Directory UMM :Data Elmu:jurnal:J-a:Journal of Empirical Finance (New):Vol7.Issue1-2.2000:

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www.elsevier.comrlocatereconbase

Visualizing time-varying correlations across

stock markets

Patrick J.F. Groenen

a,)

_{, Philip Hans Franses}

Data Theory Group, Department of Education, Leiden UniÕersity, P.O. Box 9555,

2300 RB Leiden, Netherlands

Econometric Institute, Erasmus UniÕersity Rotterdam, Rotterdam, Netherlands

Accepted 9 May 2000

Abstract

We propose a graphical method to visualize possible time-varying correlations between stock market returns. The method can be useful for observing stable or emerging clusters of stock markets with similar behavior. The graphs, which originate from applying

multidi-Ž .

mensional scaling techniques MDS , may also guide the construction of multivariate econometric models. We illustrate our method for the returns and absolute returns of 13 important stock markets.q_{2000 Elsevier Science B.V. All rights reserved.}

Keywords: Multidimensional scaling; Stock market returns; Time-varying correlations

1. Introduction and motivation

In this paper we propose a basically graphical descriptive method, which can yield insights into possible similarities across stock markets. The empirical results from our method can be helpful in guiding the design of statistical models, and these can be used to test hypotheses of interest. Additionally, our method can perhaps lead to the postulation of new hypotheses.

There are economic and statistical motivations why one would obtain insights into the correlation structure of international stock markets, while allowing for the possibility that this structure varies over time. An important economic motivation

)_{Corresponding author. Tel.:}

q31-71-527-3826; fax:q31-71-527-3865.

Ž .

E-mail address: [email protected] P.J.F. Groenen .

Ž .

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concerns the potential integration of international stock markets. For example, a relevant question is whether there is a single world factor that governs country-specific stock market behavior. Related to this, another interesting issue concerns the extent to which an emerging market can still be called emerging, where it is assumed that an emerging market displays different dynamic patterns over time

Ž . Ž .

than other markets do. Bekaert and Harvey 1995 and Bekaert et al. 1998 propose several relevant econometric methods for this issue. Our method can be seen as an additional tool to their formal methods, or perhaps as a graphical tool that can precede the analysis. Another economic motivation concerns the possibil-ity of including information on changing correlations to design more optimal portfolios.

The statistical motivations are mainly given by the fact that multivariate models for stock market returns and volatility contain a substantial number of parameters,

Ž .

see Kroner and Ng 1998 for a recent survey. The multivariate models developed

Ž . Ž . Ž .

in Bollerslev 1990 , Engle and Susmel 1993 and Engle et al. 1990 need quite a number of parameter restrictions to enable estimation. Even though Ledoit and

Ž .

Santa-Clara 1998 show that unconstrained multivariate models for volatility can be estimated for large numbers of assets, it may still be useful to visualize correlations in order to suggest some potentially plausible parameter restrictions. Indeed, as the number of variables does not necessarily limit the application of our method, it can, in principle, be used to suggest parsimonious model structures. This may be enhanced by the possible detection of a few underlying factors.

As an example of the relevance of our exercise below, consider the following. A practically tractable multivariate GARCH model imposes constant conditional

Ž .

correlations across volatility, see Bollerslev 1990 . Our empirical results for 13 international stock markets in Section 3 will show that this may not be a plausible assumption, at least not for these series. In fact, we find that there appears to be three clusters, which tend to have constant correlation in the last few years, of our sample only. We also observe that large stock market crashes seem to correspond with changing correlation structures.

The outline of our paper is as follows. In Section 2, we give the basics of the

Ž .

multidimensional scaling MDS technique, which is at the core of our empirical method and discuss the details that are relevant for our specific application. In Section 3, we apply our method for daily data on 13 stock markets, including major American, Asian, and European stock markets. We consider daily returns

Ž .

and volatility measured as absolute returns . In Section 4, we conclude our paper with some final remarks and potential topics for further research.

2. On MDS

MDS is a popular technique in several social sciences as it aims at representing

Ž .

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Ž . Ž .

Kruskal 1964 . For further reference, in our application, we consider a 13=13

Ž .

matrix of estimated correlations measured over a certain sample period . In this representation, points represent the stock markets. A small distance between two points corresponds to a high correlation between two stock markets and a large distance corresponds to low or even negative correlation. A correlation of one should lead to zero distance between the points representing perfectly correlated stock markets. In our application below, we will see that, not unexpectedly, the Dow Jones and S & P500 returns behave as such. MDS tries to estimate the distances for all pairs of stock markets to match the correlations as close as possible. MDS may thus be seen as an exploratory technique without any distributional assumptions on the data. The distances between the points in the MDS maps are generally not difficult to interpret and thus may be used to formulate more specific models or hypotheses. Also, the distance between two points should be interpreted as being the distance conditional on all the other distances.

In most practical applications, the distances are not exactly equal to one minus the relevant correlations, and hence an approximate solution needs to be found. One possibility to obtain such an approximate solution is given by minimizing the Stress function

1yr yd

Ž .

Ž

.

Ý

i j i j

i-j

StresssL X

Ž .

s m ,

Ž .

1yr

Ž

.

Ý

i j

i-j

Ž .

where ri j denotes the correlation between stock markets i and j, di j X denotes the Euclidean distance in a p-dimensional space between rows i and j of the

m=p matrix of coordinates X, andÝm _{denotes the summation over the upper}

i-j

triangular elements of the correlation matrix. The upper triangular or lower triangular part are sufficient because both 1yri j and di j are symmetric, that is,

Ž . Ž .

1yri js1yrji and di j X sd X , and both have diagonal elements equal toji

Ž . Ž .

zero, that is, 1yri is0 and d Xi i s0. Kruskal 1964 calls the numerator of Eq.

Ž .1 raw Stress. Unfortunately, the coordinates that minimize Stress cannot be

found by an analytical method and need to be computed by an iterative algorithm. The obtained representation of points is not unique in the sense that any rotation or translation of the points retains the distances. To overcome this translation freedom, we will impose that the coordinates sum to zero per dimen-sion. We will explicitly use this rotational freedom in our empirical work below. Geometrically, each variable can be thought of as a vector. Because there are only m vectors, they span an m dimensional space, where m is the number of stock markets. The Euclidean distance between the end-points of two such vectors equals two times one minus the correlation. Therefore, one minus the correlation matrix can be interpreted as a distance matrix of points in an my1 dimensional

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Ž .

space see, for example, Borg and Groenen, 1997, pp. 105–106 . In other words, an MDS solution for m vectors in my1 dimensions obviously yields a zero Stress solution.

2.1. Minimizing Stress

The minimum of Stress cannot be determined algebraically and an algorithm is needed to find the minimum. For our purposes, we use the SMACOF algorithm

ŽDe Leeuw, 1977, 1988; De Leeuw and Heiser, 1980 . The acronym SMACOF.

stands for scaling by majorizing a complicated function. The term majorization reflects a minimization method that guarantees a non-increasing series of Stress values. In all practical cases, this feature means that in each iteration, Stress is reduced until convergence is reached. The algorithm stops when the decrease in Stress between two iterations is less than a pre-specified constant, ´. By setting ´s10y6 we obtain, approximately, a three-digit accuracy of the coordinates. In

practical cases, this stopping criterion is sufficient to guarantee a local minimum of Stress. In Appendix A, we present the update formula for the coordinates. More details about SMACOF, its derivation, and a numerical example can be found in

Ž .

Borg and Groenen 1997 . The SMACOF algorithm has appeared in the

PROXS-Ž .

CAL program Busing et al., 1997 in version 10 of SPSS.

Many standard programs for MDS have an additional feature that is quite

Ž .

popular in the social sciences. These programs allow the replacement of 1_yr_{i j}

in Stress by a function of the correlations. This function is estimated simultane-ously with the coordinates. For example, this function can be restricted to be monotonous so that the order of the correlations is retained, but the specific shape of the function is free. This specific form is called ordinal MDS or nonmetric

Ž .

MDS. However, in this paper, we estimate 1yri j directly, as it allows for a more direct interpretation.

2.2. Interpreting the StressÕalue and choice of dimensionality

The next question concerns what values of Stress are acceptable at a local

Ž .

minimum. Note that the denominator of Eq. 1 is constant and thus, does not influence the location of a local minimum. However, with the inclusion of the denominator, it can be proven that at a local minimum, the value of Stress always

w x Ž

lies in the interval 0,1 irrespective of m or the specific values of ri j Borg and

. Ž .

Groenen, 1997, pp. 199–200 . For moderate-sized m say m-20 , Stress values

Ž .

smaller than 0.10 or even 0.05 are quite common. 1yStress is called the Fit and

Ž .

this measure may be interpreted as the proportion of sum of squares of 1_yr_{i j}

accounted for by the MDS model. The maximum dimensionality that can be specified for these data is my1, as Euclidean distances always fit in at most

my1 dimensions.

There are several ways to choose the dimensionality. Because MDS is an exploratory technique, the most important rule usually is whether the solution can

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be interpreted. For example, it makes no sense to require a six-dimensional solution, if only two dimensions can be interpreted. One-, two- and three-dimen-sional solutions can be represented graphically, whereas for higher dimenthree-dimen-sionality, visualization becomes more difficult. Another criterion for assessing the dimen-sionality of the solution is the so-called elbow criterion. First, a scatter plot is made of the Stress values obtained in various dimensions. Often, the dimensional-ity where an elbow occurs, defines the dimensionaldimensional-ity to be chosen. The rationale is that subsequent dimensions after the elbow only fit noise, and unlikely add substantially to the solution. A third method for choosing the dimensionality can

Ž .

rely on a version of the Akaike information criterion AIC , that is defined here as m

1yr yd

Ž .

Ž

.

Ý

i j i j

i-j AICs

Ž

m m

Ž

.

r2 log

.

m m

Ž

.

Ž

my1 p

.

yp p

Ž

.

r2 .

It should be noticed that dimensions in MDS are not nested. Non-nestedness of Stress solutions implies that, for example, the first two dimensions in a three-di-mensional solution can be different from the two dimensions found in a two-di-mensional MDS solution. Therefore, a two-ditwo-di-mensional solution cannot be ob-tained from a three-dimensional solution, and hence, a separate run of SMACOF to minimize Stress in two dimensions is necessary.

Generally, it is advised to use the classical scaling solution Torgerson, 1958;

Gower, 1966 as a start configuration, although a random start configuration could be used as well. For computational details of classical scaling, see Appendix B.

2.3. LocalÕersus global minima

The SMACOF algorithm cannot guarantee that a global minimum is obtained. With a good starting configuration, SMACOF often obtains an acceptable local

Ž .

minimum Groenen and Heiser, 1996 . We shall see below that for our applica-tion, we appear to have good initial estimates available. For MDS problems without proper initial estimates or with many local minima, special strategies are

Ž .

available Groenen and Heiser, 1996; Groenen et al., 1999 , but we will not elaborate on these here.

If only one dimension is specified, any gradient-based algorithm for minimizing Stress, such as SMACOF, fails hopelessly. The reason is that unidimensional scaling can be rewritten as a combinatorial problem, where it is the order along the dimension that determines a local minimum. For this case, we will use a special

Ž .

method based on dynamic programming Hubert and Arabie, 1986 for which it is guaranteed to find a global optimum. Note that dynamic programming for unidimensional scaling is only feasible for small- or moderate-sized MDS

prob-Ž .

lems for example m-22 because of the huge memory requirements that are of the order 2m_.

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2.4. Special cases

To get a feeling for MDS solutions, we discuss some special correlation structures and their MDS solutions. First, consider the case of all stock markets being perfectly correlated. Then one minus the correlation is zero, and a zero Stress MDS solution is obtained when all points fall in the origin. If all correlations are the same but not equal to one, then the two-dimensional MDS

Ž .

solution contains concentric circles of equally spaced points Buja et al., 1994 . For example, if there are only three points, then they lie on an equilateral triangle, four points form a square, five points form either a pentagon or a square with one point in the middle, and so on. However, in practice, one rarely encounters such special situations, and usually the MDS solution can always be interpreted directly in terms of the distances. That is, points at close distance correlate highly and points at large distance correlate less or negatively.

3. Correlation between 13 stock markets

In this section, we apply the MDS method described in Section 2 to correlations of 13 selected stock markets, including two USA markets, seven European markets and four Asian markets.

3.1. Data and research method

Our data consist of 3347 daily returns of 13 stock markets from January 2,

Ž .

1986 to October 29, 1998 to be denoted as r t . The returns are defined as the first differences of the logs of the index values. The stock markets are listed in Table 1. The data are obtained from Datastream, and they measure indexes in local

Table 1

Thirteen stock markets

Stock market Abbreviation Country

1. Brussels brus Belgium

2. Amsterdam cbs The Netherlands

3. Frankfurt dax Germany

4. New York dj USA

5. London ftse UK

6. Hong Kong hs Hong Kong

7. Madrid madrid Spain

8. Milan milan Italy

9. Tokyo nikkei Japan

10. Singapore sing Singapore

11. Standard and Poors sp USA

12. Taiwan taiwan Taiwan

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currencies. For these markets, we consider the correlations between the returns and between the absolute returns, which supposedly measure volatility.

We first compute the correlations among the 13 stock markets and then apply MDS. We shall not use all the 3347 days immediately to compute the correlations, but define periods of days, of, say, 750 days, and compute an MDS solution. In this way, we will create a sequence of graphs of the similarities of volatility across the 13 stock markets. In our implementation of the procedure in a Matlab program,

Ž .

we allow the practitioner to select the size of the window among other things and we present a shortAmovieB of the sequence of MDS plots. In this way, one can observe dynamic patterns over time.

The exact procedure we use is the following. First, we define the length of the window. Next, the correlations between the variables are computed. The resultant

Ž13=13 correlation matrix is then graphically presented by MDS. In a next step,.

we shift the sample period by one or more days. New correlations are computed for this new period and a new graphical representation is obtained by MDS. This representation defines the second frame. These steps are repeated until all data are processed.

The length of the window determines the smoothness of the temporal move-ments of the points, which represent the stock markets. The longer this period, the smaller the changes in the correlation matrix between two shifts, and the smaller are the movements of the points in the MDS representation.

3.2. Rotational inÕariance and Procrustes rotation

A well-known property of Euclidean distances is that they are rotationally invariant, which means that any rotation of the coordinates gives exactly the same distances. Thus, any MDS solution may be freely rotated without affecting Stress. For the procedure outlined above, rotational invariance implies that the points may jump all over the screen between two subsequent frames, eventhough their distances are almost the same. To avoid this, we use the rotational freedom for the solution, Xkq1, of period, Pkq1, so that it matches the solution, X , of thek

previous period, P , as closely as possible. This matching can be done by ak

Ž .

method called Procrustes rotation Cliff, 1966 .

In Procrustes rotation, the objective is to minimize the sum of squared differences between the target solution, X , and the rotated solution, X_k _k_q₁T, that is,

5 52

X_k_yX_k_q₁T ,

5 52

with T the rotation matrix to be estimated, and where A denotes the sum of squared elements of A, that is,Ý a2_{. The T that minimizes the loss equals QP}X

is is

Ž X X _.

where Q and P are orthonormal matrices that is, P PsQ QsI given by the singular value decomposition XXkXkq1sPFQX with F the diagonal matrix with non-negative singular values.

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3.3. Empirical results

For our application we choose a period covering 750 days and we shift the samples by 13 days. These settings lead to 200 frames. We first investigate whether the solutions should be unidimensional or two-dimensional. We also consider higher dimensional representations, but it turns out that the Stress value does not improve dramatically. Let us first look at how well MDS represents the

Ž .

data by considering the fit, which was defined above as 1yStress . Table 2 reports the mean and standard deviation of the fit and the AIC over the 200 periods. Clearly, the two-dimensional solutions are significantly better than the unidimensional solution if we consider the fit or the AIC values.

Individual frames of the unidimensional scaling solutions could be interpreted without much problem, but considering the frames of subsequent periods showed that the positions of the stock markets jump drastically from one end to the other and vice versa, thereby, again indicating that the unidimensional solution is too restrictive. Therefore, we now turn to the two-dimensional solutions.

For illustration, consider the two-dimensional MDS frames for the period December 14, 1987 to October 29, 1990, for the correlations between absolute

Ž .

returns see Fig. 1 . Thus, the graph gives a single frame out of the 200 frames. The solid lines represent the position of the stock markets during the periods including the previous four locations and the dashed lines represent those of the four locations before. TheseAtailsB to each point give an impression of how the MDS solutions behaved in the last 8 windows.

