Heteroskedasticity

1
Heteroskedasticity

• Distribution of Error Terms Does Not Have A
  Constant Variance.

2
Homoskedastic Errors
3
Heteroscedastic Errors

• VAR(ei)s2Z2i where Z proportionality factor

4
Heteroscedastic Errors
5
Examples Where Problem Might Arise

• Specification Error
• Cross Sectional Data with Large Variation in the
  Dependent Variable
• Improvement in Data Collection

6
Consequences of Heteroskedasticity

• Unbiased coefficients
• Variance of the Beta distribution increases
• OLS underestimates true variance and
  overestimates t-statistics

7
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.

8
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.

9
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

10
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)

11
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)

12
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