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4. Bivariate Probit/Logit Regression Models


Bivariate regression models have two equations for the two dependent variables. This chapter explains the bivariate regression model with two binary dependent variables. Like the seemingly unrelated regression model (SUR), biviriate probit/logit models assume that the “independent, identically distributed” errors are correlated (Greene 2003).

The bivariate probit model, although consuming relatively much time, is more likely to converge than the bivariate logit model. SAS supports both the bivariate probit and logit models, while STATA and LIMDEP estimate the bivariate probit model. Here we consider a model for car ownership (owncar) and housing type (offcamp).

4.1 Bivariate Probit in STATA (.biprobit)

STATA has the .biprobit command to estimate the bivariate probit model. The two dependent variables precede a set of independent variables.

. biprobit owncar offcamp income age male

Fitting comparison equation 1:
 
Iteration 0:   log likelihood = -282.96512
Iteration 1:   log likelihood = -273.84832
Iteration 2:   log likelihood = -273.81741
Iteration 3:   log likelihood = -273.81741
 
Fitting comparison equation 2:
 
Iteration 0:   log likelihood =  -54.97403
Iteration 1:   log likelihood = -45.919608
Iteration 2:   log likelihood = -43.685448
Iteration 3:   log likelihood =  -43.32265
Iteration 4:   log likelihood = -43.309675
Iteration 5:   log likelihood = -43.309654
 
Comparison:    log likelihood = -317.12707
 
Fitting full model:
 
Iteration 0:   log likelihood = -317.12707  
Iteration 1:   log likelihood = -307.15684  
Iteration 2:   log likelihood = -306.49535  
Iteration 3:   log likelihood = -306.46018  
Iteration 4:   log likelihood = -306.45493  
Iteration 5:   log likelihood = -306.45408  
Iteration 6:   log likelihood = -306.45395  
Iteration 7:   log likelihood = -306.45392  
 
Bivariate probit regression                       Number of obs   =        437
                                                  Wald chi2(6)    =      30.13
Log likelihood = -306.45392                       Prob > chi2     =     0.0000
 
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
owncar       |
      income |  -.0017168    .347905    -0.00   0.996    -.6835982    .6801645
         age |   .1492475   .0409238     3.65   0.000     .0690383    .2294568
        male |   .2594624   .1255633     2.07   0.039     .0133628     .505562
       _cons |  -2.834625   .8719679    -3.25   0.001    -4.543651   -1.125599
-------------+----------------------------------------------------------------
offcamp      |
      income |   .7519064   .8254937     0.91   0.362    -.8660316    2.369844
         age |   .5895658    .149221     3.95   0.000      .297098    .8820336
        male |   .3939644   .2834889     1.39   0.165    -.1616637    .9495925
       _cons |  -10.34593   2.947501    -3.51   0.000    -16.12293   -4.568938
-------------+----------------------------------------------------------------
     /athrho |   2.387522   27.20167     0.09   0.930    -50.92678    55.70182
-------------+----------------------------------------------------------------
         rho |   .9832658   .9027811                            -1           1
------------------------------------------------------------------------------
Likelihood-ratio test of rho=0:     chi2(1) =  21.3463    Prob > chi2 = 0.0000

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4.2 Bivariate Probit in SAS

The SAS QLIM procedure is able to estimate both the bivariate logit and probit models. You need to provide two equations that may or may not have different sets of independent variables.

PROC QLIM DATA=masil.students;
   MODEL owncar = income age male;
   MODEL offcamp = income age male;
   ENDOGENOUS owncar offcamp ~ DISCRETE(DIST=NORMAL);
RUN;

Or, simply

PROC QLIM DATA=masil.students;
   MODEL owncar offcamp = income age male /DISCRETE;
RUN;

                                       The QLIM Procedure
 
                              Discrete Response Profile of owncar
 
                       Index         Value           Frequency    Percent
 
                         1             0                   153      35.01
                         2             1                   284      64.99
 
 
                              Discrete Response Profile of offcamp
 
                       Index         Value           Frequency    Percent
 
                         1             0                    12       2.75
                         2             1                   425      97.25
 
 
                                       Model Fit Summary
 
                        Number of Endogenous Variables                 2
                        Endogenous Variable               owncar offcamp
                        Number of Observations                       437
                        Log Likelihood                        -306.45392
                        Maximum Absolute Gradient             2.16967E-6
                        Number of Iterations                          27
                        AIC                                    628.90784
                        Schwarz Criterion                      661.54730
 
Algorithm converged.
 
