The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 288 Wednesday, February 4, 2004 12:58 PM

288 The Mathematics of Financial Modeling and Investment Management

ior. If investors are risk averse, as required by the theory of investment

(see Chapter 16) then price processes must exhibit a trade off between

risk and returns. The combination of this insight with the assumption of

exponential trends yields market models with possibly diverging expo-

nential trends for prices and market capitalization.

Again, diverging exponential trends are difficult to justify in the

long run as they would imply that after a while only one entity would

dominate the entire market. Some form of reversion to the mean or

more disruptive phenomena that prevent time series to diverge exponen-

tially must be at work.

In the following sections we will proceed to describe the theory and

the estimation procedures of a number of market models that have been

proposed. After introducing general concepts of the measure of depen-

dence between random variables, we will present the multivariate ran-

dom walk model and will analyze in some detail the correlation

structure of real markets. We will introduce dimensionality reduction

techniques and multifactor models. We will then proceed to introduce

cointegration, autoregressive models, state-space models, ARCH/

GARCH models, Markov switching, and other nonlinear models.

INFINITE MOVING-AVERAGE AND AUTOREGRESSIVE

REPRESENTATION OF TIME SERIES

There are several general representations (or models) of time series. This

section introduces representations based on infinite moving averages or

infinite autoregressions useful from a theoretical point of view. In the

practice of econometrics, however, more parsimonious models such as

the ARMA models (described in the next section) are used. Representa-

tions are different for stationary and nonstationary time series. Let’s

start with univariate stationary time series.

Univariate Stationary Series

The most fundamental model of a univariate stationary time series is the

infinite moving average of a white noise process. In fact, it can be dem-

onstrated that under mild regularity conditions, any univariate station-

ary causal time series admits the following infinite moving average

representation:

∞

xt = ∑hiεti– + m

i = 0