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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.

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  • $\begingroup$ 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? $\endgroup$ – Dimitriy V. Masterov Jul 1 '15 at 21:44
  • $\begingroup$ 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 $\endgroup$ – Marquis de Carabas Jul 1 '15 at 21:56
  • $\begingroup$ 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 $\endgroup$ – Marquis de Carabas Jul 1 '15 at 22:03
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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.


Knapp LG, and TG. Seakes, (1998) “A Hausman Test for a Dummy Variable in Probit”, Applied Economics Letters 5, 321–23.

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The biprobit command will return a correlation coefficient ("rho") which is the correlation coefficient between the residuals of each of the two probits. If rho is statistically significantly different from zero, then you should estimate the two probits simultaneously. If rho is not statistically significantly different from zero, then you can stick to estimating two separate probits.

(Quoted from Marc F. Bellemare on Statalist here.)

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