The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 311 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 311

ε(t) never decay but keep on accumulating. In a trend stationary pro-

cess, on the other hand, past innovations disappear at every new step.

These considerations carry over immediately in a multidimensional

environment. Multidimensional trend stationary series will exhibit multiple

trends, in principle one for each component. Multidimensional difference-

stationary series will yield a stationary process after differencing.

Let’s now see how these concepts fit into the ARMA framework,

starting with univariate ARMA model. Recall that an ARIMA process is

defined as an ARMA process in which the polynomial B has all roots

outside the unit circle while the polynomial A has one or more roots

equal to 1. In the latter case the process can be written as

A′()∆L dxt = BL()εt

AL()= ( 1 – L)dA′() L

and we say that the process is integrated of order n. If initial conditions

are supplied, the process can be inverted and the difference sequence is

asymptotically stationary.

The notion of integrated processes carries over naturally in the mul-

tivariate case but with a subtle difference. Recall from earlier discussion

in this chapter that an ARIMA model is an ARMA model:

A ()Lxt = B ()Lεεεεt

which satisfies two additional conditions: (1) det[B(z)] has all its roots

strictly outside of the unit circle, and (2) det[A(z)] has all its roots out-

side the unit circle but with at least one root equal to 1.

Now suppose that, after differencing d times, the multivariate series

∆d xt can be represented as follows:

A′()Lxt = B′()Lεεεεt ,1 with A′()L = A ()∆L

d

In this case, if (1) B′()z is of order q and det[B′()z ] has all its roots

strictly outside of the unit circle and (2) A′()z is of order p and

det[A′()z] has all its roots outside the unit circle, then the process is

called ARIMA(p,d,q). Not all ARIMA models can be put in this frame-

work as different components might have a different order of integration.

Note that in an ARIMA(p,d,q) model each component series of the

multivariate model is individually integrated. A multivariate series is

integrated of order d if every component series is integrated of order d.