gross volailiy shocks ransmied o and gross volailiy shocks received from all oher markes. We use hese measures and he mulivariae volailiy impulse response funcions inroduced in he nex subsecion Mulivariae volailiy impulse response funcions 6

8 Hafner and Herwarz (2006) develop heir model by leing denoe an N-dimensional random vecor, so ha: P, (7) where P P ' and denoes an iid random vecor of dimension N, wih independen componens, mean zero and ideniy covariance marix. Hafner and Herwarz assume ha is measurable wih respec o he informaion se available a ime -1, 1 F 1 F. Equaion (1) implies ha, E F 0 and 1 Var. They noe ha could be he error of a VARMA process. If is a mulivariae GARCH process, hen equaion (1) may be called a srong GARCH model, according o Dros and Nijman (1993). This is convenien because i permis he modelling of news evens as appearing in he iid innovaion,. They idenify by assuming ha P is a lower riangular marix, which permis he use of a Choleski decomposiion of. They also use he fac ha independen news can ofen be idenified by means of a Jordan decomposiion, which will permi idenificaion when he innovaion vecor is non-normal. Hafner and Herwarz adop a mulivariae GARCH(p,q) model framework, given by: q i1 i ' i p vech( ) c A vech( ) B vech( ), (8) i j1 j i and use he BEKK model of Baba e al. (1985) and Engle and Kroner (1995), which is a special case of equaion (8), and is specified as: K q ' ' 0C0 Aki i i k 1 i1 K p k 1 i1 ' C A G G. (9) ki ki i ki In equaion (9), C0 is a lower riangular marix, and A ki and G ki are N N parameer marices. 2.3 Volailiy Impulse Response Funcions 7

9 Hafner and Herwarz (2006) proceed by assuming ha, a ime, some independen news is refleced in 0, and i is no specified wheher he news is good or bad. The condiional covariance marix,, is a funcion of he innovaions, 1,..., 1, he original shock, 0, and 0. Hafner and Herwarz define VIRF as he expecaion of volailiy condiional on an iniial shock and on hisory, minus he baseline expecaion ha only condiions on hisory, as given in he following: V ( ), F Evech( F ( 0 ) E vech 0 1 ) 1 (10) In equaion (10), V ) is an N * -dimensional vecor. ( 0 Hafner and Herwarz consider a VARMA represenaion of a mulivariae GARCH(p,q) model in order ' o find an explici expression for V ( 0 ), and define vech( ). They define he mulivariae GARCH(p,q) model as a VARMA(max(p,q), p) model: max( p, q) ( A B ) B u u, (11) i i i i1 j 1 p j j where u vech( ) is a whie noise vecor. From equaion (11), Hafner and Herwarz derive he VMA( ) specificaion, as follows: i0 vech( ) u, (12) i i where he * * N N marices i can be deermined recursively. The general expression for VIRF is: 1/ 2 1/ 2 ' V ( ) D ( ) D vech( I ). (13) 0 N N 0 0 N 0 0 Hafner and Herwarz (2006) consider a variey of specificaions for he baseline shock. The behaviour implied by equaion (13) is differen from radiional impulse response analysis. In (13), he impulse is an even, no odd, funcion of he shock, i is no linear in he shock, and he VIRF depends on he 8

10 hisory of he process, alhough his is via he volailiy sae a he ime he shock occurs. The decay or persisence is given by he moving average marices,, which is similar o radiional impulse response analysis. 0 Furher complicaions arise from he choice of baseline because no naural baseline exiss for 0 in VIRF, as any given baseline deviaes from he average volailiy sae. For example, a zero baseline would represen he lowes volailiy sae and volailiy forecass would increase from his baseline. Afer discussing various alernaives, Hafner and Herwarz (2006) adop he definiion given in equaion (10). In heir original analysis of exchange raes, Hafner and Herwarz examine he impac of paricular hisorical shocks ha occur in heir sample, as well as considering random shocks for heir esimaed model. In an empirical analysis of US and UK indices,we consider he onse of he GFC, which we dae as 9 Augus 2007 (GFC1), hen he dae when he financial crisis came o a head, 15 Sepember 2008, when he US governmen allowed he invesmen bank Lehman Brohers o go bankrup (GFC2). The dae 9 May 2010 marked he poin a which he focus of concern swiched from he privae secor o he public secor, and his marks he onse of he European Sovereign Deb Crisis (ESDC). We also consider random shocks in he empirical analysis. 2.4 Mulivariae GARCH Models The analysis in he paper feaures applicaions of boh he BEKK and Diagonal BEKK (DBEKK) models. The BEKK model was inroduced by Baba e al. (1985) and Engle and Kroner (1995). In he case of a model wih single lags, he BEKK recursion is: H ' ' ' ' CC Au u A B H B, (14) where H is a marix of he covariances, and C, A and B are he coefficien marices. The expression above is wrien in vech forma o generae he VIRFs, as shown below: ' ' ' ' ' ' vec H ) vec( CC ) ( A A ) vec( u u ) ( B B ) vec( H ). (15) (

