Problem Set 1: Unbiased Estimator of the Error
Variance
This assignment will guide you through the derivations needed to determine what is
a unbiased estimator of the error variance in the context of a univariate linear regression.
This assignment may be quite challenging. Good Luck!
Consider the univariate linear regression model
yt = α + βxt + ut ,
t = 1, . . . , T.
(1)
where the regressors are non-stochastic (fixed) and the disturbances have zero mean
and are uncorrelated and homoscedastic with variance equal to σ 2 , i.e., E [ut ] = 0,
C [ut , us ] = 0, t = s, and V [ut ] = E [u2t ] = σ 2 . The aim of this problem set will be to
derive that
T
1
2
uˆ2
s =
T − 2 t=1 t
ˆ t,
is an unbiased estimator of the disturbance variance σ 2 , where uˆt = yt − yˆt , yˆt = α
ˆ + βx
and α
ˆ and βˆ are the OLS estimators of α and β, respectively. The OLS estimates α
ˆ
and βˆ are given by
T
t=1
βˆ =
(yt − y¯) (xt − x¯)
T
t=1
(xt − x¯)
2
α
ˆ = y¯ − βˆx¯,
,
which can also be expressed as
T
βˆ =
T
wt yt ,
α
ˆ=
t=1
where
wt =
t=1
xt − x¯
T
t=1
qt y t ,
2
(xt − x¯)
1
,
qt =
1
− x¯ · wt .
T
Question 1 - Verify simple properties.
Verify the following 7 properties:
1.
T
t=1
wt = 0
2.
T
t=1
w t xt = 1
3.
T
t=1 qt
4.
T
t=1 qt xt
5.
T
t=1
6.
T
2
t=1 qt
7.
T
t=1 qt wt
=1
=0
1
wt2 =
T
x)2
t=1 (xt −¯
T
1
2
t=1 xt
T
T
x)2
t=1 (xt −¯
=
=
−¯
x
T
x)2
t=1 (xt −¯
As you work through the problem set, you should always look back at these 8 properties
and see which one can help you in each step.
For example, with properties 1 and 2 in hand, it is easy to show (as we did in class)
that:
T
βˆ =
T
wt yt =
t=1
T
wt (α + βxt + ut ) = β +
t=1
wt ut .
t=1
Likewise, using properties 3 and 4 it is easy to show (as we did in class) that:
T
T
T
t=1
qt ut .
qt (α + βxt...