Linear and non-linear transmission of equity return volatility: evidence from the US, Japan and Australia

Economic Modelling 17 Ž2000. 497᎐513

Linear and non-linear transmission of equity return volatility: evidence from the US, Japan and Australia 夽 ´ T. Henry b Chris Brooksa,U , Olan a

ISMA Centre, Department of Economics, The Uni¨ ersity of Reading, Reading RG6 6BA, UK Department of Economics, The Uni¨ ersity of Melbourne, Park¨ ille, Victoria 3052, Australia

b

Accepted 10 November 1999

Abstract This paper models the transmission of shocks between the US, Japanese and Australian equity markets. Tests for the existence of linear and non-linear transmission of volatility across the markets are performed using parametric and non-parametric techniques. In particular the size and sign of return innovations are important factors in determining the degree of spillovers in volatility. It is found that a multivariate asymmetric GARCH formulation can explain almost all of the non-linear causality between markets. These results have important implications for the construction of models and forecasts of international equity returns. 䊚 2000 Elsevier Science B.V. All rights reserved. JEL classifications: G12; G15 Keywords: Equity return volatility; Multivariate asymmetric GARCH; Non-linear causality; Volatility spillovers

1. Introduction A recent study of stock market volatility by Kearns and Pagan Ž1993, p. 174. 夽

This paper was written while the second author was on study leave at the ISMA Centre, University of Reading U Corresponding author. Tel.: q44-0-118-931-6768; fax: q44-0-931-4741. E-mail addresses: [email protected] ŽC. Brooks ., [email protected]. unimelb.edu.au ŽO.T. Henry.. 0264-9993r00r$ - see front matter 䊚 2000 Elsevier Science B.V. All rights reserved. PII: S 0 2 6 4 - 9 9 9 3 Ž 9 9 . 0 0 0 3 5 - 8

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suggests that ‘...there is some predictability from the past history of returns and some weak evidence that volatility is larger in a bear than a bull market’. Further evidence of such asymmetric behaviour reported by Engle and Ng Ž1993. and Glosten et al. Ž1993. inter alia lends support to such a view. The question of transmissions of volatility across markets has been dealt with by Engle et al. Ž1990., Hamao et al. Ž1990. and Karolyi Ž1995. inter alia. Theory suggests that the price of an asset is a function of the risk, or volatility of the asset. Consequently an understanding of how volatility evolves over time, and is transmitted across markets, is central to the decision making process. Moreover, optimal inference about the conditional mean of a variable requires that the conditional second moment be correctly specified. Misspecified models of stock volatility may lead to incorrect, or invalid, conclusions about stock return dynamics. The aim of this paper is to apply parametric and non-parametric techniques to model return and volatility interdependence between the US, Japanese and Australian stock markets and the direction of any such linkages. The non-parametric test for non-linear Granger causality of Hiemstra and Jones Ž1994. is used to detect possible linkages across the three markets. The results of the non-parametric tests are used to condition a parametric model of the conditional distribution of asset returns. Unlike the previous research on spillovers, our approach allows the entire variance᎐covariance structure of the model to respond in an asymmetric fashion. That is, the conditional variance Žand conditional covariance. of returns in market i can be higher when prices in market i Žor market j . are trending downwards. Booth et al. Ž1997. also allow for asymmetry in volatility, however, they assume a constant correlation form for the conditional covariance equation, which is a tenuous assumption under dynamically evolving market conditions. Brailsford Ž1996. splits the papers studying volatility spillovers into two groups. Firstly, those studying returns series and how returns relate across markets, such as Eun and Shim Ž1989., Aggarwal and Park Ž1994. and Craig et al. Ž1995.. The second grouping is formed of papers whose explicit focus is upon volatility, such as King and Wadhwani Ž1990., Karolyi Ž1995. and Koutmos Ž1996.. Lim and McNelis Ž1996. use traditional time series and neural network models to demonstrate the influence of shocks to US and Japanese equity returns on Australian equity returns. However, Lim and McNelis do not allow for asymmetric responses to innovations or more importantly, the problems associated with non-synchronous daily data. Brailsford Ž1996. examines links between the volatility of the US stock market and the volatility of the Australian and New Zealand markets using the univariate GARCH framework of Hamao et al. Ž1990.. However, the majority of recent studies of international prices and volatility focus on the US, the UK and Japan Žsee Hamao et al., 1990; Aggarwal and Park, 1994 inter alia., thus, the present paper also contributes to the literature by broadening the focus of the existing evidence. Karolyi Ž1995. studies the relationship between the US and Canadian equity markets, using the fact that the markets are open contemporaneously to circumvent the problems associated with non-synchronicity of trading and the associated correlation of price innovations. The parametric testing in this paper extends

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the work of Karolyi Ž1995. in two main ways. Firstly, we focus upon the relationship between three markets which do not trade concurrently due to time zone differences. Secondly, we generalise the framework used by Karolyi to allow for potentially asymmetric responses in the conditional variance᎐covariance structure. In addition, the use of non-parametric tests for Granger causality serves as a useful model, diagnostic in determining whether the multivariate, asymmetric GARCH model is sufficient to characterise the structure in the data. This paper is divided into four sections. Section 2 describes the data employed in the study. In Section 3 of the paper, parametric and non-parametric tests for spillovers in returns and volatility are performed. Section 4 provides a brief summary and some concluding comments.

