The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 294 Wednesday, February 4, 2004 12:58 PM

294 The Mathematics of Financial Modeling and Investment Management

that the (time-invariant) first two moments can be computed in the fol-

lowing way:

∞

cov xt ∑

i= 0

E[]xt = m

with the convention Hi= 0 if i< 0. Note that the assumption that the

Markov coefficients are an absolutely summable series is essential, oth-

erwise the covariance matrix would not exist. For instance, if the Hi

were identity matrices, the variances of the series would become infinite.

As the second moments are all constants, the series is weakly sta-

tionary. We can write the time-independent autocovariance function of

( xth– ) = HiΩΩΩΩ Hih′ –

the series, which is a n× nmatrix whose entries are a function of the lag

h, as

ΓΓΓΓ (^) x ()h =
∞

∑HiΩΩΩΩ^ H′ih–

i= 0

Under the assumption that the Markov coefficients are an abso-

lutely summable series, we can use the lag-operator Lrepresentation

and write the operator

H ()L =

∞

∑HiL

i

i= 0

so that the Wold representation of a series can be written as

xt = H ()εL + m

The concept of invertibility carries over to the multivariate case. A

multivariate stationary time series is said to be invertible if it can be rep-

resented in autoregressive form. Invertibility means that the white noise

process can be recovered as a function of the series. In order to explain

the notion of invertible processes, it is useful to introduce the generating

function of the operator H, defined as the following matrix power

series: