Multivariate Volatility and Spillover Effects in Financial Markets

Transcription

1 Mulivariae Volailiy and Spillover Effecs in Financial Markes Bernardo Veiga and Michael McAleer School of Economics and Commerce, Universiy of Wesern Ausralia Absrac: The relaionship beween volailiy and risk has been one of he main facors underlying he ineres in volailiy modelling. An imporan quesion for inernaional diversificaion is wheher shocks in one marke influence, or have spillovers ino, reurns and volailiy in oher markes. This paper ess for he exisence of volailiy spillovers among he S&P 500, FTSE 100 and Nikkei 225 sock indexes using inra-daily daa from 12/10/1992 o 7/7/2003. Exising work is exended hrough he applicaion of he vecor auoregressive moving average asymmeric generalised auoregressive condiional heeroskedasiciy (VARMA-AGARCH) model of Chan, Hoi and McAleer (2002). The resuls sugges he presence of volailiy spillovers from FTSE 100 o boh S&P 500 and Nikkei 225, and from S&P 500 o FTSE 100. Key words: Mulivariae GARCH, Asymmeries, Volailiy, Spillovers, Risk. 1. INTRODUCTION The seminal work of Tobin (1958) and Markowiz (1959) showed ha he efficiency of porfolios could be opimised by combining asses based on he correlaion in heir reurns. Grubel (1968) exended he porfolio selecion problem by considering porfolios ha conain asse holdings in oher counries, and showed ha porfolio efficiency could be improved hrough inernaional diversificaion. This has led o a vas body of research ino he degree of co-movemens among reurns in differen financial markes. Engle s (1982) research on ime-varying volailiy models has added a new dimension o he analysis of marke co-movemens. The relaionship beween volailiy and risk has been one of he main facors underlying he ineres in volailiy modelling. An imporan quesion for inernaional diversificaion is wheher shocks in one marke influence, or have spillovers ino, reurns and volailiy in oher markes. This paper ess for he exisence of price and volailiy spillovers among hree major sock marke indexes, namely S&P 500, FTSE 100 and Nikkei 225. Exising empirical research is exended hrough an applicaion of he vecor auoregressive moving average asymmeric generalised auoregressive condiional heeroskedasiciy (VARMA-AGARCH) model of Chan, Hoi and McAleer (2002). This general model has no previously been applied o es for volailiy spillovers. Chan e al. (2002) derived he necessary and sufficien condiions for sric saionariy and ergodiciy, sufficien condiions for he exisence of he log-momen and of all momens, and sufficien condiions for consisency and asympoic normaliy of he quasi-maximum likelihood esimaor (QMLE) of he VARMA-AGARCH model. Their proofs are based on he derivaion of he causal expansions, which do no require he exisence of momens. The srucural and asympoic properies of all nesed special cases follow by he imposiion of appropriae resricions, which allows he various special cases of he VARMA-AGARCH model o be esed. 2. Daa The daa used are he daily opening prices PO ) and closing prices ( PC ) from ( 12/10/1992 o 7/7/2003 for he Nikkei 225, FTSE 100 and S&P 500 sock indexes 1

2 expressed in he local currencies. A he ime of collecing he daa, his was he longes series available from DaaSream. The raionale for employing inra-daily frequency daa for modelling sock reurns and volailiy ransmission is hree-fold. Firs, marke efficiency would sugges ha news is quickly and efficienly incorporaed ino sock prices. Therefore, while informaion generaed yeserday may be significan in explaining prices oday, i is less likely ha informaion generaed las week would have any relevance oday. Second, changes in raes of reurn are news driven. Announcemens such as declaraions of war, profi forecass and changes in ineres raes are facors ha drive equiy prices in he shor run. However, since invesors have heerogeneous beliefs and expecaions, heir responses o such news can vary widely. Using daily sock daa permis an analysis of how a marke s psychology can be ransmied from one marke o anoher. Saisics Nikkei 225 FTSE 100 S&P 500 Mean Median Maximum Minimum Sd. Dev Skewness Kurosis CoV Table1: Descripive Saisics for he Volailiies of Nikkei 225, FTSE 100 and S&P 500. The plos of he volailiy of he o-c reurns of he hree indexes are given bellow: Finally, if he reurns in marke i a ime are calculaed as he log difference beween he closing prices of marke i in calendar days and -1, hese 24-hour reurns of marke i will overlap in real ime wih he 24-hour reurns of oher markes in calendar ime. The use of inra-daily daa should assis in reducing he non-synchronous rading problem, as highlighed in Scholes and Williams (1977), because he open-o-close (o-c) reurns of Tokyo do no overlap wih he o-c reurns of New York or London, while he o-c reurns in London have only a 2-hour overlap wih he o-c reurns in New York. The o-c reurns in marke i a ime ( are defined as: R = ln( PC / PO o c, i i i ) R o c, ) i where PC i and PO i are he closing and opening prices in marke i a ime, respecively. Several definiions of volailiy are available in he lieraure. This paper adops he measure of volailiy proposed in Pagan and Schwer (1999), who define he rue volailiy in o-c reurns as: NIKKEI 225_VOL FTSE 100 _VOL V 2 = ( Ro c, i E( Ro c, i )). 2

