Information Technology Reference
In-Depth Information
Random walk process is often described to students using an example of the
drunkard's walk. The drunkard wonders aimlessly, so that the direction of each step
is random and completely independent of her previous steps. In other words, the
meandering of the drunkard is described by a random walk:
x
t
−
x
t
−
1
=
ε
t
,
t
=
1
,
2
,...,
(18.29)
where
x
t
represents the position of the drunk at time
t
, and
ε
t
is a stationary white-
noise, which models the drunk's step at time
t
.
As Murray noticed [21], an unleashed puppy is another creature, whose behav-
ior reminds a random walk. Indeed, each new scent that puppy's nose comes upon
dictates a direction for the pup's next step so strongly that the last scent along with
its direction is forgotten as soon as the new scent appears. Having shown that the
puppies follow the random walk
y
t
,
t
=
1
,
2
,...
, let us represent the puppy's walk
as:
y
t
−
y
t
−
1
=
ε
t
,
t
=
1
,
2
,...,
(18.30)
where
t
is a stationary white noise (i.e., puppy's step at time
t
).
For a random walk, the best predictor of the future value is the most recently
observed one. In other words, the longer it has been since we had seen the drunk, or
the dog, the further away from the initial place, on average, they are at the moment.
As a result, even if the drunk and the dog crossed their walks at some location, as
the time goes on, they tend to wander further away from each other.
However, if the puppy belongs to the drunkard, then they will remain relatively
close to each other at all the time, similarly to the individual integrated processes that
together form a cointegrated process. Indeed, the drunk would still wonder aimlessly
in a random walk fashion, as would her puppy. However, from time to time she
would remember about her dog and call for it, the puppy would recognize her voice
and bark. They would hear each other and make their next step in each other's
direction.
The paths of the drunk and her dog are still nonstationary, but they are no longer
independent from each other. As a matter of fact, at each time, the puppy and its
master are likely to be found not far from each other. If this is true, then the distance
between two paths is stationary, and the walks of the drunk
x
t
and her dog
y
t
are said
to be
cointegrated
, i.e.,
x
t
and
y
t
are integrated
I
ε
(
1
)
, and there is a
linear combination
of
x
t
and
y
t
(with nonzero weights) that is
I
, i.e., stationary.
Mathematically, the
cointegrating relationship
between a lady and her puppy can
be written as
(
0
)
x
t
−
x
t
−
1
=
ε
t
+
c
(
y
t
−
1
−
x
t
−
1
)
,
(18.31)
−
=
ε
+
(
−
)
,
y
t
y
t
−
1
d
x
t
−
1
y
t
−
1
(18.32)
t
at time
t
=
1
,
2
,...
. Note that
ε
t
, as before, represent the stationary white noise steps
of the drunk and her dog.
Since Equation (18.31) can be easily rewritten in form of (18.17) as follows:
