1 / 12

Heteroskedasticity

- Distribution of Error Terms Does Not Have A

Constant Variance.

Homoskedastic Errors

Heteroscedastic Errors

- VAR(ei)s2Z2i where Z proportionality factor

Heteroscedastic Errors

Examples Where Problem Might Arise

- Specification Error
- Cross Sectional Data with Large Variation in the

Dependent Variable - Improvement in Data Collection

Consequences of Heteroskedasticity

- Unbiased coefficients
- Variance of the Beta distribution increases
- OLS underestimates true variance and

overestimates t-statistics

Detection of Heteroskedasticity

- All require an educated guess about Z.
- Park Test and Goldfeld-Quandt tests-- choose only

one Z - Breusch-Pagan and White tests -- choose multiple

Zs. - White test does not assume any particular form of

heteroskedasticity.

Park Test

- Identify a variable (the proportionality factor

Z) to which the error variance is related. - Obtain residuals from estimated regression

equation.

- Use these residuals to form the dependent

variable in a second equationln(ei2)

a0a1lnZiui - Test the significance of a1 with a t-test.

Goldfeld-Quandt Test

- Identify a variable (the proportionality factor

Z) to which the error variance is related. - Arrange the data set according the Z.
- Divide the sample of T observations into thirds.
- Estimate separate regressions are run on the

first third and on the last third of the data.

- Obtain the RSS for each third.
- Compute the F test-statistic to test whether the

sum of squared residuals from the last third of

the estimated equation is greater than those from

the first third. - Test Statistic is GQRSS1/RSS2

Breusch-Pagan Test

- Obtain the residuals of the estimated regression

equation. - Use the squared residuals as the dependent

variable in a secondary equation that includes

all independent variables suspected of being

related to error term.

- (ei)2a0a1Z1ia2Z2iapZpi ui
- Test the joint hypothesis that all the

coefficients in the second regression are zero.

(A Chi-Square test)

White Test

- Obtain the residuals of the estimated regression

equation - Use the squared residuals as the dependent

variable and estimate the following equation

where Xs are explanatory variables from the

original equation.

- (ei)2a0a1X1ia2X2ia3X3i a4X21i a5X22i

a6X23i a7X1i X2i a8X1iX3i a9X2i X3iui - Test the joint hypothesis that all the

coefficients are zero. (Chi-square test)

Solutions

- Redefine the variables
- Such as use of per capita variables
- Weighted Least Squares
- Divide the equation through by Z.
- Re-estimate the equation
- Heteroskedastic Corrected Errors
- Yields more accurate standard errors, though

still biased - Works for large samples