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
. a m = m proximity matrix, such as a correlation matrix, in a graphical way, see
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
m
2
1 yr yd
X
Ž .
Ž .
Ý
i j i j
i-j
Stress sL X s ,
1
Ž . Ž .
m
2
1 yr
Ž .
Ý
i j
i-j
Ž . where r
denotes the correlation between stock markets i and j, d X denotes
i j i j
the Euclidean distance in a p-dimensional space between rows i and j of the m = p matrix of coordinates X, and Ý
m
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 1 yr and d are symmetric, that is,
i j i j
Ž . Ž .
1 yr s1yr and d X sd X , and both have diagonal elements equal to
i j ji
i j ji
Ž . Ž
.
zero, that is, 1 yr s0 and d X s0. Kruskal 1964 calls the numerator of Eq.
i i i i
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 m y1 dimensional
Ž .
space see, for example, Borg and Groenen, 1997, pp. 105–106 . In other words, an MDS solution for m vectors in m y1 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 Ž
. De 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
´ s10
y6
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 1 yr directly, as it allows for a
i j
more direct interpretation. 2.2. Interpreting the Stress Õalue 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 r Borg and
i j
. 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. 1 yStress 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 m y1, as Euclidean distances always fit in at most
m y1 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
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 dimensionality, 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 dimensionality 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
2
1 yr yd
X
Ž .
Ž .
Ý
i j i j
i-j
AIC s m my1 r2 log
Ž .
Ž .
m m y1 r2
Ž .
q2 my1 pyp py1 r2 .
Ž .
Ž .
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-dimensional 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.
Ž 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. Local Õersus 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 2
m
.
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
