A volatility impulse response analysis applying multivariate GARCH models and news events around the GFC

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1 2s Inernaional Congress on Modelling and Simulaion, Gold Coas, Ausralia, 29 Nov o 4 Dec 25 A volailiy impulse response analysis applying mulivariae GARCH models and news evens around he GFC D.E. Allena, M.J. McAleerb, R. Powellc, and A.K. Singhc a Visiing Professor, School of Mahemaics and Saisics, Universiy of Sydney and Adjunc Professor, School of Business, Universiy of Souh Ausralia b Universiy Disinguished Chair Professor, Deparmen of Quaniaive Finance, Naional Tsing Hua Universiy, Taiwan, and Professor of Quaniaive Finance, Economeric Insiue, Erasmus School of Economics, Erasmus Universiy, Roerdam, The Neherlands c School of Business, Edih Cowan Universiy, Perh, WA. profallen27@gmail.com Absrac: This paper feaures an applicaion of he Hafner and Herwarz (26) approach o he analysis of mulivariae GARCH models using volailiy impulse response analysis. The daa se used feaures en years of daily reurn series for he New York Sock Exchange Index and he FTSE index from he London sock Exchange, aken from 3rd January 25 o January 3s 25. This period capures boh he Global Financial Crisis (GFC) and he subsequen European Sovereign Deb Crisis (ESDC). The aracion of he Hafner and Kerwarz (26) 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, whils avoiding ypical orhogonalizaion and ordering problems. Volailiy impulse response funcions (VIRF) provide informaion abou he impac of independen shocks on volailiy. Hafner and Herwarz s (26) VIRF exends a framework, provided by Koop e al. (996), 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. Hafner and Herwarz (26) uilise he fac ha GARCH models can be viewed as being linear in 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. Hafner and Herwarz (26) use a Jordan decomposiion of Σ in order o obain independen and idenically defined (hence i.i.d.) innovaions. One general issue in he approach is he choice of baseline volailiies. Hafner and Herwarz (26) define VIRF as he expecaion of volailiy condiional on an iniial shock and on hisory, minus he baseline expecaion ha only condiions on hisory. This makes he process endogenous, bu he choice of he baseline shock wihin he daa se sill obviously makes a difference. We explore he impac of hree differen shocks, he firs marks he onse of he GFC, which we dae as 9h Augus 27, (GFC). 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 5h Sepember 28, when he US governmen allowed he invesmen bank Lehman Brohers o go bankrup (GFC2). Our hird shock poin is May 9h 2, 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, afer Engle and Ng (993). Our modelling is underaken in he conex of a mulivariae GARCH model feauring pre-whiened reurn series, which are hen analysed via a BEKK model using a -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 our paricular reurn series. An effec previously repored by Tauchen e al., (996), who use a differen heoreical se up. Keywords: Volailiy Impulse Response Funcions, BEKK, asymmery, GFC, ESDC 8

2 Allen e al., Volailiy Impulse Responses. INTRODUCTION The similariies beween GARCH and VARMA-ype models provide a foundaion for he approach o generalize impulse response analysis, as inroduced by Sims (98), o he analysis of shocks in volailiy. Various previous approaches in he lieraure, have been made owards racing he impac of various ypes of shocks hrough ime, see Koop e al., (996); Engle and Ng, (993), Gallan e al., (993), and Lin, (997). Koop e al. (996) defined generalized impulse response funcions for he condiional expecaion using he mean of he response vecor condiional on hisory and a presen shock, compared wih a baseline ha only condiions on hisory. Hafner and Herwarz s (26) VIRF exends he framework provided by Koop e al. (996). Their approach is novel in fac is explores he condiional variance raher han he condiional mean. Given ha GARCH models can be viewed as being linear in squares, and ha mulivariae GARCH models are known o have a VARMA represenaion wih non-gaussian errors. Hafner and Hewarz (26) adop his paricular srucure o calculae condiional expecaions of volailiy analyically in heir VIRF analysis. In our 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 h Augus 27, (GFC) which 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 5 h Sepember 28 when he US governmen allowed he invesmen bank Lehman Brohers o go bankrup (GFC2). May 9 h 2 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). 2. RESEARCH METHOD AND DATA Hafner and Herwarz (26) develop heir model by leing ε denoe an N dimensional random vecor so ha: ε = Pξ, () where P P = and ξ denoes an i.i.d. random vecor of dimension N, wih independen componens, mean zero and ideniy covariance marix. They assume ha is measurable wih respec o he informaion se available a ime -, F. Equaion () implies ha E[ ε F ] =, and Var [ ε F ] =. They noe ha ε could be he error of a VARMA process. If ε is a mulivariae GARCH process hen equaion () may be called a srong GARCH model, according o Dros and Nijman (993). This is convenien because i permis he modelling of news evens as appearing in he i.i.d. innovaion ξ. They idenify ξ by assuming ha P is a lower riangular marix which permis he use of a Choleski decomposiion of. They furher use he fac ha independen news can ofen be idenified by means of a Jordan decomposiion which will permi idenificaion is when he innovaion vecor is nonnormal. They adop a mulivariae GARCH(p,q) model framework, given by: q vech( ) = c + Avech( ε ε ) + B vech( ). (2) i= i i i They hen adop he BEKK model, as discussed by Engle and Kroner (995) which is a special case of equaion (2) specified as: K q C + Aki iε i k = i= p j = k = i= j i = C A + G G. ki K p ki i ε (3) ki 9

3 Allen e al., Volailiy Impulse Responses In (3) C is a lower riangular marix and 2.. Volailiy impulse response funcions Aki and Gki are N N parameer marices. Hafner and Herwarz (26) proceed by assuming ha a ime, some independen news is refleced by ξ and i is no specified wheher i is good or bad. The condiional covariance marix is a funcion of he innovaions ξ,..., ξ, he original shock ξ and. Hafner and Herwarz (26) 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 se ou in equaion (4): In equaion (4) V ξ ) is an ( V [ ( ) ξ, F ] E[ vech( ] ( ) = E vech ) F ξ (4) * N dimensional vecor. Hafner and Herwarz (26) consider a VARMA represenaion of a mulivariae GARCH(p,q) model in order o find an explici expression for ξ ) They hen define he V ( and define η = vech( εε ). mulivariae GARCH(p,q) model as a VARMA(max(p,q)p) model: where specificaion: max( p, q) p ( Ai + Bi ) η i B ju j + i= j = η = ω + u, (5) u = η vech( ) is a whie noise vecor. From expression (5) hey derive he VMA( ) i= η = vech( ) + φ u, (6) i i Where he * N N * marices φi can be deermined recursively. The General expression for VIRF is: / 2 / 2 + V ( ξ ) φ D ( ) D vech( ξ ξ I ). (7) = N N N Hafner and Herwarz (26) consider a variey of specificaions for he baseline shock. The behavior implied by expression (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 φ, similar o radiional impulse response analysis. Furher complicaions arise from he choice of baseline, as no naural baseline exiss for ε in VIRF, because 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 (26) adop he definiion se ou in expression (4). In heir original sudy of exchange raes hey look a he impac of paricular hisorical shocks ha fall in heir sample as well as considering random shocks for heir esimaed model. We follow sui, in his sudy of US and UK indices, and consider he onse of he GFC, which we dae as 9 h Augus 27, (GFC), hen he dae when he financial crisis came o a head, 5 h Sepember 28, when he US governmen allowed he invesmen bank Lehman Brohers o go bankrup (GFC2). May 9 h 2 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.

4 Allen e al., Volailiy Impulse Responses 3. RESULTS Summary saisics for he wo index reurn series are shown in Table. Boh he NYSE and he FTSE reurn series display excess kurosis and are negaively skewed. Plos of he index values are shown in Figure. Table : Summary Saisics, using he observaions for he variable NYSERET (268 valid observaions) Mean Median Minimum Maximum Sd. Dev. C.V. Skewness Ex. kurosis % Perc. 95% Perc. IQ range Missing obs Summary Saisics, using he observaions for he variable FTSERET (268 valid observaions) Mean Median Minimum Maximum 3.92e Sd. Dev. C.V. Skewness Ex. kurosis % Perc. 95% Perc. IQ range Missing obs Noe: NYSE - Blue, FTSE Black. Figure. NYSE and FTSE Plo. Table 2 provides ess of skewness, kurosis and wheher he reurn series for he wo index series are normally disribued. The Jarque-Bera es rejecs his a beer han he % level. We uilize he Suden T Table 2. Tess of skewness, excess kurosis, and conformaion o a normal disribuion. NYSERET(*) Skewness Signif Level (Sk=). Kurosis (excess) Signif Level (Ku=). Jarque-Bera Signif Level (JB=). FTSERET(*) Skewness -.76 Signif Level (Sk=).2693 Kurosis (excess) Signif Level (Ku=). Jarque-Bera Signif Level (JB=). disribuion in our subsequen analysis. We filer he reurn series hrough an AR() and GARCH(,) processes, before proceeding o use he residuals from his in a BEKK analysis o generae he VIRF, following Hafner and Herwarz (26). Table 3 shows he resuls of he applicaion of he BEKK model. We can forecas he wo series volailiy and correlaions using he BEKK model. We forecas for days a