 
                                      Parameter Estimates
 
                                                     Standard                 Approx
            Parameter                Estimate           Error    t Value    Pr > |t|
 
            owncar.Intercept        -2.834511        0.871964      -3.25      0.0012
            owncar.income           -0.001723        0.347904      -0.00      0.9960
            owncar.age               0.149243        0.040924       3.65      0.0003
            owncar.male              0.259462        0.125563       2.07      0.0388
            offcamp.Intercept      -10.345002        2.947054      -3.51      0.0004
            offcamp.income           0.751837        0.825398       0.91      0.3624
            offcamp.age              0.589515        0.149197       3.95      <.0001
            offcamp.male             0.393859        0.283458       1.39      0.1647
            _Rho                     0.999990               0        .         .

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4.3 Bivariate Probit in LIMDEP (Bivariateprobit$)

LIMDEP has the Bivariateprobit$ command to estimate the bivariate probit model. The Lhs$ subcommand lists the two binary dependent variables, whereas Rh1$ and Rh2$ respectively indicate independent variables for the two dependent variables. In this model, you may not switch the order of dependent variables (Lhs=owncar,offcamp;) to avoid convergence problems.

BIVARIATEPROBIT;
   Lhs=offcamp,owncar;
   Rh1=ONE,income,age,male;
   Rh2= ONE,income,age,male$

Normal exit from iterations. Exit status=0.
 
+---------------------------------------------+
| FIML Estimates of Bivariate Probit Model    |
| Maximum Likelihood Estimates                |
| Model estimated: Sep 17, 2005 at 10:36:25PM.|
| Dependent variable               OFFOWN     |
| Weighting variable                 None     |
| Number of observations              437     |
| Iterations completed                 35     |
| Log likelihood function       -306.4539     |
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient  | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
          Index    equation for OFFCAMP
 Constant    -10.34508235       3.6592558   -2.827   .0047
 INCOME       .7518407011       .85274898     .882   .3780     .61683982
 AGE          .5895189160       .18572787    3.174   .0015     20.691076
 MALE         .3938599470       .29308051    1.344   .1790     .57208238
          Index    equation for OWNCAR
 Constant    -2.834513147       .84825468   -3.342   .0008
 INCOME   -.1723102966E-02      .34222451    -.005   .9960     .61683982
 AGE          .1492426338   .39739762E-01    3.755   .0002     20.691076
 MALE         .2594618946       .12565094    2.065   .0389     .57208238
          Disturbance correlation
 RHO(1,2)     .9941311591   .73338053E+09     .000  1.0000
 (Note: E+nn or E-nn means multiply by 10 to + or -nn power.)
 
          Joint Frequency Table: Columns=OWNCAR
                                 Rows   =OFFCAMP
 
(N) = Count of Fitted Values
 
                       0          1          TOTAL
 
            0         12          0            12
                  (    0)    (    0)       (    0)
 
            1        141        284           425
                  (    0)    (  437)       (  437)
 
          TOTAL      153        284           437
                  (    0)    (  437)       (  437)

SAS, STATA, and LIMDEP produce almost the same parameter estimates and standard errors with slight differences after the decimal point.

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4.4 Bivariate Logit in SAS

The QLIM procedure also estimates the bivariate logit model using the DIST=LOGIT option. Unfortunately, this model does not fit in SAS.

PROC QLIM DATA=masil.students;
   MODEL owncar = income age male;
   MODEL offcamp = income age male;
   ENDOGENOUS offcamp owncar ~ DISCRETE(DIST=LOGIT);
RUN;

Or,

PROC QLIM DATA=masil.students;
   MODEL owncar offcamp = income age male /DISCRETE(DIST=LOGIT);
RUN;


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