Detecting Correlation in Stock Market
J¨org D. Wichard, Christian Merkwirth, Maciej OgorzaÃlek
a,b,a
a
AGH University of Science and Technology
Department of Electrical Engineering
al. Mickiewicza 30
30-059 Krak´ow, Poland
b
Max-Planck-Institut f¨ur Informatik
Stuhlsatzenhausweg 85
66123 Saarbr¨ucken, Germany
Abstract
We present a new method for detecting dependencies in the stock market. In order
to find hidden correlations in the daily returns, we build cross prediction models
and use the normalized modeling error as a generalized correlation measure that
extends the concept of the classical correlation matrix.
Key words: Econophysics, Multivariate analysis, Time series analysis
PACS: 89.65.Gh, 02.50.Sk, 05.45.Tp
1 Introduction
The analysis of the the cross-correlation matrix of the returns plays an impor-
tant role in portfolio theory and financial analysis. We build the time series
of daily returns
R
i
(t) =
Y
i
(t + 1) − Y
i
(t)
Y
i
(t)
,
wherein Y
i
(t) denotes the closing-price of the i-th stock at day t. The cross-
correlation matrix of the returns is defined as
ρ
ij
=
hR
i
R
j
i − hR
i
ihR
j
i
q
hR
2
i
− hR
i
i
2
ihR
2
j
− hR
j
i
2
i
,
where the brackets indicate the time average over all trading days in the
investigated period. The analysis of ρ
ij
leads to some interesting insights in the
market dynamics. Mantegna (see Mantegna (1999)) discovered a hierarchical
Preprint submitted to Elsevier Science 25 April 2004
organization inside a portfolio of stocks by introducing a metric related to
the correlation coefficients. By definition the correlation matrix is symmetric
with respect to i and j and thus cannot be used to distinguish a symmetrical
interaction between different stocks from an asymmetric one. Following our
investigations we see strong indications that this asymmetric interaction exists
in a way that the dynamics of single stocks are leading the dynamics of others
significantly. We indicate this with a cross modeling scheme which is described
in the following section.
2 Mixed State Analysis
The scheme we introduce for market analysis is related to the “mixed state
analysis” of multivariate time series which was developed to detect weak cou-
pling between dynamical systems in the framework of chaotic synchronization
(see Wiesenfeldt et al. (2001)). This approach is based on the reconstruction
of mixed states consisting of delayed samples taken from simultaneously mea-
sured time series of both systems under investigation.
We adopted this idea and changed it for our purpose in a way that a linear
model f(
~
R
i,j
(t)) is constructed that maps the time-lagged returns of the j-th
stock together with the time-lagged returns of the i-th stock
~
R
i,j
(t) = (R
j
(t), R
j
(t − 1), . . . , R
j
(t − τ), R
i
(t − 1), . . . , R
i
(t − τ)) (1)
onto the actual returns of the i-th stock R
i
(t). The model f(·) is a linear
function that is fitted using the standard least squares approach (see for ex-
ample Hastie et al. (2001)) for multiple linear regression models, i.e. it should
minimize the residual sum of squares
P
t
(R
i
(t) − f (
~
R
i,j
(t)))
2
. We would like
to remark that this model f(·) is for sure not able to make predictions of the
returns for the next day, however it is able to find the relationship between
the actual returns R
i
(t) and R
j
(t) with resp ect to the time lagged returns,
that may contain some information about linear trends on short time scales.
If we consider a portfolio of N different stocks, we can define the N ×N-matrix
of the normalized modeling error as
cp(i, j) =
h(R
i
− f(
~
R
i,j
))
2
i
hR
2
i
− hR
i
i
2
i
, (2)
where the brackets denote the time average. The modeling error is normalized
with the variance of the time series R
i
(t) for a simple reason: A value of
cp(i, j) ≥ 1.0 indicates that the mean value hR
i
i is a more appropriate model
than f (·), which means that there is no linear dep endence in the the time series
under investigation. Smaller values of cp(i, j) give an indication that there is
at least a weak linear interrelation between the dynamics of the returns. In
2
general, the matrix cp(i, j) is not symmetric, i.e. cp(i, j) 6= cp(j, i). We define
the matrix of differences δ(i, j) as
δ(i, j) = cp(i, j) − cp(j, i). (3)
The values of δ(i, j) reflect asymmetric dependencies in the market dynamics.
If the returns of i and j are uncorrelated or they interact on the same level,
then we expect δ(i, j) ≈ 0.
For δ(i, j) > 0 we have cp(i, j) > cp(j, i) which means that the returns of
the i-th stock contain more useful information to model the returns of the
j-th stock than the other way around. In the terms of synchronization this
indicates an asymmetrical coupling strength between the two stocks.
3 Numerical Simulations
We investigate 600 trading days of the Dow-Jones Industrial Average (DJIA)
between 2-Oct-2000 and 3-Mar-2003. For all 30 stocks in the DJIA, we build
the time series of daily returns and calculate the cross-correlation matrix ρ(i, j)
(see equation 1). For the mixed state analysis we use a time lag of τ = 3 and
we calculate the matrix of the modeling error
1
as defined in equation 2 and
further the matrix of differences δ(i, j) from equation 3. The results are shown
in Figure 2. The cross-correlation matrix shows some interesting structures,
for example are there obvious clusters, there were described by Mantegna
(1999). A part of this structures can be found in the matrix of the modeling
error cp(i, j). The stocks that behave anti correlated with respect to the index
(the blue stripes in the correlation matrix) occur in cp(i, j) with an modeling
error near one. In the matrix of the error differences δ(i, j) we find the amount
of asymmetry regarding our mixed state analysis that offers a field of further
investigations. The next step will be a detailed analysis of the time dependence
of these asymmetries an the nonlinear dependencies in the stock market.
References
Hastie, T., Tibshirani, R., Friedman, J., 2001. The Elements of Statistical
Learning. Springer Series in Statistics. Springer-Verlag.
Mantegna, R., 1999. Hierarchical structure in financial markets. Eur. Phys. J.
B. 11, 193–197.
Wiesenfeldt, M., Parlitz, U., Lauterborn, W., 2001. Mixed state analysis of
multivariate time series. Int. J. Bifurcation and Chaos 11 (8), 2217–2226.
1
In order to achieve a better graphical resolution in the plots, we set the zero
diagonal elements to one.
3
AA
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SBC
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Student Version of MATLAB
AA
AXP
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HON
HPQ
IBM
INTC
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JPM
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UTX
WMT
XOM
−0.2
−0.15
−0.1
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Fig. 1. The cross-correlation matrix (top), the matrix of the normalized modeling
error cp(i, j) (middle) and the matrix δ(i, j) of the error differences as defined in
equation 3 (bottom) for 600 days of the DJIA (Ticker symbols on the left).
4
