Limited Dependent Variables

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Limited Dependent Variables

• Often there are occasions where we are interested
  in explaining a dependent variable that has only
  limited measurement
• Frequently it is even dichotomous.

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Examples

• War(1) vs. no War(0)
• Vote vs. no vote
• Regime change vs. no change

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These are often Probability Models

• E.g.
• Power disparity leads to war
• Where Yt is the occurrence (or not) of war, and
  Xt is a measure of power disparity
• We call this a Linear Probability Model

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Problems with LPM Regression

• OLS in this case is called the Linear Probability
  Model
• Running regression produces some problems
• Errors are not distributed normally
• Errors are heteroskedastic
• Predicted Ys can be outside the 0.0-1. bounds
  required for probability

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Logistic Model

• We need a model that produces probabilities
• The Logit, or cumulative logistic distribution
  offers one approach.
• Look at equation under 2 conditions
• Xi 8
• Xi -8

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Probability Ratio

• Note that
• Where

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Log Odds Ratio

• The odds ration is given by
• This model gives us a coefficient that may be
  interpreted as a change in the weighted odds of
  the dependent variable

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Estimation of Model

• We estimate this with maximum likelihood
• The significance tests are z statistics
• We can generate a Pseudo R2 which is an attempt
  to measure the percent of variation of the
  underlying logit function explained by the
  independent variables
• We test the full model with the Likelihood Ratio
  test (LR), which has a ?2 distribution with k
  degrees of freedom

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Probit

• If we3 can assume that the dependent variable is
  actually the result of an underlying (and
  unmeasurable) propensity or utility, we can use
  the cumulative normal probability function to
  estimate a Probit model
• It looks just like a logit model in practice

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The Cumulative Normal Density Function

• The normal distribution is given by
• The Cumulative Normal Density Function is

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The Standard Normal CDF

• We assume that there is an underlying threshold
  value (Ii) that if the case exceeds will be a 1
  and 0 otherwise.
• We can standardize and estimate this as

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Probit estimates

• Again, maximum likelihood estimation
• Again, a Pseudo R2
• Again, a LR ratio with k degrees of freedom

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Ordered Probit

• If the dependent variable can take on ordinal
  levels, we can extend the dichotomous Probit
  model to an n-chotomous, or ordered, Probit model
• It simply has several threshold estimated
• Ordered logit works the same way

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Multinomial Logit

• If out dependent variable takes on different
  values, but they are nominal, this is a
  multinomial logit model

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Some additional info

• Modal category is good benchmark
• Present correctly predicted

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Tobit

• Assumes 0 and scale
• E.g., the decision to incarcerate
• 0 or 1
• If 1, than how many

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Other models

• This leads to many other models
• Count models Poisson regression
• Duration/Survival/hazard models
• Censoring and truncation models
• Selection bias models

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Maximum Likelihood Estimation
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