The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 292 Wednesday, February 4, 2004 12:58 PM

292 The Mathematics of Financial Modeling and Investment Management

∞

∑ckL

CL()= AL() × BL() = k

k = 0

ck =

k

∑a^ sbks–

s = 0

We can define the left-inverse (right-inverse) of an infinite series as the oper-

ator A–1(L), such that A–1(L) ×A(L) = I. The inverse can always be com-

puted solving an infinite set of recursive equations provided that a 0 ≠0.

However, the inverse series will not necessarily be stationary. A sufficient

condition for stationarity is that the coefficients of the inverse series are

absolutely summable.

In general, it is possible to perform on the symbolic series

HL()=

∞

∑hiL

i

i = 1

the same operations that can be performed on the series

Hz()=

∞

∑hiz

i

i = 1

with z complex variable. However operations performed on a series of

lag operators neither assume nor entail convergence properties. In fact,

one can think of z simply as a symbol. In particular, the inverse does not

necessarily exhibit absolutely summable coefficients.

Stationary Univariate Moving Average

Using the lag operator L notation, the infinite moving average represen-

tation can be written as follows:







∞

∑



 hiL

i

xt = εt + m = HL()εt+ m

 (^) i = 0
Consider now the inverse series: