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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 [u2 ] = σ 2 . The aim of this problem set will be to t derive that

T

1

2

u2

ˆ

s =

T − 2 t=1 t is an unbiased estimator of the disturbance variance σ 2 , where ut = yt − yt , yt = α + βxt ,

ˆ

ˆ ˆ

ˆ ˆ

ˆ

and α and β are the OLS estimators of α and β, respectively. The OLS estimates α

ˆ

ˆ

ˆ

and β are given by

T

t=1

ˆ β =

(yt − y ) (xt − x)

¯

¯

T

t=1

(xt − x)

¯

2

α = y − β x,

ˆ ¯ ˆ¯

,

which can also be expressed as

T

ˆ β= 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

2 wt =

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

ˆ β= T

wt yt = t=1 T

wt (α + βxt + ut ) = β + t=1 wt ut .

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