11 However, a drawback of using he BEKK model is ha here are no regulariy condiions or saisical properies for full BEKK, as discussed in he nex subsecion. Chang e al. (2015) discuss sochasic processes for univariae and mulivariae condiional volailiy models, and he following subsecions draw closely on heir analysis. 2.5 Regulariy Condiions for BEKK and DBEKK The original mulivariae exension of univariae GARCH is given in Baba e al. (1985) and Engle and Kroner (1995), while a consideraion of leverage effecs and he mulivariae exension of univariae GJR is given in McAleer e al. (2009). The asymmery condiions for mulivariae GJR are given in he VARMA-AGARCH model of McAleer e al. (2009). Leverage has ypically been presened for individual equaions only, as defined by Black (1976) for univariae processes using argumens based on he deb-o-equiy raio. In order o esablish volailiy spillovers in a mulivariae framework, i is useful o define he mulivariae exension of he relaionship beween he reurns shocks and he sandardized residuals, ha is: / h, where h denoes univariae condiional volailiy. A mulivariae exension of an equaion for he condiional mean of financial reurns can be wrien as: y E( y I 1), if i is assumed ha he hree componens are m 1 vecors, where m is he number of financial asses. The mulivariae definiion of he relaionship beween and is given as: 2 D 1/, (16) 10

12 where D diag h, h,..., h ) is a diagonal marix comprising he univariae condiional volailiies. ( 1 2 m Define he condiional covariance marix of as Q. As he m 1 vecor,, is assumed o be iid for all m elemens, he condiional correlaion marix of, which is equivalen o he condiional correlaion marix of, is given by. Therefore, he condiional expecaion of (16) is defined as: Q D 1/ 2 D 1/ 2. (17) Equivalenly, he condiional correlaion marix,, can be defined as: D Q D. (18) 1/ 2 1/ 2 Equaion (17) is useful if a model of is available for purposes of esimaing (18) is useful if a model of Q is available for purposes of esimaing. Q, whereas equaion Boh equaions (17) and (18) are insrucive for a discussion of asympoic properies. As he elemens of D are consisen and asympoically normal, he consisency of consisen esimaion of esimaion of Q. As boh, whereas he consisency of Q and Q in equaion (17) depends on in equaion (18) depends on consisen are producs of marices, neiher he QMLE of asympoically normal based on he definiions given in equaions (17) and (18). Q or will be 2.6 Triangular, Hadamard and Full BEKK Wihou acually deriving he model from an appropriae sochasic process, Baba e al. (1985) and Engle and Kroner (1995) considered he full BEKK model, as well as he special cases of riangular and Hadamard (elemen-by-elemen muliplicaion) BEKK models. The specificaion of he mulivariae model is he same as he specificaion in equaion (14), namely: H ' ' ' ' CC Au u A B H B, (19) excep ha A and B are full, Hadamard or riangular marices. 11

13 Alhough esimaion of he full, Hadamard and riangular BEKK models is available in some sandard economeric and saisical sofware packages, i is no clear how he likelihood funcions migh be deermined. Moreover, he so-called curse of dimensionaliy, whereby he number of parameers o be esimaed is excessively large, makes convergence of any esimaion algorihm somewha problemaic. Jeanheau (1998) showed ha he QMLE of he parameers of he full BEKK model is consisen under a mulivariae log-momen condiion, while Come and Lieberman (2003) showed ha he QMLE are asympoically normal under he assumpion of he exisence of eighh momens. Specifically, he mulivariae log-momen condiions are difficul o verify when he marices A and B are neiher diagonal nor scalar marices, and he eighh momen condiion canno be verified for a full BEKK model. Therefore, here are as ye no verifiable asympoic properies of he full, Hadamard or riangular BEKK models. 2.7 Diagonal and Scalar BEKK Consider a vecor random coefficien auoregressive process of order one: 1 (20) where and are m 1 vecors, and is an m m marix of random coefficiens, and ~ iid ( 0, A ), ~ iid ( 0, QQ '). Technically, a vecorizaion of a full (ha is, non-diagonal or non-scalar) marix A o vec A can have dimension as high as 2 m have dimension as low as m ( m 1) / 2 m( m 1) / 2. m 2, whereas he half-vecorizaion of a symmeric marix A o vech A can 12

