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Heteroskedasticity, Moderation, and Extremity in Heterogeneous Choice Models. GARRETT GLASGOW University of California, Santa Barbara. Heterogeneous Choice Models. Uncorrected heteroskedasticity in binary and ordinal choice models will produce biased estimates.

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heteroskedasticity moderation and extremity in heterogeneous choice models

Heteroskedasticity, Moderation, and Extremity in Heterogeneous Choice Models

GARRETT GLASGOWUniversity of California, Santa Barbara

heterogeneous choice models
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.
heteroskedasticity or something else
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.
case 1 binary dependent variable

Case #1: Binary Dependent Variable

Heteroskedasticity or Moderation?

heterogeneous choice binary dependent variable
Heterogeneous Choice, Binary Dependent Variable
  • Heteroskedastic probit model:
  • As Hi increases, choice probabilities converge to 0.5.
monte carlo study
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.
monte carlo results
Monte Carlo Results
  • Heteroskedasticity and moderation can be indistinguishable in the binary dependent variable case.
case 2 ordinal dependent variable

Case #2: Ordinal Dependent Variable

Heteroskedasticity or Extremity?

heterogeneous choice ordinal dependent variable
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.
heterogeneous choice ordinal dependent variable model 2
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.
case 3 skewed ordinal dependent variable

Case #3: Skewed Ordinal Dependent Variable

Heteroskedasticity or Left-Right?

conclusions
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.