The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 306 Wednesday, February 4, 2004 12:58 PM

306 The Mathematics of Financial Modeling and Investment Management

system defined for t ≥ 0 and described by the following set of linear differ-

ence equations:

zt + 1 = Azt + But



xt = Czt + Dut + Est

where

xt =an n-dimensional vector

zt =a k-dimensional vector

ut =an m-dimensional vector

st =a k-dimensional vector

A =a k×k matrix

B =a k×m matrix

C =an n×k matrix

D =an n×m matrix

E =an n×k matrix

In the language of system theory, the variables ut are called the

inputs of the system, the variables zt are called the state variables of the

system, and the variables xt are called the observations or outputs of the

system, and st are deterministic terms that describe the deterministic

components if they exist.

The system is formed by two equations. The first equation is a

purely autoregressive AR(1) process that describes the dynamics of the

state variables. The second equation is a static regression of the observa-

tions over the state variables, with inputs as innovations. Note that in

this state-space representation the inputs ut are the same in both equa-

tions. It is possible to reformulate state space models with different,

independent inputs for the states, and the observables. The two repre-

sentations are equivalent.

The fact that the first equation is a first order equation is not restric-

tive as any AR(p) system can be transformed into a first-order AR(1)

system by adding variables. The new variables are defined as the lagged

values of the old variables. This can be illustrated in the case of a single

second-order autoregressive equation:

Xt + 1 = α 0 Xt +α 1 Xt – 1 + εt + 1

Define Yt = Xt – 1. The previous equation is then equivalent to the first-

order system: