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OLS Under Heteroskedasticity
Testing for Heteroskedasticity
Heteroskedasticity and Weighted Least Squares
Walter SosaEscudero
Econ 507. Econometric Analysis. Spring 2009
April 14, 2009
Walter SosaEscudero
Heteroskedasticity and Weighted Least Squares
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Testing for Heteroskedasticity
The Classical Linear Model:
1
Linearity:
Y
=
Xβ
+
u
.
2
Strict exogeneity:
E
(
u
) = 0
3
No Multicollinearity:
ρ
(
X
) =
K
.
4
No heteroskedasticity/ serial correlation:
V
(
u
) =
σ
2
I
n
.
Gauss/Markov Theorem:
ˆ
β
= (
X
0
X
)

1
X
0
Y
is best linear
unbiased.
ˆ
V
(
ˆ
β
) =
S
2
(
X
0
X
)

1
is an unbiased estimate of
V
(
ˆ
β
) =
σ
2
(
X
0
X
)

1
.
Walter SosaEscudero
Heteroskedasticity and Weighted Least Squares
OLS Under Heteroskedasticity
Testing for Heteroskedasticity
What happens if we drop the homoskedasticity assumption?
ˆ
β
(the OLS estimator) is still linear and unbiased (Why?).
Though linear and unbiased,
ˆ
β
is not the minimum variance
estimate (ineﬃcient).
ˆ
V
(
ˆ
β
) =
S
2
(
X
0
X
)

1
is
biased
. This makes standard ‘t’ and
‘F’ tests invalid.
Walter SosaEscudero
Heteroskedasticity and Weighted Least Squares
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Testing for Heteroskedasticity
Intuition
The presence of heteroskedastic errors should not alter the ‘central’
position of the OLS line (unbiasedness).
OLS weigths all observations equally, but in this case it makes more
sense to pay more attention to observations where the variance is
smaller.
Walter SosaEscudero
Heteroskedasticity and Weighted Least Squares
OLS Under Heteroskedasticity
Testing for Heteroskedasticity
The plan: what to do with heteroskedasticity.
1
Before abandoning OLS we will see how to
test for
heteroskedasticity.
2
Strategy 1:
Propose another more eﬃcient and unbiased
estimator for
β
(
weighted least squares (WLS)
) and a suitable
estimator for its variance.
3
Strategy 2:
Keep using OLS (it is still unbiased, though
ineﬃcient), but ﬁnd a replacement for its variance (the old
one is biased under heteroskedasticity).
Walter SosaEscudero
Heteroskedasticity and Weighted Least Squares
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Testing for Heteroskedasticity
Testing for heteroscedasticity
a) The White test
H
0
:
no heteroscedasticity,
H
A
:
there is heterocedasticity of some
form.
Consider a simple case with
K
= 3
:
Y
i
=
β
1
+
β
2
X
2
i
+
β
3
X
3
i
+
u
i
1
,...,n
Walter SosaEscudero
Heteroskedasticity and Weighted Least Squares
OLS Under Heteroskedasticity
Testing for Heteroskedasticity
Steps to implement the test:
1
Estimate by OLS, save squared residuals in
e
2
.
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This note was uploaded on 02/15/2012 for the course GEO 4167 taught by Professor Staff during the Spring '12 term at University of Florida.
 Spring '12
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