The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 295 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 295

∞

H ()z = ∑Hizi

i= 0

It can be demonstrated that, if H 0 = I, then H(0) = H 0 and the

power series H(z) is invertible in the sense that it is possible to formally

derive the inverse series,

∞

ΠΠΠΠ()z = ∑ΠΠΠΠizi

i= 0

such that

ΠΠΠΠ()zH ()z = (ΠΠΠΠ×H)()z = I

where the product is intended as a convolution product. If the coeffi-

cients ΠΠΠΠiare absolutely summable, as the process xtis assumed to be

stationary, it can be represented in infinite autoregressive form:

ΠΠΠΠ()L(xt – m)=εt

In this case the process xtis said to be invertible.

From the above, it is clear that the infinite moving average represen-

tation is a more general linear representation of a stationary time than

the infinite autoregressive form. A process that admits both representa-

tions is called invertible.

Nonstationary Series

Let’s now look at nonstationary series. As there is no very general model

of nonstationary time series valid for all nonstationary series, we have

to restrict somehow the family of admissible models. Let’s consider a

family of linear, moving-average, nonstationary models of the following

type:

t

xt = (^) ∑Hiεti– + h ()tz– 1
i= 0
where the Hiare left unrestricted and do not necessarily form an abso-
lutely summable series, h(t) is deterministic, and z–1 is a random vector
called the initial conditions, which is supposed to be uncorrelated with