Introduction to Econometrics
121
Technical Notes
Introduction to Econometrics
Lecture 12: Stationary stochastic processes
Processes and realisations
With random sampling the key concepts are the population and the sample. The population can be real, (as
when a Census of Population uses a 10% sample) or hypothetical. A hypothetical population would be
generated by a specific probability distribution function: if we say a sample is drawn from a parent
population which is N(
μ
,
σ
2
) then the population could be represented by a normal curve. In this case we
cannot measure variables for the whole (infinite) population: but we can still make inferences about the
underlying distribution and its parameters from a finite sample.
The key feature of random sampling is that each time a particular observation is ‘drawn’ for inclusion in the
sample the probability that it is drawn does not depend on earlier drawings
Prob(x
t
=a)
≠
f(x
s
), for s
≠
t
This is not true for time series, where the probability of observing a particular value may depend on the past
history of the series. For example, if the series depends on its immediate past value
Prob(x
t
=a) = f(x
t1
)
To reintroduce an appropriate concept of ‘randomness’ we need new concepts of population and sample.
Suppose we think about a time series of T observations. Then we can write the whole sequence of
observations as
{x
1
, x
2
, ...x
T
}
Now we can consider this sequence as a single sample point, and the set of all possible sequences of length
T which could (hypothetically) have been observed as being the (hypothetical) underlying population.
With random sampling the population is described in terms of an underlying probability distribution, and a
single sample point is called an observation. With time series sampling the population is described in terms
of the underlying
stochastic process
which generates the sequences, and a single sample point (a sequence)
is called a
realisation.
The general aim of time series analysis is to make inferences about an underlying
stochastic process from one (or, rarely, more) realisations of the process
Stationarity and ergodicity
Usually the data available in economics consists of a single realisation of a time series, so there is only one
sample point. If we had N independent realisations we could define what are called ‘ensemble’ estimates of
the means and variances for each observation by averaging over realisations using the usual formulae
μ
t
= N
1
Σ
i
x
it
σ
2
t
= (N1)
1
Σ
i
(x
it

μ
t
)
2
For a given observation t, x
it
and x
jt
are independent and so these averages have the conventional desirable
properties.
Suppose we have only one realisation, but this realisation has the property that its characteristics do not
depend on the observation date: specifically that
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