14 In a case where A is eiher a diagonal marix or he special case of a scalar marix, A aim, McAleer e al. (2008) showed ha he mulivariae exension of GARCH(1,1) from equaion (20), incorporaing an infinie geomeric lag in erms of he reurns shocks, is given as he diagonal BEKK (DBEKK) or scalar BEKK model, namely: Q QQ' A ' ' ' 1 1A BQ 1B, (21) where A and B are boh eiher diagonal or scalar marices. McAleer e al. (2008) showed ha he QMLE of he parameers of he diagonal or scalar BEKK models were consisen and asympoically normal, so ha sandard saisical inference on esing hypoheses is valid. Moreover, as be esimaed consisenly. Q in equaion (21) can be esimaed consisenly, in equaion (18) can also Given he above consideraions, we presen he resuls of boh full BEKK and DBEKK in he empirical analysis ha follows. We can be confiden abou he saisical properies of DBEKK when i is used o calculae VIRFs, and he imporan consideraion is wheher he wo mehods and heir associaed VIRFs, have he same implicaions for our resuls. If hey poin o he same conclusions, we can have more confidence in he resuls. 3. EMPIRICAL RESULTS Summary saisics for he wo ses of series, Oxford-Man RV series for he S&P500 and he FTSE, for a period beginning 3 January 2000 o 4 h Ocober 2016, oaling 4378 observaions, and he index reurn series for he NYSE and he FTSE, for he period 3 January 2005 o 31 December 2014, giving a oal of 2608 valid observaions, are shown in Table 1. All he series, boh he wo RV and he wo reurns series he display excess kurosis and are skewed, posiively in he case of he RV series and negaively in he case of he reurns. The ime series plos of he index values are shown in Figure 1. Table 2 provides ess of skewness, kurosis and wheher he reurn series for he wo daily realized volailiy series and wo index series are normally disribued. The Jarque-Bera (JB) es rejecs normaliy a any sandard level of significance for all series, and all display significan skewness and excess kurosis wih he excepion of he FTSE RV series, which does no show excess kurosis. 13

15 3.1 Spillover Index Resuls The resuls of he applicaion of he Diebold Yilmaz (2012) Spillover Index model are shown in Table 3. We experimened wih various lag lenghs in he VAR bu exending he lags beyond 4 did no make an appreciable difference o he Spillover Index resuls. The Durbin-Wason saisic wih a value of 2.03 suggess ha serial correlaion is no an issue. When he FTSE RV is used as he dependen variable, all he coefficiens are highly significan apar from lag 4 on he FTSE RV. If he S&P500 is he dependen variable, all coefficiens are again highly significan wih he excepion of lag 2 on S&P500 RV. In his case he Durbin-Wason saisic is 2.04 and he F saisics for boh equaions is highly significan. We can herefore proceed o he Spillover Index analysis wih confidence. Table 4 presens deails of he Spillovers across he wo series. The resuls for he daily RV series for he wo series, S&P500 and he FTSE are reasonably symmeric. Shocks o he S&P500 RV explain 83.52% of is own variabiliy, in he generalized forecas error decomposiions, whils conribuions from he FTSE RV series explain 16.5% of is variabiliy. On he reverse side of he coin, he FTSE RV series explains 79.78% of is own variabiliy wih a conribuion from he S&P500 RV of 20.2%. However, hese are average resuls across he whole sample period. To furher sharpen he resuls, we followed Diebold and Yilmaz (2012) and esimaed volailiy spillovers using 200-day rolling samples, which permis he assessmen of he exen and he naure of spillover variaion over ime via he corresponding ime series of spillover indices, which are presened graphically in he so-called oal spillover plo of Figure 2. I can be seen ha here are peaks in spillovers a he heigh of he GFC in 2008 and in 2010 a he onse of he European Sovereign deb crisis. (We analyse his furher in he nex secion in our VIRF analysis). Figure 3 shows direcional spillovers from he wo markes, and we have no included a graph of direcional spillovers o he wo markes, as his is a mirror image, given ha we are dealing wih wo markes only. Of greaer ineres is Figure 4 which shows ne spillovers as he difference beween S&P500 RV FTSE RV. Plos below zero on he graph show he periods in which he FTSE RV conribued more o S&P500 RV han vice-versa. Clearly, he predominan ne conribuion is from he S&P500 RV, as mos of he graph plos above 0, bu here are 5 disinc periods in which he reverse is he case, and noably in which coincides wih he emergence of sovereign deb problems 14

