Dear Statalist users,

I have a question which I could not clarify in my mind: What is the relationship between the coefficients obtained in a multinomial logit and a set of independent logistic regressions?

To be more accurate, I attached a very simple example below. Is there a relationship between the coefficients from estimations #1, #2, #3 and #4?


Code:
. use https://stats.idre.ucla.edu/stat/data/hsbdemo, clear
(highschool and beyond (200 cases))

. * Prog variable has three categories
. tab prog

    type of |
    program |      Freq.     Percent        Cum.
------------+-----------------------------------
    general |         45       22.50       22.50
   academic |        105       52.50       75.00
   vocation |         50       25.00      100.00
------------+-----------------------------------
      Total |        200      100.00

.
. * Generate seperate dummies for each category of the program variable
. gen general=0

. replace general=1 if prog==1
(45 real changes made)

.
. gen academic=0

. replace academic=1 if prog==2
(105 real changes made)

.
. gen vocational=0

. replace vocational=1 if prog==3
(50 real changes made)

.
. * #1 Multinomial specification  (vocational is the reference category)
. mlogit prog write, base(3)

Iteration 0:   log likelihood = -204.09667  
Iteration 1:   log likelihood = -186.05186  
Iteration 2:   log likelihood = -185.51265  
Iteration 3:   log likelihood = -185.51084  
Iteration 4:   log likelihood = -185.51084  

Multinomial logistic regression                 Number of obs     =        200
                                                LR chi2(2)        =      37.17
                                                Prob > chi2       =     0.0000
Log likelihood = -185.51084                     Pseudo R2         =     0.0911

------------------------------------------------------------------------------
        prog |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
general      |
       write |    .051801   .0225143     2.30   0.021     .0076739    .0959282
       _cons |  -2.646504   1.127438    -2.35   0.019    -4.856241   -.4367673
-------------+----------------------------------------------------------------
academic     |
       write |   .1178089   .0216189     5.45   0.000     .0754367    .1601812
       _cons |  -5.358994   1.115266    -4.81   0.000    -7.544875   -3.173113
-------------+----------------------------------------------------------------
vocation     |  (base outcome)
------------------------------------------------------------------------------

.
. * #2 Logit specifications
. logit general write

Iteration 0:   log likelihood = -106.63277  
Iteration 1:   log likelihood = -105.96906  
Iteration 2:   log likelihood = -105.96688  
Iteration 3:   log likelihood = -105.96688  

Logistic regression                             Number of obs     =        200
                                                LR chi2(1)        =       1.33
                                                Prob > chi2       =     0.2485
Log likelihood = -105.96688                     Pseudo R2         =     0.0062

------------------------------------------------------------------------------
     general |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |  -.0204257   .0176417    -1.16   0.247    -.0550028    .0141514
       _cons |  -.1689862   .9291947    -0.18   0.856    -1.990174    1.652202
------------------------------------------------------------------------------

.
. * #3 Logit specifications
. logit academic write

Iteration 0:   log likelihood = -138.37933  
Iteration 1:   log likelihood = -122.55844  
Iteration 2:   log likelihood = -122.55784  
Iteration 3:   log likelihood = -122.55784  

Logistic regression                             Number of obs     =        200
                                                LR chi2(1)        =      31.64
                                                Prob > chi2       =     0.0000
Log likelihood = -122.55784                     Pseudo R2         =     0.1143

------------------------------------------------------------------------------
    academic |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |   .0918986   .0178276     5.15   0.000     .0569572      .12684
       _cons |  -4.754081   .9582002    -4.96   0.000    -6.632119   -2.876043
------------------------------------------------------------------------------

.
. * #4 Logit specifications
. logit vocational write

Iteration 0:   log likelihood = -112.46703  
Iteration 1:   log likelihood = -99.439698  
Iteration 2:   log likelihood = -98.987386  
Iteration 3:   log likelihood = -98.986268  
Iteration 4:   log likelihood = -98.986268  

Logistic regression                             Number of obs     =        200
                                                LR chi2(1)        =      26.96
                                                Prob > chi2       =     0.0000
Log likelihood = -98.986268                     Pseudo R2         =     0.1199

------------------------------------------------------------------------------
  vocational |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       write |  -.0927851   .0191087    -4.86   0.000    -.1302375   -.0553327
       _cons |   3.624981   .9589797     3.78   0.000     1.745415    5.504547
------------------------------------------------------------------------------