Integration at cost: Evidence from volatility impulse response functions

Integration at a cost: Evidence from volatility impulse response functions Ekaterini Panopoulou∗Theologos Pantelidis† February 25, 2005 Abstract We investigate the international information transmission between the U.S. and the rest of the G-7 countries using daily stock market return data covering the last 20 years. A pre-1995 and post1995 analysis reveals that the linkages between the markets have changed substantially in the more recent era, suggesting that national markets have become more interdependent. In the majority of the countries under scrutiny, we provide evidence of direct volatility spillovers, running mainly from the US and pointing to more rapid information transmission during the recent years. We further uncover the dynamics of the volatility spillovers between the international stock markets by means of a Volatility Impulse Response Analysis. Our findings, based on three historical shocks that have caused turbulence in the stock markets, suggest that the persistence of volatility shocks has increased substantially during the post-1995 period mainly due to increased persistence and interdependence in the volatility of all markets. As a result, volatility shocks in the international stock markets nowadays perpetuate for a significant longer period compared to the pre-1995 era. JEL classification: G15, C32 Keywords: volatility spillovers, volatility impulse response functions, stock market, ARCH-BEKK Acknowledgments: WearegratefultoT.Flavin,C.HafnerandN.Pittisforhelpfulcomments and suggestions. The authors thank the EU for financial support under the “PYTHAGORAS: Funding of research groups in the University of Piraeus” through the Greek Ministry of National Education and Religious Affairs. The usual disclaimer applies. ∗Department of Banking and Financial Management, University of Piraeus, Greece and Department of Economics, National University of Ireland Maynooth. Correspondence to: Ekaterini Panopoulou, Department of Economics, National University of Ireland Maynooth, Co.Kildare, Republic of Ireland. E-mail: [email protected]. Tel: 00353 1 7083793. Fax: 00353 1 7083934. †Department of Banking and Financial Management, University of Piraeus, Greece. 1 Introduction In the wake of the stock market crash of October 1987, the study of the transmission of financial shocks across markets or countries has emerged as one of the most intensive research topics in the international finance literature in recent years. The first contributions in the so-called spillover literature came from Eun and Shim (1989) and Becker et al. (1990) who, based on either a VAR model or a set of single linear equations, tried to capture the dependence between international equity returns. In a similar way, Koch and Koch (1991) investigated the evolution of contemporaneous and lead/lag relationships among eight stock markets and concluded that regional interdependence grows over time and that the influence of the Japanese market increases at the expense of the US market. However, these studies focused on the return series and on how returns are correlated across markets, i.e. they considered only interdependence through the mean of the process. A second strand of the literature, which is growing rapidly, explicitly focuses on the volatility of equity returns, suggesting the existence of higher-order dependence stemming from the second moments. This framework is appropriate when modeling high-frequency financial time series and its importance has been recognized ever since Engle (1982) introduced the class of ARCH models. In this context, Hamao et al. (1990) employed univariate GARCH models to examine the stock markets around the 1987 US stock market crash. They found evidence of significant price-volatility spillovers from the US to the UK and Japan and from the UK to Japan for the post-crash period. In contrast, no such spillovers are found in the pre-crash period. Their results suggest that shocks that originate in the US are larger and more persistent than the UK and Japanese ones. Lin et al. (1994), using a signal extraction model with GARCH processes, found a reciprocal relationship between the price and volatility of the US and Japanese markets. Susmel and Engle (1994) analyzed the interrelationship between the US and the UK stock markets using hourly returns and did not find strong evidence of either mean or volatility spillovers between the two markets. Karolyi (1995) examined the dynamic relationship between the US and Canadian stock market returns and return volatilities using a bivariate GARCH model. He found that the effectsofshocksoriginating in the US market on the Canadian market returns and volatility are smaller and less persistent than those measured with traditional vector autoregressive models. Theodossiou 1 and Lee (1993) studied the relationship between the US, the UK, Canadian, German and Japanese stock markets using a multivariate GARCH-in-mean model and found mean and volatility spillovers between some of those markets. The authors also documented that the US is the major exporter of volatility. More recently, Koutmos and Booth (1995) examined price volatility spillovers for the US, the UK and Japan in the context of an extensive