The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 298 Wednesday, February 4, 2004 12:58 PM

298 The Mathematics of Financial Modeling and Investment Management

in modulus (that is, the roots do not lie on the unit circle), then the

operator A(L) is invertible and admits the inverse representation:

+∞ +∞

xt = A–^1 ()εL t = ∑ λiεti– , with ∑ λi +∞ <

i – = ∞ i – = ∞

In addition, if the roots are all strictly greater than 1 in modulus, then

the representation only involves positive powers of L:

+∞ +∞

xt = A–^1 ()εL t = ∑ λiεti– , with ∑ λi +∞ <

i – = ∞ i= 0

We can therefore say that, if the roots of the inverse characteristic equa-

tion of an autoregressive process are all strictly greater than 1 in modu-

lus (that is, they lie outside the unit circle), then the process is invertible

as it admits a causal infinite moving average representation.

In order to avoid possible confusion, note that the solutions of the

inverse characteristic equation are the reciprocal of the solution of the

characteristic equation defined as

Az()= zp+ a 1 zp–^1 + ...+ aP = 0

Therefore an autoregressive process is invertible with an infinite moving

average representation that only involves positive powers of the opera-

tor L if the solutions of the characteristic equation are all strictly

smaller than 1 in absolute value. This is the condition of invertibility

often stated in the literature.

Let’s now consider finite moving-average representations. A process

is called a moving average process of order q – MA(q) if it admits the

following representation:

xt = ( 1 + b 1 L+ ...+ bPLq)εt = εt+ b 1 εt– 1 + ...+ bPεtq–

In a way similar to the autoregressive case, if the roots of the equation

Bz

q

()= 1 + b 1 z+ ...+ bqz = 0

are all different from 1 in modulus, then the MA(q) process is invertible

and, therefore, admits the infinite autoregressive representation: