Chapter 10_Autocorrelated Disturbances.pdf

Ch. 10 Autocorrelated Disturbances
In a time-series setting, mon problem is autocorrelation, or serial corre-
lation of the disturbance across periods. See the plot of the residuals at Figure
on p. 251.
1 Stochastic Process
A particularly important aspect of real observable phenomena, which the random
variables concept cannot modate, is their time dimension; the concept of
random variable is essential static. A number of economic phenomena for which
we need to formulate probability e in the form of dynamic processes
for which we have discrete sequence of observations in time. The problem we
have to face is extend the simple probability model,
Φ= {f(x; θ), θ∈Θ},
to one which enables us to model dynamic phenomena. We have already moved
in this direction by proposing the random vector probability model
Φ= {f(x1, x2, ..., xT ; θ), θ∈Θ}.
The way we viewed this model so far has been as representing different char-
acteristics of the phenomenon in question in the form of the jointly distributed
.’s X1, X2, ..., XT . If we reinterpret this model as representing the same char-
acteristic but at essive points in time then this can be viewed as a dynamic
probability model. With this as a starting point let us consider the dynamic
probability model in the context of (S, F, P).
The Concept of a Stochastic Process
The natural way to make the concept of a random variable dynamic is to extend
its domain by attaching a date to the elements of the sample space S.
Definition 1:
Let (S, F, P) be a probability space and T an index set of real numbers and
define the function X(·, ·) by X(·, ·) : S × T → R. The ordered sequence of
random variables {X(·, t), t ∈ T } is called a stochastic process.
1
This definition suggests that for a stochastic process {X(·, t), t ∈ T }, for each
t ∈ T , X(·, t) represents a random variable on S. On the other hand, for each s
in S, X(s, ·) represents a function of t which we call a realization of the process.
X