The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 293 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 293

∞

Π()L = (^) ∑λiLi , Π()LHL()= I
i = 1
If the coefficients λi are absolutely summable, we can write
∞
εt = Π()Lxt = (^) ∑λiLixti–
i = 1
and the series is said to be invertible.
Multivariate Stationary Series
The concepts of infinite moving-average representation and of invert-
ibility defined above for univariate series carry over immediately to the
multivariate case. In fact, it can be demonstrated that under mild regu-
larity conditions, any multivariate stationary causal time series admits
the following infinite moving-average representation:
∞

xt = ∑Hiεti– + m

i = 0

where the Hi are n×n matrices, εt is a n-dimensional, zero-mean, white

noise process with nonsingular variance-covariance matrix Ω,and m is an

n-vector of constants. The coefficients Hi are called Markov coefficients.

This moving-average representation is called the Wold representation.

Wold representation states that any series where only the past influences

the present can be represented as an infinite moving average of white noise

terms. Note that, as in the univariate case, the infinite moving-average rep-

resentation can be written in more general terms as a sum which extends

from –∞to +∞. However a series of this type is not suitable for financial

modeling as it is not causal (that is, the future influences the present).

Therefore we consider only moving averages that extend to past terms.

Suppose that the Markov coefficients are an absolutely summable

series:

∞

Hi +∞ <

i = 0

∑

where H^2 indicates the largest eigenvalue of the matrix HH′. Under

this assumption, it can be demonstrated that the series is stationary and