Ketevani Kapanadze
Brno, 2020
Heteroskedasticity
8 Chapter
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Consequences of Heteroskedasticity for OLS
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• Consequences of heteroscedasticity for OLS
▪ OLS still unbiased and consistent under heteroscedastictiy!
▪ Also, interpretation of R-squared is not changed
▪ Heteroscedasticity invalidates variance formulas for OLS estimators
▪ The usual F-tests and t-tests are not valid under heteroscedasticity
▪ Under heteroscedasticity, OLS is no longer the best linear unbiased estimator
(BLUE); there may be more efficient linear estimators
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Heteroskedasticity-Robust Inference
after OLS Estimation
• Heteroscedasticity-robust inference after OLS
▪ Formulas for OLS standard errors and related statistics have been developed
that are robust to heteroscedasticity of unknown form
▪ All formulas are only valid in large samples
▪ Formula for heteroscedasticity-robust OLS standard error
▪ Using thes formula, the usual t-test is valid asymptotically
▪ The usual F-statistic does not work under heteroscedasticity, but
heteroscedasticity robust versions are available in most software
Also called White/Eicker standard errors. They involve the
squared residuals from the regression and from a
regression of xj on all other explanatory variables.
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• Example: Hourly wage equation
Heteroscedasticity robust standard errors may be larger
or smaller than their nonrobust counterparts. The
differences are often small in practice.
F-statistics are also often not too different.
If there is strong heteroscedasticity, differences may be larger. To be
on the safe side, it is advisable to always compute robust standard
errors.
Heteroskedasticity-Robust Inference
after OLS Estimation
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Testing for Heteroskedasticity
• Testing for heteroscedasticity
▪ It may still be interesting whether there is heteroscedasticity because then
OLS may not be the most efficient linear estimator anymore
• Breusch-Pagan test for heteroscedasticity
Under MLR.4
The mean of u2 must not vary
with x1, x2, …, xk
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• Breusch-Pagan test for heteroscedasticity (cont.)
Regress squared residuals on all explanatory
variables and test whether this
regression has explanatory power.
A large test statistic (= a high Rsquared)
is evidence against the null
hypothesis.
Alternative test statistic (= Lagrange multiplier statistic, LM obtained
by regressing residuals from unrestricted model to all explanatory
variables). Again, high values of the test statistic (= high R-squared)
lead to rejection of the null hypothesis that the expected value of u2 is
unrelated to the explanatory variables.
Testing for Heteroskedasticity
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• Example: Heteroscedasticity in housing price equations
In the logarithmic specification, homoscedasticity cannot be rejected –
benefit of using the logarithmic functional form
Heteroscedasticity
Testing for Heteroskedasticity
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• White test for heteroscedasticity
• Disadvantage of this form of the White test
▪ Including all squares and interactions leads to a large number of estimated
parameters (e.g. k=6 leads to 27 parameters to be estimated)
Regress squared residuals on all explanatory
variables, their squares, and interactions
(here: example for k=3)
The White test detects more general
deviations from heteroscedasticity than
the Breusch-Pagan test
Testing for Heteroskedasticity
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• Alternative form of the White test
• Example: Heteroscedasticity in (log) housing price equations
This regression indirectly tests the dependence of the squared residuals on the
explanatory variables, their squares, and interactions, because the predicted
value of y and its square implicitly contain all of these terms.
Testing for Heteroskedasticity
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Weighted Least Squares Estimation
• Heteroscedasticity is known up to a multiplicative constant
Transformed model
The functional form of the
heteroscedasticity is known
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• Example: Savings and income
• The transformed model is homoscedastic
• If the other Gauss-Markov assumptions hold as well, OLS applied to the
transformed model is the best linear unbiased estimator!
Note that this regression
model has no intercept
Weighted Least Squares Estimation
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• OLS in the transformed model is weighted least squares (WLS)
• Why is WLS more efficient than OLS in the original model?
▪ Observations with a large variance are less informative than observations
with small variance and therefore should get less weight
• WLS is a special case of generalized least squares (GLS)
Observations with a large
variance get a smaller weight in
the optimization problem
Weighted Least Squares Estimation
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• Unknown heteroscedasticity function (feasible GLS)
Assumed general form
of heteroscedasticity;
exp-function is used to
ensure positivity
Feasible GLS is consistent and asymptotically more efficient than OLS.
Multiplicative error (assumption:
independent of the explanatory
variables)
Use inverse values of the
estimated heteroscedasticity
funtion as weights in WLS
Weighted Least Squares Estimation
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• Example: Demand for cigarettes
• Estimation by OLS
Cigarettes smoked per day Logged income and cigarette price
Reject homo-
scedasticity
Smoking restrictions
in restaurants
Weighted Least Squares Estimation
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• Estimation by FGLS
• Discussion
▪ The income elasticity is now statistically significant; other coefficients are also
more precisely estimated (without changing qualit. results)
Now statistically significant
Weighted Least Squares Estimation
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• What if the assumed heteroscedasticity function is wrong?
▪ If the heteroscedasticity function is misspecified, WLS is still consistent under
MLR.1 – MLR.4, but robust standard errors should be computed
▪ WLS is consistent under MLR.4
▪ If OLS and WLS produce very different estimates, this typically indicates that
some other assumptions (e.g. MLR.4) are wrong
▪ If there is strong heteroscedasticity, it is still often better to use a wrong form
of heteroscedasticity in order to increase efficiency
Weighted Least Squares Estimation
Next Class
• Endogenous regressors and
instrumental variables
• Multiple Choice Quiz ☺
13.03.2020
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