### Lecture 9

```Econ107 Applied Econometrics
Topic 9: Heteroskedasticity
(Studenmund, Chapter 10)
I.
Definition and Problems
We now relax another classical assumption. This is a problem that arises often
with cross sections of individuals, households or firms. It can be a problem with
time series data, too.
Homoskedasticity exists when the variance of the disturbances is constant:
Var ( ε i ) = E ( ε i2 ) = σ 2
Assumption of equal (homo) spread (skedasticity) in the distribution of the
disturbances for all observations. Variance is a constant. Independent of anything
else, including the values of the independent variables.
Heteroskedasticity exists when the variance of the disturbances is variable:
Var ( ε i ) = E ( ε i2 ) = σ i2
The variance of the disturbances can take on a different value for each observation
in the sample. Most general specification. Takes on different values for each
observation. More often, σ i 2 may be related to one or more of the independent
variables.
Heteroskedasticity violates one of the basic classical assumptions.
Example: Suppose we estimate a cross-sectional savings function. The variance
of the disturbances may increase with disposable income, due to increased
'discretionary income'. More income has to be devoted to basic necessities.
Distributions flatten out as DI rises.
Take 3 distinct income classes (low, medium and high) – two graphs.
Other examples. Might be related to the size of some economic aggregate (e.g.,
corporation, metropolitan area, state or region).
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What happens if we use OLS in a regression with known heteroskedasticity?
(1) The estimated coefficients are still unbiased. Homoskedasticity is not a
necessary condition for unbiasedness. Same result as multicollinearity.
(2) But the OLS estimators are inefficient. This means that they're no longer
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BLUE, like in the case of serial correlation. Different from multicollinearity.
Y i = β 0 + β1 X i + ε i
with homoskedasticity:
σ
Var ( βˆ1 ) =
Σ xi2
2
with heteroskedasticity:
Σ xi2 σ i2
ˆ
Var ( β1 ) =
( Σ xi2 )2
Of course if σ i 2=σ2, we can simplify:
σ Σ xi = σ
Var ( βˆ1 ) =
( Σ xi2 )2 Σ xi2
2
2
2
Earlier formula for calculating the standard error depends on the assumption of
homoskedasticity. It’s a ‘special case’ of the more general formula. OLS
estimators will be inefficient. They’re no longer minimum variance. The
formulae for the OLS estimators of the coefficients are still the same under
heteroskedasticity.
Intuition: We want the 'best fit' possible for our regression line. Our sample
regression function should lie as close as possible to the population regression
function. OLS 'equally weights' each observation. It assumes each observation
contributes the same amount of 'information' to this estimation. With
heterogeneity that is no longer appropriate. Observations should not be equally
weighted.
Those associated with a tighter distribution of ε i contribute more information,
while those associated with a wider distribution of ε i contribute less information
about the basis of this economic behaviour. A priori observations from wider
distribution have more ‘potential error’.
By disregarding heteroskedasticity and using the OLS formula, we would produce
biased estimates of the standard errors. In general, we won't know the direction
of the bias. As a result, statistical inference would be inappropriate.
Weighted Least Squares (WLS)
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WLS essentially takes advantage of this heterogeneity in its estimators.
Assume σ i 2 is known. Transform the data by dividing by σ i :
Yi =
σi
β 0(
1
σi
ε
) + β 1( X i ) + ( i )
σi
σi
or use ‘*’ to denote the transformed variables:
*
*
*
*
Y i = β 0W i + β 1 X i + ε i
where W i * is the 'weight' given to the observation. This is just the inverse of the
standard deviation. No longer a constant term in the regression. OLS estimation
on the transformed model is WLS on the original model (denote the estimators by
βˆ0* and βˆ1* ). But why did we do this?
As a result:
Var( ε *i ) = E( ε *i 2 ) = E [( ε i / σ i )2 ] =
σ i2 = 1
σ i2
This means that the OLS estimators of the transformed data are BLUE, because
the disturbances are now homoskedastic. Not only constant, also equal to 1.
In this case βˆ0* and βˆ1* will be BLUE. The transformed model meets all the
classical assumptions, including homoskedasticity. Recall that βˆ0 and βˆ1 are
unbiased, but not efficient.
Another way to motivate the distinction between OLS and WLS is to look at the
'objective functions' of the estimation.
Under OLS we minimise the residual sum of squares:
Σ ei2 = Σ( Y i - βˆ0 - βˆ1 X i )2
e 
Σ i 
σ i 
2
But under WLS we minimise a weighted residual sum of squares:
Couple things to note:
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1. The formulae for WLS estimates of β 0 and β 1 aren’t worth committing to
memory, so don’t write them down. The key is that they look similar to the
formulae under homoskedasticity, except for the ‘weighting’ factor. True of both
2- and k-variable models. But with software packages, don’t have to know these.
Just transform data and run the regression through the origin.
2. If σ i = σ for all observations, then WLS estimators are OLS estimators. OLS
is a special case of this more general procedure.
II.
Detection
How do we know when our disturbances are heteroskedastic?
The key is that we never observe the true disturbances or the distributions from
which they are drawn. In other words, we never observe σ i (at least not unless we
see the entire population).
