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COUNT DATA MODELS

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Count data and Poisson distribution

Poisson model

Negative Binomial model

Application of count models

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Count data and Poisson distribution



Sometimes the dep var is a non-negative integer



Example:



takeover bids received by a target firm



number of unpaid credit installment



number of accidents



number of prepaid mortgage loans



<b>These are the number of incidence happened in a </b>

<b>given period of time</b>

0,1, 2,...,

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Count data and Poisson distribution



For dependent variable that is an non-negative

integer



it is an integer

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Count data and Poisson distribution



Poisson distribution describe the probability of

events occurring k times in a given period of time



The probability function is

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Properties of Poisson distribution



One parameter



Mean is equal to variance



Thus, variance increases with mean



If we model lambda as a function of explanatory

variable, we have the Poisson model

 

var

 

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Poisson model



Mean



Probability



Log-likelihood function



Estimation method: Maximum Likelihood



|



<i>X</i>

<i>i</i>

<i>i</i>

<i>E y X</i>

 



<i>e</i>







 

Pr

!

!

<i>Xi</i>
<i>i</i>

<i>k</i>

<i>X</i>

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<i>k</i>

<i>e</i>

<i>e</i>

<i>e</i>

<i>y</i>

<i>k</i>

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

_



_















1

log

<i>i</i>

log

!

<i>N</i>

<i>X</i>

<i>i</i>

<i>i</i>

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<i>i</i>

<i>L</i>

<i>e</i>



<i>y X</i>



<i>y</i>



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Interpreting Poisson model estimates



We are interested in: when X changes, how the

expected value of y changes



The marginal effect



|



<i>X</i>

<i>i</i>

<i>i</i>

<i>E y</i>

<i>X</i>

 



<i>e</i>





|



<i>i</i>

<i>i</i>

<i>X</i>

<i>i</i>

<i>E y</i>

<i>X</i>

<i>e</i>

<i>X</i>



_



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Issue in Poisson model



Under Poisson distribution, mean = variance



This means variance increases with mean



If not



variance increase at a

LOWER

rate than mean:

UNDERDISPERSION



variance increase at a

HIGHER

rate than mean:

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Negative Binomial Model



If we model



Then the mean is



And the variance



<b>This is negative binomial model</b>



Note if then the model collapse to Poisson.





 

 

Pr

!

<i>k</i>

<i>e</i>

<i>y</i>

<i>k</i>

<i>k</i>



_









|



<i>X</i>

<i>i</i>

<i>i</i>

<i>E y</i>

<i>X</i>

 



<i>e</i>







2

var

<i>y X</i>

<i>_i</i>

|

 

 

0

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Analyze the number of non-payments during a

credit contract

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Data



Dependent variable: number of nonpayment

during a credit contract



Independent variables



duration: contracting period (month)



age: (year)



collateral: dummy, 1 = with collateral



edu: schooling years (years)



banking: dummy, 1 = receiving salary via bank account



salary: monthly income (mil. VND)

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The number of nonpayments

Total 2,110 100.00

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Poisson model in Stata

_cons 2.074566 .1463598 14.17 0.000 1.787706 2.361426
married .459314 .033624 13.66 0.000 .3934121 .5252159
salary .0014747 .0045834 0.32 0.748 -.0075086 .0104581
banking -4.108164 .1516297 -27.09 0.000 -4.405353 -3.810975
edu -.0899637 .0048852 -18.42 0.000 -.0995387 -.0803888
collateral -1.540261 .0434424 -35.46 0.000 -1.625407 -1.455116
age -.035701 .0028549 -12.51 0.000 -.0412964 -.0301056
duration .0599389 .0031207 19.21 0.000 .0538225 .0660554

nonpay Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -1985.4326 Pseudo R2 = 0.6086
Prob > chi2 = 0.0000
LR chi2(7) = 6173.59
Poisson regression Number of obs = 2110
Iteration 3: log likelihood = -1985.4326

Iteration 2: log likelihood = -1985.4327
Iteration 1: log likelihood = -1985.7743
Iteration 0: log likelihood = -2066.391

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Hypothesis testing



H0: coef of all demographic variables equal zero

Prob > chi2 = 0.0000

chi2( 3) = 697.41

( 3) [nonpay]married = 0

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Prediction

npay 2110 1.891943 2.583285 .0035831 21.4108

Variable Obs Mean Std. Dev. Min Max

. sum npay

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Marginal effects

(*) dy/dx is for discrete change of dummy variable from 0 to 1

married* .1951712 .01804 10.82 0.000 .15982 .230522 .555924
salary .0006373 .00198 0.32 0.748 -.003245 .00452 11.9332
banking* -2.093989 .04643 -45.10 0.000 -2.185 -2.00298 .388152
edu -.0388793 .00305 -12.75 0.000 -.044856 -.032902 11.9995
collat~l* -.6827921 .04277 -15.96 0.000 -.766627 -.598957 .453555
age -.0154288 .00151 -10.19 0.000 -.018397 -.01246 34.9464
duration .0259036 .002 12.93 0.000 .021976 .029832 23.9223

variable dy/dx Std. Err. z P>|z| [ 95% C.I. ] X

= .43216618

y = predicted number of events (predict)
Marginal effects after poisson

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Marginal effects at a value point

