The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 305 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 305

Markov coefficients Hisatisfy the following linear difference equation

starting from q:

p

∑ AJHlj– = 0 , l> q

j= 0

Therefore, any ARMA process admits an infinite moving-average

representation whose Markov coefficients satisfy a linear difference

equation starting from a certain point. Conversely, any such linear infi-

nite moving-average representation can be expressed parsimoniously in

terms of an ARMA process.

Hankel Matrices and ARMA Models

For the theoretical analysis of ARMA processes it is also useful to

restate the above conditions in terms of the Hankel infinite matrices.^2 It

can be demonstrated that a process, stationary or not, which admits

either the infinite moving average representation

xt =

∞

∑Hiεεεε^ ti–

i= 0

or a linear moving average model

t

xt = (^) ∑ Hiεεεε (^) ti– + h ()tz
i= 0
also admits an ARMA representation if and only if the Hankel matrix
formed with the sequence of its Markov coefficients has finite rank or,
equivalently, a finite column rank or row rank.
STATE-SPACE REPRESENTATION
There is another representation of time series called state-space models.
As we will see in this section, state-space models are equivalent to ARMA
models. While the latter are typical of econometrics, state-space models
originated in the domain of engineering and system analysis. Consider a
(^2) Hankel matrices are explained in Chapter 5.