Bài 5: Mô hình Multinomial Logit

 ordinal response (ordered probit model – not covered. in this lecture).[r] (1)MULTINOMIAL LOGIT MODEL (2)Multinomial responses  logit/probit model: dependent variable = 0/1  What if more than 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  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: = Commune health center; = Public hospital; = Private hospital; = 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 _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 _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 _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 (12)Explain the estimation results  Suppose there are persons of same sex, A’s income is 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] = 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 mil VND? (*) dy/dx is for discrete change of dummy variable from 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 mil VND? (*) dy/dx is for discrete change of dummy variable from 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