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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;λi) =e-λi λi yi/yi! For λi>0
• E[yi]= Var[yi] = λi = f(xi, β)
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• λi must be positive at all times
• Therefore, we CANNOT let λi = xiβ
• Let λi = exp(xiβ)
• ln(λi) = (xiβ)
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• d ln(λi)/dxi = β
• Remember that d ln(λi) = dλi/λi
• Interpret β as the percentage change in mean outcomes for a change in x
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Problems with Poisson
• Variance grows with the mean– E[yi]= Var[yi] = λi = f(xi, β)
• Most data sets have over dispersion, where the variance grows faster than the mean
• In dr. visits sample, = 5.6, s=6.7• Impose Mean=Var, severe restriction
and you tend to reduce standard errors
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Negative Binomial Model
• Where γi = exp(xiβ) and δ ≥ 0
• E[yi] = δγi = δexp(xiβ)
• Var[yi] = δ (1+δ) γi
• Var[yi]/ E[yi] = (1+δ)
ii y
ii
iii y
yy
11
1
)1()(
)()Pr(
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• δ must always be ≥ 0• In this case, the variance grows
faster than the mean• If δ=0, the model collapses into the
Poisson• Always estimate negative binomial• If you cannot reject the null that δ=0,
report the Poisson estimates
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• Notice that ln(E[yi]) = ln(δ) + ln(γi), so
• d ln(E[yi]) /dxi = β
• 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 = 1+delta = 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)
Variable Poisson Negative Binomial
Age65 0.214 (0.026) 0.103 (0.055)
Age70 0.787 (0.026) 0.204 (0.054)
Chronic 0.500 (0.014) 0.509 (0.029)
Excel -0.784 (0.031) -0.527 (0.059)
Ln(Inc). 0.021 (0.007) 0.038 (0.016)