The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 331 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 331

EXHIBIT 12.2 Theoretical Distribution of the Eigenvalues in a Random Matrix

and Distribution of the Eigenvalues in a Sample of 500 Simulated Independent

Random Walks

If the variance-covariance matrix entries do not have a zero mean,

then the spectrum of the eigenvalues is considerably different. Malev-

ergne and Sornette^14 demonstrate that if the entries of the variance-

covariance matrix are all equal—with the obvious exception of the ele-

ments on the diagonal—then a very large eigenvalue appears while all

the others are equal to a single degenerate eigenvalue. The eigenvector

corresponding to the large eigenvalue has all components proportional

to 1, that is, its components have equal weights.

(^14) Y. Malevergne and D. Sornette, “Collective Origin of the Coexistence of Apparent
RMT Noise and Factors in Large Sample Correlation Matrices,” Cond-Mat 02/
0115, 1, no. 4 (October 2002).