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

Lab Session 7 Models for Counts

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

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) $

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)

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

“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.

Exposure Variable

Sample Selection

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

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

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

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

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

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 $

Poisson and NB with Normal Heterogeneity

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

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

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