multivariate Exponential GARCH model which can capture possible asymmetries in the volatility transmission mechanism. The authors found apart from price spillovers, extensive and reciprocal second moment interactions, which are asymmetric, i.e. negative innovations in a given market increase volatility in the next market to trade more than positive innovations. The appearance of the Asian financial crisis in 1997 revived the interest in the matter and turned the focus away from the major stock markets towards the emerging ones (see for example Ng, 2000; Caporale et al., 2001; He, 2001; Chen et al., 2002; Miyakoshi, 2003). In this study, we focus explicitly on uncovering the volatility dynamics between the US stock market and the remaining six of the G-7 countries using Volatility Impulse Response Functions for multivariate GARCH models introduced by Hafner and Herwartz (2006). In this vein, we aim at establishing the pattern of information transmission between these countries. As indicated by Ross (1989), the transmission of information to a market is related primarily to the volatility of an asset’s price changes in an arbitrage-free economy, i.e. the second moment is more important than the first one in the flow of information. In this vein, Engle et al. (1990) attribute movements in volatility to the lag with which market participants process new information. The present study makes a twofold contribution. First, we estimate a bivariate GARCH model, for which a BEKK representation is adopted (see Engle and Kroner, 1995), for each of the six countries against the US using daily returns for the last twenty years. This BEKK formulation enables us to reveal the existence of any “meteor showers”, i.e. transmission of volatility from one market to another, as well as any “heat waves”, i.e. increased persistence in market volatility (see Engle et al., 1990). Splitting our sample into two non-overlapping sub-samples of equal length, we investigate whether the efforts for more economic, monetary and financial integration have fundamentally altered the sources and intensity of volatility spillovers to the individual stock markets. Second, by using a recently developed technique, we estimate the corresponding Volatility Impulse Response Functions (VIRFs) implied by the 2 specification of each model. We then assess the impact of three historically observed shocks, i.e. the 1987 stock market crash, the 1997 Asian Financial crisis and the 2001 terrorist attack on the volatility and co-volatility of the markets. To the best of our knowledge, no other study (except for Hafner and Herwartz, 2006) has employed this innovative technique of VIRFs to study volatility dynamics in any market. More importantly, there are several reasons why VIRFs represent a convenient approach to analyse volatility spillovers. First, this technique allows the researcher to determine precisely how a shock to one market influences the dynamic adjustment of volatility to another market and the persistence of these spillover effects. Second, VIRFs depend on both the volatility state and the unexpected returns vector when the shock occurs. As a result, the asymmetric response of volatility on negative and positive “news” typically documented in the literature (see e.g. Koutmos and Booth, 1995) can easily be accommodated.1Third, contrary to typical Impulse Response Functions, this specific methodology avoids typical orthogonalization and ordering problems which would be hardly feasible in the case of highly interrelated and observed at high frequencies financial time series. The only study that is closely related to ours is the one by Leachman and Francis (1996). The authors use a two-stage procedure, i.e. they firstestimateunivariateGARCHmodels for the G-7 stock market returns and then the estimated conditional variances are used to construct a VAR system. This methodology enables them to employ the standard Impulse Response analysis and conduct Variance Decompositions in order to determine how a shock to one market influences the dynamic adjustment of volatility in the remaining markets and the persistence of these volatility spillovers. They can also quantify the relative significance of each market in generating and transmitting fluctuations to other markets. Interestingly, the authors suggest that a multivariate GARCH approach would give more efficient parameter estimates than their two-stage approach but would not enable the researcher to obtain impulse response functions, as the latter are not available for GARCH processes. It is this gap in the literature that we intend to bridge by estimating the VIRFs for the G-7 stock market returns accommodated by the aforementioned methodology. Consistent with the increased integration of capital markets already documented in the literature (see, 1Negative “news”, i.e. unexpected returns, in one market can result in a different volatility profile than positive “news”, other things being equal. 3 for example, Harvey, 1991; Bekaert and Hodrick, 1992; Campbell and Hamao, 1992), our results suggest that equity markets have become more interdependent in the post-1995 period compared with the pre-1995 period. This greater integration resulted in a significant increase in the persistence of volatility shocks for all the countries at hand. The existence of both elevated “heat waves” and “meteor showers” effects is depicted in the pattern and size of the VIRFs. The remainder of this study is organized as follows. Section 2 discusses the econometric methodology and Section 3 describes the data and presents the empirical findings for both the pre-1995 period and the post-1995 period. Section 4 offers a summary and some concluding remarks. 2 Econometric Methodology In this section, we first present the model we employ to investigate the volatility spillovers between the stock markets under scrutiny and then provide a brief description of the volatility impulse response method employed to analyse the persistence of the volatility shocks in the international stock markets. 2.1 The BEKK Model The analysis is based on a bivariate VAR(1)-GARCH(1,1) model. Let Yt=(y1t,y 2t)0be the returns vector, with y2tdenoting the US stock market and y1tone of the remaining G-7 countries. The conditional mean of the process is modeled as follows: Yt=C+M∗Yt−1+Et(1) where Cis a 2×1vector of constants and Mis a 2×2coefficient matrix and Et=(e1t,e 2t)0 is the vector of the zero-mean error terms. We allow Etto have a time-varying conditional variance, that is Var(Et|F t−1)=Htwhere Ft−1denotes the σ−field generated by all information available at time t−1. We further assume that the conditional variance, Ht, follows a bivariate GARCH(1,1) model and we, specifically, consider the following BEKK 4 representation, introduced by Engle and Kroner (1995): Et=H1/2 t∗Zt Ht=Ω∗Ω0+A∗Et−1∗E0 t−1∗A0+B∗Ht−1∗B0(2) where Ω=[ωij],i,j=1,2is a 2x2 lower triangular matrix of constants, A=[aij]and B=[bij],i,j=1,2are 2x2 coefficient matrices and Zt=(z1t,z 2t)0∼iid(   0 0   ,   10 01   ). Matrix Ameasures the extent to which conditional variances are correlated with past squared unexpected returns (i.e. deviations from the mean) and consequently captures the effects of shocks on volatility. On the other hand, matrix Bdepicts the extent to which current levels of conditional variances and covariances are related to past conditional variances and covariances. Apart from displaying sufficient generality, this model ensures that the conditional variance-covariance matrices, Ht=[hij,t],i,j=1,2,arepositivedefinite under rather weak assumptions.2 Compared to alternative multivariate GARCH representations, the BEKK model is more convenient for estimation, because it involves fewer parameters.Engle and Kroner (1995) prove that the BEKK model in (2) is second-order stationary if and only if all the eigenvalues of (A⊗A+B⊗B)are less than unity in modulus. In this case, the unconditional variance of Et,Var(Et), can easily be calculated by: vec[Var(Et)] = [I4−(A⊗A)0−(B⊗B)0]−1∗vec(Ω0Ω) where vec is the operator that stacks the columns of a square matrix.3 More in detail, the conditional variance for each equation can be expanded for the bivariate GARCH(1,1) as follows: h11,t =ω2 11 +a2 11e2 1t−1+2a11a12e1t−1e2t−1+a2 12e2 2t−1+(3) +b2 11h11,t−1+2b11b12h12,t−1+b2 12h22,t−1 h22,t =ω2 21 +ω2 22 +a2 21e2 1t−1+2a21a22e1t−1e2t−1+a2 22e2 2t−1+(4) +b2 21h11,t−1+2b21b22h12,t−1+b2 22h22,t−1 2Engle and Kroner (1995) show that Htis positive definite if at least one of Ωor Bis of full rank. 3The moment properties of multivariate GARCH processes are also examined by Hafner (2003). 5 h12,t =ω11ω21 +a11a21e2 1t−1+(a11a22 +a12a21)e1t−1e2t−1+a12a22e2 2t−1+(5) +b11b21h11,t−1+(b11b22 +b12b21)h12,t−1+b12b22h22,t−1 Suppose that we estimate a bivariate system for Canada and the US based on equations (1) - (2). In such a case, h11,t and h22,t denote the conditional variance for Canada and the US respectively, while h12,t denotes the conditional covariance between the series. Significance of any or both the elements a12,b 12 suggests that volatility in the Canadian market is affected by developments in the volatility of the US market through either the past volatility of the US market, h22,t−1, or the past squared innovations e2 2t−1(or even the cross products, e1t−1e2t−1, of past innovations). Furthermore, indirect feedbacks may exist through the past value of the conditional covariance h12,t−1. When considering the evolution of the US market volatility and its dependence on the Canadian one, the reasoning is similar and follows directly from equation (4). The contemporaneous co-movement in the volatility of the series is given by equation (5) and is a function of past squared innovations, cross products of innovations, past conditional volatilities and naturally past conditional covariance. This rich parameterization suggests that even in the case that conditional volatilities between the series are not linked directly, i.e. b12 =b21 =0, the interactions between the conditional variances is ensured by past return innovations. To cope with the excess kurtosis usually found in the estimated standardised residuals under the assumption of Gaussian innovations, we follow Bollerslev (1987) and evaluate (and maximize) the sample log-likelihood function under the assumption of t(ν)−distributed innovations. In such a case, given a sample of Tobservations, a vector of unknown parameters θand a 2×1vector of returns Yt,the bivariate BEKK model is estimated by maximizing the following likelihood function: L(θ)= T X t=1 ln(lt(θ)) (6) with lt=Γ((T+v)/2) Γ(v/2)[π(v−2)]T/2|Ht|−1/2·1+ 1 v−2E0 tH−1 tEt¸−(T+v)/2 (7) where νdenotes the degrees of freedom of the t−distribution and Γ(·)is the gamma function. This log-likelihood function is maximized using the Berndt, Hall, Hall and Hausman (1974) 6 algorithm (BHHH).4 2.2 Volatility Impulse Response Functions We now briefly describe the Volatility Impulse Response Function (VIRF) introduced by Hafner and Herwartz (2006). The authors derive the VIRF based on an alternative multivariate GARCH representation, namely the vec-representation (introduced by Engle and Kroner, 1995), given by: vech(Ht)=Q+R∗vech(Et−1∗E0 t−1)+P∗vech(Ht−1)(8) where Qis a 3×1matrix of constants, while Rand Pare 3×3coefficient matrices. vech is the operator that stacks the lower triangular part of a square matrix. It is important to note that the vec-representation given in (8) requires the estimation of 21 parameters, while the BEKK representation given in (2) has only 11 parameters. Thus, the BEKK model manages to reduce substantially the number of parameters, facilitating the estimation procedure. Obviously, the vec-model is more general than the BEKK model, since the BEKK model reduces the number of parameters by imposing some specific restrictions on the vec-model. In general, any given BEKK model has a unique equivalent vec-representation (Engle and Kroner 1995), while the converse is not true.5,6In summary, by employing the BEKK model we achieve the reduction of the number of parameters with virtually no cost in terms of the generality of the model. The derivation of the unique equivalent vec-representation of a BEKK model is straightforward.7In the rest of this paper, we assume that (8) is the equivalent vec-representation of (2). Assume that at time t=0the conditional variance is at an initial state H0and an initial 4Susmel and Engle (1994) suggest that in the case of high-frequency financial data using the t-distribution generates a more efficient estimation for conditional errors than the normal distribution. 5Two GARCH representations are equivalent if every sequence of errors {Et}generates the same sequence of conditional volatilities {Ht}for both representations. 6It is possible that a vec-model has no equivalent BEKK representation. Moreover, if there is an equivalent BEKK representation for a vec-model, this BEKK representation is not unique. 7The necessary assumptions for the equivalence of the two representations are the following: Q=   ω2 11 ω11ω21 ω2 21 +ω2 22  ,R=  a2 11 2a11a12 a2 12 a11a21 a11a22 +a12a21 a22a12 a2 21 2a21a22 a2 22  and P=  b2 11 2b11b12 b2 12 b11b21 b11b22 +b12b21 b22b12 b2 21 2b21b22 b2 22  . 7 shock Z0=(z1,0,z 2,0)0occurs. The VIRF, Vt(Z0),isthendefined as follows: Vt(Z0)=E[vech(Ht)|F t−1,Z 0]−E[vech(Ht)|F t−1] The first and third elements of Vt(Z0)(denoted as v1,t and v3,t respectively) represent the reaction of the conditional variance of the first and second variable respectively to the shock, Z0, that occurred tperiods ago. Similarly, the second element of Vt(Z0)(denoted as v2,t) represents the reaction of the conditional covariance to the shock, Z0, that occurred tperiods ago. The VIRF can easily be computed recursively based on the following relations: V1(Z0)=R∗{vech(H1/2 0Z0Z0 0H1/2 0)−vech(H0)}(9) Vt(Z0)=(R+P)∗Vt−1(Z0),t>1 The VIRF has two important differences compared to the traditional Impulse Response Function (IRF) in the conditional mean. First, the VIRF is an even function of the initial shock, that is Vt(Z0)=Vt(−Z0), contrary to the IRF that is an odd function of the initial shock. Second, the IRF is a linear function, i.e. IRF(k∗Z0)=k∗IRF(Z0),whiletheVIRF is not homogeneous of any degree. Before presenting the empirical results of this study, we briefly describe the behavior of the VIRF. First of all, let Ψ=[ψi,1]:=vech(H1/2 0Z0Z0 0H1/2 0)−vech(H0)where i=1,2,3. It is obvious that the elements of Ψare functions of the elements of the initial state H0and the elements of the shock Z0. The following three cases are of interest: Case I: Diagonal BEKK model (i.e. a12 =a21 =b12 =b21 =0) In this case, both Rand P(and thus R+P) are diagonal matrices (see footnote 7). It is easy to show that: v1,1=a2 11ψ1,1and v1,t =(a2 11 +b2 11)t−1v1,1for t>1 v2,1=a11a22ψ2,1and v2,t =(a11a22 +b11b22)t−1v2,1for t>1 v3,1=a2 22ψ3,1and v3,t =(a2 22 +b2 22)t−1v3,1for t>1 Therefore, in this particular case there are no volatility spillovers, since both v1,t and v3,t 8 10−4,2.03 ×10−4)0respectively for the Canada-US model, In this case, the initial shock is estimated to be b Z0=(b H1/2 t)−1b Et=(−9.74,−14.04)0. The corresponding initial shocks for the remaining five models are calculated in a similar way. The second shock we consider is the Asian financial crisis in 1997. More specifically, we calculate the initial shock based on the estimated models (Table 3, Panel B) for October 27, 1997. The last shock we consider is the one associated with the terrorist attack to the Twin Towers in September 2001. Since the first trading day for the US stock market after the attack was the September 17, 2001, the initial shocks are calculated based on this day. The estimated initial shocks for these three historical shocks with respect to our estimated models are reported in Table 5. Apart from the estimated parameters