# Seemingly unrelated bivariate probit for endogeneity: interpretation of Rho

I would like to estimate the effect of health insurance coverage on type of healthcare provider chosen--either public or private--at last illness using a nationally representative sample of people in Country X.

Background

In Country X, health insurance coverage is dependent on the person's occupation, i.e. civil servants participate in the civil servants' health insurance scheme, private sector employees participate in the private sector employees' health scheme, and the poor/near poor (defined by the local government) are eligible to participate in the pro-poor social health insurance scheme. Despite the availability of these schemes, a sizable proportion (>60%) still do not health insurance. Because of this, there is reason to believe that insurance participation is potentially endogenous. Failing to control for endogeneity would yield biased estimates on the effect of health insurance coverage on provider type.

Because my outcome (provider type: public/private) and potentially endogenous variable (insured: yes/no) are binary, I used the seemingly unrelated bivariate probit model (biprobit command in Stata). My reading of the documentation is that biprobit can be used as an instrumental variable approach when both the outcome and endogenous regressor are binary.

Estimating equation

I jointly estimated these two models

$provid_i=insured_i+age_i+...+province_i +\epsilon_1$ $insured_i=jobtype_i+age_i+...+province_i +\epsilon_2$

using the following Stata command:

svy: biprobit (provid=insured i.agecat [...] i.province) (insured=i.agecat [...] i.province i.rjob)

Results

             |             Linearized
|      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
provid       |
insured |  -.0720567   .4781362    -0.15   0.880    -1.009816    .8657028
agecat2 |  -.8849128   .0999153    -8.86   0.000    -1.080875   -.6889507
agecat3 |  -.8406881   .0808987   -10.39   0.000    -.9993532    -.682023
agecat4 |  -.6424246    .070328    -9.13   0.000    -.7803576   -.5044916
agecat5 |  -.4488421    .063946    -7.02   0.000    -.5742582   -.3234261
agecat6 |  -.2798056   .0596442    -4.69   0.000    -.3967846   -.1628265
urban |  -.0291482   .0627755    -0.46   0.642    -.1522687    .0939722
married |  -1.793558   .1996031    -8.99   0.000    -2.185036   -1.402081
wlth2 |  -.0618154   .0704125    -0.88   0.380    -.1999141    .0762833
wlth3 |  -.0664418   .0847278    -0.78   0.433     -.232617    .0997333
wlth4 |  -.0949232   .0951351    -1.00   0.319      -.28151    .0916635
wlth5 |   .0637338   .0860918     0.74   0.459    -.1051166    .2325841
educ_2 |   .0762313    .118413     0.64   0.520      -.15601    .3084726
educ_3 |   .2021612   .1213075     1.67   0.096    -.0357569    .4400793
educ_4 |   .4922066   .1633843     3.01   0.003      .171764    .8126491
ownshome |  -.0524788   .0402067    -1.31   0.192    -.1313355    .0263778
has_car |   .1252798   .0574899     2.18   0.029     .0125258    .2380337
lnmeanc |     .08095   .0187777     4.31   0.000     .0441218    .1177783
prov1 |   -.448546   .2548933    -1.76   0.079    -.9484635    .0513716
prov2 |   .2055194   .1028571     2.00   0.046     .0037876    .4072512
prov3 |   .1397494   .1061627     1.32   0.188    -.0684656    .3479643
prov4 |  -.1182481   .1025183    -1.15   0.249    -.3193153    .0828191
_cons |   .0763685   .3324398     0.23   0.818    -.5756396    .7283765
-------------+----------------------------------------------------------------
insured      |
agecat2 |  -.3123846   .0791789    -3.95   0.000    -.4676768   -.1570924
agecat3 |  -.2739378   .0668517    -4.10   0.000    -.4050529   -.1428228
agecat4 |  -.1623367   .0591023    -2.75   0.006    -.2782529   -.0464204
agecat5 |   .0181344    .059516     0.30   0.761    -.0985933     .134862
agecat6 |  -.0600275   .0571389    -1.05   0.294     -.172093    .0520379
urban |   .2135773   .0493271     4.33   0.000     .1168329    .3103216
married |   .1395829   .1934494     0.72   0.471    -.2398257    .5189915
wlth2 |  -.1313183   .0529009    -2.48   0.013    -.2350719   -.0275647
wlth3 |  -.2868657   .0591747    -4.85   0.000    -.4029238   -.1708075
wlth4 |  -.3304937   .0660867    -5.00   0.000    -.4601083   -.2008791
wlth5 |   -.146169   .0784983    -1.86   0.063    -.3001262    .0077883
educ_2 |  -.0329423   .1010825    -0.33   0.745    -.2311936     .165309
educ_3 |   .0602279   .1023783     0.59   0.556    -.1405647    .2610205
educ_4 |   .4550885   .1204804     3.78   0.000     .2187925    .6913844
ownshome |   .0221471   .0359649     0.62   0.538    -.0483902    .0926845
has_car |  -.0022049   .0518231    -0.04   0.966    -.1038446    .0994348
lnmeanc |  -.0137199   .0091042    -1.51   0.132    -.0315758     .004136
prov1 |   1.306906   .1138848    11.48   0.000     1.083546    1.530266
prov2 |   -.054912   .1135644    -0.48   0.629    -.2776436    .1678197
prov3 |   .2437542   .1137709     2.14   0.032     .0206175    .4668909
prov4 |   .1855854   .1336493     1.39   0.165    -.0765385    .4477093
rjob_2 |   .0339662   .0975402     0.35   0.728    -.1573375    .2252699
rjob_3 |  -.3293945   .0574735    -5.73   0.000    -.4421163   -.2166727
rjob_4 |  -.2810137   .0575245    -4.89   0.000    -.3938354   -.1681921
rjob_5 |  -.1094237   .0604152    -1.81   0.070     -.227915    .0090676
_cons |  -.6446848   .2743944    -2.35   0.019     -1.18285   -.1065201
-------------+----------------------------------------------------------------
/athrho |   .1125396   .2963974     0.38   0.704    -.4687791    .6938584
-------------+----------------------------------------------------------------
rho |   .1120669    .292675                     -.4372123     .600455
------------------------------------------------------------------------------


