Count Data Models

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Section

• Count Data Models

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Introduction

• Many outcomes of interest are integer counts
• Doctor visits
• Low work days
• Cigarettes smoked per day
• Missed school days
• OLS models can easily handle some integer models

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• Example
• SAT scores are essentially integer values
• Few at tails
• Distribution is fairly continuous
• OLS models well
• In contrast, suppose
• High fraction of zeros
• Small positive values

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• OLS models will
• Predict negative values
• Do a poor job of predicting the mass of
  observations at zero
• Example
• Dr visits in past year, Medicare patients(65)
• 1987 National Medical Expenditure Survey
• Top code (for now) at 10
• 17 have no visits

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• visits Freq. Percent
  Cum.
• -----------------------------------------------
• 0 915 17.18 17.18
• 1 601 11.28 28.46
• 2 533 10.01 38.46
• 3 503 9.44 47.91
• 4 450 8.45 56.35
• 5 391 7.34 63.69
• 6 319 5.99 69.68
• 7 258 4.84 74.53
• 8 216 4.05 78.58
• 9 192 3.60 82.19
• 10 949 17.81 100.00
• -----------------------------------------------
• Total 5,327 100.00

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Poisson Model

• yi is drawn from a Poisson distribution
• Poisson parameter varies across observations
• f(yi?i) e-?i ?i yi/yi! For ?igt0
• Eyi Varyi ?i f(xi, ß)

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• ?i must be positive at all times
• Therefore, we CANNOT let ?i xiß
• Let ?i exp(xiß)
• ln(?i) (xiß)

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• d ln(?i)/dxi ß
• Remember that d ln(?i) d?i/?i
• Interpret ß as the percentage change in mean
  outcomes for a change in x

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Problems with Poisson

• Variance grows with the mean
• Eyi Varyi ?i f(xi, ß)
• Most data sets have over dispersion, where the
  variance grows faster than the mean
• In dr. visits sample, ? 5.6, s6.7
• Impose MeanVar, severe restriction and you tend
  to reduce standard errors

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Negative Binomial Model

• Where ?i exp(xiß) and d 0
• Eyi d?i dexp(xiß)
• Varyi d (1d) ?i
• Varyi/ Eyi (1d)

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• d must always be 0
• In this case, the variance grows faster than the
  mean
• If d0, the model collapses into the Poisson
• Always estimate negative binomial
• If you cannot reject the null that d0, report
  the Poisson estimates

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• Notice that ln(Eyi) ln(d) ln(?i), so
• d ln(Eyi) /dxi ß
• Parameters have the same interpretation as in the
  Poisson model

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In STATA

• POISSON estimates a MLE model for poisson
• Syntax
• POISSON y independent variables
• NBREG estimates MLE negative binomial
• Syntax
• NBREG y independent variables

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Interpret results for Poisson

• Those with CHRONIC condition have 50 more mean
  MD visits
• Those in EXCELent health have 78 fewer MD visits
• BLACKS have 33 fewer visits than whites
• Income elasticity is 0.021, 10 increase in
  income generates a 2.1 increase in visits

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Negative Binomial

• Interpret results the same was as Poisson
• Look at coefficient/standard error on delta
• Ho delta 0 (Poisson model is correct)
• In this case, delta 5.21 standard error is
  0.15, easily reject null.
• Var/Mean 1delta 6.21, Poisson is
  mis-specificed, should see very small standard
  errors in the wrong model

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Selected Results, Count ModelsParameter
(Standard Error)