The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 286 Wednesday, February 4, 2004 12:58 PM

286 The Mathematics of Financial Modeling and Investment Management

approximation of equity price processes and presents serious problems

of estimation in the case of a large number of processes.

■ Factor models. Factor models address the problem of estimation in the

case of a large number of processes. In a factor model there are correla-

tions only among factors and between each factor and each time series.

Factors might be exogenous or endogenously modeled.

■ State-space models. State-space models describe factors as autoregres-

sive processes. They work in stationary and nonstationary environ-

ments. In the latter case, state-space models are equivalent to

cointegrated models.

■ Cointegrated models. In a cointegrated model there are portfolios

which are described by autocorrelated, stationary processes. All pro-

cesses are linear combinations of common trends that are represented

by the factors.

The above models are all linear. However, nonlinearities are at work

in financial time series. One way to model nonlinearities is to break down

models into two components, the first being a linear autoregressive model

of the parameters, the second a regressive or autoregressive model of

empirical quantities whose parameters are driven by the first. This is the

case with most of today’s nonlinear models (e.g., ARCH/GARCH mod-

els), Hamilton models, and Markov switching models.

There is a coherent modeling landscape, from correlated random

walks and factor models to the modeling of factors, and, finally, the

modeling of nonlinearities by making the model parameters vary. Before

describing models in detail, however, let’s present some key empirical

facts about financial time series.

STYLIZED FACTS OF FINANCIAL TIME SERIES

Most sciences are stratified in the sense that theories are organized on

different levels. The empirical evidence that supports a theory is gener-

ally formulated in a lower level theory. In physics, for instance, quan-

tum mechanics cannot be formulated as a standalone theory but needs

classical physics to give meaning to measurement. Economics is no

exception. A basic level of knowledge in economics is represented by the

so-called stylized facts. Stylized facts are statistical findings of a general

nature on financial and economic time series; they cannot be considered

raw data insofar as they are formulated as statistical hypotheses. On the

other hand, they are not full-fledged theories.