The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 300 Wednesday, February 4, 2004 12:58 PM

300 The Mathematics of Financial Modeling and Investment Management

Conversely, if all the roots of the polynomial B(z) are strictly greater

than 1, then the ARMA(p,q) process can be expressed as an autoregres-

sive process:

AL

εt

()

= -------------xt

BL()

Note that in the above discussions every process was centered—that

is, it had zero constant mean. As we were considering stationary pro-

cesses, this condition is not restrictive as the eventual nonzero mean can

be subtracted.

Note also that ARMA stationary processes extend through the

entire time axis. An ARMA process, which begins from some initial con-

ditions at starting time t =0, is not stationary even if its roots are

strictly outside the unit circle. It can be demonstrated, however, that

such a process is asymptotically stationary.

Nonstationary Univariate ARMA Models

So far we have considered only stationary processes. However, ARMA

equations can also represent nonstationary processes if some of the

roots of the polynomial A(z) are equal to 1 in modulus. A process

defined by the equation

AL()xt = BL()εt

is called an Autoregressive Integrated Moving Average (ARIMA) process

if at least one of the roots of the polynomial Ais equal to 1 in modulus.

Suppose that λbe a root with multiplicity d. In this case the ARMA rep-

resentation can be written as

A′()L(I– λL) ()ε

d

xt = BLt

AL()= A′()L(I– λL)d

However this formulation is not satisfactory as the process Ais not

invertible if initial conditions are not provided; it is therefore preferable

to offer a more rigorous definition, which includes initial conditions.

Therefore, we give the following definition of nonstationary integrated

ARMA processes.