1. Lecture 4: Count Data Models
Jacek Wallusch _________________________________ Introduction to Econometrics
Lecture 4:
Count Data Models

2. Count Data ____________________________________________________________________________________________
definitions
Count
1. to name numbers in regular order;
2. to have a specified value [a touchdown counts for six points]*;
Integer
1. anything complete in itself; entity;
2. any positice or negative whole number or zero;
ItE: 4
Webster’s New World College Dictionary
* That’s Webster as well!!!

3. Count Data ____________________________________________________________________________________________
Example
Doctor Visits: number of doctor visits per one respondent within a specified period of time
explanatory variables:
socioeconomic – gender, age, squared age, income
health insurance status indicators – insured, not insured
recent health status measure – number of illnesses in past 2 weeks, number of days of reduced activity
long-term health status measure – health questionnaire score
ItE: 4
Example: Cameron and Trivedi 1998

4. Count Data ____________________________________________________________________________________________
Example
Doctor Visits: see cont_docvisits.xls
exlaining variable: avg. = std.dev. = 7.395
skew = kurt =
ItE: 4
Example: Katchova 2013

5. Count Data ____________________________________________________________________________________________
Examples
Other Topics:
research and development: patents granted, domestic patents, foreign patents etc.
sales: number of items sold
quality control: number of defective items produced
Important note: time structure and/or cross-section
ItE: 4

6. Count Data ____________________________________________________________________________________________
Background
Poisson distribution and rare events
Poisson density:
expected value and variance:
Warning: equidispersion property is often violated in practice, i.e.
ItE: 4
m – intensity (rate) coefficient, t – exposure (length of time during which the events are recorded)

7. Estimation ____________________________________________________________________________________________
coefficients
Poisson distribution and rare events
The model:
left-hand variable y: number of occurences of the event of interest (e.g. doctor visits)
mean coefficient:
ItE: 4
b – coefficient vector, x – vector of linearly independent regressors

8. Conditional probability that the left hand variable = y
Poisson model
Poisson distribution and estimated probabilities
Using the estimated mean to calculate probabilities:
condition
Conditional probability that the left hand variable = y
ItE: 4
b – coefficient vector, x – vector of linearly independent regressors

9. Other Distributions ____________________________________________________________________________________________
negative binomial
Important limitation: Overdispersion
Solution: negative binomial model
mean coefficient:
variance:
ItE: 4
r – number of trials at wich success occurs, p – probability of success

10. Distributions ____________________________________________________________________________________________
examples
Randomly generated distributions
descriptive statistics:
min
max 9
average
3.024
4.495
variance
2.884
2.286
ItE: 4

11. Results ____________________________________________________________________________________________
Interpretation
Marginal Effects
Exponential conditional mean:
differentiation:
Procedures:
ItE: 4
(1) average response after aggregating over all individuals vs. (2) response for the individual with average characteristics

12. Methods of Estimatios ____________________________________________________________________________________________
Overdispersion
GRETL
Test for overdispersion:
null hypothesis: no overdispersion
interpretation: rejection of the null suggests the need for different distribution
ItE: 4

13. Methods of Estimatios ____________________________________________________________________________________________
Overdispersion
GRETL
NEG BIN 2:
a-coefficient: measure of heterogeneity between individuals
NEG BIN 1:
conditional variance: scalar multiple (g) of conditional mean
ItE: 4