Heteroskedasticity, Moderation, and Extremity in Heterogeneous Choice Models

1
Heteroskedasticity, Moderation, and Extremity in
Heterogeneous Choice Models

• GARRETT GLASGOWUniversity of California, Santa
  Barbara

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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.

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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.

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Case 1 Binary Dependent Variable

• Heteroskedasticity or Moderation?

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Heterogeneous Choice, Binary Dependent Variable

• Heteroskedastic probit model
• As Hi increases, choice probabilities converge to
  0.5.

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Binary Dependent Variable With Heteroskedasticity
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Binary Dependent Variable With Moderation
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Monte Carlo Study

• Generated 1000 data sets, 1000 observations each.
  y XB e. y 1 if ygt0, 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.

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Monte Carlo Results

• Heteroskedasticity and moderation can be
  indistinguishable in the binary dependent
  variable case.

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Case 2 Ordinal Dependent Variable

• Heteroskedasticity or Extremity?

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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.

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Ordinal Dependent Variable With Heteroskedasticity
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Ordinal Dependent Variable With Extremity
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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.

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Example 1 Working and Motherhood
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Distribution of Warm by Gender
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Example 2 Race and Ambivalence
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Distribution of Quota by Ambivalence
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Case 3 Skewed Ordinal Dependent Variable

• Heteroskedasticity or Left-Right?

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Skewed Ordinal Dependent Variable With
Heteroskedasticity
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Race and Ambivalence, Model 2
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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.