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5 Additional Topics in Basic
Regression Analysis
• 1) Heteroskedasticity/BLUE
• 2) Measuring Model Fit
• 3) Changing Units of Measurement
• 4) Dummy Variables
• 5) Omitted Variable Bias
Heteroskedasticity/BLUE  I
• Our regression model is:
• Recall that
u
i
represents variables other than
X
i
that
affect
Y
i
.
•
u
i
is a random variable, just like
X
i
and
Y
i
. Let’s
think about its variance.
• More specifically, let’s think about its
conditional
variance,
i.e.
01
=
ii
i
YX
u
β
++
( )
Var u X
c
=
Heteroskedasticity/BLUE – II
• What exactly is the conditional variance:
• It tells us the what the variance of
u
i
is
given
that
X
i
equals some number c, i.e.
–Va
r
(
u
i
 X
i
=
5) is the variance of
u
i
if
X
i
=
5.
r
(
u
i
 X
i
=
20) is the variance of
u
i
if
X
i
=
20.
• Sometimes Var(
u
i
 X
i
=c) is the same for every
possible value of
X
i
(i.e. for every possible c) Other
times, Var(
u
i
 X
i
=c) varies depending on the
particular value of
X
i
(i.e c).
( )
Var u X
c
=
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View Full Document Heteroskedasticity/BLUE  III
•D
e
f
n
:
• 1) If Var(
u
i
 X
i
=c) is the same for every value c (i.e. the
variance of
u
i
does not depend on
X
i
), then
u
i
is
homoskedastic
.
• 2) If Var(
u
i
 X
i
=c) does depend on c, then
u
i
is
heteroskedastic
.
• Example: #WebsiteHits =
β
0
+
β
1
Advertising$ +
u
• There are two important implications of this distinction
between homoskedasticity and heteroskedasticity.
Heteroskedasticity/BLUE  IV
• If
u
i
is
homoskedastic,
then:
• 1) The OLS estimator
β
1
is BLUE, i.e. it is the
B
est,
L
inear, conditionally
U
nbiased
E
stimator.
– “Best” means that the OLS estimator
β
1
is the most
efficient
of these estimators, i.e. the one with the smallest
variance.
• 2) We can actually use a slightly simpler formula
for Var(
β
1
) and SE(
β
1
) (again, you don’t need to
remember this exact formula)
()
2
1
11
1
2
1
1
ˆ
1
2
ˆˆ
ˆ
Var
SE
Var
1
n
i
i
n
i
i
u
n
n
XX
n
β
ββ
=
=
−
==
−
∑
∑
Heteroskedasticity/BLUE  V
• If
u
i
is
heteroskedastic,
then:
• 1) The OLS estimator
β
1
is not BLUE, i.e. there are
potentially more efficient other estimators out there
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This note was uploaded on 05/26/2008 for the course ECON 103 taught by Professor Sandrablack during the Spring '07 term at UCLA.
 Spring '07
 SandraBlack
 Econometrics

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