5 Allen e al., Volailiy Impulse Responses he end of our ime series and use a window of 4 daily observaions o fi he model. The resuls are shown in Figure 2. Table 3. BEKK model Variable Coeff Sd Error T-Sa Signif Consan LNYSERET{} Consan LFTSERET{} C(,) C(2,) C(2,2) e A(,) A(,2) A(2,) A(2,2) B(,) B(,2) B(2,) B(2,2) Shape Figure 2. day forecass based on a BEKK model The recen experience of relaively high volailiies cause he increase in he wo forecas volailiies, whils he correlaion ends owards he average over he sub-sample. Plos of he VIRFs are shown in Figure 3, panels A and B. The VIRF impulse responses for Augus 9 h 27, 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, and 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 o he impac of he EU deb crisis on May 5 h 2, in which he NYSE iniial response is jus over 5, whils he FTSE response a he same poin in ime is nearly 2, suggesing ha, as migh be expeced, he EU deb crisis had a larger impac in London han in New York. These shocks have been prediced using a baseline of zero. The 27 shocks ake a period of abou 6 monhs o work hrough, whils he 2 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 longer, in relaion o he 2 shock, again, perhaps in response o he choice of baseline. Thus, he choice of baseline remains a key issue in he implemenaion of VIRF analysis. 2

6 Allen e al., Volailiy Impulse Responses Panel A: Baseline 9 h Augus 27 and May 5 h 2 Panel B: Baseline 28 Sepember 5 h and May 5 h 2 Figure 3. VIRF Panel B of Figure 3 conrass he Sepember 5 h 28 GFC impac wih he May 5 h 2 EU deb crisis once again, and he choice of baselines mirrors ha made in Panel A. The impac of he shock in 28, a he heigh of he GFC, is relaively higher han previously, in boh New York and London. On he NYSE i approaches 25, whils on he FTSE i is even higher, approaching 4, and he shocks in boh markes ake longer o die ou han in 27, aking 9 monhs o ge back o equilibrium. The covariance approaches 2 and remains a high levels for 6-7 monhs. The 2 May 5 h graphs are he same as in Panel A and included for he purposes of direc comparison. Given ha we are considering VIRF in he conex of sock marke indices i seemed appropriae o consider leverage effecs via he inroducion of he separae consideraion of he impac of negaive shocks. The asymmeric BEKK model esimaed is shown in Table 4 (for he sake of breviy only he mulivariae GARCH and asymmeric erms are repored). Figure 4 shows he VIRF (Again, for he sake of breviy only Sepember 28 and May 2 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 8 and he LSE variance increases o 5, in Sepember 28. The duraion of he response for boh 28 and 2 is reduced o 3 monhs for boh he variances and covariances. 4. CONCLUSION In his paper we have applied he Hafner and Herwarz (26) VIRF analysis o en years of daily reurn series aken from he New York Sock Exchange Index, and he London Sock Exchange FTSE index, for a period from 3rd January 25 o January 3s 25. An aracive feaure of VIRF analysis of he effecs 3

7 Allen e al., Volailiy Impulse Responses of shocks on volailiy hrough ime, is ha he shocks are reaed as being endogenous. However, we also noe ha he choice of he baseline for he shock makes a difference. A conribuion of his paper is o consider leverage effecs, which are well documened in he empirical analysis of sock markes, see e.g. Engle and Ng (993). We show ha he impac of negaive shock is larger, bu of shorer duraion, han ha implied by a symmeric reamen of shocks. Figure 4. VIRF based on Asymmeric BEKK (responses o negaive price movemens). Table 4. Asymmeric BEKK model based on disribuion. Variable Coeff Sd Error T-Sa Signif A(,) A(,2) A(2,) A(2,2) B(,) B(,2) B(2,) B(2,2) D(,) D(,2) D(2,) D(2,2) Shape ACKNOWLEDGEMENTS The auhors are graeful o Thomas Doan and Esima for assisance wih Ras coding. The usual cavea applies o any errors. REFERENCES Engle, R.F., Ng, V.K., (993). Measuring and esing he impac of news on volailiy. Journal of Finance 48, Engle, R.F., Kroner, K.F., (995). Mulivariae simulaneous generalized ARCH. Economeric Theory, Dros, F., Nijman, T., (993). Temporal aggregaion of GARCH processes. Economerica 6, Gallan, A.R., Rossi, P.E., Tauchen, G., (993). Nonlinear dynamic srucures. Economerica 6, Hafner, C. M and H. Herwarz, (26). Volailiy impulse responses for mulivariae GARCH models: An exchange rae illusraion, Journal of Inernaional Money and Finance, 25, Koop, G., Pesaran, M.H., Poer, S.M., (996). Impulse response analysis in nonlinear mulivariae models. Journal of Economerics 74, Lin, W.-L., (997). Impulse response funcion for condiional volailiy in GARCH models. Journal of Business & Economic Saisics 5, Sims, C., (98). Macroeconomics and realiy. Economerica 48, -48. Tauchen, G., H. Zhang, and M. Liu, (996). Volume, volailiy and leverage, a dynamic analysis, Journal of Economerics, 74,