of the BEKK models and the corresponding initial shocks, the calculation of the VIRFs requires the initial state of volatility, H0,assuggested by equation (9). To make our findings invariant to the choice of initial state, we select the last day of our sample, i.e. October 8, 2004 and employ this estimated conditional variancecovariance matrix in both sub—periods.15 This allows a direct comparison of the VIRFs between the two sub-periods under consideration. We first investigate the size of the effect of each shock on the conditional variance of the series under examination. Table 7 reports the maximum value and the 1-step ahead value of the VIRF divided by the initial conditional variance. As expected, the effect of the 1987 crash on the conditional volatility dynamics is substantially greater than the corresponding effects stemming from the other two shocks considered in this analysis. For example, in the case of the US, the crash of 1987 induces a rise in volatility that is 6.57 and 17.18 times the initial volatility for the pre-1995 and post-1995 period respectively.16 The corresponding figures for the Asian financial crisis are 1.06 and 2.78 for the periods examined. Even milder effects are prevalent when the third shock is considered.17 The increased intensity of volatility spillovers during the recent years is easily shown when 15The initial states employed for each pair of countries are presented in Table 6. Alternatively, the estimated unconditional variance matrix of Etcould be employed as an initial state. In such a case, our results would be qualitatively similar to the ones reported. 16The analysis for the US is based on the Canada-US model, although quantitatively similar results are drawn from the rest of the bivariate models. 17In this case, if a similar shock occurred in the pre-1995 period, the increase in the US volatility would be just 0.53 times the initial volatility, while in the case of the post-1995 period, the volatility increase would be 1.38 times the initial volatility. 15 the path of the impulse responses in the volatility of each country is considered. Figures 2-4 plot the VIRFs for the three shocks respectively. In most cases, the VIRF is maximised the day after the shock. However, in some cases the effect of the initial shock gradually increases, reaching its maximum value after many days or even weeks. In general, the impulses are declining starting from a high level of volatility. In some cases, however, there is evidence of an initial shock amplification which increases further the initial effect. Eventually, in all cases the VIRFs resume their declining path towards zero. We now focus on the estimated half-life of the volatility shocks, which are reported in Table 8 (Panels A, B and C for the three shocks respectively) for all six pairs of countries and the two sub-periods.18 Apparently, the half-lives of volatility shocks paint a similar picture pointing to more persistent shocks in the more recent era. For example, for the pre-1995 period and for the 1987 crash, the estimated half-lives for the US, the country from which the crash originated, range from 48 to 91 days, while if the shock occurred in the post-1995 period, the respective figures would be 98 days to 219 days. The half-lives for the rest of the countries are lower than the US in the first sub-period ranging from 13 days (UK and France) to 41 days (Japan). Surprisingly, our findings from the recent era suggest that a similar to the “1987 crash” shock would induce volatility spillovers that would last for significantly longer period nowadays compared to the pre-1995 period. Our results for the other two shocks are qualitative similar to those of the first shock, so we do not discuss them separately to save space. In summary, the empirical findings of this volatility impulse response experiment suggest that the increased integration of the international stock markets during the post-1995 period has also caused an increase in the persistence of volatility shocks. As a result, similar volatility shocks can perpetuate for a significant longer period nowadays compared to the pre-1995 era. 4 Conclusions There is extensive empirical work in the literature with respect to interdependencies between financial markets and more specifically, national stock markets. This paper focuses on second18Note that the half-lives in Table 8 are calculated after the Initial Shock Amplification has been deducted. 16 order interdependencies, i.e. linkages through the conditional variances of the series. The analysis was performed using daily closing stock index data from the G-7 stock markets for thelast20years. ByadoptingabivariateBEKKrepresentation and splitting our sample into two 10-year sub-samples, we first examined whether stock market linkages between the US and the remaining of the G-7 countries have changed during the recent years. As a second step, we employed a new technique developed by Hafner and Herwartz (2006) and estimated the Volatility Impulse Response Functions (VIRFs) related to each pair of our countries. This technique enabled us to quantify the size and the persistence of three historical shocks that have caused turbulence in the stock markets. Furthermore, the significantly different structure of stock markets in the preand post-1995 periods allowed comparisons that shed some light into the current behavior of stock markets. Our empirical findings can be summarised as follows. We confirmed the established view that the US stock market is the major volatility exporter country. Specifically, there is evidence of significant volatility spillovers from the US to Canada, France and Germany during the pre-1995 period. For the same period, the rest of the G-7 countries, i.e. Italy, Japan and the UK appear secluded and invulnerable to shocks originating in the US. On the other hand, our findings for the more recent period point to increased integration between the markets. Specifically, the smaller of the G-7 countries, i.e. Canada, France, Germany and Italy mainly import volatility from the US. A more important finding, however, is the evidence in favor of bidirectional volatility spillovers between the US and Japan, as well as the US and the UK. Our results suggest that shocks originating in the UK affect positively the US stock market while the Japanese ones influence the US market negatively, inducing lower levels of volatility. Our VIRFs analysis of three historical shocks, namely the 1987 crash, the 1997 Asian financial crash and the 2001 terrorist attack provided useful insights with respect to the size and persistence of volatility shocks. We specifically found evidence in favor of increased amplitude and duration of volatility spillovers in the post-1995 sample compared to the pre-1995 one. This intensity of shocks mainly stems from the increased interdependence and persistence of the equity market volatilities documented in the recent era. Consequently, had a shock similar to the one of the 1987 crash occurred in the more recent years, the time required for this shock to die out would have been extremely longer nowadays compared to the pre-1995 period. 17 The method employed here can also be applied to other cases that involve high frequency data, mainly financial data, to examine linkages and uncover the volatility dynamics between the series under examination. Volatility spillovers between exchange rate markets or between stock markets and exchange rates can be detected and quantified through the VIRFs. Another promising route for further investigation may be the extension of this bivariate analysis to a higher order one, allowing for interactions among three or more countries. Both these extensions will be the object of our future work. References [1]Becker, K.G., J.E. Finnerty and M. Gupta (1990). The intertemporal relation between the U.S. and Japanese stock markets. Journal of Finance, 45, 1297-1306. [2]Bekaert, G. and R.J. Hodrick (1992). Characterizing predictable components in excess returns on equity and foreign exchange markets. Journal of Finance, 47, 467-470. [3]Berndt, E.K., B.H. Hall, R.E. Hall and J.A. Hausman (1974). Estimation and inference in non-linear structural models. Annals of Economic and Social Measurement, 69, 542-547. [4]Billio, M. and L. Pelizzon (2003). Volatility spillovers before and after EMU in European stock markets. Journal of Multinational Financial Management, 13, 323-340. [5]Bollerslev, T. (1987). A conditional heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 69, 542-547. [6]Campbell, Y. and Y. Hamao (1992). Predictable stock returns in the United States and Japan: A study of long-term capital market integration. Journal of Finance, 47, 43-69. [7]Caporale, G.M., N. Pittis and N. Spagnolo (2001). Volatility transmission and financial crises. DP02-2001, Centre for Monetary and Financial Economics, South Bank University, London, forthcoming at the Journal of Economics and Finance. [8]Chen, G., M. Firth and O.M. Rui (2002). Stock market linkages: Evidence from Latin America. Journal of Banking and Finance, 26(6), 1113-1141. 18 [9]Cheung Y.M. and L.K. Ng (1996). A causality-in-variance test and its application to financial market prices. Journal of Econometrics, 72, 33-48. [10]Engle, R.F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica, 50, 987-1008. [11]Engle, R.F., T. Ito and W.L. Lin (1990). Meteor showers or heat wages? Heteroskedastic intra-daily volatility in the foreign exchange market. Econometrica, 58, 525-542. [12]Engle, R.F. and K.F. Kroner (1995). Multivariate simultaneous generalized ARCH. Econometric Theory, 11, 122-150. [13]Eun, C. and S. Shim (1989). International transmission of stock market movements. Journal of Financial and Quantitative Analysis, 24, 241-256. [14]Flavin T., M. Hurley and F. Rousseau (2002). Explaining stock market correlation: A gravity model approach. The Manchester School, 70, 87-106. [15]Hafner, C.M. (2003). Fourth moment structure of multivariate GARCH models. Journal of Financial Econometrics, 1, 26-54. [16]Hafner, C. and H. Herwartz (2006). Volatility Impulse Response Functions for Multivariate GARCH Models: An Exchange Rate Illustration. Journal of International Money and Finance, forthcoming. [17]Hamao, Y., R. Masulis and V. Ng (1990). Correlations in price changes and volatility across international stock markets. Review of Financial Studies, 3, 281-307. [18]Harvey, R. (1991). The world price of covariance risk. Journal of Finance, 46, 111-158. [19]He, L.T. (2001). Time variation paths of international transmission of stock volatility — US vs. Hong Kong and South Korea. Global Finance Journal, 12(1), 79-93. [20]Karolyi, A. (1995). A multivariate GARCH model of international transmission of stock returns and volatility. Journal of Business, Economics and Statistics, 13, 11-25. [21]Karolyi, A. and R.M. Stulz (1996). Why do markets move together? An investigation of US-Japan stock return comovements. Journal of Finance, 51, 951-986. 