Limited Dependent VariablesVariables
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Transcript of Limited Dependent VariablesVariables
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Limited Dependent Variables
Often there are occasions where we are
interested in explaining a dependentvariable 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 Xtis a measure of power disparity
We call this a Linear Probability Model
ttt eXBaY 1
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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 true probabilities
The Logit, or cumulative logistic distribution offers one
approach.
This produces a sigmoid curve.
Look at equation under 2 conditions: Xi = +
Xi
= -
)( 211
1iXBBi e
Y
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Sigmoid curve
http://en.wikipedia.org/wiki/Logistic_function -
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Probability Ratio
Note that
Where
Z
Z
ZXBBie
eee
Pii
11
11
1)_( 21
ii XBBZ 21
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Log Odds Ratio
The logit is the log of the odds ratio, and is givenby:
This model gives us a coefficient that may beinterpreted as a change in the weighted odds ofthe dependent variable
ii
i
ii XBBZ
PPL 21
1ln
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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 tomeasure the percent of variation of the underlyinglogit function explained by the independentvariables
We test the full model with the Likelihood Ratiotest (LR), which has a 2 distribution with k degreesof freedom
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Neural Networks
The alternate formulation is representative of asingle-layer perceptron in an artificial neural
network.
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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
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The Cumulative Normal Density
Function
The normal distribution is given by:
The Cumulative Normal Density Function is:
2
2
2
)(
22
1)(
X
eXf
0 2
2
2
)(
22
1)(
XX
eXF
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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
iXBB
zi dzeIF21 2
2/
2
1)(
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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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Assumptions of Models
All Ys 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
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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 values
estimated
Ordered logit works much the same way
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Multinomial Logit
If our dependent variable takes on different
values, but they are nominal, this is a
multinomial logit model
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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.
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Stata
Use Leadership Change data
(1992 cross section) 1992-Stata
http://localhost/var/www/apps/conversion/tmp/Data/logit_data_st9_1992.xlshttp://localhost/var/www/apps/conversion/tmp/Data/Logit_Data_ST9_1992.dtahttp://localhost/var/www/apps/conversion/tmp/Data/Logit_Data_ST9_1992.dtahttp://localhost/var/www/apps/conversion/tmp/Data/Logit_Data_ST9_1992.dtahttp://localhost/var/www/apps/conversion/tmp/Data/Logit_Data_ST9_1992.dtahttp://localhost/var/www/apps/conversion/tmp/Data/logit_data_st9_1992.xls -
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Test different models
Dependent variable Leadership change
Examine distributiontables ledchan1
Independent variables
Try different
Try corrand then (pwcorr)
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Try the following
regress ledchan1 grwthgdp hlthexp i l l i t_f pol i ty2
logit ledchan1 grwthgdp hl thexp il l i t_f pol ity2
logistic ledchan1 grwthgdp hl thexp il l i t_f poli ty2
probit ledchan1 grwthgdp hl thexp i l l i t_f poli ty2
ologit ledchan1 grwthgdp hl thexp il l i t_f pol ity2
oprobit ledchan1 grwthgdp hl thexp il l i t_f poli ty2
mlogit ledchan1 grwthgdp hl thexp i l l i t_f poli ty2
tobit ledchan1 grwthgdp hl thexp il l i t_f poli ty2, ul l l
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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?
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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