Replicate ivprobit results using gsem

[Lukas Lang]

Hi,

I would like to run two equivalent commands using ivprobit and gsem, but I am failing to obtaining the same results.

Here is my attempt:

clear all
webuse laborsup
ivprobit fem_work (other_inc = male_educ) fem_educ kids, first
gsem    (fem_work <- other_inc fem_educ kids L@1, probit) ///
        (other_inc <- fem_educ kids male_educ), ///
        var(L@1)

These are the results:

. clear all

. webuse laborsup

. ivprobit fem_work (other_inc = male_educ) fem_educ kids, first

Fitting exogenous probit model

Iteration 0:   log likelihood = -344.63508  
Iteration 1:   log likelihood = -252.10819  
Iteration 2:   log likelihood = -252.04529  
Iteration 3:   log likelihood = -252.04529  

Fitting full model

Iteration 0:   log likelihood = -2368.2142  
Iteration 1:   log likelihood = -2368.2062  
Iteration 2:   log likelihood = -2368.2062  

Probit model with endogenous regressors                 Number of obs =    500
                                                        Wald chi2(3)  = 163.88
Log likelihood = -2368.2062                             Prob > chi2   = 0.0000

----------------------------------------------------------------------------------------------
                             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------------------+----------------------------------------------------------------
fem_work                     |
                   other_inc |  -.0542756   .0060854    -8.92   0.000    -.0662028   -.0423485
                    fem_educ |    .211111   .0268648     7.86   0.000     .1584569    .2637651
                        kids |  -.1820929   .0478267    -3.81   0.000    -.2758315   -.0883542
                       _cons |   .3672086   .4480724     0.82   0.412    -.5109971    1.245414
-----------------------------+----------------------------------------------------------------
other_inc                    |
                    fem_educ |   .3351866   .2825972     1.19   0.236    -.2186937     .889067
                        kids |   .8329056   .5475666     1.52   0.128    -.2403052    1.906116
                   male_educ |   2.845253    .282746    10.06   0.000     2.291081    3.399425
                       _cons |   9.872562   5.029193     1.96   0.050     .0155242     19.7296
-----------------------------+----------------------------------------------------------------
                  /athrho2_1 |   .3907859   .1509443     2.59   0.010     .0949404    .6866313
                   /lnsigma2 |   2.813383   .0316228    88.97   0.000     2.751404    2.875363
-----------------------------+----------------------------------------------------------------
 corr(e.other_inc,e.fem_work)|   .3720375   .1300518                      .0946562    .5958136
              sd(e.other_inc)|   16.66621   .5270318                      15.66461    17.73186
----------------------------------------------------------------------------------------------
Wald test of exogeneity (corr = 0): chi2(1) = 6.70        Prob > chi2 = 0.0096
Instrumented: other_inc
 Instruments: fem_educ kids male_educ

. gsem    (fem_work <- other_inc fem_educ kids L@1, probit) ///
>                 (other_inc <- fem_educ kids male_educ), ///
>                 var(L@1)

Fitting fixed-effects model:

Iteration 0:   log likelihood = -2373.0212  
Iteration 1:   log likelihood = -2371.4789  
Iteration 2:   log likelihood = -2371.4753  
Iteration 3:   log likelihood = -2371.4753  

Refining starting values:

Grid node 0:   log likelihood = -2377.7146

Fitting full model:

Iteration 0:   log likelihood = -2377.7146  
Iteration 1:   log likelihood = -2371.4764  
Iteration 2:   log likelihood = -2371.4747  
Iteration 3:   log likelihood = -2371.4747  