In this output, one can see that $\rho$, which is the correlation between $\epsilon_1$ and $\epsilon_2$ is not significant (t=0.38, p=0.70). I have the following interpretation questions:

1. Is it correct for me to conclude that there is no evidence that insurance is endogenous in this case?

2. Cameron and Trivedi (2009) state that in the non-SUR biprobit model, if $\rho==0$, then the joint model collapses into two separate models for $provider$ and $insured$. In the case of the SUR biprobit that I used, can I report the results of a regular probit model instead of the results of the bivariate probit that I fitted?

Reference

Cameron AC, Trivedi PK. Microeconometrics Using Stata. College Station, TX, USA: Stata Press; 2009.

• Doesn't biprobit usually do a likelihood ratio test that $\rho=0$ at the end, comparing the likelihood of the full bivariate model with the sum of the log likelihoods for the univariate probit models? Jul 1, 2015 at 21:44
• Sorry, I neglected to mentioned that I used the svy prefix (now corrected in the question). When you use the svy prefix, Stata does not spit out the LR test of $\rho=0$. However, the p-value for the LR test is equivalent to the p-value on /athrho in the model output Jul 1, 2015 at 21:56
• For example, without svy, the coefficient on /athrho was 0.121 ( t =0.74, p =0.462). For the Wald test of $\rho=0$, $\chi^2$ was 0.54, with a p-value of 0.4616 Jul 1, 2015 at 22:03

Knapp and Seaks (1998) show that a likelihood-ratio test of whether $\rho=0$ can be used as a Hausman endogeneity test. I think that logic carries over to survey data. This means that the separate probits are probably OK, though I would report the results of the test (or the confidence interval).
I am not sure if I would call a recursive biprobit a SUR estimator.