Limited Dependent Variables

1
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.

2
Examples

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

3
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

4
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

5
Logistic Model

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

6
Sigmoid curve
7
Probability Ratio

• Note that
• Where

8
Log Odds Ratio

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

9
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

10
Neural Networks

• The alternate formulation is representative of a
  single-layer perceptron in an artificial neural
  network.

11
Probit

• If we can assume that the dependent variable is
  actually the result of an underlying (and
  immeasurable) propensity or utility, we can use
  the cumulative normal probability function to
  estimate a Probit model
• Also, more appropriate if the categories (or
  their propensities) are likely to be normally
  distributed
• It looks just like a logit model in practice

12
The Cumulative Normal Density Function

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

13
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

14
Probit estimates

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

15
Assumptions of Models

• All Ys are in 0,1 set
• They are statistically independent
• No multicollinearity
• The P(Yi1) is normal density for probit, and
  logistic function for logit

16
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 values estimated
• Ordered logit works much the same way

17
Multinomial Logit

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

18
Some additional info

• The Modal category is good benchmark
• Present correctly predicted
• This can be calculated and presented.
• This, when compared to the modal category, gives
  us a good indication of fit.

19
Stata

• Use Leadership Change data
• (1992 cross section) 1992-Stata

20
Test different models

• Dependent variable Leadership change
• Examine distribution
• tables ledchan1
• Independent variables
• Try different
• Try corr and then (pwcorr)

21
Try the following

• regress ledchan1 grwthgdp hlthexp illit_f polity2
• logit ledchan1 grwthgdp hlthexp illit_f polity2
• logistic ledchan1 grwthgdp hlthexp illit_f
  polity2
• probit ledchan1 grwthgdp hlthexp illit_f polity2
• ologit ledchan1 grwthgdp hlthexp illit_f polity2
• oprobit ledchan1 grwthgdp hlthexp illit_f
  polity2
• mlogit ledchan1 grwthgdp hlthexp illit_f polity2
• tobit ledchan1 grwthgdp hlthexp illit_f polity2,
  ul ll

22
Tobit

• Assumes a 0 value, and then a scale
• E.g., the decision to incarcerate
• 0 or 1
• (Imprison or not)
• If Imprison, than for how many years?

23
Other models

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