Generalized structural equation model                      Number of obs = 500

Response: fem_work
Family:   Bernoulli
Link:     Probit  

Response: other_inc
Family:   Gaussian
Link:     Identity

Log likelihood = -2371.4747

 ( 1)  [fem_work]L = 1
 ( 2)  [/]var(L) = 1
----------------------------------------------------------------------------------
                 | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------+----------------------------------------------------------------
fem_work         |
       other_inc |  -.0540839   .0057625    -9.39   0.000    -.0653782   -.0427896
        fem_educ |    .307729   .0376262     8.18   0.000     .2339831    .3814749
            kids |  -.2908398   .0669661    -4.34   0.000     -.422091   -.1595886
               L |          1  (constrained)
           _cons |  -.6589946   .4799027    -1.37   0.170    -1.599587    .2815973
-----------------+----------------------------------------------------------------
other_inc        |
        fem_educ |   .3351866   .2825972     1.19   0.236    -.2186937     .889067
            kids |   .8329056   .5475666     1.52   0.128    -.2403052    1.906116
       male_educ |   2.845253    .282746    10.06   0.000     2.291081    3.399425
           _cons |   9.872562   5.029193     1.96   0.050     .0155242     19.7296
-----------------+----------------------------------------------------------------
           var(L)|          1  (constrained)
-----------------+----------------------------------------------------------------
 var(e.other_inc)|   277.7625   17.56725                      245.3799    314.4187
----------------------------------------------------------------------------------

Can anybody help with this?

[Lukas Lang]

Hi,

I have made some progress, but I have not sorted my real problem.

It seems we can induce gsem to produce results that are equivalent to ivprobit in the case of one endogenous variable as follows:

clear all
webuse laborsup
ivprobit fem_work (other_inc = male_educ) fem_educ kids, first
gsem    (fem_work <- other_inc fem_educ kids L@1, probit) ///
        (other_inc <- fem_educ kids male_educ L), ///
        var(L@1)
nlcom    (fem_educ : _b[fem_work:other_inc] / sqrt(2)) ///
        (fem_educ : _b[fem_work:fem_educ] / sqrt(2)) ///
        (kids : _b[fem_work:kids] / sqrt(2)) ///
        (cons : _b[fem_work:_cons] / sqrt(2))

The results are:

. clear all

. webuse laborsup

. ivprobit fem_work (other_inc = male_educ) fem_educ kids, first

Fitting exogenous probit model

Iteration 0:   log likelihood = -344.63508  
Iteration 1:   log likelihood = -252.10819  
Iteration 2:   log likelihood = -252.04529  
Iteration 3:   log likelihood = -252.04529  

Fitting full model

Iteration 0:   log likelihood = -2368.2142  
Iteration 1:   log likelihood = -2368.2062  
Iteration 2:   log likelihood = -2368.2062  

Probit model with endogenous regressors                 Number of obs =    500
                                                        Wald chi2(3)  = 163.88
Log likelihood = -2368.2062                             Prob > chi2   = 0.0000

-------------------------------------------------------------------------------
              | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
fem_work      |
    other_inc |  -.0542756   .0060854    -8.92   0.000    -.0662028   -.0423485
     fem_educ |    .211111   .0268648     7.86   0.000     .1584569    .2637651
         kids |  -.1820929   .0478267    -3.81   0.000    -.2758315   -.0883542
        _cons |   .3672086   .4480724     0.82   0.412    -.5109971    1.245414
--------------+----------------------------------------------------------------
other_inc     |
     fem_educ |   .3351866   .2825972     1.19   0.236    -.2186937     .889067
         kids |   .8329056   .5475666     1.52   0.128    -.2403052    1.906116
    male_educ |   2.845253    .282746    10.06   0.000     2.291081    3.399425
        _cons |   9.872562   5.029193     1.96   0.050     .0155242     19.7296
--------------+----------------------------------------------------------------
   /athrho2_1 |   .3907859   .1509443     2.59   0.010     .0949404    .6866313
    /lnsigma2 |   2.813383   .0316228    88.97   0.000     2.751404    2.875363
--------------+----------------------------------------------------------------
corr(e.othe~c,|
   e.fem_work)|   .3720375   .1300518                      .0946562    .5958136
sd(e.other_~c)|   16.66621   .5270318                      15.66461    17.73186
-------------------------------------------------------------------------------
Wald test of exogeneity (corr = 0): chi2(1) = 6.70        Prob > chi2 = 0.0096
Instrumented: other_inc
 Instruments: fem_educ kids male_educ

. gsem    (fem_work <- other_inc fem_educ kids L@1, probit) ///
>         (other_inc <- fem_educ kids male_educ L), ///
>         var(L@1)

Fitting fixed-effects model:

Iteration 0:   log likelihood = -2373.0212  
Iteration 1:   log likelihood = -2371.4789  
Iteration 2:   log likelihood = -2371.4753  
Iteration 3:   log likelihood = -2371.4753  

Refining starting values:

Grid node 0:   log likelihood = -2377.8837

Fitting full model:

Iteration 0:   log likelihood = -2377.8837  
Iteration 1:   log likelihood = -2375.2652  
Iteration 2:   log likelihood = -2368.7263  (not concave)
Iteration 3:   log likelihood = -2368.3357  
Iteration 4:   log likelihood = -2368.2452  
Iteration 5:   log likelihood = -2368.2063  
Iteration 6:   log likelihood = -2368.2061  

Generalized structural equation model                      Number of obs = 500

Response: fem_work
Family:   Bernoulli
Link:     Probit  

Response: other_inc
Family:   Gaussian
Link:     Identity

Log likelihood = -2368.2061

 ( 1)  [fem_work]L = 1
 ( 2)  [/]var(L) = 1
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
fem_work     |
   other_inc |  -.0767588   .0086034    -8.92   0.000    -.0936212   -.0598964
    fem_educ |   .2985556   .0379894     7.86   0.000     .2240978    .3730133
        kids |  -.2575158   .0676328    -3.81   0.000    -.3900736    -.124958
           L |          1  (constrained)
       _cons |   .5193852   .6334912     0.82   0.412    -.7222347    1.761005
-------------+----------------------------------------------------------------
other_inc    |
    fem_educ |   .3351903   .2825977     1.19   0.236    -.2186911    .8890717
        kids |   .8329029   .5475682     1.52   0.128    -.2403109    1.906117
   male_educ |    2.84525   .2827427    10.06   0.000     2.291084    3.399415
           L |   8.769337   3.097587     2.83   0.005     2.698178     14.8405
       _cons |   9.872564   5.029133     1.96   0.050     .0156446    19.72948
-------------+----------------------------------------------------------------
       var(L)|          1  (constrained)
-------------+----------------------------------------------------------------
var(e.othe~c)|   200.8641   54.24256                      118.3147     341.009
------------------------------------------------------------------------------

. nlcom   (fem_educ : _b[fem_work:other_inc] / sqrt(2)) ///
>                 (fem_educ : _b[fem_work:fem_educ] / sqrt(2)) ///
>                 (kids : _b[fem_work:kids] / sqrt(2)) ///
>                 (cons : _b[fem_work:_cons] / sqrt(2))

    fem_educ: _b[fem_work:other_inc] / sqrt(2)
    fem_educ: _b[fem_work:fem_educ] / sqrt(2)
        kids: _b[fem_work:kids] / sqrt(2)
        cons: _b[fem_work:_cons] / sqrt(2)

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    fem_educ |  -.0542767   .0060835    -8.92   0.000    -.0662002   -.0423532
    fem_educ |   .2111107   .0268625     7.86   0.000     .1584611    .2637603
        kids |  -.1820912   .0478236    -3.81   0.000    -.2758237   -.0883587
        cons |   .3672608   .4479459     0.82   0.412     -.510697    1.245219
------------------------------------------------------------------------------

In the other_inc equation, we can add a latent variable (L) and we can then transform the gsem coefficients to obtain the ivprobit coefficients

However, why is this trick not working when we have multiple endogeonus variables (which is my actual problem)?

See the following example:

gen new_iv=runiform(1, 2)
ivprobit fem_work (other_inc fem_educ = male_educ new_iv) kids, first
gsem    (fem_work <- other_inc fem_educ kids L@1, probit) ///
        (other_inc <- kids male_educ new_iv L) ///
        (fem_educ <- kids male_educ new_iv L), ///
        var(L@1)

In this case, the results are:

. gen new_iv=runiform(1, 2)

. ivprobit fem_work (other_inc fem_educ = male_educ new_iv) kids, first

Fitting exogenous probit model

Iteration 0:   log likelihood = -344.63508  
Iteration 1:   log likelihood = -252.08506  
Iteration 2:   log likelihood = -252.02331  
Iteration 3:   log likelihood = -252.02331  

Fitting full model

Iteration 0:   log likelihood = -3561.6983  
Iteration 1:   log likelihood = -3561.6822  
Iteration 2:   log likelihood = -3561.6822  

Probit model with endogenous regressors                 Number of obs =    500
                                                        Wald chi2(3)  = 176.15
Log likelihood = -3561.6822                             Prob > chi2   = 0.0000

-------------------------------------------------------------------------------
              | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
fem_work      |
    other_inc |  -.0502853   .0216314    -2.32   0.020    -.0926821   -.0078885
     fem_educ |   .3279882   .2941496     1.12   0.265    -.2485345    .9045108
         kids |  -.1813605    .058115    -3.12   0.002    -.2952639   -.0674571
        _cons |  -1.232472   4.602739    -0.27   0.789    -10.25368    7.788731
--------------+----------------------------------------------------------------
other_inc     |
         kids |   .8361891   .5488621     1.52   0.128    -.2395608    1.911939
    male_educ |   2.828898   .2833297     9.98   0.000     2.273582    3.384214
       new_iv |   1.353638   2.586518     0.52   0.601    -3.715845    6.423121
        _cons |   12.09435    5.12694     2.36   0.018     2.045734    22.14297
--------------+----------------------------------------------------------------
fem_educ      |
         kids |    .053571   .0866261     0.62   0.536     -.116213     .223355
    male_educ |  -.0292372   .0447175    -0.65   0.513    -.1168819    .0584075
       new_iv |   .5076314   .4082263     1.24   0.214    -.2924774     1.30774
        _cons |   11.53806   .8091772    14.26   0.000     9.952105    13.12402
--------------+----------------------------------------------------------------
   /athrho2_1 |   .3262499   .2833922     1.15   0.250    -.2291886    .8816885
   /athrho3_1 |  -.3523941   1.081658    -0.33   0.745    -2.472405    1.767616
   /athrho3_2 |   .0518104   .0447214     1.16   0.247    -.0358419    .1394626
    /lnsigma2 |   2.814514   .0316228    89.00   0.000     2.752535    2.876494
    /lnsigma3 |    .968268   .0316228    30.62   0.000     .9062885    1.030247
--------------+----------------------------------------------------------------
corr(e.othe~c,|
   e.fem_work)|   .3151472   .2552464                     -.2252583    .7072642
corr(e.fem~uc,|
   e.fem_work)|  -.3384971   .9577213                     -.9858601    .9433476
corr(e.fem~uc,|
  e.other_inc)|    .051764   .0446015                     -.0358266    .1385654
sd(e.other_~c)|   16.68507   .5276282                      15.68233    17.75192
sd(e.fem_educ)|   2.633379   .0832748                      2.475119    2.801759
-------------------------------------------------------------------------------
Wald test of exogeneity: chi2(2) = 8.27                   Prob > chi2 = 0.0160
Instrumented: other_inc fem_educ
 Instruments: kids male_educ new_iv

. gsem    (fem_work <- other_inc fem_educ kids L@1, probit) ///
>         (other_inc <- kids male_educ new_iv L) ///
>         (fem_educ <- kids male_educ new_iv L), ///
>         var(L@1)

Fitting fixed-effects model:

Iteration 0:   log likelihood =   -3567.19  
Iteration 1:   log likelihood = -3565.6477  
Iteration 2:   log likelihood = -3565.6441  
Iteration 3:   log likelihood = -3565.6441  

Refining starting values:

Grid node 0:   log likelihood = -3592.1953

Fitting full model:

Iteration 0:   log likelihood = -3592.1953  (not concave)
Iteration 1:   log likelihood = -3570.3565  (not concave)
Iteration 2:   log likelihood =  -3568.212  (not concave)
Iteration 3:   log likelihood = -3567.4249  (not concave)
Iteration 4:   log likelihood = -3566.8534  (not concave)
Iteration 5:   log likelihood = -3566.4715  (not concave)
Iteration 6:   log likelihood = -3566.0964  (not concave)
Iteration 7:   log likelihood = -3564.8077  
Iteration 8:   log likelihood = -3563.1887  
Iteration 9:   log likelihood = -3562.4282  
Iteration 10:  log likelihood = -3561.7669  
Iteration 11:  log likelihood = -3561.7567  
Iteration 12:  log likelihood = -3561.7565  

Generalized structural equation model                      Number of obs = 500

Response: fem_work
Family:   Bernoulli
Link:     Probit  

Response: other_inc
Family:   Gaussian
Link:     Identity

Response: fem_educ
Family:   Gaussian
Link:     Identity

Log likelihood = -3561.7565

 ( 1)  [fem_work]L = 1
 ( 2)  [/]var(L) = 1
------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
fem_work     |
   other_inc |  -.0767468   .0085912    -8.93   0.000    -.0935854   -.0599083
    fem_educ |   .2610255   .0424455     6.15   0.000     .1778339    .3442172
        kids |  -.2541954   .0672316    -3.78   0.000     -.385967   -.1224238
           L |          1  (constrained)
       _cons |   .9647083   .5966083     1.62   0.106    -.2046224    2.134039
-------------+----------------------------------------------------------------
other_inc    |
        kids |   .8395442   .5487988     1.53   0.126    -.2360818     1.91517
   male_educ |   2.830935   .2830593    10.00   0.000     2.276149     3.38572
      new_iv |   1.084682    2.49373     0.43   0.664    -3.802939    5.972303
           L |   8.881188    3.04594     2.92   0.004     2.911255    14.85112
       _cons |   12.46174   5.062753     2.46   0.014     2.538929    22.38456
-------------+----------------------------------------------------------------
fem_educ     |
        kids |   .0536665   .0866259     0.62   0.536     -.116117    .2234501
   male_educ |  -.0291795   .0447179    -0.65   0.514    -.1168249     .058466
      new_iv |   .4999718   .4078124     1.23   0.220    -.2993257    1.299269
           L |   .2529253    .233157     1.08   0.278    -.2040539    .7099046
       _cons |   11.54853   .8086922    14.28   0.000     9.963521    13.13354
-------------+----------------------------------------------------------------
       var(L)|          1  (constrained)
-------------+----------------------------------------------------------------
var(e.othe~c)|   199.5276   53.95235                      117.4459    338.9755
var(e.fem~uc)|   6.870726   .4463382                      6.049321    7.803666
------------------------------------------------------------------------------

. nlcom   (fem_educ : _b[fem_work:other_inc] / sqrt(2)) ///
>                 (fem_educ : _b[fem_work:fem_educ] / sqrt(2)) ///
>                 (kids : _b[fem_work:kids] / sqrt(2)) ///
>                 (cons : _b[fem_work:_cons] / sqrt(2))

    fem_educ: _b[fem_work:other_inc] / sqrt(2)
    fem_educ: _b[fem_work:fem_educ] / sqrt(2)
        kids: _b[fem_work:kids] / sqrt(2)
        cons: _b[fem_work:_cons] / sqrt(2)

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    fem_educ |  -.0542682   .0060749    -8.93   0.000    -.0661749   -.0423616
    fem_educ |   .1845729   .0300135     6.15   0.000     .1257475    .2433983
        kids |  -.1797433     .04754    -3.78   0.000    -.2729199   -.0865667
        cons |   .6821518   .4218658     1.62   0.106    -.1446899    1.508994
------------------------------------------------------------------------------

Help would be immensely appreciated!

Thank you,

Lukas

[Lukas Lang]

I am sorry, I have been working on this for the last couple of days and I am now stuck.

Anyone who can help with this?

Or, anyone you know who can help with this?

Many thanks,

Lukas