12-FinEcon-Model Sel Page 329 Wednesday, February 4, 2004 12:59 PM
Financial Econometrics: Model Selection, Estimation, and Testing 329dence though the correlation coefficient is zero. Uncorrelated variables
are not necessarily independent. If the variables X,Y have a nonlinear
dependence relationship, then the correlation coefficient might become
meaningless.^11RANDOM MATRICES
Modeling log prices of equity portfolios as a set of correlated arithmetic
random walks is only a rough approximation in the sense that this
model, when estimated, has poor forecasting ability. A key reason is
that the full variance-covariance matrix is unstable. This fact can be
ascertained in different ways. A simple test is the computation of the
variance-covariance matrix over a moving window. If one performs this
computation on a broad set of equity price processes such as the S&P
500, the result is a matrix that fluctuates in a nearly random way
although the average correlation level is high, in the range of 15 to
17%. Exhibit 12.1 illustrates the amount of fluctuations in a correlation
matrix estimated over a moving window. The plot represents the aver-
age when the sampling window moves.
An evaluation of the random nature of the variance-covariance
matrix was proposed by Laloux, Cizeau, Bouchaud, and Potters^12
using the Random Matrices Theory (RMT). This theory was developed
in the 1950s in the domain of quantum physics.^13 A random matrix is
the variance covariance matrix of a set of independent random walks.
As such, its entries are a set of zero-mean independent and identically
distributed variables. The mean of the random correlation coefficients
is zero as these coefficients have a symmetrical distribution in the range
[–1,+1].
Interesting results can be demonstrated in the case that both the
number of sample points M and the number N of time series tend to
infinity. Suppose that both T and N tend to infinity with a fixed ratioQ = M ⁄ N ≥ 1(^11) See Paul Embrechts, Filip Lindskog, and Alexander McNeil, “Modelling Depen -
dence with Copulas and Applications to Risk Management,” Chapter 8 in S. Rachev
(ed.), Handbook of Heavy Tailed Distributions in Finance (Amsterdam: Elsevier/
North Holland, 2003).
(^12) L. Laloux, P. Cizeau, J.-P. Bouchaud, and M. Potters, “Noise Dressing of Financial
Correlation Matrices,” Physics Review Letter 83 (1999), pp. 1467–1470.
(^13) M.L. Mehta, Random Matrix Theory (New York: Academic Press, 1995).