The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 337 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 337

It is clear that we can approximately represent each series Xi as a

linear combination of the factors plus a small uncorrelated noise. In fact

we can write

p n p

Xi = ∑αiFi + ∑ αiPi = ∑αiFi + ε

i = 1 i = p + 1 i = 1

where the last term is a noise term. Therefore to implement PCA one

computes the eigenvalues and the eigenvectors of the variance-covari-

ance matrix and chooses the eigenvalues significantly different from

zero. The corresponding eigenvectors are the weights of portfolios that

form the factors. Criteria of choice are somewhat arbitrary.

Note that PCA works either on the variance-covariance matrix or on

the correlation matrix. The technique is the same but results are gener-

ally different. PCA applied to the variance-covariance matrix is sensitive

to the units of measurement, which determine variances and covariances.

This observation does not apply to returns, which are dimensionless

quantities. However, if PCA is applied to prices and not to returns, the

currency in which prices are expressed matters; one obtains different

results in different currencies. In these cases, it might be preferable to

work with the correlation matrix.

We have described PCA in the case of time series, which is the rele-

vant case in econometrics. However PCA is a generalized dimensionality

reduction technique applicable to any set of multidimensional observa-

tions. It admits a simple geometrical interpretation which can be easily

visualized in the three-dimensional case. Suppose a cloud of points in the

three-dimensional Euclidean space is given. PCA finds the planes that cut

the cloud of points in such a way as to obtain the maximum variance.

Suppose that there is a strict factor structure, which means that

returns exactly follow the model

r = a + Bf + εεεε

with

E[εεεεf] = 0

E[εεεεεεεε' f] = Σ

The matrix B can be obtained diagonalizing the variance-covariance

matrix. In general, the structure of factors will not be strict and one will

try to find an approximation by choosing only the largest eigenvalues.