(*) dy/dx is for discrete change of dummy variable from 0 to 1

married* .0236672 .00443 5.34 0.000 .014985 .03235 1
salary .0000948 .0003 0.31 0.754 -.000498 .000688 30
banking* -3.845207 .33501 -11.48 0.000 -4.50182 -3.18859 1
edu -.0057814 .00102 -5.68 0.000 -.007777 -.003786 16
collat~l* -.0504903 .00878 -5.75 0.000 -.067693 -.033288 0
age -.0022943 .00044 -5.21 0.000 -.003158 -.001431 34
duration .0038519 .00069 5.59 0.000 .002501 .005203 24

variable dy/dx Std. Err. z P>|z| [ 95% C.I. ] X

= .0642636

y = predicted number of events (predict)
Marginal effects after poisson

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Marginal effects at a value point

(*) dy/dx is for discrete change of dummy variable from 0 to 1

married* .1960826 .01812 10.82 0.000 .160569 .231597 .555924
salary .0006403 .00199 0.32 0.748 -.00326 .004541 11.9332
banking* -2.103768 .04655 -45.19 0.000 -2.19501 -2.01252 .388152
edu -.0390608 .00306 -12.75 0.000 -.045065 -.033057 11.9995
collat~l* -.6859805 .04296 -15.97 0.000 -.770188 -.601773 .453555
age -.0155008 .00152 -10.19 0.000 -.018483 -.012519 34.9464
duration .0260245 .00202 12.90 0.000 .022071 .029978 24

variable dy/dx Std. Err. z P>|z| [ 95% C.I. ] X

= .43418424

y = predicted number of events (predict)
Marginal effects after poisson

means used for age collateral edu banking salary married

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Negative binomial model

Likelihood-ratio test of alpha=0: chibar2(01) = 3.5e-05 Prob>=chibar2 = 0.498

alpha 1.21e-08 1.89e-06 5.9e-141 2.5e+124

/lnalpha -18.2266 155.4434 -322.8902 286.4369

_cons 2.074564 .1463599 14.17 0.000 1.787704 2.361424
married .4592988 .033624 13.66 0.000 .393397 .5252006
salary .0014748 .0045834 0.32 0.748 -.0075086 .0104581
banking -4.108135 .1516278 -27.09 0.000 -4.40532 -3.81095
edu -.0899611 .0048852 -18.41 0.000 -.099536 -.0803861
collateral -1.540237 .0434421 -35.45 0.000 -1.625382 -1.455092
age -.0356999 .0028549 -12.50 0.000 -.0412953 -.0301044
duration .0599365 .0031207 19.21 0.000 .05382 .0660529

nonpay Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -1985.4326 Pseudo R2 = 0.4782
Dispersion = mean Prob > chi2 = 0.0000

LR chi2(7) = 3639.28
Negative binomial regression Number of obs = 2110

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Application of Count Models

Dione&Vanasse (1992) Automobile Insurance

Ratemaking in the Presence of Asymmetrical

<i>Information. J of Applied Econometrics 7: 149-65.</i>



Quebec drivers 1982-83, for insurers to classify

drivers.



dep var: number of accidents reported by police



indep var: driver’s characteristics [age, gender,

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Application of Count Models



Greene (1994) Accounting for Excess Zeros and Sample

Selection in Poisson and Negative Binomial Regression

<i>Models. Working Paper, New York University.</i>



data of credit card applicants



dep var: number of derogatory reports



indep var:



income, expenditure



age

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Application of Count Models



Dione et al. (1996) Count Data Models for a Credit

<i>Scoring System. J of Empirical Finance 3: 303-25.</i>



data of 4,700 clients granted credit by a bank



dep var: number of unpaid monthly payments during

the contracting period



indep var:



income



age



duration of contracting period



marital status

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Application of Count Models



Jaggia&Thosar (1993) Multiple Bids as a Consequence of Target

<i>Management Resistance : A Count Data Approach. Review of </i>

<i>Qualitative Finance and Accounting 3: 447-57.</i>



data of 126 firms that were targets of tender offers 1978-85



dep var: number of bids after the initial bid received



indep var:



legal defense



real restructuring



financial restructuring



white knight



initial bid premium



institutional holdings



size

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