16 in Europe and again in The impac of condiional volailiy a specific daes will be explored in he nex secion. 3.2 Mulivariae GARCH analysis The presence of excess skewness leads us o employ he Suden disribuion which is used in he subsequen GARCH analysis. We filer he reurn series hrough an AR(1) process before proceeding o use he subsequen residuals in a mulivariae BEKK analysis o generae he VIRF, as in Hafner and Herwarz (2006). Table 5 shows he resuls of he applicaion of he filers, and Table 6 gives he diagnosics for he residuals. The applicaion of he AR(1) model appears o whien he residuals, and he Ljung-Box Q saisics for serial correlaion sugges ha correlaion is no a problem. The Jarque-Bera (JB) es srongly rejecs normaliy for he shocks, so we conduc he subsequen analysis using he - disribuion. 3.1 Resuls from BEKK analysis Table 7 shows he resuls of he applicaion of he BEKK model. We can forecas he volailiy and correlaions for he wo series using he BEKK model. We forecas for 100 days a he end of he ime series and use a window of 400 daily observaions o fi he model. The resuls are shown in Figure 5. The recen experience of relaively high volailiies cause he increase in he wo forecas volailiies, while he correlaion ends owards he mean over he sub-sample. Plos of he VIRFs are shown in Figure 6, Panels A and B. The VIRF impulse responses for 9 Augus 2007, as shown in Panel A, use he variance a ha poin in ime as he baseline. The iniial response for he NYSE is scaled a jus under When his is compared o he impulse response of he FTSE in he UK, he response is even larger a jus over These have been compued using a baseline of he esimaed volailiy sae, so hey are excess over he prediced covariance. They can be conrased wih he impac of he EU deb crisis on 5 May 2010, in which he NYSE iniial response is jus over 1500, while he FTSE response a he same poin in ime is nearly 2000, suggesing ha, as migh be expeced, he EU deb crisis had a larger impac in London han i had in New York. 15

17 These shocks have been prediced using a baseline of zero. The 2007 shocks ake a period of abou 6 monhs o work hrough, while he 2010 shocks ake a longer period of 8-9 monhs, bu his may well reflec he choice of a lower baseline. The covariances show a dramaic spike in response o boh shocks bu remain higher for longer, in relaion o he 2010 shock, possibly in response o he choice of baseline, as menioned above. Thus, he choice of baseline remains a key issue in he implemenaion of VIRF analysis. Panel B of Figure 6 conrass he 15 Sepember 2008 GFC impac wih he 5 May 2010 EU deb crisis once again, and he choice of baselines mirrors ha in Panel A. The impac of he shock in 2008, a he heigh of he GFC, is relaively higher han previously, in boh New York and London. On he NYSE i approaches 25000, while on he FTSE i is even higher, approaching 40000, and he shocks in boh markes ake longer o die ou han hey did in 2007, aking 9 monhs o reurn o equilibrium. The covariance approaches and remains a high levels for 6-7 monhs. The 5 May 2010 graphs are he same as in Panel A, and are included for he purpose of a direc comparison. Given ha we are considering VIRF in he conex of sock marke indices, i seems appropriae o consider asymmery effecs via he inroducion of he separae consideraion of he impac of negaive shocks. The esimaes of he BEKK and asymmeric BEKK- models are shown in Tables 7 and 9, and he eigenvalues from BEKK- and asymmeric BEKK- are given in Tables 8 and 10, respecively (for he sake of breviy, only he mulivariae GARCH and asymmeric erms are repored in he ables). The analysis is broadly similar as described above. Figure 6 shows he VIRF (for he sake of breviy only Sepember 2008 and May 2010 are considered). The key difference in he resuls, when compared o he previous analysis, is ha he VIRFs are larger and of shorer duraion. For example, he NYSE variance increases o 8000 and he FTSE variance increases o 15,000 in Sepember The duraion of he response for boh 2008 and 2010 is reduced o 3 monhs for boh he variances and covariances. However, in Secion 2.3 in his paper noed ha we can be confiden abou he saisical properies of DBEKK when i is used o calculae VIRFs, which is no he case for full BEKK. The key finding is wheher he wo mehods and heir associaed VIRFs have he same implicaions for he empirical resuls. If he empirical resuls lead o he same conclusions, we can have greaer confidence in he 16

