1. Heteroskedasticity

3. What does it mean? The variance of the error term is not constant
What are its consequences? The least squares results
are no longer efficient and t tests and F tests results may be misleading
How can you detect the problem? Plot the residuals against each of the regressors or use one of the more formal tests
How can I remedy the problem? Respecify the model – look for other missing variables; perhaps take logs or choose some other appropriate functional form; or make sure relevant variables are expressed “per capita”

4. Consumption function example (cross-section data): credit worthiness as a missing variable?

5. The Homoskedastic Case

6. The Heteroskedastic Case

7. Causes
Model misspecification - omitted variable or improper functional form.
Learning behaviors across time
Changes in data collection or definitions.
Outliers or breakdown in model.
Frequently observed in cross sectional data sets where demographics are involved (population, GNP, etc).

8. The consequences of heteroskedasticity
OLS estimators are still unbiased (unless there are also omitted variables)
However OLS estimators are no longer efficient or minimum variance
The formulae used to estimate the coefficient standard errors are no longer correct
so the t-tests will be misleading (if the error variance is positively related to an independent variable then the estimated standard errors are biased downwards and hence the t-values will be inflated)
confidence intervals based on these standard errors will be wrong

9. Detecting heteroskedasticity
Visual inspection of scatter diagram or the residuals
Goldfeld-Quandt test
suitable for a simple form of heteroskedasticity
Breusch-Pagan test
a test of more general forms of heteroskedastcity

10. Plot residuals against one variable at a time

11. Goldfeld-Quandt test
Order the n cases by the X that you think is correlated with ei2.
Drop a section of c cases out of the middle (one-fifth is a reasonable number).
Run separate regressions on both upper and lower samples.

12. Goldfeld-Quandt test (cont.)
Do F-test for difference in error variances F has (n - c - 2k)/2 degrees of freedom for each

13. Breusch-Pagan-Godfrey Test
Estimate model with OLS
Obtain
Construct variables
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14. Breusch-Pagan-Godfrey Test (cont.)
Regress pi on the X (and other?!) variables
Calculate
Note that
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15. White’s Generalized Heteroskedasticity test
Estimate model with OLS and obtain residuals
Run the following auxiliary regression
Higher powers may also be used, along with more X’s
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16. White’s Generalized Heteroskedasticity test (cont.)
Note that
The degrees of freedom is the number of coefficients estimated minus 1.
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17. Remedies Respecification of the model
Include relevant omitted variable(s)
Express model in log-linear form or some other appropriate functional form
Where respecification won’t solve the problem use robust Heteroskedastic Consistent Standard Errors (due to Hal White, Econometrica 1980)

18. Iteratively weighted least squares (IWLS)
Obtain estimates of ei2 using OLS
Use these to get "1st round" estimates of σi
Using formula above replace wi with 1/ si and obtain new estimates for a and ß.
Adjust data
Use these to re-estimate
Repeat Step 3-5 until a and ß converge.
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