volatility and cross correlation across stock markets

8/7/2019 Volatility and Cross Correlation Across Stock Markets

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VOLATILITY AND CROSS CORRELATION ACROSS MAJOR STOCK MARKETS

Latha Ramchand

Department of FinanceUniversity of HoustonHouston, TX 77204-6282

Tel: (713) 743-4769

Raul SusmelDepartment of FinanceUniversity of HoustonHouston, TX 77204-6282

Tel: (713)-743-4763

Fax: (713) 743-4789

November, 1997

ABSTRACT

Several papers have documented the fact that correlations across major stock marketsare higher when markets are more volatile - this is done by comparing unconditionalcorrelations over sub- periods or by using conditional correlations that are timevarying. In this paper we examine the relation between correlation and variance in a

conditional time and state varying framework. We use a switching ARCH (SWARCH)technique that does two things. One, it enables us to model variance as state varying.Two, a bivariate SWARCH model allows us to go from conditional variance to statevarying covariances and correlations and hence test for differences in correlationsacross variance regimes. We find that the correlations between the U.S. and otherworld markets are on average 2 to 3.5 times higher when the U.S. market is in a highvariance state as compared to a low variance regime. We also find that, compared to aGARCH framework, the portfolio choices resulting from our SWARCH model lead tohigher Sharpe ratios.

JEL: C53, G15

We would like to thank Dave Blackwell, Jim Hamilton, Andy Thompson and Jia Hefor helpful comments and the support of the Texas Time Series Institute.


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Title: "VOLATILITY AND CROSS CORRELATION ACROSS MAJOR STOCKMARKETS"

ABSTRACT

Several papers have documented the fact that correlations across major stock marketsare higher when markets are more volatile - this is done by comparing unconditionalcorrelations over sub- periods or by using conditional correlations that are timevarying. In this paper we examine the relation between correlation and variance in aconditional time and state varying framework. We use a switching ARCH (SWARCH)technique that does two things. One, it enables us to model variance as state varying.Two, a bivariate SWARCH model allows us to go from conditional variance to state

varying covariances and correlations and hence test for differences in correlationsacross variance regimes. We find that the correlations between the U.S. and otherworld markets are on average 2 to 3.5 times higher when the U.S. market is in a highvariance state as compared to a low variance regime. We also find that, compared to aGARCH framework, the portfolio choices resulting from our SWARCH model lead tohigher Sharpe ratios.


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I. INTRODUCTION

A variety of papers have documented the fact that correlations across major

stock markets change over time. Makridakis and Wheelwright (1974) and Bennett and

Kelleher (1988) find that international correlations are unstable over time. King,

Sentana and Wadhwani (1994) use monthly stock returns and find that covariances

change over time. Kaplanis (1988) compares matrices of monthly returns of ten

markets and rejects the null hypothesis of constant correlations. Using Chow tests for

the years 1972, 1980 and 1987, Koch and Koch (1991) report higher correlations in

more recent years.

While these studies find that correlations have changed over time, there is also

evidence that correlations tend to increase during unstable periods. King and

Wadhwani (1990) and Bertero and Mayer (1990) find that international correlation

tends to increase during periods of market crises. Longin and Solnik (1995) use a

bivariate GARCH model and find that the correlations between the major stock

markets rise in periods of high volatility. To avoid the non-synchroneity problem,

Karolyi and Stulz (1995) construct an index of interlisted Japanese stocks trading as

American Depository Receipts in New York and examine the pattern of covariances

between this index and American returns. They find that covariances are high when

returns on the national indices are high and when "markets move a lot." These papers

suggest that while variances and covariances across markets are changing over time,

the spillover effects are also a function of the magnitude of volatility shocks. In other

words, variances, covariances and correlations could be both time and state varying.

The papers cited above examine changes in correlations by either comparing


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unconditional correlations across various sub periods or by examining conditional

time varying correlations. Our objective here is to test the hypothesis that

correlations across major stock markets depend on the variance regime of returns in a

framework where the variance regime is endogenously determined. In other words,

given the evidence that spillover or contagion effects are stronger during large

market swings, we test to see if market correlations change across variance states. Our

methodology differs from that commonly used in this literature. Previous work in

modelling variance has relied on the ARCH and GARCH family of models as the

technique of choice. As pointed out by Lamoureux and Lastrapes (1990), these models

however, are seriously affected by the presence of structural breaks. For example, for

U.S. weekly stock returns, Hamilton and Susmel (1994) show that ARCH models are

inadequate when the data is characterized not so much by persistent shocks but by

structural breaks leading to switches in variance regimes. Cai (1994) and Hamilton

and Susmel (1994) propose a switching ARCH or SWARCH model that incorporates

the fact that volatility is both time and state variant. This formulation reduces the

impact of persistent shocks to volatility since it allows for changing volatility states.

To begin with, we employ a univariate SWARCH model to determine if

domestic (U.S) market variance is state dependent. Second, in order to capture the

interdependence between variance and correlations, we propose a bivariate SWARCH

model that makes correlations a function of the variance regime. Our work is closely

related to Longin and Solnik (1995) who address the same issue by using a GARCH

(1,1) along with different parameterizations of the correlation coefficient. One such

parameterization that is relevant here is based on a threshold model for correlations


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i.e. correlations are made to depend on the size of past shocks. They find that when

shocks to the U.S. market are larger in absolute value than the unconditional U.S.

standard deviation, the correlation of the US with the German, French and Swiss

market increases. Our analysis differs from theirs in two respects: One, our model

permits us to model correlation as dependent on the variance regime of the US market

where this variance regime is endogenously determined.1 Two, Longin and Solnik

(1995) use monthly while we use weekly data.2

While the SWARCH technique could potentially accommodate many states of

variance, we find that a two state formulation is a parsimonious way to capture the

shifts in variance. In fact a two state formulation is able to capture in a statistical and

economic sense, the changes in variance regimes, while a three state regime is

rejected. Furthermore, we find that the correlation between the U.S. and other equity

markets is 2.5 to 3 times larger when the U.S. market is in a high variance state.

Our results have implications for portfolio diversification strategies. If for

instance, markets are highly correlated during high variance regimes and vice versa,

then the benefits to diversification are reduced. Using the predictions of our bivariate

two state SWARCH for the U.S. and the Europe, Asia and Far East markets, we

construct mean variance optimal portfolios that reflect the time and state varying

nature of the covariance structure. Our portfolios lead to higher Sharpe ratios than

1In Longin and Solnik (1995), the volatility thresholds are determined outside of the model.

2Weekly data is less noisy than daily data; on the other hand monthly data is less informative than

weekly. For these reasons we use weekly as opposed to daily or monthly data.


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those using a bivariate GARCH formulation. In the high variance state since

correlations are on average 3 times higher, optimal portfolios are fully invested in

either the domestic or the foreign market.3 It is in the low variance state when the

correlation across markets is low that optimal portfolios are more diversified.

The rest of this paper is organized as follows: Section II presents the data;

Section III discusses the SWARCH technique and reports results using a univariate

SWARCH model; Section IV generalizes to a bivariate SWARCH model and examines

the evidence on the time and state dependencies of the variance and covariance

structure across different equity markets; Section V uses the results from the bivariate

SWARCH model to construct mean variance efficient portfolios and compares the

SWARCH with the GARCH technique; Section VI concludes.

II. DATA

We use weekly (Thursday to Thursday) stock returns of major equity markets

around the world compiled by Morgan Stanley Capital International Perspective. We

use weekly as opposed to monthly data since we need enough observations to be able

to estimate the different states but without the noise of daily data. The country indices

account for at least 80% of each country's stock market capitalization. The data cover

the period January 1980 through January 1990 and are in terms of dollars, for a total of

522 observations. Table 1 reports univariate statistics for the various indices. The

3French and Poterba (1991) and Tesar and Werner (1992) report that portfolio compositions based on

actual data indicate a strong "home bias".


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coefficients of skewness and kurtosis reveal nonnormality in the data. This result is

confirmed by the Jarque-Bera normality test, for all the series. The Ljung-Box

Q-statistics along with the autocorrelations indicate significant autocorrelations in all

markets except the U.K. The LM-ARCH univariate tests, ARCH(4), reveal ARCH

effects for thirteen of the series.4

III. THE SWITCHING ARCH (SWARCH) MODEL - THE UNIVARIATE CASE

Consider an AR(1)-GARCH(p,q) process for a variable yt:

[--- Unable To Translate Box ---]

Such models have found a wide variety of applications in the finance literature and

their appeal lies in their ability to capture the time varying nature of variance.5

GARCH models, however, do not adequately capture structural shifts in the data that

are caused by low probability events like the Crash of 1987, recessions, changes in the

policy of the Federal Reserve and so on. Diebold (1986) and Lamoureux and Lastrapes

(1990) argue that the usual high persistence found in ARCH models is due to the

presence of structural breaks. Schwert (1990), Nelson (1991) and Engle and Mustafa

(1992) show that standard ARCH models are not flexible to describe events like the

Crash of 1987. Cai (1994), Brunner (1991) and Hamilton and Susmel (1994) modify the

4 We also performed a multivariate ARCH test, including lagged U.S. squared returns and lagged

regional squared returns. With the exception of Denmark, we found evidence for a time-varying

variance in all series.

5See Bollerslev, Chou and Kroner (1992) for an exhaustive review of this literature.


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ARCH specification to account for such structural changes in data and propose a

Switching ARCH (SWARCH) model where the variance of the process is modelled as:


where the subscript st

denotes the state of the economy at time t. The constant _0,st

,

captures the structural shift parameter and the autoregressive coefficients _i,st,st-i

depend on the current and lagged state of the economy. For instance, a shift from a

low to a high variance state, would be captured in a change in the _i's. Hamilton and

Susmel (1994) allow for K states and propose the SWARCH(K,q) model:


where the _'s are scale parameters that capture the change in regime. One of the _'s is

unidentified and, hence, _1

is set equal to 1. Following Hamilton (1989), maximum

likelihood estimation is straightforward.6

Such a model also requires a formulation of the probability law that causes the

economy to switch among regimes. One simple specification is that the state of the

economy is the outcome of a K-state Markov chain that is independent of yt

for all t:

Prob(st

= j|st-1

= i, st-2

= k,..., yt, y

t-1, y

t-2,...) = Prob (s

t= j|s

t-1= i) = p

ij. Under this

specification the transition probabilities, the pij's, are constant. For example, if the

economy was in a high variance state last period (st=2), the probability of changing to

6See Susmel (1994) for a discussion of the SWARCH technique and possible variants.


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the low variance state (st=1) is a fixed constant p

21.

As a byproduct of the maximum likelihood estimation, Hamilton (1989) shows

that we can make inferences about the particular state of the security at any date. The

"filter probabilities," p(st,s

t-1|y

t,y

t-1,..,y

-3), denote the conditional probability that the

state at date t is st, and that at date t-1 was s

t-1. These probabilities are conditional on

the values of y observed through date t. The "smooth probabilities,"

p(st|y

T,y

T-1,...,y

-3), on the other hand are inferences about the state at date t based on

data available through some future date T (end of sample). For a two state

specification, the smooth probabilities at time t are represented by a 2x1 vector

denoting the probability estimates of the two states. That is, the smooth probabilities

represent the ex-post inference made by an econometrician about the state of the

security at time t, based on the entire time series.

