The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 289 Wednesday, February 4, 2004 12:58 PM

Financial Econometrics: Time Series Concepts, Representations, and Models 289

where the hi are coefficients and εt–i is a one-dimensional zero-mean

white-noise process. This is a causal time series as the present value of

the series depends only on the present and past values of the noise pro-

cess. A more general infinite moving-average representation would

involve a summation which extends from –∞to +∞. Because this repre-

sentation would not make sense from an economic point of view, we

will restrict ourselves only to causal time series.

A sufficient condition for the above series to be stationary is that the

coefficients hi are absolutely summable:

∞

∑

h^2 < ∞

i

i = 0

Also, in general it can be demonstrated that given any stationary pro-

cess xi, if the sequence of coefficients hi is absolutely summable, then the

process

∞

yi = ∑hixi

i = 1

is stationary.

The Lag Operator L

Let’s now simplify the notation by introducing the lag operator L. The

lag operator L is an operator that acts on an infinite series and produces

another infinite series shifted one place to the left. In other words, the

lag operator replaces every element of a series with the one delayed by

one time lag:

Lx()t = xt – 1

The n-th power of the lag operator shifts a series by n places:

L

n

()xt =xtn–

Negative powers of the lag operator yield the forward operator F,

which shifts places to the right. The lag operator can be multiplied by a

scalar and different powers can be added. In this way, linear functions

of different powers of the lag operator can be formed as follows: