Bài 10: Mô hình Count Data

 Under Poisson distribution, mean = variance  This means variance increases with mean..  If not.[r]

COUNT DATA MODELS

Count data and Poisson distribution Poisson model

Negative Binomial model Application of count models

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

 These are the number of incidence happened in a

given period of time

0,1, 2, ,

Count data and Poisson distribution

 For dependent variable that is an non-negative

integer

 it is an integer

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

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 

Poisson model

 Mean

 Probability

 Log-likelihood function

 Estimation method: Maximum Likelihood

 |  Xi

i

E y X   e 

    Pr ! ! Xi i k X e

k e e

e y k k k           1

log i log !

N

X

i i i

i

L e  y X  y



Interpreting Poisson model estimates

 We are interested in: when X changes, how the

expected value of y changes

 The marginal effect

 |  Xi

i

E y X   e 

 | 

i

i X

i

E y X

e X

 



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:

Negative Binomial Model

 If we model

 Then the mean is  And the variance

 This is negative binomial model

 Note if then the model collapse to Poisson.

      Pr ! k e y k k     

 |  Xi

i

E y X   e 

  2

var y Xi |   

0

Analyze the number of non-payments during a credit contract

Data

 Dependent variable: number of nonpayment

during a credit contract

 Independent variables

 duration: contracting period (month)  age: (year)

 collateral: dummy, = with collateral  edu: schooling years (years)

 banking: dummy, = receiving salary via bank account  salary: monthly income (mil VND)

The number of nonpayments

Total 2,110 100.00

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

Hypothesis testing

 H0: coef of all demographic variables equal zero

Prob > chi2 = 0.0000 chi2( 3) = 697.41 ( 3) [nonpay]married = 0

Prediction

npay 2110 1.891943 2.583285 .0035831 21.4108 Variable Obs Mean Std Dev Min Max sum npay

Marginal effects

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

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

Marginal effects at a value point

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

married* .0236672 00443 5.34 0.000 .014985 .03235 salary 0000948 .0003 0.31 0.754 -.000498 000688 30 banking* -3.845207 33501 -11.48 0.000 -4.50182 -3.18859 edu -.0057814 00102 -5.68 0.000 -.007777 -.003786 16 collat~l* -.0504903 00878 -5.75 0.000 -.067693 -.033288 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

Marginal effects at a value point

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

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

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

Application of Count Models

Dione&Vanasse (1992) Automobile Insurance Ratemaking in the Presence of Asymmetrical

Information J of Applied Econometrics 7: 149-65.

 Quebec drivers 1982-83, for insurers to classify

drivers.

 dep var: number of accidents reported by police  indep var: driver’s characteristics [age, gender,

Application of Count Models

 Greene (1994) Accounting for Excess Zeros and Sample

Selection in Poisson and Negative Binomial Regression Models Working Paper, New York University.

 data of credit card applicants

 dep var: number of derogatory reports  indep var:

 income, expenditure  age

Application of Count Models

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

Scoring System J of Empirical Finance 3: 303-25.

 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

Application of Count Models

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

Management Resistance : A Count Data Approach Review of

Qualitative Finance and Accounting 3: 447-57.

 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