To begin with, we model the U.S. market using a SWARCH framework to test

for changes in variance states or regimes. A univariate SWARCH(2,1) model with two

states and one autoregressive coefficient in the variance equation is the preferred

specification to model U.S. (domestic) stock return variance.7 Following Lo and

MacKinlay (1990), we use an AR(1) model for the mean return equation. In particular,

the model used is:


where rt

measures weekly U.S. stock returns. The state variable st

can take on two

7Other variants of the SWARCH model were tried with no qualitative change in results.


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values depending on the variance regime.

Table 2 reports the results of the SWARCH(2,1) model. Testing the hypothesis

that _2=1 would constitute a Wald test of regime switching. In Table 2 the coefficient

estimate of _2

suggests that variance in the high variance regime is more than six

times that in the low variance regime. Furthermore using a standard ratio test, this

difference in variances is significant. Testing the hypothesis of _2=1 using standard

tests, however, is an informal test of the null hypothesis of no-regime switching. This

is because under the null of no-switching, we have unidentified parameters. For

example, in the two-state case, under the null hypothesis of no regime-change, the

parameters describing the high variance state are not identified. For these reasons,

standard likelihood ratios are inappropriate and can only be used as a rough

approximation.

Hence, we use a test procedure proposed by Hansen (1992) to test the null

hypothesis of no-regime switching. Hansen (1992) proposes a simple test based on the

supremum of a reparameterized likelihood ratio statisitic. This test produces an upper

bound for the likelihood ratio test. Using Hansen's (1992) likelihood ratio test we

obtain an upper bound for the p-value of the null hypothesis of no-regime switching

(_2=1) equal to 0.034.

Table 2 also reports the log likelihood function of the SWARCH(2,1) (L2,1),

SWARCH(2,2) (L2,2

) and the SWARCH(3,2) (L3,2

) models. Standard likelihood ratio


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tests cannot reject the SWARCH(2,1).8 As a diagnostic check, we report Ljung-Box

Q-statistics for standardized residuals and for squared standardized residuals. None of

the Q-statistics is significant at the 5% level. Figure 1 plots on the first panel the

weekly returns data for the U.S., on the second panel the smoothed probabilities, Prob

(st=1|y

T,y

T-1,...), that the market was in state 1 at time t, and on the third panel the

smoothed probabilities, Prob (st=2|y

T,y

T-1,...), that the market was in state 2 at time t.

Based on Hamilton's (1989) classification system, wherein an observation

belongs to state i if the smooth probability Prob (st=i|y

T,y

T-1,...) is higher than .5, the

returns data is divided into two regimes, the high variance and the low variance

regimes. Next, using the U.S. market as the home market and based on the state of the

variance of the home market return, the correlation between the home and foreign

markets is calculated in the two states. These results are reported in Table 3. These

results suggest that except in the case of Italy and Sweden, higher correlations

between the U.S. and other major stock markets are associated with periods of high

domestic (U.S.) variance and vice versa. The correlations are from 1.02 to 2.63 times

higher in the high variance state. The last column of Table 3 also provides an

indication of the proportion of times that a positive foreign market return would have

hedged a negative U.S. return when the U.S. is in a high variance regime. For instance,

8Again, we should point out that the usual asymptotic distribution theory does not hold for this case,

because under the null hypothesis of K-1 states, the parameters that describe the Kth state are

unindentified. The improvement in the likelihood value is so small, that we take it as a strong

indication of a two state model.


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when the home (US) market is in the high variance state, a positive return on the U.K.

market would hedge a negative return in the home market about 11% of the time.

The univariate model however does not explicitly model correlations and hence does

not test for statistical significance of the differences in the correlations across variance

regimes. The bivariate formulation in the next section addresses these issues.

IV. A BIVARIATE SWARCH MODEL

Following Longin and Solnik (1995), in the bivariate setting we use two series

at a time, one being the U.S. stock market return and the other a foreign market

return. The foreign markets we choose are the same four OECD markets examined in

Tesar and Werner (1992) viz. Japan, the U.K., Germany and Canada. Even if we assume

that markets in each country are characterized by two variance regimes, we now have

four states to model. For instance, with the U.S. and Japan in a system, we have the

following four primitive states, st*:

st*=1: U.S. - Low variance, Japan - Low variance

st*=2: U.S. - Low variance, Japan - High variance

st*=3: U.S. - High variance, Japan - Low variance

st*=4: U.S. - High variance, Japan - High variance

As it was assumed for the univariate case, the probability law that causes the

economy to switch among states is given by a K*=4 state Markov chain, P*, with a

typical element given by Prob(st* = j|s

t-1* = i) = p

ij*. As discussed in Hamilton and


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Lin (1996) and Susmel (1996), this specification is very general and encompasses

different interactions among the variance states of both countries. In other words, the

transition probabilities, the pij*'s, could be restricted to fit different assumptions about

the underlying variance states. For example, focusing on p24

*, if the variance states of

the U.S and Japan are independent, then, p24

* = p12

US p22

JAP. On the other hand, if the

variance states of Japan and the U.S. are exactly the same (i.e., pij

US = pij

JAP), then, p24

*

= 0.