From Fig. 1, several conclusions can be drawn. First, the two American stock markets Standard and Poors and Dow Jones correlate in a similar way with the other stock markets because these two are located almost on top of each other. Secondly, we see that the London FTSE index has moved from the center to the lower left corner, whereas Brussels has moved more upwards, indicating that these stock markets became increasingly dissimilar during that period in terms of their volatility. Another striking feature is the movement of Taiwan, whose volatility has followed much different patterns than the American stock markets, because the

Table 2

Mean m and standard deviation s of fit and AIC for 200 periods for one- and two-dimensional solutions of the return and volatility data

Dim Fit AIC

m s m s

Returns 1 0.845 0.018 y128.37 17.74

2 0.943 0.008 y196.83 18.00

Volatility 1 0.811 0.018 y113.58 16.14

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Fig. 1. Frames of the MDS solution for the absolute returns for January 2, 1986 to November 17, 1988

Župper panel and the period December 14, 1987 to October 29, 1990 lower panel . The solid lines. Ž .

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distance between these points increased substantially. Also, the Hong Kong index moved away from the American stock markets and the Singapore stock market has moved towards Hong Kong and away from the Nikkei index and Taiwan. Apparently, the similarity between Singapore and TaiwanrNikkei has decreased while that of Singapore and Hong Kong increased. In a dynamic representation of the frames, a refined exploration of the relations of the stock markets should be possible.

The frames discussed above are only snapshots of the total number of possible pictures. In Section 3.4, we will present more frames of this AmovieB and show how the stability of these solutions can be assessed.

3.4. Stability of the solutions

To investigate the stability of our representations, we apply the moving blocks

Ž .

bootstrap for time series Efron and Tibshirani, 1993, Section 8.6 . The bootstrap assesses stability by repeatedly sampling, say B times, with replacements from the original data, analyzing each of the B bootstrap samples, and comparing the results. The moving block bootstrap for time series is different from the ordinary bootstrap in the sense that blocks of k subsequent days are sampled from the original data to preserve longitudinal effects. For our data, we set the block length

k to 40, such that 710 different blocks are possible.

The bootstrap results in B configurations, so each stock market can be represented B times. Instead of showing all B points for the 13 stock markets, we represent the cloud of bootstrap points for each stock market by the ellipse

Ž .

covering 90% of the points in the cloud see Meulman and Heiser, 1983 . The center of each ellipse is the centroid of the B bootstrap points. Fig. 2 shows the 90% confidence regions obtained by the bootstrap using Bs100 for the returns for January 2, 1986 to November 17, 1988. In this figure, the individual locations

Ž .

of the bootstrap points for ‘hs’ Hong Kong are also given. It can be seen that exactly 10 points fall outside the ellipse and that a large part of the ellipse in the lower right corner is empty. In this case, the ellipse does not model the cloud of bootstrap points very well. This will be true for other points as well if a high coverage percentage such as 95% or 99% is required. Therefore, we use a reasonable coverage percentage of 90%.

As indicated before, MDS solutions may be freely rotated without affecting the Stress. Furthermore, the sum of variances of the coordinates over the dimensions may be seen as the scale of the MDS map. This scale reflects the fit of the solution

ŽDe Leeuw and Heiser, 1980 and is of less importance for substantive interpreta-.

tion, because distances are interpreted relative to each other. For this reason, the scale of the bootstrap solutions are not of great interest. Therefore, we optimally dilate, rotate, and translate each bootstrap solution optimally to the original

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Fig. 2. Bootstrap stability results of the solution for January 2, 1986 to November 17, 1988 for correlations on returns. The ellipses present 90% confidence regions obtained by the bootstrap. For stock market ‘hs’, all the bootstrap sample points are plotted.

1970 , so that each of the B bootstrap solutions for period P matches the X ask k

closely as possible. Applying these transformations is necessary to avoid trivially large confidence regions due to the rotational invariance of MDS.

Figs. 3–5 show the frames for three characteristic periods, modeling the correlations between returns in the upper panels and correlations between absolute returns in the lower panels. The ellipses give the stability of a stock market by the bootstrap using Bs100. For most graphs, these regions are small enough to suggest a reasonable confidence in the MDS solutions.

There are several empirical conclusions one can draw from the graphs in Figs. 3–5, and we will mention just a few here. First of all, going from the first to the last graph, one can clearly observe that there seem to emerge three clusters, which show similar behavior within those clusters and different behavior across the clusters. These clusters are the US, the European, and the Asian countries. Hence, there does not seem to be a single world market, but perhaps there are three important regional markets. Notice that this holds for the returns and for the absolute returns. This last observation would match with standard financial theory,

Ž . Ž .

which tells us that higher lower volatility corresponds with higher lower returns. Indeed, if this would be the case, one would expect to see similar patterns over time across returns and volatility.

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Fig. 3. Solution for January 2, 1986 to November 17, 1988 for correlations on returns with Stress

Ž . Ž .

0.0567 upper panel and correlations on volatilities with Stress 0.0550 lower panel . The ellipses present 90% confidence regions obtained by the bootstrap.

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Fig. 4. Solution for December 10, 1990 to October 25, 1993 for correlations on returns with Stress

Ž . Ž .

0.0654 upper panel and correlations on volatilities with Stress 0.0739 lower panel . The ellipses present 90% confidence regions obtained by the bootstrap. The lines represent the last eight positions before.

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Fig. 5. Solution for November 15, 1995 to September 30, 1998 for correlations on returns with Stress

Ž . Ž .

0.0384 upper panel and correlations on volatilities with Stress 0.0578 lower panel . The ellipses present 90% confidence regions obtained by the bootstrap. The lines represent the last eight positions before.

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Comparing the returns with the volatility, we see that during the period

Ž .

January, 1986 to November, 1988 Fig. 3 , the returns of Taiwan correlated reasonably high with Madrid, Brussels, and Milan, but this was not so for the

Ž .

volatility. For the period December, 1990 to October, 1993 Fig. 4 , we see that in terms of volatility, the London FTSE index is much closer to the Japan Nikkei index as compared to the returns solution. The three clusters start to emerge more clearly in the returns solution.

Another conclusion concerns the behavior of the Taiwanese stock market index. Figs. 4 and 5 show that Taiwan behaved differently during the second part of the 1980s, that it came closer to Japan in later years, and it became integrated into the Asian cluster in the last years of the sample. Hence, Taiwan seems to have transformed from a type of an emerging market to a more mature market.

A third conclusion is that dramatic events like the October 19, 1987 crash, and the Asian crisis, which may have started with the fall of the Thai baht, July 2,

1997 seem to correspond with some form of clustering behavior. No clear clusters can be observed for samples, which include the 1987 crisis and its aftermath. However, during the beginning of the 1990s, one can observe that clusters of

countries tend to appear. When the Asian crisis gets included in the samples Fig.

5 , one can observe an even larger tendency of clustering. The Asian stock markets behaved similarly during the beginning of the 1990s.

4. Concluding remarks

In this paper, we proposed simple graphical tools to visualize time-varying correlations between stock market behavior. We illustrated our MDS-based method on returns and absolute returns of 13 stock markets. We found that throughout the years, three clusters of similarly behaving stock markets have emerged, and also that Taiwan can hardly be considered as an emerging market anymore.

There are several issues relevant for further research. A first issue concerns applying our method to alternative data sets, with perhaps different sampling frequencies, to see how informative the method can be in other cases. A second issue concerns taking the graphical evidence seriously and incorporating it in an econometric time series model to see if it can improve empirical specification strategies. A last issue concerns the construction of a measure to indicate for how many variables our method is still empirically tractable.

Acknowledgements

We would like to thank three anonymous referees for their useful comments. A previous version of this paper was presented at the Fall Meeting of the Dutch–

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Ž .

Flemish Classification Society VOC , November 13, 1998. We would like to thank the participants for their helpful suggestions.

Appendix A. The update of the SMACOF algorithm

In Section 2.1, an outline of the SMACOF algorithm for MDS was presented. Here, we present the formula to produce the update Xk at iteration k from the previous configuration Xky1, which forms the heart of the SMACOF algorithm.

Let the m=m matrix, Bky1, have elements bi j with

1yr

°

i j

y if i/j and di j

Ž

Xky1

.

di j

Ž

Xky1

.

~

0 if i/j and d

Ž

.

_s0

bi js i j ky1

Ý

bi l if isj.

¢

ls1 , l/i

Then the update formula of SMACOF becomes

X smy1

B X

k ky1 ky1 .

Appendix B. Classical scaling

If no good initial estimate X of the coordinates is available, then the classical0

Ž .

scaling solution Torgerson, 1958; Gower, 1966 may be used. The classical scaling solution is the default initial configuration in many MDS programs. The classical scaling method is based on the observation that the matrix of squared Euclidean distances DŽ2. can be expressed in matrix algebra as

DŽ2.

sh1Xq1hXy2XXX,

Ž

B.1

.

where h is the vector with the diagonal elements of XXX and 1 is a vector of ones.

Ž .

Pre- and post-multiplying both sides of Eq. B.1 by the centering matrix JsIy

my1₁₁X

gives JDŽ2._J

sy2JXXXJ,

Ž

B.2

.

where the terms h1X and 1hX disappear because of the centering matrix J. If we

Ž . Ž2.

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X _Ž _.

sXX . Now, X or a rotation of X can be found by computing the

eigendecom-position

y1r2JDŽ2.

JsQGQX,

Ž

B.3

.

where Q is the matrix of eigenvectors with QXQsQQXsI and G is the diagonal matrix of eigenvalues. Choosing XsQG1r2 _{reconstructs an X which has the}

desired distances.

To compute the classical scaling solution, we replace DŽ2.

by the matrix

Ž₁₁X _.Ž2. _Ž _.2

yR which has elements 1yri j . Then, we compute the

eigendecomposi-Ž . 1r2

tion in Eq. B.3 and compute XsQG . Finally, only the first p columns of X are retained to construct the classical scaling solution in p dimensions.

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Hubert, L.J., Arabie, P., 1986. Unidimensional scaling and combinatorial optimization. In: de Leeuw,

Ž .

J., Heiser, W.J., Meulman, J.J., Critchley, F. Eds. , Multidimensional Data Analysis. DSWO Press, Leiden, pp. 181–196.

Kroner, K.F., Ng, V.K., 1998. Modeling asymmetric comovements of asset returns. Review of Financial Studies 11, 817–844.

Kruskal, J.B., 1964. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27.

(18)

Ledoit, O., Santa-Clara, P., 1998. Estimating large conditional covariance matrices with an application

Ž .

to international stock markets UCLA working paper .

Meulman, J.J., W.J. Heiser, 1983. The display of bootstrap solutions in multidimensional scaling, unpublished manuscript.

Schonemann, P.H., Carroll, J.D., 1970. Fitting one matrix to another under choice of a central dilation¨

and rigid motion. Psychometrika 35, 245–256.

(1)

Fig. 4. Solution for December 10, 1990 to October 25, 1993 for correlations on returns with Stress

Ž . Ž .

(2)

Fig. 5. Solution for November 15, 1995 to September 30, 1998 for correlations on returns with Stress

Ž . Ž .

(3)

Comparing the returns with the volatility, we see that during the period

Ž .

January, 1986 to November, 1988 Fig. 3 , the returns of Taiwan correlated reasonably high with Madrid, Brussels, and Milan, but this was not so for the

Ž .

A third conclusion is that dramatic events like the October 19, 1987 crash, and the Asian crisis, which may have started with the fall of the Thai baht, July 2,

countries tend to appear. When the Asian crisis gets included in the samples Fig.

5 , one can observe an even larger tendency of clustering. The Asian stock markets behaved similarly during the beginning of the 1990s.

4. Concluding remarks

Acknowledgements

We would like to thank three anonymous referees for their useful comments. A previous version of this paper was presented at the Fall Meeting of the Dutch–

(4)

Ž .

Flemish Classification Society VOC , November 13, 1998. We would like to thank the participants for their helpful suggestions.

Appendix A. The update of the SMACOF algorithm

Let the m=m matrix, Bky1, have elements bi j with 1yr

°

i j

y if i/j and di j

Ž

Xky1

.

di j

Ž

Xky1

.

~

0 if i/j and d

Ž

.

_s0

bi js i j ky1

Ý

bi l if isj.

¢

ls1 , l/i

Then the update formula of SMACOF becomes X smy1

B X k ky1 ky1 .

