The Mathematics of Financial Modelingand Investment Management

18-MultiFactorModels Page 547 Wednesday, February 4, 2004 1:10 PM

Multifactor Models and Common Trends for Common Stocks 547

studies that appear to demonstrate that equity price processes are not

linear in the sense that their DGP is a nonlinear function. Volatility clus-

tering and structural breaks are the most widely cited nonlinear effects.

This lead to the development of nonlinear models for portfolio manage-

ment; nonlinear dynamics and universal approximation schemes for

DGPs such as neural networks have been widely described. However,

tests for low-dimensional nonlinear dynamics have not given consis-

tently positive results. Despite a period of intense experimentation dur-

ing the 1990s, the techniques of nonlinear dynamics have not been

successful in describing price processes.

Nevertheless, approximation schemes remain a subject of study and

experimentation. Vector support machines based on the Vapnik-Cher-

vonenkis theory of learning (see Chapter 12) are one of the latest addi-

tions to a long series of adaptive methods. By their nature, adaptive

methods produce nonlinear DGPs that change continuously. While gen-

eral conclusions are difficult, many experiments have confirmed that

nonlinear approximation schemes have some predictive power—some

trading strategies based on them have been profitable. However, most

efforts are now confined to proprietary trading systems.

Two classes of nonlinear methods that have received a lot of atten-

tion, at both the theoretical and practical levels, are (1) ARCH-GARCH

methods and (2) Markov switching and multiplicative state-space meth-

ods. Both are based on splitting the model into two parts: one part is a

linear regressive or autoregressive model, the other an autoregressive

model that drives the first.

The ARCH model (described in Chapter 12) was initially proposed

to model the clustering of volatility. Its generalization, the GARCH fam-

ily of models, applies to processes such as financial time series that

exhibit volatility clustering. The GARCH(m,q) model represents the

observed process, for example equity returns, as a sequence of IID vari-

ables multiplied by a coefficient which obeys an ARMA(m,q) model as

follows:

rt = σtεt

m q

σ^22

t = ∑αiσti– + ∑βirtj–

i = 1 j = 1

The GARCH(m,q) model can be further generalized to multivariate

processes by modeling not only the process’s volatility but the entire

variance-covariance matrix. In this form the model is known as multi-