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Bài 5: Mô hình Multinomial Logit

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(1)

MULTINOMIAL LOGIT MODEL




(2)

Multinomial responses



logit/probit model: dependent variable = 0/1


What if more than 2 categories?



Example: long term effect of the exposure to



radiation may be



1 – dead of cancer




(3)

Multinomial responses



Example: choice of health care providers



1 – public hospital



2 – private hospital/clinic


3 – “lang y”



4 – self-treatment



Other examples:



choice of car (Y = Toyota, Honda, Suzuki, Mazda, KIA…)


choice of specialization at university



choice of occupation





(4)

Multinomial logistic regression model



Multinomial logit model (MNL) is used to analyze



the relationship between categorical variables and


other explanatory variables



Notice:



nominal response (MNL)



ordinal response (ordered probit model – not covered




(5)

The dependent variable



The occurrence of an alternative j for individual i


Probability of occurrence of each alternative



1, 2, 3,...,



i



Y

J



i1


i2


i3



iJ




1

probability =


2

probability =


3

probability =


...

...



probability =



i



p


p



Y

p



J

p




(6)

The logit (log-odds ratio)


Logit


i1


i1


i2


i1


i3


i1


1

log


2

log


3

log


...

...


0


i



iJ


i


iJ


i


iJ


i

iJ


p


Y

h


p


p


Y

h


p


p


Y

h


p



Y

J

h










(7)

Modelling the logits



i1


i1

1


i2


i1

2



i3


i1

3


1

log


2

log


3

log


...

...


0


i

i


iJ


i

i


iJ


i

i


iJ


i

iJ


p



Y

h

X



p


p



Y

h

X



p


p



Y

h

X



p




Y

J

h




(8)

Modeling the probabilities



1
2
3

i1


1


i2


1


i3


1


iJ


1



1

probability =



2

probability =



3

probability =




(9)

Maximum likelihood estimation



MNL model is estimated by maximizing the



log-likelihood function



1

1




log

ln



N

J



ij

ij


i

j



L

y

p









0

if j is NOT chosen



1

if j is chosen




(10)

Data



id

case

choice

thunhap

gioitinh



1

1

2

12

1



1

2

4

12

1



2

1

3

21

1



3

1

17

0



3

2

17

0




3

3

17

0





Choice: 1 = Commune health center; 2 = Public hospital; 3 = Private hospital; 4 = Lang y;


5 = Individual health care provider




(11)

Estimate MNL in Stata


mlogit choice thunhap gioitinh



(choice==2 is the base outcome)



_cons -2.872579 .1263704 -22.73 0.000 -3.12026 -2.624898
gioitinh .0954035 .1621446 0.59 0.556 -.222394 .413201
thunhap 1.97e-06 4.44e-07 4.43 0.000 1.10e-06 2.84e-06
5



_cons -3.795684 .3483009 -10.90 0.000 -4.478342 -3.113027
gioitinh .1885005 .3481328 0.54 0.588 -.4938273 .8708283
thunhap -8.18e-06 4.17e-06 -1.96 0.050 -.0000164 -1.22e-08
4



_cons -1.763979 .0821101 -21.48 0.000 -1.924912 -1.603046
gioitinh .2087203 .0979244 2.13 0.033 .016792 .4006485
thunhap 1.81e-06 4.19e-07 4.32 0.000 9.90e-07 2.63e-06
3




_cons -1.180521 .1060245 -11.13 0.000 -1.388325 -.9727166
gioitinh .1822304 .1061043 1.72 0.086 -.0257303 .3901911
thunhap -9.39e-06 1.29e-06 -7.29 0.000 -.0000119 -6.86e-06
1



(12)

Explain the estimation results



Suppose there are 2 persons of same sex, A’s



income is 1 mil VND higher than that of B, so



A:



B:



i1



1

1

1

1



log

i

i

i



iJ


p



X

thunhap

gioitinh



p

  




1



1

1

1



log

A

A

A



AJ



p



thunhap

gioitinh


p

 





1



1

1

1



log

B

A

1

B



BJ



p




(13)

Explain the estimation results



the estimated coefficient indicates the responses



of log-odds ratio for a unit change in explanatory



variable



1



1

1



1



1


log

log

log



B


B

A

BJ



A


BJ

AJ



AJ



p



p

p

p



p



p

p



p




(14)

Hypothesis testing



Test the null hypothesis



1

2

1



H0:

j

 

...

J

0



Prob > chi2 = 0.0000


chi2( 4) = 82.49


( 4) [5]thunhap = 0




(15)

Hypothesis testing


Kiểm định giả thuyết



1



H0:

0



Prob > chi2 = 0.0000


chi2( 1) = 53.17


( 1) [1]thunhap = 0




(16)

Kiểm định



Test the null hypothesis: all coefs in [1] = 0



Prob > chi2 = 0.0000


chi2( 2) = 56.38


( 2) [1]gioitinh = 0



( 1) [1]thunhap = 0



. test [1]



Prob > chi2 = 0.0000


chi2( 2) = 56.38


( 2) [1]gioitinh = 0



( 1) [1]thunhap = 0




(17)

Marginal effect



What happens to the probability of choosing [1] if



income increase by 1 mil VND?



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




gioitinh* .0132477 .00985 1.34 0.179 -.006067 .032563 .527194


thunhap -.0009239 .0001 -8.99 0.000 -.001125 -.000723 82.9054



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



= .10632185



y = Pr(choice==1) (predict, p outcome(1))


Marginal effects after mlogit




(18)

Marginal effect



For a female with income 500 mil VND/year, the




probability of choosing [1] if income increase by 1


mil VND?



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




gioitinh* .0002118 .00023 0.91 0.364 -.000246 .000669 1


thunhap -.0000202 .00001 -2.23 0.026 -.000038 -2.4e-06 500



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



= .00199241



y = Pr(choice==1) (predict, p outcome(1))


Marginal effects after mlogit




(19)

Prediction



Predict the probability of choosing private



hospitals/clinic



bvtu 3475 .1502158 .0348196 .1121532 .6523073



Variable Obs Mean Std. Dev. Min Max


. sum bvtu







×