18 empirical resuls. In Secion 3.2 we presen he empirical resuls and VIRFs from a diagonal BEKK (DBEKK) analysis. 3.2 Resuls from DBEKK The DBEKK model has valid saisical properies and regulariy condiions, so we can be confiden in he empirical resuls. I has o be borne in mind ha DBEKK has fewer parameers, so is VIRFs are simpler han are hose for full BEKK. We esimae DBEKK using he same procedure as discussed previously, and use a -disribuion and include asymmery. The asymmeric DBEKK model esimaed using a -disribuion (DBEKK-) is much beer behaved, as can be seen in Table 10. All he coefficiens apar from one ha are shown in Table 5 are significan. The eigenvalues shown in Table 11 are sable, given ha all are less han one. Figure 8 shows he impulse responses generaed by he asymmeric DBEKK model esimaed using a disribuion (DBEKK-). The resuls in Panel A reflec he fac ha he 9 Augus 2007 VIRF has a baseline calculaed on he shock a ha poin in ime, while he 15 Sepember 2008 shock has a baseline of zero. The resuls are consisen wih he previous BEKK esimaes in ha he asymmeric DBEKK model produces negaive shocks ha las for only 3 monhs in duraion. The 2008 shocks again are larger in LFTSERET han on NYSERET. Panel B in Figure 8 is consruced in a similar manner. The 9 Augus 2007 VIRF is calculaed on he shock a ha poin in ime, while he 15 Sepember 2008 shock is calculaed using a zero baseline. Consisen wih he previous resuls, he shocks have a hree-monh duraion, and heir relaive sizes are he same as previously calculaed, revealing ha boh he BEKK and DBEKK resuls are enirely consisen. In order o complee he analysis, we also calculae a DBEKK model wihou asymmeries and presen he resuls in Tables and in Figure 9. All he coefficiens for he DBEKK model, wihou asymmeries, as shown in Table 12, are highly significan. The eigenvalues, as shown in Table 13, are closer o one han for he DBEKK model wih asymmeries, as repored in Table 10, suggesing ha he sandard BEKK model is less sable. 17

19 In Figure 9, for purposes of comparison, we depic he VIRFs for he GFC2 period and he Euro deb crisis. The VIRFs in Figure 9 are consisen wih he previous analysis using he full BEKK model wihou asymmeries. The impac of he 2008 shock is larger in London han in New York, using he shock a ha poin in ime as a baseline. A similar paern is observed in he 2010 Euro-deb shock. Once again, we observe, ignoring he asymmeries, he duraion of he shock is much longer, and now exends o eigheen monhs in all figures before equilibrium is re-esablished. This is more han double he duraions of he VIRFs recorded for he full BEKK model wihou asymmeries, bu he relaive duraions remain consisen. 4. CONCLUSION In his paper we have applied wo differen mehods based on VAR and impulse response analysis o examine volailiy spillovers beween he New York and he London sock markes. We analysed daily RV esimaes aken from he Oxford-Man Realised Library running from he beginning of 2000 odae using he Diebold and Yilmaz (2012) Spillover Index. The analysis revealed ha boh he S&P500 and he FTSE conribued around 20% in erms of spillovers o he RV of he oher marke. Figure 2 revealed ha oal spillovers across he wo markes peaked in 2008 and in 2010, whils Figure 4 showing ne spillovers, revealed ha hough he predominan direcion of spillovers was from he S&P500 RV o he FTSE RV, here were sill 5 periods in which he direcion of spillovers was reversed, he mos recen being in 2010 and This firs porion of he analysis concenraed on RV series, as analysed in a VAR and generalized impulse response framework. The second porion of he analysis used he Hafner and Herwarz (2006) Volailiy Impulse Response Funcion (VIRF) approach o examine en years of daily reurn series from he New York Sock Exchange Index, and he London Sock Exchange FTSE 100 index, for he period 3 January 2005 o 31 January An aracive feaure of VIRF analysis of he effecs of shocks on volailiy hrough ime is ha he shocks are reaed as endogenous. In his analysis he focus is on shocks o condiional volailiy, as opposed o RV. Given ha we are operaing in a mulivariae GARCH framework, we can accommodae asymmery effecs, and sudy posiive and negaive shocks separaely, a luxury no afforded by our daily RV series. An imporan difference in his porion of he analysis is ha we use a paricular poin in ime for he commencemen of our condiional volailiy modelling. However, we also noe ha he choice of he 18

20 baseline for he shock makes a considerable difference. A useful conribuion of his paper is o consider asymmeric effecs, which are well documened in he empirical analysis of sock markes (see, for example, Engle and Ng (1993)). We showed ha he impacs of negaive shocks are larger, bu of shorer duraion, han hose implied by a symmeric reamen of shocks. Our empirical analysis is based on applicaion of he full BEKK model, for which no verifiable asympoic properies exis, as well as he diagonal BEKK (DBEKK) model, which is no so consrained. The empirical resuls our consisen and sugges ha he inclusion of asymmeries is imporan when VIRF analysis is applied o sock marke daa. I was found ha he responses o negaive shocks are deeper and of shorer duraion han he responses o posiive shocks. The empirical resuls of boh he BEKK and DBEKK models are srongly consisen wih each oher. The resuls of our analysis are no necessarily good news for invesors. Volailiy spillovers increase in imes of crises, making hedging more difficul, and he response is paricularly sharp, hough more shor lived, as revealed by he VIRF analysis, o negaive shocks. Acknowledgemens For financial suppor, he firs auhor wishes o hank he Ausralian Research Council and he second auhor wishes o acknowledge he Ausralian Research Council and he Naional Science Council, Minisry of Science and Technology, Taiwan. The auhors are graeful o Tom Doan and Esima for helpful assisance wih RATS coding. We are graeful o he wo anonymous reviewers for very helpful commens and suggesions. 19

21 REFERENCES Aler, A., and A. Beyer, (2013) The Dynamics of Spillover Effecs during he European Sovereign Deb Turmoil, European Cenral Bank Working Paper Series, No Baba, Y., R.F. Engle, D. Kraf and K.F. Kroner (1985), Mulivariae simulaneous generalized ARCH, Unpublished manuscrip, Deparmen of Economics, Universiy of California, San Diego, CA, USA. Black, F. (1976), Sudies of sock marke volailiy changes, in Proceedings of he American Saisical Associaion, Business and Economic Saisics Secion, Washingon, DC, USA, 1976, pp Chang, C.-L. Y.-Y. Li and M. McAleer (2015), Volailiy spillovers beween energy and agriculural markes: A criical appraisal of heory and pracice, Economeric Insiue Research Paper EI , Erasmus School of Economics, Erasmus Universiy Roerdam. Come, F. and O. Lieberman (2003), Asympoic heory for mulivariae GARCH processes, Journal of Mulivariae Analysis, 84, Diebold, F.X. and K. Yilmaz, (2009), Measuring Financial Asse Reurn and Volailiy Spillovers, Wih Applicaion o Global Equiy Markes, Economic Journal, 119, Diebold, F.X. and K. Yilmaz, (2012), Beer o give han o receive: Predicive direcional measuremen of volailiy spillovers, Inernaional Journal of Forecasing, 28(1), Diebold, F.X. and K. Yilmaz, (2014), On he Nework Topology of Variance Decomposiions: Measuring he Connecedness of Financial Firms, Journal of Economerics, 182, Diebold, F.X. and K. Yilmaz, (2016), Trans-Alanic Equiy Volailiy Connecedness: U.S. and European Financial Insiuions, , Journal of Financial Economerics, 14, Engle, R.F. and V.K. Ng (1993), Measuring and esing he impac of news on volailiy, Journal of Finance, 48, Engle, R.F. and K.F. Kroner (1995), Mulivariae simulaneous generalized ARCH, Economeric Theory, 11, Engle, Rober F., Takaoshi Io and Wen-Ling Lin (1990), Meeor Showers or Hea Waves? Heeroskedasic Inra-Daily Volailiy in he Foreign Exchange Marke, Economerica, 58, Dros, F. and T. Nijman (1993), Temporal aggregaion of GARCH processes, Economerica, 61, Gallan, A.R., P.E. Rossi and G. Tauchen (1993), Nonlinear dynamic srucures, Economerica, 61,