For example, take our original example of the savings function. If we had the
entire census of 4 million Singaporeans we’d be able to calculate it. We’d know
how the dispersion in the disturbances varies with disposable income. But in
samples we have to make an educated guess.
We consider 4 diagnostic tests or indicators.
1. A Priori Information
This might be 'anticipated' (e.g., based on past empirical work). Check the
relevant literature in this area. Might show clear and persistent evidence of
heteroskedastcity. For example, check both domestic and overseas studies of
savings regressions. Is it a commonly reported problem in this empirical work?
Key is that you see it coming. Remainder of the tests are ‘post-mortems’.
2. Graphical Methods
Key: We'd like information on u i 2, but all we ever see are e i 2. We want to know
whether or not these squared residuals exhibit any 'systematic pattern'. With
homoskedasticity we'll see something like this:
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No relationship between e i 2 and the explanatory variable.
Even if we get this pattern, we can't rule out the possibility of heteroskedasticity.
We may have to plot these squared residuals against other explanatory variables.
The same could be done for the squared residuals and the fitted value.
Alternatively, with
heteroskedasticity what we'll see
are patterns like this:
3. Park Test
The Park test is just a
‘formalisation’ of the plotting of
the squared residuals against
another variable (often one of the
explanatory variables).
Use a two-step procedure:
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1. Run OLS on your regression. Retain the squared residuals.
Assume that:
σ i2 = σ 2 Z iβ
This implies a 'log-log' linear relationship between the squared residuals and Z i .
2. Estimate the following:
ln ei2 = ln σ 2 + β ln Z i + u i
= α + β ln Z i + u i
Test H 0 : β=0. If null is rejected, this suggests that heteroskedasticity is present.
You need to choose which variables might be related to the squared residuals
(often an independent variable is used). If β>0, then upward-sloping curved
relationship. If β<0, then downward-sloping curved relationship.
One problem is that rejection of H 0 is “... sufficient, but not a necessary condition
for heteroskedasticity”. Another problem is that this test imposes an assumed
relationship between a particular variable and the squared residuals.
4. White Test
This gets around the problems of the Park test that the disturbances are likely to be
heteroskedastic.
Use a three-step procedure:
1. Run OLS on your regression (assume X i2 and X i3 are the two independent
variables). Retain the squared residuals.
2. Estimate the following auxiliary regression:
2
ei = α 0 + α 1 X 2 i + α 2 X 3 i + α 3 X 2 i
2
+ α 4 X 3i + α 5 X 2 i X 3i + u i
2
3. Test the overall significance of the auxiliary regression. To do this, use nR22 .
Under the null of homoskedasticity, nR22 follows the chi-square distribution with
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degrees of freedom equal to the number of slope coefficients in the auxiliary
regression, where n is the sample size, R22 is the coefficient of determination of the
auxiliary regression.
III.
Remedial Measures
What do you do when heteroskedasticity is suspected?
(i) When σ i is known, transform data by dividing both dependent and independent
variables by σ i and run OLS. This is the weighted least squares procedure.
Not a very interesting situation. This information is rarely available.
(ii) When σ i is unknown, determine the ‘likely form’ of the heteroskedasticity.
Transform the data accordingly, and run weighted least squares.
Two Examples:
Suppose we have the following regression for a cross section of cities:
CRi = β 0 + β 1 EXPi + β 2 POPi + ε i
where:
CR i = Per capita crime rate.
EXP i = Per capita expenditures on police.
POP i = Population.
The first slope coefficient picks up the ‘effectiveness’ of police expenditures at the
margin (negative). The second says that crime might increase with the size of the
metropolitan population (positive).
(1) Suppose we suspect that:
Var ( ε i ) = E ( ε i2 ) = σ 2 POPi
where σ2 is a constant.
Transform the data and estimate the following:
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CRi
=
POPi
β0
+ β1
POPi
-9
EXP1
POPi
+ β2
POPi
+
POPi
εi
POPi
or
CRi
POPi
= β0
1
+ β 1 EXPi + β 2 POPi + u i
POPi
POPi
The residuals are now homoskedastic. (The proof left to you as an exercise.)
This doesn’t change the interpretation of the coefficients. Dividing both sides by
the same variable.
(2) Suppose we now suspect that:
Var ( ε i ) = E ( ε i2 ) = σ 2 Cˆ Ri
2
where Cˆ Ri is the fitted value. Transform the data and estimate the following:
1
CRi
POPi
= β0
+ β 1 EXPi + β 2
+ ui
Cˆ Ri
Cˆ Ri
Cˆ Ri
Cˆ Ri
The residuals are now homoskedastic.
Operationally, this second example requires two steps:
(1) Run OLS on the original equation with the untransformed data (recall that the
estimated coefficients are still unbiased, although they are inefficient).
(2) Transform the data by dividing the dependent and independent variables with
these fitted values and estimate as above.
(iii). Using heteroskedastic-corrected (HC or White) standard errors.
Heteroskedasticity does not cause bias of OLS estimates but impacts the standard
errors. HC technique directly adjusts the standard errors of OLS estimates to take
account of heteroskedasticity.
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IV. Questions for Discussion: Q10.13
V. Computing Exercise: Johnson, Ch10, 1-5
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