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.

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

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

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

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 = -∞

Sigmoid curve

Probability Ratio Note that Where

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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?

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