1. Heteroskedasticity The Problem:
Classical assumption of homoskedasticity is violated:
V(ei) - not constant
When does this happen?
Time Series
Yields
Prices
Cross Section
consumption patterns
firm size - performance (revenue/sales)

2. Food Expenditures versus Income (Before Taxes)
Data Source: 1996 Statistics Canada expenditure survey

3. Heteroskedasticity - Consequences
1) The OLS estimators:
Estimators: Linear unbiased
Not best (minimum variance)
2) More Problems: OLS variance estimate is incorrect:
3) True variance:

4. Ignoring Heteroskedasticity - Using OLS1)
2) V(b2) - is not correct
3) Hypothesis tests, confidence intervals, F-tests
all invalid
3) Using V(b2)* - correct variance estimator is not best

5. 1) Graphical Methods - subjective (residual plots)
DIAGNOSTICS
1) Graphical Methods - subjective (residual plots)
> focus on the OLS residuals
> is there a pattern ?
METHOD
Plot residuals
Are there patterns?
2) Empirical Tests:
Park Test
Goldfeld Quandt Test
Breusch-Pagan Test
White Test

6. 1) Graphical Methods: Income and Expenditures

7. PARK 2 STAGE TEST THE TEST
R. E. Park Estimation with Heteroscedastic Error Terms, Econometrica,
Vol. 34, No. 4. (Oct., 1966), p. 888.
Park proposed the following general hypothesis:
The variance of the disturbance increases as the independent variable increases.
THE TEST
In order to test this hypothesis he proposed the following specific model:

8. if  0 => heteroscedastic
PARK 2 STAGE TEST
STEP 1
Run a regression and compute the squared OLS residuals
STEP 2
Run the regression equation above: i.e
if  0 => heteroscedastic
if  = => homoscedastic

9. Issues with the Park Test
1 ) Functional Form
2) What happens if vi is Heteroscedastic ?
3) Power of the Test (Type II error)
Accept Ho:  = (homoscedastic) ??
=> Perhaps choose   0.10 (significance)

10. Park Test: Example with Income and Food Expenditures
Data: Statistics Canada 1996 Expenditure Survey
1000 observations were selected at random and then the data were trimmed
so that income was a minimum of $5000 and a maximum of $300,000.
This left 938 observations.
Two versions of the Park Test were run:
1) Regress natural log of the squared residuals on the natural log of income before taxes.
2) Regress the squared residuals on income
. predict err, residuals
. gen e2 = err^2
. gen le2 = ln(e2)
. gen libt = ln(ibt)

11. Park Test: Example with Income and Food Expenditures
. regress le2 libt
Source | SS df MS Number of obs =
F( 1, 936) =
Model | Prob > F =
Residual | R-squared =
Adj R-squared =
Total | Root MSE =
le2 | Coef. Std. Err t P>|t| [95% Conf. Interval]
libt |
_cons |
Conclusion ?

12. Park Test: Example with Income and Food Expenditures
. regress e2 ibt
Source | SS df MS Number of obs =
F( 1, 936) =
Model | e e Prob > F =
Residual | e e R-squared =
Adj R-squared =
Total | e e Root MSE = 2.0e+07
e2 | Coef. Std. Err t P>|t| [95% Conf. Interval]
ibt |
_cons |
Conclusion ?

13. White Test (1980) Test statistic: STEP 1: STEP 2: Test H0:
Regress OLS residuals on regressors + squares + X-products
STEP 2: Test H0:
Ho: all slope coefficients = 0 (Jointly)
e.g. Homoscedastic residuals
Test statistic: