The Mathematics of Financial Modelingand Investment Management

12-FinEcon-Model Sel Page 333 Wednesday, February 4, 2004 12:59 PM

Financial Econometrics: Model Selection, Estimation, and Testing 333

the eigenvalues of a random matrix with the exception of a few large

eigenvalues.

These considerations lead to adopting models where the correlation

structure is concentrated in a number of factors. A model for asset log

prices which is compatible with the findings on the correlation matrices

is the generic multifactor model that we can write as follows:

x = a + Bf + εεεε

where x is the n-vector of the process to be modeled, f is a k-vector of

common factors with k << n, a is an n-vector of constants, B is an n×k

matrix and ε is an n-vector of random disturbances such that:

E[εεεεf] = 0

E[εεεεεεεε′ f] = Σ

The key advantage of multifactor models, that we discuss in Chap-

ter 18, is that the number of factors is generally much smaller than the

number of variables, thus implementing a substantial dimensionality

reduction. Note that in the above form, a multifactor model is a static

regression model, not a dynamic econometric model; it describes the

static regression relationship of the process variables on factors.

As explained in the previous chapter, state-space models combine a

multifactor regression model with an autoregressive model for the fac-

tors. This combination of autoregressive models for the factors and of

multifactor regressive models for the process variables result in impor-

tant families of dynamic models including models of cointegrating rela-

tionships.

The latter point raises an important issue in modern econometrics.

In principle, the variables x can be any sort of economic or financial

quantities. However, multifactor models were developed and are used

mainly in the context of financial econometrics. In that context, the

variables x generally represent returns. This is by no means the only

possible or useful interpretation of factor models. In fact, cointegration

models are effectively multifactor models whose main variables are log

prices and whose factors are the common trends.

There are therefore two different interpretations for and uses of fac-

tor models in financial econometrics. The most widely used factor mod-

els are models of returns such that factorization implements a

dimensionality reduction. However, more recently factor models—either

as cointegrated models of returns and prices or, equivalently, as state-