Heteroskedasticity, Moderation, and Extremity in Heterogeneous Choice Models

1. Heteroskedasticity, Moderation, and Extremity in Heterogeneous Choice Models GARRETT GLASGOWUniversity of California, Santa Barbara

2. Heterogeneous Choice Models • Uncorrected heteroskedasticity in binary and ordinal choice models will produce biased estimates. • Heteroskedasticity may also be of substantive interest. • Heterogeneous choice models developed to model this heteroskedasticity.

3. Heteroskedasticity or Something Else? • Unfortunately, in some cases heterogeneous choice models will produce results that look like heteroskedasticity when the error term is actually homoskedastic. • I consider three cases here: a binary dependent variable, an ordinal dependent variable, and a skewed ordinal dependent variable.

4. Case #1: Binary Dependent Variable Heteroskedasticity or Moderation?

5. Heterogeneous Choice, Binary Dependent Variable • Heteroskedastic probit model: • As Hi increases, choice probabilities converge to 0.5.

6. Binary Dependent Variable With Heteroskedasticity

7. Binary Dependent Variable With Moderation

8. Monte Carlo Study • Generated 1000 data sets, 1000 observations each. y* = XB + e. y = 1 if y*>0, y = 0 otherwise. • First condition: half of observations have larger error variance multiplied by 2 (heteroskedasticity) • Second condition: half of observations have additional variable = –X/2 (moderation). • Estimated heteroskedastic probit under both conditions.

9. Monte Carlo Results • Heteroskedasticity and moderation can be indistinguishable in the binary dependent variable case.

10. Case #2: Ordinal Dependent Variable Heteroskedasticity or Extremity?

11. Heterogeneous Choice, Ordinal Dependent Variable • Heteroskedastic ordered probit model: • As Hi increases, choice probabilities converge to 0.5 for extreme categories, 0 for middle categories.

12. Ordinal Dependent Variable With Heteroskedasticity

13. Ordinal Dependent Variable With Extremity

14. Heterogeneous Choice, Ordinal Dependent Variable, Model 2 • Modified heteroskedastic ordered probit model: • As Hi increases, choice probabilities converge to 1/M for each choice category. Variance in the observed rather than latent variable.

15. Example #1: Working and Motherhood

16. Distribution of “Warm” by Gender

17. Example #2: Race and Ambivalence

18. Distribution of “Quota” by Ambivalence

19. Case #3: Skewed Ordinal Dependent Variable Heteroskedasticity or Left-Right?

20. Skewed Ordinal Dependent Variable With Heteroskedasticity

21. Race and Ambivalence, Model 2

22. Conclusions • Distinguishing heteroskedasticity from other effects on the choice probabilities is difficult. • Several models considered, but all results could be explained by effects other than heteroskedasticity. • Perhaps this is a problem that must be solved through theory and measurement rather than a statistical model.