My Texts

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 = ? + ?xt + 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 ? 2 , i.e., E [ut ] = 0,
C [ut , us ] = 0, t = s, and V [ut ] = E [u2t ] = ? 2 . The aim of this problem set will be to
derive that
T
1
2
u?2
s =
T ? 2 t=1 t
? t,
is an unbiased estimator of the disturbance variance ? 2 , where u?t = yt ? y?t , y?t = ? + ?x
and ? and ?? are the OLS estimators of ? and ?, respectively. The OLS estimates and ?? are given by
T
t=1?? =(yt ? y?) (xt ? x?)
T
t=1(xt ? x?)2? = y? ? ??x?,,which can also be expressed as
T?? =Twt yt ,?=t=1where
wt =t=1xt ? xT
t=1qt y t ,2(xt ? x?)1,qt =1
? x? ยท wt .
T Question 1 – Verify simple properties.
Verify the following 7 properties:
1.T
t=1wt = 02.T
t=1w t xt = 13.T
t=1 qt4.T
t=1 qt xt5.T
t=16.T
2
t=1 qt7.T
t=1 qt wt=1
=0
1wt2 =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?? =Twt yt =
t=1Twt (? + ?xt + ut ) = ? +
t=1wt ut .
t=1Likewise, using properties 3 and 4 it is easy to show (as we did in class) that:
TTTt=1qt ut .qt (? + ?xt…

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