3 H r = W + A ε + C I ( η ) ε s + l= 1 l l l = 1 l = 1 BlH l r l l l S&P 500_VOL Table 1 gives he descripive saisics for he volailiy of he o-c reurns of he hree indexes. Each of he volailiy series exhibis clusering, which needs o be capured by an appropriae model. Furhermore, all series appear o conain a number of observaions which migh legiimaely be regarded as ouliers. The volailiy in all series appears o increase dramaically around 1997, corresponding o he Asian economic and financial crises. This increase in volailiy persiss unil he end of he sample, and is likely o have been affeced by he Sepember 11, 2001 erroris aacks and he conflics in Afghanisan and Iraq. I is ineresing o noe ha his increase in volailiy is much more pronounced for S&P 500 and FTSE 100 han for Nikkei 225, which may arise because he USA and UK have been more direcly involved in he war on error han has Japan. 3. Model This paper uses he vecor auoregressive moving average asymmeric generalised auoregressive condiional heeroskedasiciy, or VARMA-AGARCH, model of Chan, Hoi and McAleer (2002) o es for he exisence of volailiy spillovers. The general model is given by: Y = E ) + ε ( Y F 1 Φ ( L)( Y µ ) = Ψ( L) ε ε = Dη where H = h,..., h )', W = ω,..., ω )', ( 1 m ( 1 m ( 1/ D = diag h i ), η = ( η 1,..., η m )', ε = ( ε1,..., ε m ), A l, C l and B are m m marices wih ypical l elemens α, γ and ij ij β, respecively, for ij i, j = 1,..., m, I( η ) = diag( I( ηi )) is an m m marix, p Φ ( L) = I Φ1 L... Φ L and Ψ... m q ( L) = I m Ψ1 L Ψq L are polynomials in L, he lag operaor, F is he pas informaion available o ime, I is he m m m ideniy marix, and I ( η i ) is an indicaor funcion, given as: 1, ε i 0 I( η i ) = 0, ε i > 0 The ime subscrips in he model correspond o rading ime and no calendar ime. For example, in he condiional mean and condiional variance models for FTSE 100, he informaion se of raders in London includes he pas informaion from London as well as informaion from Tokyo for he same calendar day, and informaion from New York for he previous calendar day. The coefficiens ij and ij for i j measure he exen o which he lagged uncondiional shock and lagged condiional variance in marke j, respecively, influence he condiional variance in marke i. An aracive feaure of he VARMA- AGARCH model is is abiliy o capure mulivariae asymmeries concerning he impac of posiive and negaive uncondiional shocks o marke i on he condiional variance in marke i hrough he coefficien i. If i is posiive, i implies ha negaive shocks increase he condiional volailiy in marke i o a larger exen han posiive shocks. 4. Mehod There is no overlap beween rading hours in he Tokyo marke and he oher wo markes. However, a wo-hour overlap in rading exiss p 3

4 beween London and New York. In order o simplify he analysis, i is assumed ha he hree markes do no overlap. Non-overlapping rading implies ha he mean and variance in each marke can be condiioned upon any informaion which has already been observed in a paricular marke, as well as in he oher wo markes. The possible biases arising from overlapping rading hours beween he London and New York sock markes have been invesigaed in Hamao e al. (1990) and Koumos and Booh (1995), where i has generally been found ha he parameer esimaes are no significanly affeced. Due o he non-synchronous naure of he inra-daily daa used in his paper, join esimaion is no appropriae. The sequenial esimaion procedure used for nonsynchronous daa is as follows: (1) For each financial index reurn series, he univariae GARCH (1,1) model wih a VAR mean specificaion is esimaed, and he uncondiional shocks and sandardized residuals of he hree financial index reurns are saved. (2) For each reurn, he univariae AR(1)- GARCH(1,1) and AR(1)-AGARCH(1,1) models are esimaed, including he lagged squared uncondiional shocks and he lagged condiional variances of he oher wo financial indexes, where he lags refer o rading raher han calendar ime. The ess of inerdependence and asymmery are valid under he null hypohesis of independen and symmeric effecs, so ha sep (2) is valid under he join null hypohesis. The primary purpose of he srucural and asympoic heory is o demonsrae ha such esing is sraighforward and valid. This is in conras o, for example, he univariae and mulivariae EGARCH models, for which he asympoic heory has ye o be esablished. 5. Resuls The resuls repored in Table 2 sugges ha all markes experience posiive and significan reurns spillovers from he oher wo markes, such ha a posiive (negaive) shock o one marke increases (decreases) reurns in he following wo markes o open. Reurns Consan MA(1) Nikkei (- 1) FTSE (-1) Table 2: Condiional Mean Generally, he more recen is informaion in rading ime, he more likely will ha informaion be significan in explaining he condiional mean. In he condiional mean equaion for Nikkei 225 (S&P 500), he mos recen informaion in rading ime used o explain he reurns a ime are he reurns o S&P 500 (FTSE 100) on he previous (same) calendar day. The mos recen reurn has a larger impac on he condiional mean han on he reurns o FTSE 100 (Nikkei 225) on he previous (same) calendar day, or he own one-period lagged reurns. For FTSE 100, he bigges impac comes from he reurns o S&P 500 on he previous calendar day, followed by he own lagged reurns, while he smalles impac arises from he reurns o Nikkei 225 on he same calendar day. I is ineresing o noe ha he own lagged reurns are no significan for Nikkei 225 and FTSE 100, suggesing ha he own long run persisence is dominaed by spillover effecs. As expeced, informaion generaed by S&P 500 has he sronges spillover effecs on he reurns of he oher wo markes, while Nikkei 225 has he weakes spillover effecs. The shor run persisence of shocks o index i in he same marke is given by ( i i ), where i is he shor run persisence of posiive shocks and i i is he shor run persisence of negaive shocks. The empirical resuls repored in Table 3 show ha he condiional volailiy of Nikkei 225 is affeced by boh is shor run posiive and negaive shocks. The condiional volailiy of FTSE 100 and S&P 500 are affeced only by heir own shor run negaive S&P (-1) Nikkei FTSE S&P Noes: 1. The hree enries for each parameer are heir respecive esimae, he asympoic -raio and he Bollerslev-Wooldridge (1992) robus -raio. 2. Nikkei (-1), FTSE (-1) and S&P (-1) denoe he lagged reurns for each index. 3. Enries in bold are significan a he 5% level. 4

5 Reurns N F S N F S Nikkei FTSE S&P Noes: 1. The hree enries for each parameer are heir respecive esimae, he asympoic -raio and he Bollerslev-Wooldridge (1992) robus -raio. 2. The parameers in he condiional variance equaion associaed wih S&P, Nikkei and FTSE reurns are denoed by subscrips S, N and F, respecively. 3. Enries in bold are significan a he 5% level. shocks, while he condiional volailiy of S&P 500 is affeced by shor run shocks o FTSE 100 and by is own shor run negaive shocks. The long run persisence of shocks o index i in he same marke is given by ( i i i. All indexes are affeced by he long run persisence of own shocks. Table 3 shows ha he condiional volailiy of Nikkei 225 is negaively affeced by he long run persisence of shocks from FTSE 100. The condiional volailiy of FTSE 100 is posiively affeced by he long run persisence of shocks from S&P 500. Finally, he condiional volailiy of S&P 500 is negaively affeced by he long run persisence of shocks from FTSE 100. Table 3 Condiional Variance reac differenly o posiive and negaive shocks. A posiive suggess ha he condiional volailiy of each marke increases more from a negaive han from a posiive shock, which is consisen wih he sylised fac ha volailiy reacs more srongly o bad news han o good news. The resuls repored in his paper differ from hose available in he lieraure. For example, Hamao e al. (1990) find evidence of volailiy spillovers from S&P 500 o boh FTSE 100 and Nikkei 225, and from Nikkei 225 o FTSE 100. Theodossiou and Lee (1993) repor evidence of volailiy spillovers from S&P 500 o all oher indexes. In comparison, his paper finds evidence of volailiy spillovers from For he condiional variance equaion, he iming of mulivariae effecs does no appear o be a significan facor. The sronges effec always comes from he sum of he own ARCH and GARCH effecs, which in real ime consiues he mos disan informaion. A comparison of he wo mulivariae effecs shows ha, for FTSE 100, he mos disan mulivariae effec has a sronger effec han he mos recen, while he opposie is rue for S&P 500. In fac, he mos recen mulivariae ARCH and GARCH effecs for he condiional variance equaions of Nikkei 225 and FTSE 100 are no significan a he 5% level. The asymmeric coefficien is posiive and significan for S&P 500, FTSE 100 and Nikkei 225 in Table 3, suggesing ha hese markes FTSE 100 o boh S&P 500 and Nikkei 225, and from S&P 500 o FTSE 100. Hamao e al. (1991) repor evidence ha volailiy spillover effecs can be ime varying. If his is correc, hen he differences in he repored findings can be aribued o differences in he samples used in he empirical sudies. Hamao e al. (1990) use daa for he period 01/04/1985 o 31/05/1988, while Theodossiou and Lee (1993) use daa from 11/01/1980 o 27/12/1991. A second possible explanaion lies in he use of only asympoic -raios in hese wo sudies. As can be seen in Table 2, he asympoic - raios are ypically higher in absolue value han heir robus counerpars, which is likely 5

