The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 310 Wednesday, February 4, 2004 12:58 PM

310 The Mathematics of Financial Modeling and Investment Management

Zt =∆Yt =∆(∆Xt) =(Xt – Xt – 1 )–(Xt – 2 – Xt – 3 )

=Xt –Xt – 1 – Xt – 2 +Xt – 3

Differencing can be written in terms of the lag operator as

∆Xtd =( 1 – L)dXt

A difference stationary series is a series that is transformed into a

stationary series by differencing. A difference stationary series can be

written as

∆Xt μ ε+ = ()t

Xt = Xt – 1 ++με()t

where ε(t) is a zero-mean stationary process and μis a constant. A trend

stationary series with a linear trend is also difference stationary, if spac-

ings are regular. The opposite is not generally true. A time series is said

to be integrated of order n if it can be transformed into a stationary

series by differencing n times.

Note that the concept of integrated series as defined above entails

that a series extends on the entire time axis. If a series starts from a set

of initial conditions, the difference sequence can only be asymptotically

stationary.

There are a number of obvious differences between trend stationary

and difference stationary series. A trend stationary series experiences

stationary fluctuation, with constant variance, around an arbitrary

trend. A difference stationary series meanders arbitrarily far from a lin-

ear trend, producing fluctuations of growing variance. The simplest

example of difference stationary series is the random walk.

An integrated series is characterized by a stochastic trend. In fact, a

difference stationary series can be written as

t – 1

Xt =μt + ∑ε() s +ε()t

s + 0

The difference Xt –X*t between the value of a process at time t and

the best affine prediction at time t – 1 is called the innovation of the pro-

cess. In the above linear equation, the stationary process ε(t) is the inno-

vation process. A key aspect of integrated processes is that innovations