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

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!!!

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

Count Data ____________________________________________________________________________________________ Example Doctor Visits: see cont_docvisits.xls exlaining variable: avg. = 6.823 std.dev. = 7.395 skew = 4.176 kurt = 46.744 ItE: 4 Example: Katchova 2013

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

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)

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

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

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

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

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

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

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