The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 296 Wednesday, February 4, 2004 12:58 PM

296 The Mathematics of Financial Modeling and Investment Management

the white noise process. The essential differences of this linear model

with respect to the Wold representation of stationary series are:

■ The presence of a starting point and of initial conditions.

■ The absence of restrictions on the coefficients.

■ The index twhich restricts the number of summands.

The first two moments of a linear process are not constant. They can be

computed in a way similar to the infinite moving average case:

t

cov(xtxth– )= ∑HiΩΩΩΩH' ih– + h ()tvar ()zh′

i= 0

E[]xt = mt = h ()tE[]z

Let’s now see how a linear process can be expressed in autoregres-

sive form. To simplify notation let’s introduce the processes εεεε ̃ t and x ̃ t

and the deterministic series h ̃ ()t defined as follows:

εεεε ̃ t = 

εεεεtif t> 0

x ̃ t= 

xtif t> 0

h ̃ ()= 

htif t> 0

t

0if t<^0 0if t<^0 0if t<^0

It can be demonstrated that, due to the initial conditions, a linear pro-

cess always satisfies the following autoregressive equation:

ΠΠΠΠ()Lxt =εεεεt + ΠΠΠΠ()Lh × ()tz– 1

A random walk model

t

xt = xt– 1 + εt = εt+ ∑εti–

i= 1

is an example of a linear nonstationary model.

The above linear model can also represent processes that are nearly

stationary in the sense that they start from initial conditions but then

converge to a stationary process. A process that converges to a station-

ary process is called asymptotically stationary.

We can summarize the previous discussion as follows. Under mild

regularity conditions, any causal stationary series can be represented as