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