2. Data description Weekly data on the closing values of the Australian All Ordinaries index, PA ,t , the Standard and Poor’s 500 index, PU,t , and the Japanese Nikkei-Dow Index, PJ,t were collected for the period from 01r01r1980 to 22r06r1998.1 The weekly frequency was chosen to avoid the problems associated with returns calculated using daily opening prices as discussed by Hamao et al. Ž1990., or with returns calculated using daily opening and closing prices as documented by Stoll and Whaley Ž1990. and Lin et al. Ž1994.. Moreover, such spillover effects attributable to non-synchronous trading hours are less likely to be manifest in weekly data. Consequently, any evidence of spillovers detected in our data is less likely to be spurious. A potential difficulty with the type of approach which we adopt is in the choice of countries for analysis. In theory, one could include data from the markets of all countries which may comprise investors’ portfolios, and allow for linkages between them all in order to prevent a problem akin to ‘missing variable bias’. However, multivariate GARCH models are prone to the ‘curse of dimensionality’. Including additional variables in the state vector greatly increases the number of parameters to be estimated. Given the non-linear structure of the model and the computationally intensive nature of the estimation we restrict our analysis to the US, Japan, and Australia. Fig. 1 displays the index series Pi ,t and the corresponding returns, R i ,t . The time series plots clearly illustrate the differing conditions across markets over the sample period. The US and Australian markets experienced unprecedented growth in the 1990s. In contrast the Japanese market has stagnated since the crash of 1990. At the time the data were collected ŽJune 1998., the Nikkei-Dow was at approximately 40% of its 1989 peak. The maximum value in the data series is 38 915, but since late 1992 the Nikkei-Dow has traded in the 15᎐20 000 range. 1

Australian variables are distinguished by the use of an A subscript. The corresponding US and Japanese variables carry a subscripted U and J, respectively.

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Fig. 1. The data.

The data were transformed into continuously compounded daily returns on each index calculated as R i ,t s 100 = log Ž PitrPity1 ., for i s A,U,J. Visual inspection of Fig. 1 suggests that the returns data display the volatility clustering phenomenon associated with GARCH processes. Large Žsmall. shocks of either sign tend to follow large Žsmall. shocks. Table 1 displays the summary statistics for the data and the correlation matrix for the returns. It is well known that non-synchronous trading may induce a first

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Table 1 Summary statistics for the dataa Mean RU,t RJ ,t RA ,t

RU,t RJ ,t RA ,t

⌽

0.236 0.085 0.158 Q Ž5. 12.9267 w0.024x 7.745 w0.171x 35.496 w0.000x RU,t 9.0224 w0.029x

Correlation matrix RU,t RU,t 1.000 RU,t 0.248 RU,t 0.302

Var

Skew

Kurt

N Ž2.

5.547 7.630 6.587

y2.473 y0.339 y3.499

36.472 4.611 49.755

w0.000x w0.000x w0.000x

AŽ5. 16.789 w0.005x 107.986 w0.000x 3.771 w0.583x

N. Sign 2.858 w0.004x 1.745 w0.081x y1.262 w0.207x

N. Size y1.891 w0.059x y3.437 w0.001x y1.965 w0.049x

P. Size y2.340 w0.019x y3.155 w0.002x 0.919 w0.358x

RJ ,t 13.996 w0.003x

RA ,t 4.406 w0.220x

RU,t

RU,t

1.000 0.317

1.000

Marginal significance levels displayed as w.x. N Ž2. is the Bera and Jarque Ž1980. test for normality distributed as ␹ 2 Ž2.. Q Ž5. is a Ljung᎐Box test for fifth order serial correlation in R i t . A Ž5. is Engle’s test for fifth order ARCH, distributed as ␹ 2 Ž5.. N.Sign, N.Size and P.Size refer to Negative Sign and Negative and Positive Size bias tests, respectively. ⌽ is the joint test for size and sign bias suggested by Engle and Ng Ž1993., distributed as ␹ 2 Ž3.. a

order moving average error in stock index returns Žsee Scholes and Williams, 1977 and Gallant et al., 1992 inter alia for further details.. The serial correlation in the returns, R i ,t , may indeed be as a result of stale prices, although if US or Japanese returns do indeed cause Australian returns the serial correlation in R A t could be due to variable omission. The approach taken here is to explicitly model the MA error in our tests. The data also fail to satisfy the no-ARCH null hypothesis of a fifth order LM test for ARCH and a Ljung᎐Box test for fifth order serial correlation of the squared return data. The estimated unconditional density functions for R i ,t are skewed to the left, and markedly leptokurtic when compared with the standard normal distribution. This is reinforced by the Bera᎐Jarque tests for normality, which are significant for any standard level of confidence. Fat tails and non-normal distributions are a common feature of financial data. The maximum likelihood methods commonly used to obtain parameter estimates for the GARCH class of models are predicated on the assumption of normality. However, estimators that are more robust to departures from gaussianity have been recently proposed Žsee

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Weiss, 1986, Bollerslev and Wooldridge, 1992, and Lee and Hansen, 1994, inter alia.. Our approach is to use the Bollerslev and Wooldridge Ž1992. estimator of the variance᎐covariance matrix of parameters to allow for the non-normality in the data. Table 1 also presents the test for asymmetry in returns volatility, proposed by Engle and Ng Ž1993.. Define Ni ,ty1 as an indicator dummy that takes the value of 1 if R i ,ty1 - 0 and the value of zero otherwise. Likewise, defining Pi ,ty1 as 1 y Ni ,ty1 , then the Engle and Ng Ž1993. joint test for asymmetry in variance may be based on the regression R 2i ,t s ␾ i ,0 q ␾ i ,1 Ni ,ty1 q ␾ i ,2 Ni ,ty1 R i ,ty1 q ␾ i ,3 Pi ,ty1 R i ,ty1 q ¨ i ,t

Ž1.

where ¨ i ,t is a white noise disturbance term. Significance of the parameter ␾ i,1 indicates the presence of sign bias. That is, positive and negative innovations in R i ,t affect future volatility differently to the prediction of the model. Similarly significance of ␾ i,2 or ␾ i,3 would suggest size bias, where not only the sign, but also the magnitude of innovation in R i ,t is important. A joint test for sign and size bias, based upon the Lagrange Multiplier Principle may be performed as T.R 2 from the estimation of Eq. Ž1.. The results of the test suggest that the conditional volatility of the returns series may be sensitive to both the size and sign of shocks to volatility. Any candidate model may be rejected if positive and negative innovations affect future volatility differently to the prediction of the model.

3. Testing for linear and non-linear causality in returns and variances The aim of this study is to produce a time series model for the returns which captures the stylised features of the data, in particular allowing for linkages and volatility spillovers between markets. To this end, we make extensive use of a non-linear Granger causality Žhereafter NLGC. test due to Hiemstra and Jones Ž1994.. The test, a modification of that proposed by wBaek and Brock Ž1992., hereafter BBx, is essentially a multivariate version of the BDS test Žsee Brock et al., 1996., which should have power against a variety of non-linear data generating processes Žsee the Monte Carlo results of Hiemstra and Jones, 1993.. This is an important generalisation, for there is no reason why the causality should be of the linear type, and Brock Ž1991. suggests that linear Granger causality tests will have low power against many types of non-linear causality. In particular, if the causality is in the conditional variance rather than the conditional mean, a situation which is of relevance here, then linear causality tests will typically fail to pick this up. The Hiemstra and Jones modification of the BB test improves the small sample properties of the test and also allows a removal of the unrealistic assumption of BB that both series to which the test is applied are i.i.d. under the null. Hiemstra and Jones Ž1993. also show that their modified test is robust to the presence of structural breaks in the series. A potentially serious difficulty with an application of semi-non-parametric tests,

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such as the NLGC, is that rejection of the null hypothesis of no non-linear causality unfortunately provides the researcher with little information as to the appropriate functional form for the resulting non-linear model Žsee Brooks, 1999 and Brooks and Heravi, 1999 for descriptions of the issues involved and their likely empirical impact.. This problem is akin to those involving an application of the BDS test in the univariate context, and can to a certain extent be mitigated by employing other, more specific tests, such as the ARCH test used here, in combination with the NLGC test. An extensive derivation of the test is given in Brooks Ž1998., and therefore only a summarised version is given here; readers are referred to this or the original papers for further details. Using a notation which closely follows Hiemstra and Lx Ly Jones Ž1994., and letting, X tm , X tyL x , YtyL y denote a lead vector for X of length m, and lag vectors for X and Y of length Lx and Ly, respectively, i.e. X tm s Ž X t , X tq1 , . . . , X tqmy1 . ,m s 1,2, . . . ;t s 1,2, . . .

Ž2.

Lx Ž . X tyL x s X tyL x , X tyL xq1 , . . . , X ty1 , Lx s 1,2, . . . ;t s Lx q 1, Lx q 2, . . .

Ž3.

Ly Ž . YtyL y s YtyL y ,YtyL yq1 , . . . ,Yty1 , Ly s 1,2, . . . ;t s Ly q 1, Ly q 2, . . .

Ž4.

Then for given values of m, Lx, and Ly all 1 and e ) 0, if Y does not strictly Granger cause X, we may write Lx Lx 5 Ly 5 5 Ly Pr Ž 5 X tm y X sm 5 - e < 5 X tyL x y X syL x - e, YtyL y y YsyL y - e . Lx Lx 5 . s Pr Ž 5 X tm y X sm 5 - e < 5 X tyL x y X syL x - e

Ž5.

where PrŽ.. denotes a probability measure and 5 ⭈ 5 denotes a distance measure Žin this case the supremum norm.. The absence of Granger causality implies that the probability that the lead vectors are within distance e is the same whether we have information about the distances between the Yt lag vectors or not. Letting  x t 4 and  yt 4 denote the actual realisations of the process and I Ž A, B,e . denoting an indicator function which takes the value 1 if the vectors A and B are within a distance e of each other and zero otherwise, then the estimates of the relevant correlation integrals for computing the test statistics can be expressed as C1 Ž m q Lx, Ly,e,n . '

2 n Ž n y 1.

mq L x mqL x . Ý Ý I Ž x tyL x , x syL x ,e ⭈

t-s

Ly Ly I Ž ytyL y , y syL y ,e .

C2 Ž Lx, Ly,e,n . '

2 n Ž n y 1.

Lx Lx Ly Ly . Ý Ý I Ž x tyL x , x syL x ,e ⭈ I Ž ytyL y , y syL y ,e .

t-s

Ž6.

Ž7.

C. Brooks, O.T. Henry r Economic Modelling 17 (2000) 497᎐513

504

C3 Ž m q Lx,e,n . '

C4 Ž Lx,e,n . '

2 n Ž n y 1.

2 n Ž n y 1.

mq L x mqL x . Ý Ý I Ž x tyL x , x syL x ,e

Ž8.

t-s

Lx Lx . Ý Ý I Ž x tyL x , x syL x ,e

Ž9.

t-s

for t, s s max Ž Lx, Ly . q 1,..., T y m q 1, n s T q 1 y m y max Ž Lx, Ly .. Under the null hypothesis that yt does not Granger cause x t , then wHiemstra and Jones Ž1994, appendixx show that the test statistic

'n

ž

C1 Ž m q Lx, Ly,e,n . C2 Ž Lx, Ly,e,n .

y

C3 Ž m q Lx,e,n . C4 Ž Lx,e,n .

/

Ž 10 .

is asymptotically distributed as a normal variate with mean zero and variance that is a complicated function of Ž m, Lx, Ly, e, n.. The causality test is a portmanteau test that will also have power against linear data generating processes. Therefore, the test is run on the residuals of a vector autoregressive moving average ŽVARMA. model comprising the two series, X t , Yt that are under consideration. The lag lengths for these linear filters are determined in each case using the multivariate version of the Hannan and Quinn Ž1979. and Schwarz Ž1978. information criteria. Results derived from an application of the test to the raw returns series are given in Table 2 for values of er␴ equal to one-half, one, and three-halves the standard deviation of the data, as recommended by the authors of the test, and for lags Ž Lx s Ly . of 1 and 4. Other values of the lag parameter from 1 to 8 were also employed, but are not shown due to space constraints, since the results were qualitatively unchanged. The most important feature of these results is the strong evidence of causality running from the US and Japan to Australia, and weak evidence from Japan to the Table 2 Non-linear Granger causality tests from raw data Žpre-filtered through a VAR model. a US

␧r␴ US JAP AUS

JAP

0.5

1.0

1.5

᎐

᎐

᎐

y2.238U y1.669 2.243U 3.643U

y1.583 y1.774 2.083U 3.415U

y0.918 y2.205U 0.596 y0.226

AUS

0.5

1.0

1.5

0.5

1.0

1.5

y0.326 y0.985 ᎐

y1.997U y1.682 ᎐

0.597 y0.045 ᎐

1.516 1.473

1.157 y0.316

y0.142 y1.276 0.666 y0.697 ᎐

y0.396 y1.807 0.325 y0.878 ᎐

y0.597 y0.712 y0.145 y0.522 ᎐

1.451 2.279U

a The columns refer to tests of causality from a country while the rows represent the causality to that country. Cell entries give the value of the test statistic, which is asymptotically distributed as a standard normal under the null. First and second entries in each cell represent results for the number of lags Ž Lx s Ly . equal to 1 and 4, respectively. An asterisk denotes a test statistic that exceeds the normal critical value at the 5% level.

