(2.9)
We once again underscore the main point (hoping to drive it home) by quot-
ing our constant refrain pertaining to Weiner filtering: The preceding mod-
els are all constructed using a linear combination of past values of the white
noise series.An important consequence of that fact is that the sum of two in-
dependent ARMA series is also ARMA.
The Random Walk Process
An important and special ARMA series that merits discussion is the random
walk. The random walk has been studied extensively by scientists from var-
ious disciplines. Phenomena ranging from the movement of molecules to
fluctuations of stock prices have been modeled as random walks. Let us
therefore discuss this in some detail.
A random walk is an AR(1) series with a= 1. From the definition of an
AR series given, the value of the time series at time tis therefore
yt=et+et–1+et–2+... = et+yt–1 (2.10)In words, the random walk is essentially a simple sum of all the white noise
realizations up to the current time. The AR representation provides an al-
ternate way to look at the random walk. It is the value of the time series one
time step ago plus the white noise realization at the current time step. The
white noise realization at the current time step in the case of the random
walk is known as the innovation. Figure 2.4 is a picture of the random walk
generated using the white noise series in Figure 2.1.
Let us now begin to examine some properties of the random walk. What
do we expect the variance of the random walk to be at time t? Applying the
formulas from the appendix in Chapter 1 on the MA(∞) (infinity) represen-
tation of the random walk, along with the fact that white noise drawings are
uncorrelated, we have
(2.11)
Since these random white noise drawings all have the same variance, the
variance of the random walk at any time tis clearly
(2.12)Note that in this case the variance depends on the time instant, and it in-
creases linearly with time t. (If the variance increases linearly with t, then the
var()yttt= var()εvar()yttt t=var()εε ε+var()−− 12 +var()++L var()ε 1++ + +...+[]εβε βεtt11 2 2−−t βεqtq−yy ytt t=++...+[]αα 11 −−2 2 αptpy−+22 BACKGROUND MATERIAL