# Econometrics Exercise

## Econometrics Exercise

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
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...