22 Gerd, H., A. Lunde, N. Shephard and K. Sheppard (2009) Oxford-Man Insiue's realized library, Oxford-Man Insiue, Universiy of Oxford. Hafner, C.M. and H. Herwarz (2006), Volailiy impulse responses for mulivariae GARCH models: An exchange rae illusraion, Journal of Inernaional Money and Finance, 25, Jeanheau, T. (1998), Srong consisency of esimaors for mulivariae ARCH models, Economeric Theory, 14, Jin, X., S.X. Lin, and M. Tamvakis, (2012). Volailiy ransmission and volailiy impulse response funcions in crude oil markes, Energy Economics, 34(6), Kloessner, S. and S. Wagner, (2012): Exploring All VAR Orderings for Calculaing Spillovers? Yes, We Can! - A Noe on Diebold and Yilmaz (2009), Journal of Applied Economerics, 29 (1), Koop, G., M.H. Pesaran and S.M. Poer (1996). Impulse response analysis in nonlinear mulivariae models. Journal of Economerics, 74, Koumos, G. and G.G. Booh (1995), Asymmeric volailiy ransmission in inernaional sock markes, Journal of Inernaional Money and Finance, 14, Le Pen, Y., B. Sévi, (2010), Volailiy ransmission and volailiy impulse response funcions in European elecriciy forward markes, Energy Economics, 32(4), Lin, W.-L. (1997), Impulse response funcion for condiional volailiy in GARCH models, Journal of Business & Economic Saisics, 15, McAleer, M., S. Hoi and F. Chan (2009), Srucure and asympoic heory for mulivariae asymmeric condiional volailiy, Economeric Reviews, 28, Olson, E., A.J. Vivian, and M.E. Wohar, (2014), The relaionship beween energy and equiy markes: Evidence from volailiy impulse response funcions. Energy Economics, 43, Panopoulou, E., and T. Panelidis, (2009) Inegraion a a cos: evidence from volailiy impulse response funcions, Applied Financial Economics, 19(11), Pesaran, M.H. and Shin, Y. (1998), Generalized Impulse Response Analysis in Linear Mulivariae Models, Economics Leers, 58, Ross, S.A. (1989), Informaion and volailiy: The no-arbirage Maringale approach o iming and resoluion irrelevancy, Journal of Finance, 44, 1-17 Sims, C. (1980), Macroeconomics and realiy, Economerica 48, Tauchen, G., H. Zhang, and M. Liu (1996), Volume, volailiy and leverage: A dynamic analysis, Journal of Economerics, 74,

23 Table 1 Summary Saisics, using he observaions for he variable SP500rv10 (4307 valid observaions) Mean Median Minimum Maximum e e Sd. Dev. C.V. Skewness Ex. kurosis % Perc. 95% Perc. IQ range Missing obs e e Summary Saisics, using he observaions for he variable FTSErv10 (4309 valid observaions) Mean Median Minimum Maximum e e Sd. Dev. C.V. Skewness Ex. kurosis % Perc. 95% Perc. IQ range Missing obs e Summary Saisics for (2608 valid observaions) NYSERET (2608 valid observaions) Mean Median Minimum Maximum Sd. Dev. C.V. Skewness Ex. kurosis % Perc. 95% Perc. IQ range Missing obs Summary Saisics for (2608 valid observaions) FTSERET Mean Median Minimum Maximum e Sd. Dev. C.V. Skewness Ex. kurosis % Perc. 95% Perc. IQ range Missing obs

