LECTURE 11: HETEROSKEDASTICITY
HETEROSKEDASTICITY
Homoskedasticity:
Var(
ε
i
) =
σ
2
Heteroskedasticity:
Var(
ε
i
) =
σ
i
2
What happens if we violate this assumption of the CLRM?
OLS still unbiased but not minimum variance (i.e. not BLUE)
Our estimates of the variance will be biased
Because of this, our usual hypothesis testing routine is unreliable.
Why is OLS inefficient?
2
i
β
,
β
ε
ˆ
min
1
0
With OLS, we weight each
2
i
ε
ˆ
equally, whether it comes from a population
with a large variance or with a small variance.
Ideally, we would like to give more weight to observations coming from
populations with smaller variances, as this would enable us to estimate the
PRF more accurately.
What do we do? WEIGHTED LEAST SQUARES
DETECTION OF HETEROSKEDASTICITY
No surefire method; generally, we don’t know true
σ
2
i
and often only have a few
observations for each X.
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Priors
More likely in crosssectional data
Firms (example, firm size and investment)
Individuals (earning and savings, education and earnings)
2.
Graphical examination of residuals
Plot
i
ε
ˆ or
2
i
ε
ˆ against X to see if there is a pattern
Plot
i
ε
ˆ or
2
i
ε
ˆ against
i
Y
ˆ
if there are multiple X’s
3.
More rigorous tests
There are a number of tests that involve regressing forms of the
residual on X’s.
One test:
WHITE’S GENERAL HETEROSKEDASTICITY
TEST
a) Y
i
=
β
0
+
β
1
X
1i
+
β
2
X
2i
+
ε
i
b) Estimate with OLS, obtain residuals
i
ε
ˆ
c) Run the following auxiliary regression:
i
i
2
i
1
5
2
i
2
4
2
i
1
3
i
2
2
i
1
1
0
2
i
v
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 Winter '07
 SandraBlack
 Regression Analysis, Yi, standard errors, Zi Zi Zi Zi Zi Zi, Zi Zi Zi, CLRM assumptions

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