(PRF), the corresponding disturbances are still unobservable.
If the disturbances were observable, an unbiased estimator of
2
would be obtained as (Erlat, 1992, pp.3233):
2
2
1
1
T
t
t
u
s
T
However as it has emphasized above,
t
u
’s are
unobservable
and we
need
sample
counterparts for them, which are called residuals (
ˆ
t
u
).
Thus, we will substitute
ˆ
t
u
’s for
t
u
’s and we will calculate the
variance of these
ˆ
t
u
’s. However, this is a bit problematic. Note th
at
ˆ
t
u
’s are
OLS residuals
. Hence, they must obey the OLS estimator
rules of:
(1)
1
ˆ
0
T
t
t
u
, and
(2)
1
ˆ
0
T
t
t
t
u X
Hence, to call a series of observations as OLS residuals, these two
conditions must be met. Now, the question here is:
how many of
ˆ
t
u
’s
are random
or
how many
ˆ
t
u
are free to vary
? This is an important
question since in statistics, the
variance is calculated by dividing the
sum of squared deviations from mean to the number of observations
that are free to vary
.
To digest this crucial issue let us give an example. Suppose we have a
sample size of 5 (i.e., T=5):
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ECON 301 (01)  Introduction to Econometrics I
March, 2012
METU  Department of Economics
Instructor: Dr. Ozan ERUYGUR
email:
[email protected]
Lecture Notes
19
t=1
t=2
t=3
t=4
t=5
1
ˆ
u
may happen
randomly
2
ˆ
u
may happen
randomly
3
ˆ
u
may happen
randomly
If
4
ˆ
u
happen
randomly, there
is no guarantee
that
1
ˆ
0
T
t
t
t
u X
holds.
If
5
ˆ
u
happen
randomly, there
is no guarantee
that
1
ˆ
0
T
t
t
u
holds.
As can be seen from this example that the number of values of
ˆ
t
u
that
are free to vary is not 5 but instead 3! In other words, in this example
we are free to choose the first three numbers at random, but the fourth
and fifth must be chosen so that
1
ˆ
0
T
t
t
t
u X
and
1
ˆ
0
T
t
t
u
conditions
hold, thus the number of values of
ˆ
t
u
that are free to vary is not 5 but
instead 3. We call the number of observations that are free to vary as
degrees of freedom
2
.
For a regression model with
k
independent variables and
1
intercept
term, we can generalize the degrees of freedom as
df=Tk1.
Hence, an
unbiased estimator
of
2
can be calculated as:
2
2
1
ˆ
ˆ
(
)
ˆ
1
T
t
t
t
u
u
T
k
Note that
ˆ
0
t
u
since
1
ˆ
0
T
t
t
u
condition of OLS estimation. Hence,
the
unbiased estimator
of
2
can be rewritten as:
2
The concept of
degrees of freedom
is central to the principle of estimating statistics of populations from
samples of them.
ECON 301 (01)  Introduction to Econometrics I
March, 2012
METU  Department of Economics
Instructor: Dr. Ozan ERUYGUR
email:
[email protected]
Lecture Notes
20
2
2
1
ˆ
ˆ
1
T
t
t
u
T
k
where
ˆ
t
u
is the OLS residual, and it is given by
ˆ
ˆ
t
t
t
u
Y
Y
.
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 Spring '10
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 Econometrics, Standard Deviation, Variance, Probability theory, Cauchy distribution, Dr. Ozan Eruygur

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