Appendix B. Classical scaling

If no good initial estimate X of the coordinates is available, then the classical0

Ž .

DŽ2.

sh1Xq1hXy2XXX,

Ž

B.1

.

where h is the vector with the diagonal elements of XXX and 1 is a vector of ones.

Ž .

Pre- and post-multiplying both sides of Eq. B.1 by the centering matrix JsIy

my1₁₁X

gives JDŽ2._J

sy2JXXXJ,

Ž

B.2

.

where the terms h1X and 1hX disappear because of the centering matrix J. If we

Ž . Ž2.

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X _Ž _.

sXX . Now, X or a rotation of X can be found by computing the eigendecom-position

y1r2JDŽ2.

JsQGQX,

Ž

B.3

.

where Q is the matrix of eigenvectors with QXQsQQXsI and G is the diagonal matrix of eigenvalues. Choosing XsQG1r2 _{reconstructs an X which has the}

desired distances.

To compute the classical scaling solution, we replace DŽ2.

by the matrix

Ž₁₁X _.Ž2. _Ž _.2

yR which has elements 1yri j . Then, we compute the

eigendecomposi-Ž . 1r2

tion in Eq. B.3 and compute XsQG . Finally, only the first p columns of X are retained to construct the classical scaling solution in p dimensions.

References

Bekaert, G., Harvey, C.R., 1995. Time-varying world market integration. Journal of Finance 50, 403–444.

Bekaert, G., Harvey, C.R., Lumsdaine, R.L., 1998. Dating the integration of world equity markets,

Ž .

unpublished manuscript Stanford University .

Bollerslev, T., 1990. Modeling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH approach. Review of Economics and Statistics 72, 498–505.

Borg, I., Groenen, P.J.F., 1997. Modern Multidimensional Scaling: Theory and Applications. Springer, New York.

Ž .

program for individual differences scaling with constraints. In: Bandilla, W., Faulbaum, F. Eds. , SoftStat’97. Lucius and Lucius, Stuttgart, pp. 67–74.

Cliff, N., 1966. Orthogonal rotation to congruence. Psychometrika 31, 33–42.

De Leeuw, J., Heiser, W.J., 1980. Multidimensional scaling with restrictions on the configuration. In:

Ž .

Krishnaiah, P.R. Ed. , Multivariate Analysis vol. V North-Holland, Amsterdam, pp. 501–522. Engle, R.F., Susmel, R., 1993. Common volatility in international equity markets. Journal of Business

and Economic Statistics 11, 167–176.

Engle, R.F., Ng, V., Rothschild, M., 1990. Asset pricing with a factor ARCH covariance structure: empirical estimates for treasury bills. Journal of Econometrics 45, 213–238.

Gower, J.C., 1966. Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325–338.

Groenen, P.J.F., Heiser, W.J., 1996. The tunneling method for global optimization in multidimensional scaling. Psychometrika 61, 529–550.

Groenen, P.J.F., Heiser, W.J., Meulman, J.J., 1999. Global optimization in least-squares multidimen-sional scaling by distance smoothing. Journal of Classification 16, 225–254.

Hubert, L.J., Arabie, P., 1986. Unidimensional scaling and combinatorial optimization. In: de Leeuw,

Ž .

J., Heiser, W.J., Meulman, J.J., Critchley, F. Eds. , Multidimensional Data Analysis. DSWO Press, Leiden, pp. 181–196.

Kroner, K.F., Ng, V.K., 1998. Modeling asymmetric comovements of asset returns. Review of Financial Studies 11, 817–844.

Kruskal, J.B., 1964. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika 29, 1–27.

(6)

Ledoit, O., Santa-Clara, P., 1998. Estimating large conditional covariance matrices with an application

Ž .

to international stock markets UCLA working paper .

Meulman, J.J., W.J. Heiser, 1983. The display of bootstrap solutions in multidimensional scaling, unpublished manuscript.

Schonemann, P.H., Carroll, J.D., 1970. Fitting one matrix to another under choice of a central dilation¨ and rigid motion. Psychometrika 35, 245–256.

Directory UMM :Data Elmu:jurnal:J-a:Journal of Empirical Finance (New):Vol7.Issue1-2.2000:

Visualizing time-varying correlations across

stock markets

Patrick J.F. Groenen

, Philip Hans Franses

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