1. Discrete Choice Modeling
William Greene
Stern School of Business
New York University
Lab Sessions

2. Lab Session 7
Models for Counts

3. Data Set healthcare.lpj Data for this session are
Refer to healthcare.lim for full list of the variables.
This is an unbalanced panel. The group counter is already in the data set. Use ;PDS=_Groupti for panel models

4. Dependent Variables: Count
DOCVIS = count of visits to the doctor
HOSPVIS = count of visits to the hospital.
There are outliers. It helps to use truncated or censored samples
(1) Truncated Data
SAMPLE ; All $
REJECT ; DocVis > 10 $ before using or
REJECT ; HospVis > 10 $ before using.
Then, if using a panel data estimator, use
REGRESS ; Lhs = One ; Rhs = One ; Str = ID ; Panel $
to create the _GROUPTI count variable
(2) Censored Data
CREATE ; DocVis10 = Min(10,DocVis) ; Hosp10 = Min(10,HospVis) $

5. Models for Count Data Basic models Poisson and negative binomial
POISSON ; Lhs = y ; Rhs = One,… $
NEGBIN ; Lhs = y ; Rhs = One,… $
Many extensions
Various heterogeneity forms
Panel data
Random parameters and latent class
Zero inflation
Sample selection
Censoring and truncation
Numerous others… (some, far from all, shown below)

6. Robust Covariance Matrix
“Robust” sandwich estimator is appropriate for the Poisson and other loglinear models
POISSON ; … ; ROBUST $

7. “Offset” Variables Poisson (and NB) mean is a rate per unit of time.
In the sample, all observations should be observed (exposed) for the same length of time.
Else, the appropriate model is as shown below.

8. Exposure Variable

9. Sample Selection

10. Estimating a Selection Model
? Selection Equation
Probit ; Lhs = … ; Rhs = …
; Hold $
? Main Regression Equation
Poisson ; Lhs = … ; Rhs = …
; Selection $

11. NegBin with Heterogeneity in Alpha
In the negative binomial model, the overdispersion parameter is α, with the model assumption
λ = exp(β’x)
E[y|x] = λ
Var[y|x] = λ[1+ α λ]
We allow α to be heterogeneous:
α = exp(δ’z)
Use ; Hfn = … variables in z

12. Censoring and Truncation
Censoring limit C
Right (upper): Values larger than C are set to Ci. The largest value in the sample is C. Use ;LIMIT=C;MAXIMUM
Left (lower): Values less than C are set equal to C. The smallest value in the sample is C. Use ;LIMIT=C
Truncation limit C
Right (upper): Values greater than or equal to C have been discarded. The largest value in the sample is C-1. Use ;LIMIT=C;TRUNCATION;UPPER
Left (lower): Values less than or equal to C have been discarded. The smallest value in the sample is C+1. Use ;LIMIT=C;TRUNCATION

13. Zero Inflation Two regime, latent class model Reduced form:
Prob[Regime 1 => y=0] = q
Prob[y = j|Regime 2] = Poisson or NegBin, λ=exp(β’x)
Reduced form:
Prob[y=0] = q + (1-q)P(0)
Prob[y=j > 0] = (1-q)P(j)
Regime Models:
q = Probit or Logit
Structures:
ZIP: Probit or Logit F(γ’z) z can be any set of variables
ZIP-tau: Probit or Logit F(τ β’x) – same β’x as above

14. ZIP and ZIP-tau Models ;ZIP Logit, ZIP-tau
;ZIP = Normal: Probit ZIP-tau
;ZIP [=Normal] ; Rh2 = variables in z
Alternative Models
Default is Poisson
;MODEL = NegBin
;MODEL = Gamma

15. Hurdle Model Two Part Model:
Prob[y=0] = Logit or Probit using β’x from the count model or γ’z as specified with ;RH2=list
Prob[y=j|j>0] = Truncated Poisson or NegBin
Two part decision: Drug or alcohol use, for example
POISSON ; … ; Hurdle $
POISSON ; … ; Hurdle ; Rh2 = List $

16. Poisson and NB with Normal Heterogeneity

17. Fixed and Random Effects
Poisson or NegBin
;PDS=setting
Fixed Effects
Default is a conditional esitmator
;FEM uses the unconditional estimator
The two are algebraically identical but use different algorithms
Random Effects
Use ; RANDOM
Can be fit as a random parameters model with just a random constant

18. Random Parameters Poisson ; LHS = dependent variable
; RHS = independent variable(s)
; PDS = setting (may be ;PDS=1)
; RPM ; PTS = number of Points
; Halton (for smarter integration method)
; Correlated if desired to fit correlated
parameters model
; FCN = variable(n) , variable(n), …
to indicate which parameters are random

19. Latent Class Poisson (or NEGBIN) ; LHS = dependent variable
; RHS = independent variable(s)
; LCM for a latent class model
; LCM = variables if probabilities are heterogeneous
; PDS = setting (may be ;PDS=1)
; PTS = number of latent classes