1. Qualitative and Limited Dependent Variable Models
ECON 6002
Econometrics
Memorial University of Newfoundland
Qualitative and Limited Dependent Variable Models
Chapter 16
Adapted from Vera Tabakova’s notes

2. Chapter 16: Qualitative and Limited Dependent Variable Models
16.1 Models with Binary Dependent Variables
16.2 The Logit Model for Binary Choice
16.3 Multinomial Logit
16.4 Conditional Logit
16.5 Ordered Choice Models
16.6 Models for Count Data and extensions
16.7 Limited Dependent Variables
Principles of Econometrics, 3rd Edition

3. 16.6 Models for Count Data
When the dependent variable in a regression model is a count of the number of occurrences of an event, the outcome variable is y = 0, 1, 2, 3, … These numbers are actual counts, and thus different from the ordinal numbers of the previous section. Examples include:
The number of trips to a physician a person makes during a year.
The number of fishing trips taken by a person during the previous year.
The number of children in a household.
The number of automobile accidents at a particular intersection during a month.
The number of televisions in a household.
The number of alcoholic drinks a college student takes in a week.
Principles of Econometrics, 3rd Edition

4. 16.6 Models for Count Data
If Y is a Poisson random variable, then its probability function is This choice defines the Poisson regression model for count data.
(16.27)
“rate”
Also equal
To the variance
(16.28)
Principles of Econometrics, 3rd Edition

5. 16.6.1 Maximum Likelihood Estimation
If we observe 3 individuals: one faces no event, the other two two events each:
Principles of Econometrics, 3rd Edition

6. 16.6.2 Interpretation in the Poisson Regression Model
So now you can calculate the predicted probability
of a certain number y of events
Principles of Econometrics, 3rd Edition

7. 16.6.2 Interpretation in the Poisson Regression Model
(16.29)
You may prefer to express this marginal effect as a %:
Principles of Econometrics, 3rd Edition

8. 16.6.2 Interpretation in the Poisson Regression Model
If there is a dummy
Involved, be careful,
remember
Which would be identical
to the effect of a dummy
In the log-linear model
we saw under OLS
Principles of Econometrics, 3rd Edition

9. Extensions: overdispersion
Under a plain Poisson the mean of the count is assumed to be equal to
the average (equidispersion)
This will often not hold
Real life data are often overdispersed
For example:
a few women will have many affairs and many women will have few
a few travelers will make many trips to a park and many will make few
etc.
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10. Extensions: overdispersion
use "C:\bbbECONOMETRICS\Rober\GRAD\GROSMORNE.dta", clear
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11. Extensions: negative binomial
Under a plain Poisson the mean of the count is assumed to be equal to
the average (equidispersion)
The Poisson will inflate your t-ratios in this case, making you think that your model works better than it actually does 
Or use a Negative Binomial model instead (nbreg) or even a Generalised Negative Binomial (gnbreg) , which will allow you to model the overdispersion parameter as a function of covariates of our choice
You can also test for overdispersion, to test whether the problem is significant
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12. Extensions: negative binomial
sum visits
Variable | Obs Mean Std. Dev Min Max
visits |
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13. Extensions: negative binomial
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14. Extensions: excess zeros
Often the numbers of zeros in the sample cannot be accommodated
properly by a Poisson or Negative Binomial model
They would underpredict them too
There is said to be an “excess zeros” problem
You can then use hurdle models or zero inflated or zero augmented
models to accommodate the extra zeros
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15. Extensions: excess zeros
Often the numbers of zeros in the sample cannot be accommodated
properly by a Poisson or Negative Binomial model
They would underpredict
them too
nbvargr
Is a very useful
command
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16. Extensions: excess zeros
You can then use hurdle models or zero inflated or zero augmented
models to accommodate the extra zeros
They will also allow you to have a different process driving the value of the
strictly positive count and whether the value is zero or strictly positive
EXAMPLES:
Number of extramarital affairs versus gender
Number of children before marriage versus religiosity
In the continuous case, we have similar models (e.g. Cragg’s Model) and an example is that of size of Insurance Claims from fires versus the age of the building
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17. Extensions: excess zeros
You can then use hurdle models or zero inflated or zero augmented
models to accommodate the extra zeros
Hurdle Models A hurdle model is a modified count model in which there are two processes, one generating the zeros and one generating the positive values. The two models are not constrained to be the same. In the hurdle model a binomial probability model governs the binary outcome of whether a count variable has a zero or a positive value. If the value is positive, the "hurdle is crossed," and the conditional distribution of the positive values is governed by a zero-truncated count model.
Example: smokers versus non-smokers, if you are a smoker you will smoke!
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18. Extensions: excess zeros
Hurdle Models In Stata Joseph Hilbe’s downloadable ado hplogit will work, although it does not allow for two different sets of variables, just two different sets of coefficients
See also hnblogit
Example: smokers versus non-smokers, if you are a smoker you will smoke!
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19. Extensions: excess zeros
You can then use hurdle models or zero inflated or zero augmented
models to accommodate the extra zeros
Zero-inflated models (initially suggested by D. Lambert) attempt to account for excess zeros in a subtly different way.
In this model there are two kinds of zeros, "true zeros" and excess zeros.
Zero-inflated models estimate also two equations, one for the count model and one for the excess zero's.
The key difference is that the count model allows zeros now. It is not a truncated count model, but allows for “corner solutions”
Example: meat eaters (who sometimes just did not eat meat that week) versus vegetarians who never ever do
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20. Extensions: excess zeros
webuse fish
We want to model how many fish are being caught by fishermen at a state park.
Visitors are asked how long they stayed, how many people were in the group, were there children in the group and how many fish were caught.
Some visitors do not fish at all, but there is no data on whether a person fished or not.
Some visitors who did fish did not catch any fish (and admitted it ) so there are excess zeros in the data because of the people that did not fish.
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21. Extensions: excess zeros
. histogram count, discrete freq
Lots of zeros!
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22. Extensions: excess zeros
Vuong test
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23. Extensions: excess zeros
Vuong test
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24. Extensions: truncation
Count data can be truncated too (usually at zero)
So ztp and ztnb can accommodate that
Example: you interview visitors at the recreational site, so they all made at least that one trip
In the continuous case we would have to use the truncreg command
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25. Extensions: truncation
This model works much better and showcases the bias in the previous estimates:
Smaller now estimated Consumer Surplus
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26. Extensions: truncation
This model works much better and showcases the bias in the previous estimates:
Now accounting for overdispersion
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27. Extensions: truncation and endogenous stratification
Example: you interview visitors at the recreational site, so they all made at least that one trip
You interview patients at the doctors’ office about how often they visit the doctor
You ask people in George St. how often the go to George St…
Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially
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28. Extensions: truncation and endogenous stratification
Then you are oversampling “frequent visitors” and biasing your estimates, perhaps substantially
It turns out to be supereasy to deal with a Truncated and Endogenously Stratified Poisson Model (as shown by Shaw, 1988):
Simply run a plain Poisson on “Count-1” and that will work (In STATA: poisson on the corrected count)
It is more complex if there is overdispersion though 
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29. Extensions: truncation and endogenous stratification
Supereasy to deal with a Truncated and Endogenously Stratified Poisson Model
Much smaller now estimated Consumer Surplus
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30. Extensions: truncation and endogenous stratification
Endogenously Stratified Negative Binomial Model (as shown by Shaw, 1988; Englin and Shonkwiler, 1995):
Even after accounting for overdispersion, CS estimate is relatively low
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31. Extensions: truncation and endogenous stratification
How do we calculate the pseudo-R2 for this model???
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32. Extensions: truncation and endogenous stratification
GNBSTRAT will also allow you to model the overdispersion parameter in this case, just as gnbreg did for the plain case
See also Hilbe’s CENPOIS: Stata module to estimate censored maximum likelihood Poisson regression models
And in general take a good look at:
Hilbe, J. (2011). Negative Binomial Regression, 2nd ed. Cambridge, UK: Cambridge University Press.
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33. Extensions: endogeneity
Sample selection models and endogenous switching (ssm and espoisson)
(See also movestay would work for a continuous dependent variable in a similar setting)
Endogenous treatment models
Mtreatnb allows for a multinomial treatment
(from Stata Help: mtreatnb fits a treatment-effects model that considers the effects of an endogenously chosen multinomial treatment on another endogenous count outcome, conditional on two sets of independent variables. The treatment variable is modeled via a multinomial logit and the outcome via a negative binomial regression. The model is fitted using maximum simulated likelihood. The simulator uses Halton sequences.)
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34. Extensions: multivariate models
bivariate poisson, and my personal favourite, at least for the name: the SUPREME model
King, G. A seemingly unrelated Poisson regression model Sociological Methods and Research, 1989, 17, 235–255
bivariate NB (seemingly unrelated negative binomial)
Hausman et al. (1984) and a bit more flexible in: Winkelmann, R. Seemingly unrelated negative binomial regression Oxford Bulletin of Economics and Statistics, 2000, 62,
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35. Extensions: mixed-effects Poisson
xtmepoisson (from STATA help) fits mixed-effects models for count responses.
Mixed models contain both fixed effects and random effects. The fixed effects are analogous to standard regression coefficients and are estimated directly.
The random effects are not directly estimated (although they may be obtained postestimation) but are summarized according to their estimated variances and covariances.
Random effects may take the form of either random intercepts or random coefficients, and the grouping structure of the data may consist of multiple levels of nested groups
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36. Extensions: mixed-effects Poisson
xtmepoisson (from STATA help) fits mixed-effects models for count responses.
The distribution of the random effects is assumed to be Gaussian. The conditional distribution of the response given the random effects is assumed to be Poisson
Because the log likelihood for this model has no closed form, it is approximated by adaptive Gaussian quadrature.
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37. Extensions: finite mixture models
AKA Latent Class Models
Fmm
See examples in the works by Deb and Trivedi for medical care (see Cameron & Trivedi MMA and MUS)
And, again Hilbe (2011)
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38. Extensions: panels and pseudo panels
Xtpoisson, xtnb
Xtgee in general
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39. NOTE: what is the exposure
Count models often need to deal with the fact that the counts may be measured over different observation periods, which might be of different length (in terms of time or some other relevant dimension)
For example, the number of accidents are recorded for 50 different intersections. However, the number of vehicles that pass through the intersections can vary greatly. Five accidents for 30,000 vehicles is very different from five accidents for 1,500 vehicles.
Count models account for these differences by including the log of the exposure variable in model with coefficient constrained to be one.
The use of exposure is often superior to analyzing rates as response variables as such, because it makes use of the correct probability distributions
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40. Keywords binary choice models logistic random variable censored data
logit
conditional logit
log-likelihood function
count data models
marginal effect
feasible generalized least squares
maximum likelihood estimation
Heckit
multinomial choice models
identification problem
multinomial logit
independence of irrelevant alternatives (IIA)
odds ratio
ordered choice models
index models
ordered probit
individual and alternative specific variables
ordinal variables
Poisson random variable
individual specific variables
Poisson regression model
latent variables
probit
likelihood function
selection bias
limited dependent variables
tobit model
linear probability model
truncated data
Principles of Econometrics, 3rd Edition

41. References Hoffmann, 2004 for all topics
Long, S. and J. Freese for all topics, most of all for postestimation and reporting tricks
Cameron and Trivedi’s book for count data
Winkelmann’s 2008 book on count data is free as an ebook from the QEII
Hilbe (2011) for NB related models and count data models in general
Cameron&Trivedi’s MUS and MMA
Greene’s Econometric Analysis
Agresti, A. (2001) Categorical Data Analysis (2nd ed). New York: Wiley