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US, but virtually no evidence of causality from Australia to the other two markets. That Australia cannot influence the US or Japan is hardly surprising given the relative sizes of the equity markets. The purpose of the current paper is to specify a time series model that adequately captures these features of the data. Consider the VARMA Ž m,n. ᎐GARCH Ž p,q . model, R t s ␮ q Ł Ž L . R ty1 q ⌳Ct q ⌰ Ž L . ␧ t

Ž 11.

Ý t < ⍀ ty1 ; N Ž 0, Ht . Where R t is the 3 = 1 vector of returns, ␮ is the 3 = 1 vector of intercept coefficient, ⌸ Ž L. is a vector autoregressive lag polynomial, ⌰ Ž L. represents the vector moving average structure, ⌳ is the 3 = 1 vector of parameters associated with the 1987 crash dummy Ct and ␧ t is the 3 = 1 vector of residuals. The lag orders of the VARMA, m and n, were chosen using the Schwarz Ž1978. and Hannan and Quinn Ž1979. Information Criteria. A VARMA Ž1,1. was deemed optimal. Consider the VARMA Ž1,1. for RU,t , R J,t and R A ,t RU ,t ␥U ␮U R J ,t s ␮ J q ␾ U ␮A R A ,t ␺U

␥J ␾J ␺J

␥A ␾A ␺A

RU ,ty1 ␭U R J ,ty1 q ␭ J R A ,ty1 ␭A

␧ U ,ty1 q ␧ J ,ty1 ␧ A ,ty1

Ct ␪U Ct q ␪ J Ct ␪A

␧ U ,ty1 ␧ J ,ty1 ␧ A ,ty1 Ž 12.

Where ␮ i , ␥ i , ␾ i , ␺ i , ␭ i and ␪ i represent parameters to be estimated and ␧ i ,t represents an innovation for i s U, J, A. Ct is a dummy variable taking the value of 1 in October 1987 and zero otherwise. The test of linear Granger causality from R J,t to RU,t is simply a test of the restriction H0 :␥ J s 0 in Eq. Ž12.. The null hypothesis is therefore, of no causality. Similarly, the test of whether the Standard and Poors index is caused by the All Ordinaries index is a test of the restriction H0 :␥A s 0. Such restrictions are easily tested using Wald statistics. The parameterisation for Ht conditional upon ⍀ ty1 , the information set, allows each element of the conditional variance᎐covariance matrix, Ht to depend on q lags of the squares and cross products of the elements of ␧ t as well as p lags of the element of Ht . Defining h t as ¨ ecŽ Ht . where ¨ ec is the vector operator that stacks the columns of a matrix, the GARCH Ž1,1. ¨ ec model may be written as X

¨ ec Ž Ht . s h t s C0 q A1¨ ec Ž ␧ ty1 ␧ ty1 . q B1 h ty1

Ž 13 .

where there are three variance and three covariance equations in Ht . Restricting the matrices A1 and B1 to be diagonal gives the model proposed by Bollerslev et al. Ž1988. where each element of the conditional variance᎐covariance X matrix h jk ,t depends on past values of itself and past values of ␧ j,t ␧ j,t . Such a

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parameterisation rules out spillovers in variance, which may be a tenuous assumption. The log likelihood for Eq. Ž13. for a sample of T observations is given by T

Ls

Ý Lt

Ž 14 .

ts1

Lt s

n 2

ln Ž 2␲ . y

1 2

ln < Ht < y

1 2

ln ␧ tX Hy1 t ␧t

However, maximisation of Eq. Ž14. is formidable. There are 78 parameters to be estimated in the conditional variance᎐covariance structure of the trivariate GARCH Ž1,1. ¨ ec model. Additionally, it is necessary for Ht to be positive definite for all values of ␧ t in the sample. The difficulty of checking, let alone imposing such a restriction led Engle and Kroner Ž1995. to propose the parameterisation Eq. Ž15. X

X

X

X ) ) ) ) Ht s C0) C0) q A11 ␧ ty1 ␧ ty1 A11 q B11 Hty1 B11

Ž 15.

where ) c11 C0) s 0 0

␤ 11) s ␤ 21) ␤ 31)

