Multivariate Volatility Impulse Response Analysis of GFC News Events

Transcription

1 TI /III Tinbergen Insiue Discussion Paper Mulivariae Volailiy Impulse Response Analysis of GFC News Evens David E. Allen 1 Michael McAleer 2 Rober Powell 3 AbhayK. Singh 3 1 Universiy of Sydney, and Universiy of Souh Ausralia, Ausralia; 2 Naional Tsing Hua Universiy, Taiwan, Erasmus Universiy Roerdam, Tinbergen Insiue, he Neherlands, and Compluense Universiy of Madrid, Spain; 3 Edih Cowan Universiy, Perh, Ausralia.

2 Tinbergen Insiue is he graduae school and research insiue in economics of Erasmus Universiy Roerdam, he Universiy of Amserdam and VU Universiy Amserdam. More TI discussion papers can be downloaded a hp:// Tinbergen Insiue has wo locaions: Tinbergen Insiue Amserdam Gusav Mahlerplein MS Amserdam The Neherlands Tel.: +31(0) Tinbergen Insiue Roerdam Burg. Oudlaan PA Roerdam The Neherlands Tel.: +31(0) Fax: +31(0) Duisenberg school of finance is a collaboraion of he Duch financial secor and universiies, wih he ambiion o suppor innovaive research and offer op qualiy academic educaion in core areas of finance. DSF research papers can be downloaded a: hp:// Duisenberg school of finance Gusav Mahlerplein MS Amserdam The Neherlands Tel.: +31(0)

3 Mulivariae Volailiy Impulse Response Analysis of GFC News Evens* David E. Allen School of Mahemaics and Saisics Universiy of Sydney and School of Business Universiy of Souh Ausralia Michael McAleer Deparmen of Quaniaive Finance Naional Tsing Hua Universiy, Taiwan and Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam, The Neherlands and Tinbergen Insiue, The Neherlands and Deparmen of Quaniaive Economics Compluense Universiy of Madrid, Spain Rober Powell School of Business Edih Cowan Universiy Perh, Ausralia AbhayK. Singh School of Business Edih Cowan Universiy Perh, Ausralia July 2015 * 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, Taiwan. The auhors are graeful o Tom Doan and Esima for helpful assisance wih RATS coding.

4 Allen e al., A mulivariae volailiy impulse response analysis Absrac This paper applies he Hafner and Herwarz (2006) (hereafer HH) approach o he analysis of mulivariae GARCH models using volailiy impulse response analysis. The daa se feaures en years of daily reurns series for he New York Sock Exchange Index and he FTSE 100 index from he London Sock Exchange, from 3 January 2005 o 31 January This period capures boh he Global Financial Crisis (GFC) and he subsequen European Sovereign Deb Crisis (ESDC). The aracion of he HH approach is ha i involves a novel applicaion of he concep of impulse response funcions, racing he effecs of independen shocks on volailiy hrough ime, while avoiding ypical orhogonalizaion and ordering problems. Volailiy impulse response funcions (VIRF) provide informaion abou he impac of independen shocks on volailiy. HH s VIRF exends a framework provided by Koop e al. (1996) for he analysis of impulse responses. This approach is novel because i explores he effecs of shocks o he condiional variance, as opposed o he condiional mean. HH use he fac ha GARCH models can be viewed as being linear in he squares, and ha mulivariae GARCH models are known o have a VARMA represenaion wih non-gaussian errors. They use his paricular srucure o calculae condiional expecaions of volailiy analyically in heir VIRF analysis. A Jordan decomposiion of Σ is used o obain independen and idenically disribued innovaions. A general issue in he approach is he choice of baseline volailiies. VIRF is defined as he expecaion of volailiy condiional on an iniial shock and on hisory, minus he baseline expecaion ha condiions on hisory. This makes he process endogenous, bu he choice of he baseline shock wihin he daa se makes a difference. We explore he impac of hree differen shocks, he firs marking he onse of he GFC, which we dae as 9 Augus 2007 (GFC1). This 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 a 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 hird shock is 9 May 2010, which marked he poin a which he focus of concern swiched from he privae secor o he public secor. A furher conribuion of his paper is he inclusion of leverage, or asymmeric effecs. Our modelling is underaken in he conex of a mulivariae GARCH model feauring pre-whiened reurn series, which are hen analysed using boh BEKK and diagonal BEKK models wih he -disribuion. 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, in he conex of he reurn series. Keywords: Volailiy impulse response funcions (VIRF), BEKK, DBEKK, Asymmery, GFC, ESDC. JEL: C22, C32, C58, G32. 2

