Standard error for the correlation coefficient (rho) in a bivariate probit model

[Stein Monteiro]

Hey STATA users,

I have been having this problem for awhile. It might be more of a theoretical issue. But any help would be appreciated.

I fit the bivariate probit model:

biprobit (y1 = x1 x2 x3) (y2 = x1 x2 x3)

From this we get a correlation coefficient estimate at the very end of the table labelled rho. If I want to obtain a good estimate of the standard error of rho, how could I go about doing this?

Thank you!

[Carlo Lazzaro]

Stein:
you can go -bootstrap-:

. use http://www.stata-press.com/data/r14/school

. biprobit private vote years logptax loginc

Fitting comparison equation 1:

Iteration 0:   log likelihood = -31.967097 
Iteration 1:   log likelihood = -31.452424 
Iteration 2:   log likelihood = -31.448958 
Iteration 3:   log likelihood = -31.448958 

Fitting comparison equation 2:

Iteration 0:   log likelihood = -63.036914 
Iteration 1:   log likelihood = -58.534843 
Iteration 2:   log likelihood = -58.497292 
Iteration 3:   log likelihood = -58.497288 

Comparison:    log likelihood = -89.946246

Fitting full model:

Iteration 0:   log likelihood = -89.946246 
Iteration 1:   log likelihood = -89.258897 
Iteration 2:   log likelihood = -89.254028 
Iteration 3:   log likelihood = -89.254028 

Bivariate probit regression                     Number of obs     =         95
                                                Wald chi2(6)      =       9.59
Log likelihood = -89.254028                     Prob > chi2       =     0.1431

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
private      |
       years |  -.0118884   .0256778    -0.46   0.643    -.0622159    .0384391
     logptax |  -.1066962   .6669782    -0.16   0.873    -1.413949    1.200557
      loginc |   .3762037   .5306484     0.71   0.478     -.663848    1.416255
       _cons |  -4.184694   4.837817    -0.86   0.387    -13.66664    5.297253
-------------+----------------------------------------------------------------
vote         |
       years |  -.0168561   .0147834    -1.14   0.254    -.0458309    .0121188
     logptax |  -1.288707   .5752266    -2.24   0.025    -2.416131   -.1612839
      loginc |    .998286   .4403565     2.27   0.023     .1352031    1.861369
       _cons |  -.5360573   4.068509    -0.13   0.895    -8.510188    7.438073
-------------+----------------------------------------------------------------
     /athrho |  -.2764525   .2412099    -1.15   0.252    -.7492153    .1963102
-------------+----------------------------------------------------------------
         rho |  -.2696186   .2236753                     -.6346806    .1938267
------------------------------------------------------------------------------
LR test of rho=0: chi2(1) = 1.38444                       Prob > chi2 = 0.2393

  
. bootstrap e(rho), reps(200) : biprobit private vote years logptax loginc
(running biprobit on estimation sample)

Bootstrap replications (200)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
..................................................    50
..................................................   100
..................................................   150
..................................................   200

Bootstrap results                               Number of obs     =         95
                                                Replications      =        200

      command:  biprobit private vote years logptax loginc
        _bs_1:  e(rho)

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |  -.2696186   .2602223    -1.04   0.300     -.779645    .2404077
------------------------------------------------------------------------------




. estat bootstrap, all

Bootstrap results                               Number of obs     =         95
                                                Replications      =        200

      command:  biprobit private vote years logptax loginc
        _bs_1:  e(rho)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _bs_1 |  -.26961864   .0369066   .26022232    -.779645   .2404077   (N)
             |                                      -.6513465   .4084706   (P)
             |                                      -.6601551   .2706797  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval

.

[Stein Monteiro]

This is great!

Thanks a lot Carlo, works perfectly!