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 Y
t
is the occurrence (or not) of war, and X
t
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:
X
i
= +
∞
X
i
=

∞
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 R
2
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
χ
2
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 (I
i
) 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(Y
i
=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
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