) c12 ) c 22 0

) ) c13 ␣ 11 ) ) ) c 23 , A11 s ␣ 21 ) ) c 33 ␣ 31

␤ 12) ) ␤ 22 ) ␤ 32

) ␣ 12 ) ␣ 22 ) ␣ 32

) ␣ 13 ) ) ␣ 23 , B11 ) ␣ 33

␤ 13) ) ␤ 23 ) ␤ 33

Following Glosten et al. Ž1993. we capture asymmetry in the variance᎐covariance structure using a threshold term in the variance. Define the threshold ␰ i ,t s min Ž ␧ i ,t ,0., for i s U, J, A, then the BEKK model in Eq. Ž15. may be extended to allow for asymmetric responses as X

X

X

X

X X ) ) ) ) ) ) Ht s C0) C0) q A11 ␧ ty1 ␧ ty1 A11 q B11 Hty1 B11 q D 11 ␰ ty1 ␰ ty1 D 11

where

␦ 11) ) ) D 11 s ␦ 21 ) ␦ 31

) ␦ 12 ) ␦ 22 ) ␦ 32

) ␦ 13 ) ␦ 23 ) ␦ 33

␰ U ,t and ␰ t s ␰ J ,t ␰ A ,t

Ž 16.

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The asymmetric BEKK 2 parameterisation requires estimation of 33 parameters in the variance᎐covariance structures and guarantees Ht positive definite. Kroner and Ng Ž1995. analyse the asymmetric properties of time-varying covariance matrix models, identifying three possible forms of asymmetric behaviour. Firstly, the covariance matrix displays own variance asymmetry if h i ,t Ž h j,t ., the conditional variance of R i ,t Ž R j,t ., is affected by the sign of the innovation in R i ,t Ž R j,t .. Secondly, cross-variance asymmetry implies that the sign of an innovation to R j,t Ž R i . affects the conditional variance of R i ,t Ž R j,t .. Thirdly, if the covariance of the returns is affected by the sign of the innovation in return for any market, then the model is said to display covariance asymmetry. The symmetric BEKK model Eq. Ž15. is given as a special case of Eq. Ž16. where ␦m ,n s 0, for all m s 1, 2, 3, and n s 1, 2, 3. Table 3 reports parameter estimates for Eq. Ž16. and relevant diagnostic statistics suggesting that the model is a reasonable conditional data characterisation. The Bollerslev and Wooldridge Ž1992. QMLE estimator was used to obtain robust estimates of the parameter variance matrix. The standardised residual series appear free from serial correlation. Similarly, the conditional variance estimates satisfy the moment condition h i ,t s EŽ ␧ i2,t . of the Pagan᎐Sabau statistic. A likelihood ratio test restricting the asymmetric BEKK model to have symmetric variance᎐covariance structure was not satisfied for the data. The estimated degree of persistence for each of the conditional variance series is reasonably high. However, in all cases the slope coefficient obtained from a regression of h m m ,t on h m m ,ty1 was insignificantly different from unity using either standard or Dickey᎐Fuller critical values. We conclude that the estimates are consistent with weak stationarity and that the series are possibly integrated in variance ŽIGARCH.. The Wald tests for Granger causality in the mean also appear to support the hypothesis that events in the US cause events in Australia, with no evidence of feedback. There also is no evidence to suggest that returns to the Japanese market influence, or are influenced by, either the US or Australian markets. The signifi) ) ) cance of the off-diagonal elements in A11 , B11 , and D 11 is suggestive of spillovers ) in variance. In particular the significance of the parameters in the D 11 matrix suggests that the volatility spillovers depend upon not only the size, but also the sign of the innovation in returns. As a further diagnostic of Eq. Ž16., pairwise applications of the NLGC test on the standardised residuals, z i ,t , for i s U, J, A, the squared standardised residuals, 2 z i,t , and the conditional variances, h m m ,t for m s 1, 2, 3, were performed. Use of the NLGC test on data which have been pre-filtered using the multivariate GARCH model enables one to determine whether the posited model is sufficient to describe the relationship between the series. Any significant evidence of NLGC in the raw or linearly filtered data should disappear when applied to the residuals

2

The acroynm BEKK reflects the fact that drafts of Engle and Kroner Ž1995. were written with Yoshi Baba and Dennis Kraft.

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of an appropriately specified GARCH model. Failure to accept the no-causality null hypothesis would constitute statistical evidence that Eq. Ž16. was incorrectly specified. This line of analysis is in a similar vein to the use of the univariate BDS test Žsee Brock et al., 1996. on raw data and on GARCH models Žsee, for example, Brooks, 1996 or Hsieh, 1989 inter alia.. The results are presented in panels A, B, and C, respectively of Table 4. It is evident that, with very few exceptions, there is little evidence of non-linear Granger causality in either the conditional means or the conditional variances of the series. Our findings in the multivariate context are very much in the spirit of these previous papers which have employed the BDS tests as a diagnostic for univariate GARCH models. That is, we find significant evidence of NLGC in the raw data, which are very strongly reduced when an appropriate GARCH model, in this case a multivariate asymmetric GARCH model, is fitted to the data.

