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
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
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 . .
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 . .
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)
+---------------------------------------------+
| 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.
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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