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: