The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 308 Wednesday, February 4, 2004 12:58 PM

308 The Mathematics of Financial Modeling and Investment Management

t

xt = ∑Hiεεεεti– + h ()tz

i = 0

It can be demonstrated that this system admits the state-space repre-

sentation:

zt + 1 = Azt + Bεεεεt



xt = Czt + Dεεεεt

if and only if its Hankel matrix is of finite rank. In other words, a time

series which admits an infinite moving-average representation and has a

Hankel matrix of finite rank can be generated by a state-space system

where the inputs are the noise. Conversely, a state-space system with

white-noise as inputs generates a series that can be represented as an

infinite moving-average with a Hankel matrix of finite rank. This con-

clusion is valid for both stationary and nonstationary processes.

Equivalence of State-Space and ARMA Representations

We have seen in the previous section that a time series which admits an

infinite moving-average representation can also be represented as an

ARMA process if and only if its Hankel matrix is of finite rank. There-

fore we can conclude that a time series admits an ARMA representation

if and only if it admits a state-space representation. ARMA and state-

space representations are equivalent.

To see the equivalence between ARMA and state-space models, con-

sider a univariate ARMA(p,q) model

p q

xt = ∑φtxti– + ∑ψjεtj– , ψ 0 = 1

i = 1 j = 0

This ARMA model is equivalent to the following state-space model

xt = Czt

zt = Azt–1 + εt

where

C = [φ 1 ... φp 1 ψ 1 ... ψq]