The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 297 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 297

an infinite moving average of a white noise process. If the series can also

be represented in an autoregressive form, then the series is said to be

invertible. Nonstationary series do not have corresponding general rep-

resentations. Linear models are a broad class of nonstationary models

and of asymptotically stationary models that provide the theoretical

base for ARMA and state-space processes that will be discussed in the

following sections.

ARMA REPRESENTATIONS

The infinite moving average or autoregressive representations of the pre-

vious section are useful theoretical tools but they cannot be applied to

estimate processes. One needs a parsimonious representation with a

finite number of coefficients. Autoregressive moving average (ARMA)

models and state-space models provide such representation; though

apparently conceptually different, they are statistically equivalent.

Stationary Univariate ARMA Models

Let’s start with univariate stationary processes. An autoregressive pro-

cess of order p – AR(p) is a process of the form:

xt + a 1 xt– 1 + ...+ aPxtP– = εt

which can be written using the lag operator as

p

AL()xt = ( 1 + a 1 L+ ...+ aPL)xt = xt+ a 1 Lx+ ...+ aPL xtP= εt

p

t –

Not all processes that can be written in autoregressive form are sta-

tionary. In order to study the stationarity of an autoregressive process,

consider the following polynomial:

Az()= 1 + a 1 z+ ...+ aPzp

where zis a complex variable.

The equation

Az()= 1 + a 1 z+ ...+ aPzp= 0

is called the inverse characteristic equation. It can be demonstrated that

if the roots of this equation, that is, its solutions, are all different from 1