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...