24 Table 2 Tess of Skewness, Excess Kurosis, and Normaliy, Base Series S&P500 RV Skewness Signif Level (Sk=0) 0.0 Kurosis (excess) Signif Level (Ku=0) Jarque-Bera Signif Level (JB=0) 0.0 FTSERET RV Skewness Signif Level (Sk=0) 0.0 Kurosis (excess) Signif Level (Ku=0) Jarque-Bera Signif Level (JB=0) 0.0 NYSERET(*100) Skewness Signif Level (Sk=0) 0.0 Kurosis (excess) Signif Level (Ku=0) 0.0 Jarque-Bera Signif Level (JB=0) 0.0 FTSERET(*100) Skewness Signif Level (Sk=0) Kurosis (excess) Signif Level (Ku=0) 0.0 Jarque-Bera Signif Level (JB=0)

25 VAR/Sysem - Esimaion by Leas Squares Daily(5) Daa From 2000:01:07 To 2016:07:07 Usable Observaions 4305 Dependen Variable SP500RV Mean of Dependen Variable Sd Error of Dependen Variable Sandard Error of Esimae Sum of Squared Residuals Durbin-Wason Saisic Table 3 VAR analysis of RV Series Variable Coeff Sd Error T-Sa Signif ************************************************************************************ 1. SP500RV{1} SP500RV{2} SP500RV{3} SP500RV{4} FTSERV{1} FTSERV{2} FTSERV{3} FTSERV{4} Consan F-Tess, Dependen Variable SP500RV Variable F-Saisic Signif ******************************************************* SP500RV FTSERV Dependen Variable FTSERV Mean of Dependen Variable Sd Error of Dependen Variable Sandard Error of Esimae Sum of Squared Residuals Durbin-Wason Saisic Variable Coeff Sd Error T-Sa Signif ************************************************************************************ 1. SP500RV{1} SP500RV{2} SP500RV{3} SP500RV{4} FTSERV{1} FTSERV{2} FTSERV{3} FTSERV{4} Consan F-Tess, Dependen Variable FTSERV Variable F-Saisic Signif ******************************************************* SP500RV FTSERV

26 Table 4 Spillover Index SP500 RV FTSE RV From Ohers SP500 RV FTSE RV Conribuion o ohers

27 Table 5 AR(1) and preliminary GARCH(1,1) analysis of reurn series NYSE Variables Coefficien -saisic Significance Consan LNYSERET(1) GARCH(1,1) C A B FTSE Consan e LFTSERET(1) C e A B Table 6 Residual diagnosics ARCH-LM(1) JB Q(10) Q(20) LNYSERET (0.004) (0.000) (0.437) (0.235) LFTSERET (0.967) (0.000) (0.823) (0.528) 26

28 Table 7 BEKK Variable Coefficien Sandard Error -saisic Significance Consan LNYSERET{1} Consan LFTSERET{1} C(1,1) C(2,1) C(2,2) e A(1,1) A(1,2) A(2,1) A(2,2) B(1,1) B(1,2) B(2,1) B(2,2) Shape Table 8 Eigenvalues from BEKK Var JB p-value All

29 Table 9 Asymmeric BEKK- Variable Coefficien Sandard Error -saisic A(1,1) A(1,2) A(2,1) A(2,2) B(1,1) B(1,2) B(2,1) B(2,2) D(1,1) D(1,2) D(2,1) D(2,2) Shape Significance 28

30 Table 10 Asymmeric DBEKK- Variable Coefficien Sandard Error Mean Model LNYSERET -saisic Consan LNYSERET(1) Mean Model LFTSERET Consan LFTSERET(1) C(1,1) C(2,1) C(2,2) A(1) A(2) B(1) B(2) D(1) D(2) Shape Significance Table 11 Eigenvalues from Asymmeric BEKK , , , 0 Var JB p-value All

31 Table 12 DBEKK- wihou Asymmeries Variable Coefficien Sandard Error Mean Model LNYSERET -saisic Consan LNYSERET(1) Mean Model LFTSERET Consan LFTSERET(1) C(1,1) C(2,1) C(2,2) A(1) A(2) B(1) B(2) Shape Significance Table 13 Eigenvalues from BEKK , , , 0 Var JB p-value All

32 Figure 1 Plos of FTSE and NYSE values, plus S&P500 and FTES RV Noe: NYSE - Blue, FTSE Black. S&P500 Realised Volailiy (RV) SP500rv FTSE Realised Volailiy (RV FTSErv

33 40 Figure 2. T oal Volailiy Spillovers, Two M arkes Figure 4 Ne Pairwise Volailiy Spillovers SP500 RV-FTSE RV 32