19 [22]Koch, P.D. and T.W. Koch (1991). Evolution in dynamic linkages across daily national stock indexes. Journal of International Money and Finance, 10, 231-251. [23]Koutmos, G. and G.G. Booth (1995). Asymmetric volatility transmission in international stock markets. Journal of International Money and Finance, 14, 747-762. [24]Leachman, L.L. and B. Francis (1996). Equity market return volatility: Dynamics and transmission of the G-7 countries. Global Finance Journal, 7(1), 27-52. [25]Lin, W.-L., R.F. Engle and T. Ito (1994). Do bulls and bears move across borders? International transmission of stock returns and volatility. Review of Financial Studies, 7 (3), 507-538. [26]Miyakoshi, T. (2003). Spillovers of stock return volatility to Asian equity markets from Japan and the US. International Financial Markets, Institutions and Money, 13, 383-399. [27]Ng, A. (2000). Volatility spillover effects from Japan and the US to the Pacific-Basin. Journal of International Money and Finance, 19, 207-233. [28]Ross, S.A. (1989). Information and volatility: The no-arbitrage Martingale approach to timing and resolution irrelevancy. Journal of Finance, 44, 1-17. [29]Susmel, R. and R.F. Engle (1994). Hourly volatility spillovers between international equity markets. Journal of International Money and Finance, 13(1), 3-25. [30]Theodossiou, P. and U. Lee (1993). Mean and volatility spillovers across major national stock markets: Further empirical evidence. Journal of Financial Research, 16, 337-350. 20 21 Table 1: Summary Descriptive Statistics Panel A: Full Sample (31/12/84-8/10/04) Canada France Germany Italy Japan UK US Mean 0.00025 0.00035 0.00028 0.00029 9.59E-05 0.00044 0.00034 Median 0.00049 0.00046 0.00032 0.00037 0.00000 0.00055 0.00025 Maximum 0.08874 0.08289 0.08769 0.07099 0.12883 0.07231 0.09095 Minimum -0.12111 -0.08430 -0.11494 -0.10678 -0.13823 -0.14047 -0.22899 Std. Dev. 0.00960 0.01355 0.01453 0.01304 0.01602 0.01150 0.01093 Skewness -1.17403 -0.22157 -0.29519 -0.36396 0.11682 -0.59935 -2.07463 Kurtosis 17.7069 5.88741 7.16838 6.90276 7.63023 10.6913 47.1517 Panel B: First subsample (31/12/84-31/12/94) Canada France Germany Italy Japan UK US Mean 0.00016 0.00044 0.00031 0.00027 0.00048 0.00054 0.00033 Median 0.00037 0.00049 0.00000 0.00034 0.00053 0.00054 0.00029 Maximum 0.08874 0.08289 0.08769 0.07099 0.12883 0.07231 0.09095 Minimum -0.12111 -0.08430 -0.11494 -0.10678 -0.13823 -0.14047 -0.22899 Std. Dev. 0.00800 0.01317 0.01351 0.01397 0.01560 0.01179 0.01045 Skewness -2.07996 -0.36377 -0.50435 -0.39156 -0.01657 -1.01165 -4.80774 Kurtosis 43.4971 7.11751 10.3723 7.59009 10.1313 15.5090 108.379 Panel C: Second subsample (1/1/95-8/10/04) Canada France Germany Italy Japan UK US Mean 0.00034 0.00028 0.00027 0.00029 -0.00025 0.00035 0.00035 Median 0.00064 0.00041 0.00053 0.00045 -0.00043 0.00056 0.00017 Maximum 0.04690 0.06198 0.06837 0.05592 0.12354 0.05797 0.05574 Minimum -0.09033 -0.07362 -0.08559 -0.07543 -0.06592 -0.05886 -0.07114 Std. Dev. 0.01088 0.01389 0.01542 0.01213 0.01639 0.01124 0.01136 Skewness -0.80306 -0.10832 -0.16263 -0.31626 0.22749 -0.16387 -0.10826 Kurtosis 8.96719 4.94645 5.24439 5.42687 5.72985 5.29419 6.25458 22 Table 2: Unrestricted Estimated GARCH(1,1)-BEKK Models Panel A: 1st subsample (31/12/84-31/12/94) Panel B: 2nd subsample (1/1/95-8/10/04) c 11 α11 α12 b11 b 12 d.f. c11 α11 α12 b11 b 12 d.f. c 21 c 22 α21 α22 b21 b 22 Eigenvalues (s.e.) c21 c 22 α21 α22 b21 b 22 Eigenvalues (s.e.) Canada 0.0013* 0.2596* -0.0173 0.9387* 0.0149* 0.9885 4.8333* Canada 0.0007* 0.2125* 0.0314 0.9747* -0.0074 0.9963 7.3849* (0.0001) (0.0291) (0.0183) (0.0116) (0.0060) 0.9710 (0.3160) (0.0001) (0.0183) (0.0185) (0.0043) (0.0050) 0.9961 (0.5860) 0.0004* 0.0006* 0.0656* 0.1355* -0.029* 0.9938* 0.9692 LL 0.0005* 0.0006* 0.0344 0.2301* -0.0074 0.9705* 0.9940 LL (0.0001) (0.0002) (0.0331) (0.0193) (0.0128) (0.0062) 0.9625 17066.3 (0.0002) (0.0001) (0.0208) (0.0213) (0.0054) (0.0055) 0.9939 17029.1 France 0.0036* 0.2614* 0.0545* 0.9213* -0.0012 0.9890 5.6514* France 0.0013* 0.2167* -0.0271 0.9692* 0.0110 0.9977 8.4343* (0.0004) (0.0280) (0.0221) (0.0160) (0.0078) 0.9458 (0.4033) (0.0002) (0.0189) (0.0235) (0.0057) (0.0071) 0.9886 (0.8219) 0.0004 0.0007* -0.0110 0.1533* -0.0019 0.9830* 0.9425 LL -0.0003 0.0008* 0.0106 0.2315* 0.0018 0.9682* 0.9883 LL (0.0002) (0.0001) (0.0146) (0.0130) (0.0073) (0.0026) 0.9214 15138.7 (0.0002) (0.0001) (0.0164) (0.0186) (0.0051) (0.0053) 0.9798 15977.8 Germany 0.0028* 0.2738* 0.0423* 0.9339* 0.0044 0.9881 5.2204* Germany 0.0007* 0.2205* 0.0384 0.9733* -0.0055 0.9997 8.4185* (0.0003) (0.0257) (0.0207) (0.0109) (0.0067) 0.9593 (0.3353) (0.0002) (0.0165) (0.0227) (0.0043) (0.0069) 0.9932 (0.8173) -0.0001 0.0008* -0.0064 0.1415* 0.0041 0.9828* 0.9566 LL -0.0003 0.0009* 0.0017 0.2415* 0.0021 0.9656* 0.9931 LL (0.0001) (0.0001) (0.0137) (0.0123) (0.0055) (0.0027) 0.9423 15197.8 (0.0002) (0.0001) (0.0122) (0.0162) (0.0034) (0.0047) 0.9869 15921.2 Italy 0.0021* 0.2436* 0.0132 0.9584* -0.0044 0.9896 5.6541* Italy 0.0017* 0.2409* 0.0068 0.9572* 0.0027 0.9964 8.6927* (0.0003) (0.0223) (0.0163) (0.0076) (0.0046) 0.9786 (0.4018) (0.0002) (0.0205) (0.0180) (0.0074) (0.0053) 0.9863 (0.8834) 0.0002 0.0007* 0.0082 0.1477* -0.0027 0.9836* 0.9784 LL 0.0001 0.0007* 0.0236 0.2293* -0.0067 0.9718* 0.9855 LL (0.0001) (0.0001) (0.0107) (0.0121) (0.0036) (0.0022) 0.9780 14978.5 (0.0002) (0.0001) (0.0180) (0.0165) (0.0064) (0.0042) 0.9745 16219.0 Japan 0.0025* 0.3437* 0.0593* 0.9281* -0.0104 0.9898 5.2258* Japan 0.0020* 0.2072* 0.0069 0.9696* 0.0041 0.9946 8.3958* (0.0002) (0.0235) (0.0279) (0.0089) (0.0074) 0.9808 (0.3588) (0.0003) (0.0170) (0.0249) (0.0050) (0.0063) 0.9902 (0.8265) 0.0002 0.0007* 0.0243* 0.1537* -0.008* 0.9827* 0.9648 LL -0.0002 0.0007* -0.026* 0.2308* 0.0049 0.9709* 0.9893 LL (0.0002) (0.0001) (0.0100) (0.0122) (0.0038) (0.0024) 0.9634 14890.7 (0.0002) (0.0001) (0.0132) (0.0153) (0.0039) (0.0038) 0.9837 15242.6 UK 0.0031* 0.2693* 0.0061 0.9239* 0.0057 0.9864 6.0113* UK 0.0013* 0.2186* -0.095* 0.9574* 0.0320* 0.9976 9.0356* (0.0004) (0.0304) (0.0250) (0.0164) (0.0073) 0.9497 (0.4041) (0.0001) (0.0170) (0.0191) (0.0065) (0.0073) 0.9776 (0.8716) 0.0004 0.0008* -0.0049 0.1656* -0.0020 0.9796* 0.9485 LL -0.000* 0.0008* 0.0565* 0.2343* 0.0003 0.9623* 0.9649 LL (0.0002) (0.0001) (0.0188) (0.0135) (0.0090) (0.0033) 0.9281 15438.0 (0.0002) (0.0002) (0.0182) (0.0209) (0.0075) (0.0072) 0.9506 16512.5 Notes: Standard errors are reported in parentheses. An asterisk indicates significance at the 5% level. d.f. refers to degrees of freedom of the t-distribution. LL refers to the value of the log-likelihood function. 23 Table 3: Restricted Estimated GARCH(1,1)-BEKK Models Panel A: 1st subsample (31/12/84-31/12/94) Panel B: 2nd subsample(1/1/95-8/10/04) c 11 α11 α12 b11 b 12 d.f. c11 α11 α12 b11 b 12 d.f. c 21 c 22 α21 α22 b21 b 22 Eigenvalues (s.e.) c21 c 22 α21 α22 b21 b 22 Eigenvalues (s.e.) Canada 0.0010* 0.2161* 0.9567* 0.0099* 0.9923 5.0970* Canada 0.0007* 0.1896* 0.0495* 0.9796* -0.0116* 0.9968 7.3827* (0.0001) (0.0170) (0.0073) (0.0026) 0.975 (0.3232) (0.0001) (0.0132) (0.0164) (0.0028) (0.0042) 0.9956 (0.5843) 0.0007* 0.1580* 0.9835* 0.975 LL 0.0006* 0.0007* 0.2555* 0.9652* 0.9939 LL (0.0001) (0.0125) (0.0022) 0.9619 17065.9 (0.0002) (0.0001) (0.0169) (0.0044) 0.9939 17026.8 France 0.0036* 0.2692* 0.0424* 0.9218* 0.9886 5.6243* France 0.0015* 0.2108* 0.9696* 0.0064* 0.9962 8.4791* (0.0004) (0.0263) (0.0197) (0.0143) 0.9459 (0.3983) (0.0002) (0.0162) (0.0047) (0.0020) 0.9900 (0.8214) 0.0002* 0.0008* 0.1465* 0.9834* 0.9459 LL 0.0008* 0.2375* 0.9694* 0.9900 LL (0.0001) (0.0001) (0.0122) (0.0023) 0.9221 15137.2 (0.0001) (0.0148) (0.0036) 0.9846 15975.4 Germany 0.0028* 0.2667* 0.0545* 0.9373* 0.9881 5.2190* Germany 0.0009* 0.2233* 0.0187* 0.9726* 0.9964 8.4130* (0.0003) (0.0221) (0.0152) (0.0089) 0.9603 (0.3334) (0.0002) (0.0145) (0.0075) (0.0033) 0.9959 (0.8164) 0.0009* 0.1445* 0.9835* 0.9603 LL 0.0009* 0.2358* 0.9700* 0.9959 LL (0.0001) (0.0109) (0.0022) 0.9496 15196.3 (0.0001) (0.0146) (0.0036) 0.9957 15919.6 Italy 0.0021* 0.2426* 0.9596* 0.9898 5.6196* Italy 0.0017* 0.2335* 0.9597* 0.0044* 0.9964 8.6295* (0.0003) (0.0216) (0.0072) 0.9796 (0.3938) (0.0002) (0.0193) (0.0068) (0.0017) 0.9859 (0.8451) 0.0008* 0.1435* 0.9845* 0.9795 LL 0.0008* 0.2336* 0.9705* 0.9859 LL (0.0001) (0.0121) (0.0022) 0.9795 14977.0 (0.0001) (0.0154) (0.0037) 0.9755 16217.8 Japan 0.0025* 0.3386* 0.9317* 0.9891 5.1422* Japan 0.0021* 0.2112* 0.9682* 0.0060* 0.9936 8.4674* (0.0003) (0.0231) (0.0085) 0.9827 (0.3418) (0.0003) (0.0171) (0.0052) (0.0025) 0.9893 (0.8384) 0.0008* 0.1569* 0.9821* 0.9681 LL 0.0008* -0.0115* 0.2323* 0.9712* 0.9874 LL (0.0001) (0.0128) (0.0025) 0.9681 14885.1 (0.0001) (0.0058) (0.0151) (0.0036) 0.9874 15240.8 UK 0.0028* 0.2601* 0.9355* 0.9851 6.0111* UK 0.0014* 0.2192* -0.0943* 0.9575* 0.0317* 0.9976 9.0840* (0.0004) (0.0270) (0.0130) 0.9583 (0.3949) (0.0002) (0.0168) (0.0190) (0.0063) (0.0070) 0.9781 (0.8752) 0.0003* 0.0009* 0.1631* 0.9791* 0.9583 LL -0.0006* 0.0008* 0.0561* 0.2347* 0.9626* 0.9658 LL (0.0001) (0.0001) (0.0130) (0.0028) 0.9428 15437.0 (0.0002) (0.0002) (0.0145) (0.0160) (0.0043) 0.9513 16511.2 Notes: See Table 2. 24 Table 4: Likelihood Ratio Tests Canada France Germany Italy Japan UK 1st subsample LR-stat. 0.8529 3.1034 3.0371 2.8405 11.2143 2.0793 d.f. 6 6 7 8 8 7 p-value 0.9906 0.7958 0.8815 0.9440 0.1898 0.9553 2nd subsample LR-stat. 4.4864 4.8151 3.1584 2.4012 3.6086 2.6975 d.f. 5 5 5 6 6 2 p-value 0.4817 0.4389 0.3969 0.8794 0.7295 0.2596 Notes: The null hypothesis tested is: Restricted Model preferred to Unrestricted Model. Table 5: Historical Shocks Canada France Germany Italy Japan UK Crash 1987 Z10 -9.74 -3.99 -2.46 -3.63 -7.28 -8.63 Z20 -14.04 -16.85 -16.97 -17.39 -17.32 -15.74 Asian Crisis Z10 -8.83 0.68 -0.41 -0.82 -1.37 0.96 Z20 -4.24 -7.47 -7.07 -6.82 -6.85 -7.05 Twin Towers Z10 1.16 0.68 -0.41 -2.37 -1.97 2.79 Z20 -5.27 -7.47 -7.07 -4.65 -4.88 -5.76 Table 6: Initial State H0 Canada France Germany Italy Japan UK h11,0 0.7390 1.0380 1.0517 0.5925 1.4174 0.5321 h12,0 0.3397 0.4019 0.4491 0.1299 0.2504 0.1942 h22,0 0.5724 0.5587 0.5553 0.5486 0.5272 0.5993 Notes: Figures are expressed in 10-4.