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 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 = +∞
Xi = -∞

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 χ2 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 Y’s are in {0,1} set
They are statistically independent
No multicollinearity
The P(Yi=1) 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) 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