5 Allen e al., A mulivariae volailiy impulse response analysis 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. Hafner and Herwarz s (2006) Volailiy Impulse Response Funcions (VIRFs) 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. 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 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 remainder of he paper is as follows. In Secion 2 he research mehods and daa are discussed, including volailiy impulse response funcions, mulivariae GARCH models, he regulariy condiions for BEKK and diagonal BEKK (DBEKK) models, he riangular, Hadamard and full BEKK 3

6 Allen e al., A mulivariae volailiy impulse response analysis 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 Hafner and Herwarz (2006) develop heir model by leing denoe an N-dimensional random vecor, so ha: P, (1) 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 1 F F. Equaion (1) implies ha, E F 1 0 and 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( ), (2) 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 (2), and is specified as: K q ' ' 0C0 Aki i i k 1 i1 K p k 1 i1 ' C A G G. (3) ki ki 4 i ki

7 Allen e al., A mulivariae volailiy impulse response analysis In equaion (3), C0 is a lower riangular marix, and A ki and G ki are N N parameer marices. 2.1 Volailiy Impulse Response Funcions 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 (4) In equaion (4), 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 ), and define vech( ). They define he mulivariae ( 0 GARCH(p,q) model as a VARMA(max(p,q), p) model: max( p, q) ( A B ) B u u, (5) i i i i1 j 1 p j j where u vech( ) is a whie noise vecor. From equaion (5), Hafner and Herwarz derive he VMA( ) specificaion, as follows: i0 vech( ) u, (6) i i where he * * N N marices i can be deermined recursively. The general expression for VIRF is: 5

8 Allen e al., A mulivariae volailiy impulse response analysis 1/ 2 1/ 2 ' V ( ) D ( ) D vech( I ). (7) 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 (7) is differen from radiional impulse response analysis. In (7), he impulse is an even, no odd, funcion of he shock, i is no linear in he shock, and he VIRF depends on he 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 (4). 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. 2.2 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, (8)

9 Allen e al., A mulivariae volailiy impulse response analysis 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 ). (9) ( 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. 2.3 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: 7

10 Allen e al., A mulivariae volailiy impulse response analysis 2 D 1/, (10) 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 (10) is defined as: Q D 1/ 2 D 1/ 2. (11) Equivalenly, he condiional correlaion marix,, can be defined as: D Q D. (12) 1/ 2 1/ 2 Equaion (11) is useful if a model of is available for purposes of esimaing Q, whereas equaion (12) is useful if a model of Q is available for purposes of esimaing. Boh equaions (11) and (12) 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 (11) depends on in equaion (12) depends on consisen are producs of marices, neiher he QMLE of asympoically normal based on he definiions given in equaions (11) and (12). Q or will be 2.4 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 (8), namely: 8

11 Allen e al., A mulivariae volailiy impulse response analysis H ' ' ' ' CC Au u A B H B, (13) excep ha A and B are full, Hadamard or riangular marices. 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. 2.5 Diagonal and Scalar BEKK Consider a vecor random coefficien auoregressive process of order one: 1 (14) where and are m 1 vecors, and is an m m marix of random coefficiens, and ~ iid ( 0, A ), ~ iid ( 0, QQ '). 9

12 Allen e al., A mulivariae volailiy impulse response analysis 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 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 (14), 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, (15) 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 (15) can be esimaed consisenly, in equaion (12) 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 index reurn series for he period 3 January 2005 o 31 December 2014, giving a oal of 2608 valid observaions, are shown in Table 1. Boh he NYSE and he FTSE reurn series display excess kurosis and are negaively skewed. The ime series plos of he index values are shown in Figure 1. 10

13 Allen e al., A mulivariae volailiy impulse response analysis Table 2 provides ess of skewness, kurosis and wheher he reurn series for he wo index series are normally disribued. The Jarque-Bera (JB) es rejecs normaliy a any sandard level of significance. For his reason, he Suden disribuion is used in he subsequen 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 3 shows he resuls of he applicaion of he filers, and Table 4 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 4 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 2. 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 3, 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. 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 11

14 Allen e al., A mulivariae volailiy impulse response analysis of baseline, as menioned above. Thus, he choice of baseline remains a key issue in he implemenaion of VIRF analysis. Panel B of Figure 3 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 5 and 7, and he eigenvalues from BEKK- and asymmeric BEKK- are given in Tables 6 and 9, respecively (for he sake of breviy, only he mulivariae GARCH and asymmeric erms are repored in he ables). The analysis is broadly similar as descrived above. Figure 4 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 empirical resuls. In Secion 3.2 we presen he empirical resuls and VIRFs from a diagonal BEKK (DBEKK) analysis. 3.2 Resuls from DBEKK 12

15 Allen e al., A mulivariae volailiy impulse response analysis 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 8. All he coefficiens apar from one ha are shown in Table 5 are significan. The eigenvalues shown in Table 9 are sable, given ha all are less han one. Figure 5 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 5 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 6. All he coefficiens for he DBEKK model, wihou asymmeries, as shown in Table 10, are highly significan. The eigenvalues, as shown in Table 11, are closer o one han for he DBEKK model wih asymmeries, as repored in Table 6, suggesing ha he sandard BEKK model is less sable. In Figure 6, for purposes of comparison, we depic he VIRFs for he GFC2 period and he Euro deb crisis. The VIRFs in Figure 6 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 13

16 Allen e al., A mulivariae volailiy impulse response analysis 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 he Hafner and Herwarz (2006) Volailiy Impulse Response Funcion (VIRF) analysis o 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. However, we also noe ha he choice of he 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. 14

17 Allen e al., A mulivariae volailiy impulse response analysis REFERENCES 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, 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, 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, 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, Koop, G., M.H. Pesaran and S.M. Poer (1996). Impulse response analysis in nonlinear mulivariae models. Journal of Economerics, 74, 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, 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,

18 Allen e al., A mulivariae volailiy impulse response analysis Table 1 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 Table 2 Tess of Skewness, Excess Kurosis, and Normaliy NYSERET(*100) Skewness Signif Level (Sk=0) 0 Kurosis (excess) Signif Level (Ku=0) 0 Jarque-Bera Signif Level (JB=0) 0 FTSERET(*100) Skewness Signif Level (Sk=0) Kurosis (excess) Signif Level (Ku=0) 0 Jarque-Bera Signif Level (JB=0) 0 16

19 Allen e al., A mulivariae volailiy impulse response analysis Table 3 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 4 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) 17

20 Allen e al., A mulivariae volailiy impulse response analysis Table 5 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 6 Eigenvalues from BEKK Var JB p-value All

21 Allen e al., A mulivariae volailiy impulse response analysis Table 7 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 19

22 Allen e al., A mulivariae volailiy impulse response analysis Table 8 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 9 Eigenvalues from Asymmeric BEKK , , , 0 Var JB p-value All

23 Allen e al., A mulivariae volailiy impulse response analysis Table 10 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 11 Eigenvalues from BEKK , , , 0 Var JB p-value All

24 Allen e al., A mulivariae volailiy impulse response analysis Figure 1 Noe: NYSE - Blue, FTSE Black. Figure day forecass based on BEKK 22

25 Allen e al., A mulivariae volailiy impulse response analysis Figure 3 VIRF Panel A: Baselines 9 Augus 2007 and 5 May 2010 VIRF Panel B: Baselines 15 Sepember 2008 and 5 May

26 Allen e al., A mulivariae volailiy impulse response analysis Figure 4 VIRF Asymmeric BEKK (responses o negaive price movemens) 24

27 Allen e al., A mulivariae volailiy impulse response analysis Figure 5 VIRF Asymmeric DBEKK- Panel A Panel B 25

28 Allen e al., A mulivariae volailiy impulse response analysis Figure 6 VIRF for GFC2 and Euro Deb crisis using DBEKK- 26