The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 299 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 299

+∞ +∞

ε –

t =^ B

(^1) ()ε=

L t ∑ πiεti– , with ∑ πi +∞ <

i – = ∞ i= 0

In addition, if the roots of B(z) are strictly greater than 1 in modulus,

then the autoregressive representation only involves past values of the

process:

+∞ +∞

ε –

t =^ B

(^1) ()ε=

L t ∑πiεti– , with ∑ πi +∞ <

i= 0 i= 0

As in the previous case, if one considers the characteristic equation,

Bz()= zq+ b 1 zq–^1 + ...+ bq = 0

then the MA(q) process admits a causal autoregressive representation if

the roots of the characteristic equation are strictly smaller than 1 in

modulus.

Let’s now consider, more in general, an ARMA process of order p,q.

We say that a stationary process admits a minimal ARMA(p,q) repre-

sentation if it can be written as

xt+ a 1 xt– 1 + apxtp– = b 1 εt+ ...+ bqεtq–

or equivalently in terms of the lag operator

AL()xt = BL()εt

where εtis a serially uncorrelated white noise with nonzero variance, a

= b 0 = 1, ap≠0, bq≠0, the polynomials Aand Bhave roots strictly

greater than 1 in modulus and do not have any root in common.

Generalizing the reasoning in the pure MA or AR case, it can be

demonstrated that a generic process, which admits the ARMA(p,q) rep-

resentation A(L)xt = B(L)εtis stationary if both polynomials Aand B

have roots strictly different from 1. In addition, if all the roots of the

polynomial A(z) are strictly greater than 1 in modulus, then the

ARMA(p,q) process can be expressed as a moving average process:

BL()

xt = -------------εt

AL()

0