The system can be written as:


where t

= [rxt,ry

t] is a 2x1 vector of returns, =[ex

t,ey

t] is a 2x1 vector of disturbances,

which follow a bivariate normal distribution, with zero mean and a time varying

conditional covariance matrix Ht (for notational convenience, we drop the dependence

of Ht

on the states of the economy). A = [ax,a

y] and B = [b

x, b

y] are 2x1 vectors. The

variance covariance matrix Ht

tracks the variances of the two series in the following

way:


where x stands for the domestic return and y for the foreign return series. We also

need to keep track of the covariances in the different states. To keep the system

tractable and to facilitate convergence of the maximum likelihood procedure, we

generalize the model proposed by Bollerslev (1990) and assume that correlations in


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states 1 and 2 are equal; similarly, the correlations in states 3 and 4 are the same i.e.

the correlations are a function only of the state of the domestic return. 9. The

covariances are set up such that:


Hence, we have a bivariate SWARCH (2, 1, 2) system, i.e., 2 states for each series, 1

autoregressive coefficient in the variance equation for each series, and 2 covariances.

We can rewrite the above equations as:


This specification allows the series rxt and ryt to be related through the off-diagonal

elements of Ht

and through the nonlinearities associated with dependent states.

Estimation is done using maximum likelihood. The results from this SWARCH

regression are reported in Table 4: Panel A documents the results for the U.S. and

Japan, Panel B for the U.S. and the U.K., Panel C for the U.S. and Germany and Panel D

for the U.S. and Canada. The results in Panel A suggest that one can indeed talk about

two variance regimes. The parameter estimates for _x

and _y

indicate that the variance

in the high variance state is on average 6 times that in the low variance regime for the

U.S. and 3 times for Japan. Importantly, the null that these coefficients equal one is

9Our correlation structure is parameterized such that correlations between the domestic and any foreign

market change only when the domestic market switches to a different volatility regime. Even with this

simple structure we have 24 parameters to estimate. A more general structure would allow changes in

correlation when either market in the system shifts to a different state; this however would increase the

number of parameters to be estimated to 38.


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rejected. The ARCH coefficients in the variance equation are not significant for both

the U.S. and the Japanese markets.

The correlation coefficients in the two regimes are significantly different from

zero; also the correlation between the U.S. and the Japanese market when the U.S.

market is in the high variance regime is about 1.94 times the correlation when the U.S.

market is characterized by low variance. We use a likelihood ratio test, LR-_, to

determine if the correlations differ across the two regimes. To calculate the LR-_, we

also estimate the bivariate SWARCH model with a constant correlation across

regimes. This model with a constant correlation coefficient has a likelihood function

equal to -2325.3. Then, LR-_=2x(2325.3-2324.0)=2.6, which has a p-value of 0.27.

Therefore, we fail to reject the hypothesis of equal correlations at the 5% level.

One reason for this might be the fact that the high variance state is composed of

few observations. Longin and Solnik's (1995) point estimates, however, suggest that

large negative U.S. shocks decrease the correlation coefficient by almost two-thirds

while the large positive U.S. shocks marginally increase the correlation coefficient.

However, this assymetric effect is not statistically significant, which is consistent with

our results. The last row in Table 4 indicates that when the U.S. market is in a high

variance state, a positive Japanese return can counter a domestic negative return

22.22% of the time. The first panel in Figure 2 plots the smoothed probabilities that

the bivariate system was in state 1 at time t and similarly panels 2 through 4 do the

same for the other states. If we integrate the smoothed probabilities on the first (last)

two panels of Figure 2, we obtain the smoothed probability that the U.S. stock market

was in the low (high) variance state at time t. Similarly, if we integrate the smoothed


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probabilities in the first (second) panel and the third (fourth) panel we obtain the

smoothed probability that the Japanese stock market was in the low (high) variance

state at time t. In terms of variance states, the U.S. stock market is a more stable

market than the Japanese stock market. The predominant joint state is the first one,

st*=1, where both stock markets are in the low variance state. The expected duration of

this state is 1/(1-p11

)=30.5 weeks. The expected durations of the other three states,

st*=2, s

t*=3, and s

t*=4, are 8.1, 4.5, and 7.4 weeks, respectively.

The results in Table 4 Panel B for the U.S. and the U.K. suggest that the

correlations differ by a factor of three across the two states. Also, the correlations are

significantly different across the two states ( the LR-_ statistic is 21.6, p-value of less

than .0001).10 The last row of Panel B reinforces these results on the poor hedge that

the U.K. market offers to a U.S. investor, i.e. only 11.11% of the time does the U.K.

market make up for negative domestic returns when the U.S. market is in a high

variance state. Panel C examines US verus German returns. Once again, the _

coefficients for both series are significantly different from one suggesting that

variance is indeed characterized by two states in these markets. The correlations are

also larger, 1.87 times larger in the high compared to the low volatility regime. The

LR-_ has a p-value of 0.06, and hence we are unable to reject the null of equal

10This finding differs from Longin and Solnik (1995) who do not find evidence of an asymmetric effect

of large shocks in the correlation coefficient.


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correlations at the 5% level.11 The results in Table 4 Panel D suggest that the U.S. and

Canada have similar switching parameters. The correlation coefficient is .68 in the

U.S. low variance state and .85 in the U.S. high variance state. The LR-_ has a p-value of

0.005, and, therefore, we are able to reject the null of equal correlations at less than the

1% significance level.1213

Overall these results suggest that modelling variance as both a time varying

and state varying phenomenon contributes both in a statistical as well as economic

sense to our understanding of the underlying return process. Correlations across

major world stock markets are time and state dependent. In all cases examined here,

the correlations between the U.S. and foreign stock markets is 2 to more than 3 times

11Once again the inability of the LR-_ to reject the null hypothesis of constant correlation may be due to

the fewness of observations in the second high variance regime. Longin and Solnik (1995) find similar

point estimates for the correlation coefficient between the U.S. and Germany. They however find a

significant difference between the two estimated correlation coefficients.

12This result differs from Longin and Solnik (1995) who do not find any statistical difference between

the correlation coefficient for these markets under large or small U.S. shocks.

13 Also, a likelihood ratio test rejects the independent volatilty states model with a p-value of .000016.

Therefore, there is strong evidence that the volatility states in the U.S. and in Canada are not

independent, which might not be surprising, given the integration of stock markets of both countries

post-1981, reported by Mittoo (1992).


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higher when the U.S. market is in a high variance regime than otherwise.14 Longin

and Solnik (1995) find smaller differences between the correlation coefficients when

the U.S. experienced large versus small shocks.15

V. A COMPARISON OF SWARCH AND GARCH PORTFOLIO WEIGHTS

FORECASTS

Jorion (1985) points out that the benefits of international diversification arise

more on account of risk reduction than increase in mean returns. The SWARCH

technique by introducing state in addition to time varying variance ought to improve

14 We also estimated correlations using monthly returns, from January 1970 to August 1990 expressed

in local currency. The correlation coefficients for Japan and the U.S. were .26 and 0.57 for the low and

high U.S. variance states respectively. Similarly, the correlation coefficients for the U.K. and the U.S.

were .42 and .72; for Germany and the U.S. .05 and .87, and for Canada and the U.S. .61 and .69 in the

low and high US variance states respectively. With the exception of Germany and the U.S., the monthly

local currency results are in the same range as the weekly U.S. dollar results.

15To see how sensitive our results are to the October 1987 crash, we also ran the regressions dummying

out the week of the October 1987 crash. We find that the second state changes significantly only for the

U.K. In a two state representation, for the U.K. market, the week of the October 1987 crash dominates

the second state, therefore, the second state plays the role of a dummy variable. A dummy variable for

that week corresponds to an "unusual variance state," and therefore allows the U.K. series to display

"low" and "high" variance, like in the other markets. The correlation coefficients using the October 1987

dummy are .30 and .78, for the U.S. low and high volatility states respectively. Also, they are

significantly different from each other.


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the precision of the variance covariance estimates and hence improve the performance

of portfolios formed based on this technique. In this section, we illustrate the use of a

SWARCH model in a dynamic portfolio choice problem and compare the

performance of a SWARCH investor with the performance of a GARCH investor. In

order to construct optimal (in this case minimum variance) portfolio weights, a

multivariate SWARCH specification that models all equity markets simultaneously is

called for. However, in the interests of tractability, we chose to stay with a bivariate

formulation using the U.S. and the Europe, Asia and Far East (EAFE) index. The model

we use to construct minimum variance portfolios is based on the exponential utility

function.16 We assume that investors maximize:


where W0

denotes current wealth, W is expected wealth and _ is the coefficient of

absolute risk aversion. In terms of weights we can rewrite the above as:


where wt

denotes the nx1 vector of weights, _t

is a nx1 vector of expected returns, _tis

the (nxn) variance covariance matrix of returns, i is an nx1 vector of ones. Using the

first order conditions from this maximization we obtain the following optimal

weights:


In our framework, both _t, _

tand hence the weights w

t, are time and state varying. _

t

16French and Poterba (1993) and Tesar and Werner (1994) use the same specification to examine the

evidence on international diversification.


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is a 2x1 vector of predicted means. We use an AR(1) model to predict _t. _

tis the

predicted variance using all available information at time t-1 and is estimated as

follows: First, based on the Markov process of the bivariate SWARCH for the U.S. and

the EAFE, we obtain 16 variance forecasts. Second, we weight each variance forecast

by its respective filter probability to arrive at the variance covariance matrix at time t

for each state.

We also estimate a constant correlation bivariate GARCH(1,1) model, as

proposed by Bollerslev (1990), for the same series. The likelihood ratios for the

SWARCH(2,1,2) and the bivariate GARCH(1,1) are equal to -2172.1 and -2203.8

respectively.17 We also estimated Hansen's (1992) test to formally test the null

hypothesis of no regime-switching for the bivariate system. We reject the null

hypothesis of no-regime switching with a p-value of less than 1%. The estimated

SWARCH correlation coefficients are .37 when the U.S. market is in the low variance

17We also estimated a Threshold-GARCH(1,1) model, similar to the one used by Longin and Solnik

(1995). Under this T-GARCH specification _=_0+_

01S

t-1, where S

t-1is a dummy variable that

takes the value of 1 if the estimated conditional variance of the U.S. market is greater

than its unconditional variance and 0 otherwise. The value of the likelihood function

using this T-GARCH model is -2190.8, which is substantially lower than the SWARCH

likelihood function. The estimated coefficients for _0

and _01

are 0.354 (.048) and 0.210

(.065), respectively -standard errors are in parentheses.


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state and .71, when the U.S. market is in the high variance state.18 The LR-_ has a

p-value of 1%. The estimated GARCH correlation coefficient is .42.

Next, we compare the Sharpe ratios from the SWARCH and GARCH strategies.

Panel A of Table 5 reports the Sharpe ratio using the SWARCH strategy19 - the mean

Sharpe ratio for such a portfolio is 0.1858 with a standard deviation 0.0844. Panel B of

Table 5 does the same using the GARCH results --the Sharpe ratio using the

GARCH(1,1) technique is 0.1789 with a standard deviation of 0.0903.20 We should

point out, however, that given the distribution of Sharpe ratios, both Sharpe ratios are

not statistically different from each other. Figure 3 and Figure 4 plot the optimal

portfolio weights for the U.S. using the SWARCH and GARCH models respectively.

The GARCH Sharpe ratios are more leptokurtic than the SWARCH Sharpe ratios.

When the economy is in the low variance state, SWARCH and GARCH Sharpe ratios

are similar. Also examined were the Sharpe ratios for 1987 using both techniques and

18 We also estimate the SWARCH model with four correlation coefficients, one for each state - the main

results are unchanged. To save space, we do not report the complete SWARCH and GARCH estimation

results.

19In all these calculations of the weights and the Sharpe ratios, _ was set = 3. Note that changing _ does

not affect the optimal weights, ceteris paribus.

20GARCH weights are more volatile than SWARCH weights. Hence, this comparison is favorable to

the GARCH(1,1) model since we are not taking into account the transaction costs involved every time

the portfolio is rebalanced.


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the SWARCH technique results in a ratio of 0.1733 versus 0.1461 using a GARCH

formulation. The reason for this difference is because it is in the high variance regime

that the predictions from the SWARCH model are more valuable than a non-state

varying framework.2122

These results show that at least as far as the U.S. investor is concerned, the

benefits of diversification change depending on the state of the variance structure. For

example, during the U.S. high variance state, the correlation between U.S. and EAFE

returns significantly increases, but since the EAFE variance is below the U.S. variance,

the EAFE weight significantly increases.23 The results presented above could also

potentially explain the "home bias" in portfolios reported in French and Poterba

21To determine the sensitivity of our results to initial conditions, we also compared the Sharpe ratio

using a bivariate SWARCH versus a bivariate GARCH for the U.S. and the EAFE by starting from 1981

rather than 1980 i.e. we omit the first fifty observations from our original sample. The results are very

similar to those in Table 7. The SWARCH model results in a Sharpe ratio of 0.1859 versus 0.1764 for the

GARCH.

22We also examine the sensitivity of our results to the October 1987 crash using a dummy for the week

of the crash. We find that the two state results still holds. i.e. the two-state results are not an artifact of

the crash alone. The Sharpe ratio using the dummy is slightly higher at 0.1962 which is to be expected

since the week of the crash falls in the high variance regime.

23These results support Value-at-Risk models where regulators impose a large multiplicative factor

because they are concerned that periods of high turbulence will be characterized by high correlations

and, hence, diversification will not work well in these states.


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(1991), Adler and Jorion (1994), and Tesar and Werner (1992). For instance, if investors

forecast that during high variance periods the correlations increase, but the foreign

market variance increases more than the domestic variance, the portfolio weights

would show an "ex-post" home bias. For example, in Figure 3, during the U.S. high

variance state an EAFE investor displays a "home bias." The general results point to

the fact that modelling variances and covariances in a time and state varying

framework has implications for portfolio choice decisions.

VI. CONCLUSION

This paper examines the relation between time and state varying variance and

correlations between the U.S. and foreign equity markets. We find that variance is

indeed time and state varying and as a result the covariance structure between

markets is also changing over time. Specifically, the covariance is such that during

periods of high U.S. variance, foreign markets become more highly correlated with

the U.S. market. This has implications for portfolio diversification strategies. While

our model specifications were chosen with tractability in mind, it is possible to

generalize our results to a multivariate SWARCH model that can include all major

equity markets. This we leave for future research.


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Table 1

Univariate statistics on major market returns 1980:1990

MARKET MEAN SD SK _ _ 1 Q(4) ARCH(4) JB

A.- EUROPEANAUSTRIA .24* 2.64 0.53 1.47 .25 39.22* 81.61* 70.2*BELGIUM .23* 2.71 0.08 3.88 .13 21.8* 4.91 319.7*DENMARK .31 2.76 0.23 1.30 .12 7.54 3.33 40.0*FRANCE .25 3.16 -0.98 5.41 .16 18.82* 28.81* 704.6*GERMANY .24 2.81 -0.18 1.89 .14 14.30* 41.22* 77.5*ITALY .34* 3.73 -0.82 5.49 .04 14.48* 28.96* 698.5*NETHERLANDS .27* 2.61 -0.72 4.13 .08 13.31* 2.93* 406.1*

NORWAY .17 3.46 -0.90 4.91 .18 27.11* 57.84* 581.1*SPAIN .21 3.24 -0.76 8.76 .08 4.18 60.11* 1681.3*SWEDEN .44* 2.92 -0.50 2.86 .14 13.03* 53.06* 194.7*SWITZER .27 2.53 -0.90 8.40 .16 16.12* 1.08* 1563.5*UK .25 3.03 -1.07 7.29 .00 6.09 0.33* 1202.3*

B.- FAR EASTHONG KONG .17 4.97 -1.69 12.91 .15 14.57* 6.31 3800.4*JAPAN .46* 2.73 0.05 0.94 .10 8.57 15.94* 18.8*SINGAPORE .20 3.48 -2.32 17.93 .18 19.13* 144.61* 7322.8*AUSTRALIA .16 3.70 -2.07 15.55 .18 22.26* 156.61* 5512.5*

C.- NORTH AMERICACANADA .15 2.64 -0.76 5.53 .16 16.15* 20.83* 697.0*US .22* 2.25 -1.03 8.04 .05 2.21 7.42 1464.8*

D.- WORLDINDEX .28* 1.91 -1.09 8.36 .12 9.40 10.00* 1589.9*EAFE .34* 2.18 -0.55 3.68 .13 12.39* 13.58* 6740.6

Notes:Singapore also includes Malaysia.

* significant at the 5% level.SK: skewness coefficient._: Kurtosis coefficient._1: first order autocorrelation coefficient.

Q(4): Ljung-Box statistic with 4 lags.ARCH(4): ARCH test with 4 lags for the own squared returns.JB : Jarque-Bera (1980) normality test, which follows a chi-squared distribution, with twodegrees of freedom.


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Table 2 - SWARCH (2,1) estimation for U.S. returns


Estimate Std.error

a 0.2504 0.0889*

b 0.0045 0.0480

_0 3.1489 0.3120*

_1 0.0584 0.0878

_2 6.329 1.7339**

L2,1: -1112.61

L2,2: -1112.62

L3,2: -1108.89Q(12): 17.39Q2(12): 9.76

Notes:Lk,q: Log likelihood of SWARCH(K,q).

Q(j): Ljung-Box (1978) statistic with j lags for standardized residuals.Q2(j): Ljung-Box (1978) statistic with j lags for standardized squared residuals.* indicates significance at the 5% level.

** indicates significance at 5% where the null hypothesis is that _2 = 1.


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Table 3

Correlations between the U.S. and other markets in the two variance regimes

High variance Low variance % Hedging

UK 0.7251 0.3124 11.11%

Germany 0.4275 0.2546 22.22%

Japan 0.2400 0.2344 22.22%

Canada 0.7983 0.6386 2.78%

Spain 0.2775 0.1481 19.44%

France 0.5471 0.2079 22.22%

HongKong 0.3208 0.2144 22.22%

Australia 0.2400 0.2344 19.45%

Denmark 0.3103 0.1980 16.67%

Belgium 0.3937 0.2114 22.22%

Italy 0.1160 0.1632 30.55%

Netherlands 0.7382 0.4454 19.44%

Norway 0.4735 0.2873 27.78%

Singapore 0.3762 0.2105 19.44%

Sweden 0.1730 0.2095 13.88%

Switzerland 0.5763 0.3318 13.88%EAFE 0.5098 0.3474 11.11%

World 0.8628 0.7658 0.00%

EAFE is the Europe, Asia and Far East index.


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Table 4 - Panel AU.S. - Japan: Bivariate SWARCH (2,1,2) results


Estimate Std.error

ax 0.2458 0.0864*

ay 0.4140 0.1110*

bx -0.0108 0.0437

by 0.1322 0.0442*

_0x 3.2300 0.2850*

_0y 4.1277 0.5891*

_1x 0.0055 0.0367

_1y 0.0670 0.0699

_x 6.4425 1.6110**

_y 3.2886 0.5730**

_1 0.2364 0.0482*

_2 0.4597 0.1190*

% hedging in state 2 22.22%

L2,1 - US = -1112.6

L2,1 - JAP = -1232.9

L2,1,2- US and JAP = -2324.0QUS(12) = 12.83

Q2US(12) = 9.38

QJAP(12) = 7.31

Q2JAP(12) = 6.91

LR-_ = 2.6 (0.273)Notes: Lk,q: Log likelihood of SWARCH(K,q).Lk,l,q: Log likelihood of bivariate SWARCH(K,l,q).

Qi(j): Ljung-Box (1978) statistic with j lags for standardize residuals for series i.

Q2i(j): Ljung-Box (1978) statistic with j lags for standardize squared residuals for series i.

LR-_ = Likelihood ratio test for the null hypothesis _1=_2 (p-value in parenthesis).


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Table 4 - Panel BU.S. - U.K.: Bivariate SWARCH (2,1,2) results


Estimate Std.error

ax 0.2632 0.0852*

ay 0.2393 0.1210*

bx -0.0482 0.0401

by 0.0200 0.0431

_0x 3.0052 0.2885*

_0y 6.5380 0.5521*

_1x 0.0499 0.0552

_1y 0.0000

_x 7.1465 1.4901**

_y 4.6783 0.9523**

_1 0.3045 0.0450*

_2 0.9095 0.0418*


L2,1 - US = -1112.6

L2,1 - UK = -1280.3

L2,1,2- US and UK = -2342.2QUS(12) = 20.45

Q2US(12) = 7.31

QUK(12) = 12.66

Q2UK(12) = 3.95






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Table 4 - Panel CU.S. - Germany: Bivariate SWARCH (2,1,2) results


Estimate Std.error

ax 0.2302 0.0868*

ay 0.1534 0.1105

bx -0.0292 0.0434

by 0.1235 0.0433*

_0x 3.1673 0.3089*

_0y 4.7724 0.5793*

_1x 0.0262 0.0507

_1y 0.0200 0.0597

_x 6.1759 1.4367**

_y 3.5021 0.6474**

_1 0.2862 0.0472*

_2 0.5356 0.1052*


L2,1 - US = -1112.6

L2,1 - GER = -1249.4

L2,1,2- US and GER = -2330.8QUS(12) = 15.14

Q2US(12) = 6.69

QGER(12) = 8.68

Q2GER(12) = 7.25





Table 4 - Panel D

U.S. - Canada: Bivariate SWARCH (2,1,2) results[--- Unable To Translate Box ---]

Estimate Std.error

ax 0.2423 0.0856*

ay 0.1678 0.0924

bx -0.0200 0.0367


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by 0.1666 0.0346*

_0x 3.2318 0.2860*

_0y 3.3700 0.2926*

_1x

0.0207 0.0434

_1y 0.000

_x 4.6754 0.7671**

_y 4.7300 0.5469**

_1 0.6772 0.0265*

_2 0.8497 0.0329*


L2,1 - US = -1112.6

L2,1 - CAN = -1170.7L2,1,2- US and CAN = -2096.7

QUS(12) = 16.64

Q2US(12) = 7.09

QCAN(12) = 4.53

Q2CAN(12) = 9.31






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Table 5Panel A - Sharpe ratio using a bivariate SWARCH - U.S. and EAFE (1980-1990)

Sharpe Ratio

Mean 0.1858

Standard Deviation 0.0844

Skewness -0.6738

Excess Kurtosis 0.8044

Jarque-Bera test 95.0733

N 518

Panel B - Sharpe ratio using a bivariate GARCH(1,1) - U.S. and EAFE (1980-1990)

Sharpe Ratio

Mean 0.1789

Standard Deviation 0.0903

Skewness 0.6114

Execess Kurtosis 4.0242

Jarque-Bera test 1430.3947

N 518


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References

Adler, M. and P. Jorion (1992), "Foreign Portfolio investment," in The new PalgraveDictionary of Money and Finance, eds. P. Newman, M. Milgate and J. Eatwell.

London: MacMillan.

Bennett, P. and J. Kelleher (1988), "The International Transmission of Stock PriceDisruption in October 1987, Federal Reserve Bank of New York Quarterly Review, 12,17-33.

Bertero, E. and C. Mayer (1989), "Structure and Performance: Global Interdependenceof Stock Markets around the Crash of October 1987," Center for Economic PolicyResearch, discussion paper No. 307.

Bicksler, J. L. (1974), "World, national and industry factors in equity returns," Journal

of Finance, 29, 395-398.

Bollerslev, T. (1990), "Modelling the Coherence in Short-Run Nominal Exchange Rates:A Multivariate Generalized ARCH Approach," Review of Economics and Statistics, 72,498-505.

Bollerslev, T., R.Y. Chou and K. Kroner, (1992), "ARCH Modelling in Finance: AReview of the Theory and Empirical Evidence", Journal of Econometrics, 69, 542-547

Brunner, A.D. (1991), "Testing for Structural Breaks in U.S. Post-war Inflation Data,unpublished manuscript, Board of Governors of the Federal Reserve System,

Washington, D.C.

Cai, Jun (1994), "A Markov Model of Unconditional Variance in ARCH", Journal ofBusiness and Economic Statistics, 12, 309-316.

Davies, R.B. (1977), "Hypothesis Testing When a Nuisance Parameter is Present Onlyunder the Alternative," Biometrika, 64, 247-254,

Diebold, F.X. (1986), "Modelling the Persistence of Conditional Variances: AComment," Econometric Reviews, 5, 51-56.

Engle, R.F. and C. Mustafa (1992), "Implied ARCH Models from Options Prices,"Journal of Econometrics, 52, 289-311.

French, K. and J.M. Poterba (1991), "International Diversification and InternationalEquity Markets," American Economic Review, 81, 222-226.

Hamilton, J.D. (1989), "A New Approach to the Economic Analysis of NonstationaryTime Series and the Business Cycle," Econometrica, 57, 357-384.


33/34

Hamilton, J.D. (1996), "Specification Testing in Markov-Switching Time SeriesModels," Journal of Econometrics, 70, 127-157.

Hamilton, J.D. and G. Lin (1996), "Stock Market Volatility and the Business Cycle,"

Journal of Applied Econometrics, 11, forthcoming.

Hamilton, J.D. and Susmel R. (1994), "Autoregressive Conditional Heteroscedasticityand Changes in Regime", Journal of Econometrics, 64, 307-333.

Hansen, B.E. (1992), "The Likelihood Ratio Test under Non-standard Conditions:Testing the Markov Trend Model of GNP," Journal of Applied Econometrics, 7,S61-S82.

Heston, S.L., and K.G. Rouwenhorst, (1994), "Does industrial structure explain thebenefits of international diversification?", Journal of Financial Economics, 36, 3-28.

Jorion Philippe, (1985), "International Portfolio Diversification with Estimation Risk,"Journal of Business, 3, 259-278.

Karolyi, A. and Stulz R.M.,(1995), "Why do Markets Move Together?", Working Paper,Ohio State University

King, M. and Wadhwani S. (1990),"Transmission of Volatility Between Stock Markets",The Review of Financial Studies, 3, 5-33

King, M., E. Sentana and Wadhwani S. (1994),"Volatility and links between National

Stock Markets", Econometrica, 62, 901-934

Lamoureux, C.G. and W.D. Lastrapes (1990), "Persistence in Variance, StructuralChange and the GARCH Model", Journal of Business and Economic Statistics, 5,121-129.

Lo, Andrew W. and Craig MacKinlay (1990), "Data-Snooping Biases in Tests ofFinancial Asset Pricing Models", Review of Financial Studies, v3(3), 431-468

Longin, F. and B. Solnik (1995), "Is the correlation in international equity returnsconstant: 1960-1990?," Journal of International Money and Finance, 14, 3-23.

Makridakis, S. G. and S. C. Wheelwright (1974), "An analysis of the interrelationshipsamong the major world stock exchanges," Journal of Business, Finance andAccounting, 1, 195-216.

Mittoo, U.R. (1992), "Additional Evidence on Integration in the Canadian StockMarket," Journal of Finance, 47, 2035-2054.


34/34

Nelson, D.B. (1991), "Conditional Heteroskedasticity in Asset Returns: A NewApproach," Econometrica, 59, 347-370.

Roll, R., (1992), "Industrial Structure and the Comparative Behavior of InternationalStock Market Indices", Journal of Finance, 47, 3-42

Swchert, G.W. (1990), "Stock Returns and Real Activiy: A Century of Evidence," Journalof Finance, 45, 1237-1258.

Susmel, R. (1996), "Switching Volatility in International Equity Markets," unpublishedmanuscript, Dept. of Finance, University of Houston.

Tesar, L.L. and I. Werner (1992), "Home Bias and the Globalization of Securities," NBERWorking Paper no. 4218. Cambridge, Mass.: National Bureau of Economic Research.

volatility and cross correlation across stock markets

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