6 due o he inclusion of exreme observaions in he samples. Thus, he asympoic -raios rejec he null hypohesis of no spillover effecs more frequenly han do heir robus counerpars. 6. Conclusion In his paper, open-o-close reurns of he FTSE 100, S&P 500 and Nikkei 225 sock indexes were used o es for reurns and volailiy spillovers. The VARMA-AGARCH model of Chan, Hoi and McAleer (2002) was used o model he mulivariae condiional volailiies and o es for he exisence of volailiy spillovers. Open-o-close reurns were inended o reduce he degree of overlap among he reurns, and hence reduce he measuremen errors inheren in using nonsynchronous daa. Due o he non-synchronous naure of he sock marke daa, a sequenial esimaion procedure was used. Tesing for reurns and volailiy spillovers across markes is imporan for a variey of heoreical and empirical problems. As exising ess have largely been based on symmeric models, which are ypically misspecified, asymmeric mulivariae models were esimaed o es for he presence of asymmeries. The VARMA-AGARCH model and sequenial esimaion permi valid saisical inference because he srucural and saisical properies of he model have been esablished in Chan, Hoi and McAleer (2002). Significan evidence of reurns spillovers were found across all pairs of sock indexes, as well as volailiy spillovers from FTSE 100 o boh S&P 500 and Nikkei 225, and from S&P 500 o FTSE 100. These resuls differ from hose in he lieraure, where volailiy originaing in S&P 500 has generally been found o have spillover effecs o all oher indexes, and Nikkei 225 volailiy has been found o have spillovers effecs o FTSE Acknowledgemens The auhors wish o hank Felix Chan and Suheijla Hoi for helpful commens and suggesions. The firs auhor wishes o acknowledge a Universiy Posgraduae Award and an Inernaional Posgraduae Research Scholarship a he Universiy of Wesern Ausralia. The second auhor wishes o hank he Ausralian Research Council for financial suppor. 8. References Bollerslev, T. and J.M. Wooldrige (1992), Quasi-Maximum Likelihood Esimaion of Dynamic Models wih Time Varying Covariances, Economeric Reviews, 11, Chan, F., S. Hoi and M. McAleer., Srucure and Asympoic Theory for a Mulivariae Asymmeric Volailiy: Empirical Evidence for Counry Risk Raings, Paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, Ausralia, July Engle, R.F., Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, , Grubel, H.G., Inernaionally Diversified Porfolios: Welfare Gains and Capial Flows, American Economic Review, 58, , Hamao, Y.R., R.W. Masulis and V.K. Ng., Correlaions in Price Changes and Volailiy Across Inernaional Sock Markes, Review of Financial Sudies, 3, , Hamao, Y.R., R.W. Masulis and V.K. Ng., The Effec of he 1987 Sock Crash on Inernaional Financial Inegraion, in W.T. Ziemba, W. Bailey and Y.R. Hamao (eds.), Japanese Financial Marke Research, Elsevier, , Koumos, G. and G.G. Booh., Asymmeric Volailiy Transmission in Inernaional Sock Markes, Journal of Inernaional Money and Finance, 14, , Markowiz, H., Porfolio Selecion: Efficien Diversificaion of Invesmens, Blackwell, New York Pagan, A.R., and G.W. Schwer., Alernaive Models for Condiional Sock Volailiy, Journal of Economerics, 45, , Scholes, M., and J. Williams., Esimaing Beas from Nonsynchronous Daa, Journal of Financial Economics, 5, , Theodossiou, P. and U. Lee., Mean and Volailiy Spillovers across Major Naional Sock Markes: Furher Empirical Evidence, Journal of Financial Research, 16, ,

7 Tobin, J., Liquidiy Preference as a Behavior Towards Risk, Review of Economic Sudies, 25, 65-86,