Table 3 BEKK model estimates of US and Australian stock returns a Conditional mean equations Rt s ␮ q⌸ Ž L. Rty1 q ⌳Ct q ␧t q ⌰ty1

␮U 0.257 Ž4.120.

RU,ty1 y0.158 Žy3.779.

RJ ,ty1 y0.024 Žy1.299.

RA ,ty1 0.015 Ž0.497.

Ct y15.019 Žy12.302.

␧U,ty1 0.003 Ž0.006.

␮J

RU,ty1 0.018 Ž0.589.

RJ ,ty1 0.152 Ž3.549.

RA ,ty1 y0.004 Žy0.147.

Ct y3.968 Ž3.105.

␧J ,ty2 y0.206 Ž4.612.

0.097 Ž3.517.

RU,ty1 0.088 Ž3.347.

RJ ,ty1 0.006 Ž0.311.

RA ,ty1 0.485 Ž14.084.

Ct y7.341 Ž3.105.

␧A ,ty2 y0.488 Ž17.846.

Residual diagnostics Mean ␧U,t 0.022

Variance 0.995

Skew y0.267

Kurt 1.897

Q Ž5. 5.718 w0.335x 4.791 w0.442x 11.857 w0.037x

PyS 0.556 w0.573x 0.003 w0.997x 0.033 w0.967x

RU,t

RJ ,t

0.131 Ž2.040.

␮J RA ,t

␧J ,t

y0.013

0.989

y0.364

1.826

␧A,t

y0.001

1.010

y0.101

0.884

Tests for linear non-causality in return Variable Causal variables RU,t RU,t RJ ,t RA ,t

0.348 w0.555x 11.022 w0.001x

RJ ,t 1.687 w0.194x

0.097 w0.756x

RA ,t 0.247 w0.619x 0.022 w0.883x

C. Brooks, O.T. Henry r Economic Modelling 17 (2000) 497᎐513

509

Table 3 Ž Continued. Conditional ¨ ariance᎐co¨ ariance structure X X X X Ht s C0 ) C0 ) q A11 ) ␧ty1 ␧ ty1 A11 ) q B11 ) Hty1 B11) 0.097 Ž 1.803 . U C0

ˆ s

0

y0.198

ˆ s

0.838

Ž y5.558 .

Ž 10.491 .

0.106

y0.065 Ž y0.541 .

Ž 0.916 .

0

U B11

y0.415

U A11

ˆ

s

y0.101

0

y0.929

y0.121

y0.687

y0.107

y0.924

y0.043

0.633

0.140

Ž 15.453 .

Ž 3.610 .

y0.372

Ž y24.594 .

Ž y0.566 .

Ž y4.351 .

Ž y2.570 .

Ž y53.733 .

Ž y5.367 .

0.115

0.238

0.043

0.077

Ž 3.175 .

Ž 15.678 .

Ž 0.821 .

Ž 1.131 .

y0.010

y0.081

0.007

Ž y0.379 .

y1.199 Ž y4.389 .

Ž y1.123 .

Ž 0.304 .

y0.070

y0.203

Ž y2.608 .

Persistence comparison hU,t hJ ,t 0.840 0.898 Ž48.021. Ž63.465.

Ž y5.624 .

U D11

ˆ s

Ž 17.468 .

Ž y1.879 .

Ž y8.899 .

y0.300 Ž y3.371 .

0.065

0.455

0.093

Ž 1.714 .

Ž 9.384 .

Ž 2.213 .

y0.141

y0.095

Ž y1.284 .

Ž 1.692 .

0.140 Ž 1.936 .

hA ,t 0.870 Ž54.615.

a Notes: Robust t-ratios displayed as Ž... Marginal significance levels displayed as w.x. Q Ž5. are Ljung᎐Box tests for fifth order serial correlation in z i t . P-S is the Pagan-Sabau Ž1990. moment-based specification test for GARCH models

4. Summary and conclusions This paper sought to examine the relationship between the US, Japanese and Australian stock markets, and in particular to test for evidence that both the returns and the volatility of returns on equities in one market are ‘Granger-caused’ by events in the overseas markets. In order to isolate clearly the direction of causality, a multivariate model of US, Japanese and Australian stock returns and volatility was estimated using the parameterisation suggested by Ng and Kroner Ž1995.. The results demonstrate strong evidence that the return on Australian equities is caused by events in the US equity market. There is little evidence to suggest reverse causality between the two markets, which is unsurprising given the relative sizes of the markets. However, the estimated variance᎐covariance matrix of returns is both time varying and asymmetric. This implies that, not only the magnitude, but also the sign of the innovation in returns determines the spillover. Therefore, the evidence suggests that Australian equity markets will be more volatile when US markets are trending downwards. Gruen and Shuetrim Ž1994. and de Roos and Russell Ž1996. document the transmission of US business cycle fluctuations to Australia. The evidence in the current paper suggests the popularly held view that ‘when the US sneezes Australia catches pneumonia’ is equally valid in terms of the equity markets. Similar results

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510

Table 4 Non-linear Granger causality tests for residuals, squared residuals, and conditional variances from the asymmetric Garch modela Country ␧r␴

US 0.5

JAP 1.0

Panel A: results on residuals US ᎐ ᎐ JAP AUS

0.683 0.082 1.431 y2.361U y1.519 y2.357U y1.288 y2.236U

1.5 ᎐

0.316 y0.982 0.391 ᎐ 0.967 y1.307 y1.218 y0.728 y1.285

Panel B: results on squared residuals US ᎐ ᎐ ᎐ JAP AUS

0.425 1.224 y0.238 2.361U y1.516 y2.361U y1.284 y2.224U

0.5

y0.822 0.345 y1.332 y0.750

0.982 y1.212 0.382 0.621 y1.332 y2.628U

Panel C: results on conditional ¨ ariances US ᎐ ᎐ ᎐ y0.321 y0.987 JAP y1.624 y0.985 y1.323 ᎐ y2.345U y0.547 y0.745 AUS y1.846 y1.587 y0.984 y0.357 y2.007U y1.781 y0.752 y0.675

AUS 1.0

1.5

y1.268 y1.943 ᎐ y1.376 y2.254U

y0.973 2.006U y1.621 0.788 ᎐ y0.842 y0.342 y1.321 ᎐ y0.774

1.810 2.988U y0.099 0.087 y1.921 y0.562 y0.927 y0.127 ᎐ ᎐

y0.368 y0.982 y0.636 y0.927 0.273 0.878

y0.767 y0.687 y0.384 0.572 0.238 y0.056 0.624 ᎐ 1.245

1.642 y0.235 0.858 y0.749

0.261 0.384 ᎐

y0.982 y0.032 y0.081 0.068 ᎐ y0.783 y0.359 y0.862 ᎐ y0.742

y1.486 y0.723

0.5

1.0

᎐

1.5

᎐

y0.648 0.475 y0.279 0.238 y0.157 y0.167 0.048 y0.782 ᎐ ᎐

a The columns refer to tests of causality from a country while the rows represent the causality to that country. Cell entries give the value of the test statistic, which is asymptotically distributed as a standard normal under the null. First and second entries in each cell represent results for the number of lags Ž Lx s Ly . equal to 1 and 4, respectively. An asterisk denotes a test statistic that exceeds the normal critical value at the 5% level.

concerning the dominance of US financial markets over those of the rest of the world, have been observed in a different context by Clare and Thomas Ž1992.. They found that the US term structure has more predictive power for UK bond returns than the UK term structure. It is difficult to explain the asymmetric spillover of returns across markets. Ross Ž1989. uses a no arbitrage economy to demonstrate that fluctuations in price changes are related to the flow of new information arriving to the market. Engle et al. Ž1990. argue that changes in variance reflect delays due to information processing and co-ordination of policy. Cheung and Ng Ž1996. argue that this relation between information arrival and volatility provides insights into the patterns of causality between two economic time series. Such insights may be useful in the construction of more adequate models of the underlying data generating process.

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Additionally, such models of the mechanism through which price movements are transmitted around the world may be used to derive implications for the pricing of securities, for hedging and trading policies, and to direct regulators of financial markets. The results are, to an extent, in contrast with the existing literature, which in general studies the US, Japanese and UK equity markets. Lin et al. Ž1994. suggest that there is a reciprocal relationship between the mean and variance of the US and Japanese markets. Here there is little evidence of such a reciprocal relationship. There is evidence to support the hypothesis that shocks to the US markets spillover to the Australian markets. However, this evidence is more consistent with one-way causality in mean return and variance, with events in the US markets influencing events in the Australian market. There is no significant evidence of a leadrlag link between the US and Japanese markets. These results are not necessarily inconsistent with the weak form of the efficient markets hypothesis. Evidence of a statistically significant leadrlag relationship does not directly imply excess Žrisk- and transactions cost-adjusted. returns will exist. The long-run availability of such excess returns is the condition which would have to be fulfilled for a violation of the efficient markets hypothesis.

Acknowledgements The authors would like to thank without implication an anonymous referee of this journal, Simon Burke and Salih Neftci for helpful comments. The software used to compute the non-linear Granger causality test was generously provided by Craig Hiemstra. The usual disclaimer applies to any remaining errors or omissions.

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