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I am trying to decide between 3-5 different ways of calculating the
AME of treatment from a biprobit with endogenous treatment.

At the bottom, there is a toy example using the private school
attendance data that captures the problem with the code and output.
The model seems to fit pretty well.

Here's a short summary. I am calculating the AME with

1) margins, which gives an AME of .103 with 95% CI of (-.701, .907)
2) AME by hand #1, AME of .095 with (.092, 0.098)
3) AME by hand #2, AME of .095 with (.092, 0.098)
4) bootstrapped ATE using user-written biprobittreat: AME/ATE of .095
(-.501,.691)   (Normal CI)
(-.593,.352)   (Percentile CI)
(-.572,.372)   (Bias-Corrected CI)

Methods (1)-(3) were suggested by Austin Nichols in his presentation
on binary regression
(http://www.stata.com/meeting/chicago11/materials/chi11_nichols.pdf).
biprobittreat, the GoF test, and the papers are available from Richard
Chiburis' site (https://webspace.utexas.edu/rcc485/www/code.html).

All the AMEs are close, but the CIs are fairly different. Method (1)
is convenient, but probably wrong. Methods (2) and (3) use finite
difference and give identical results. The bootstrapped AME matches
(2) and (3), but the confidence intervals are much closer to what
margins produces.

Two questions. Which CIs would you prefer and why? Is the regression
of the ME on constant in (2) and (3) the right way to construct the
AME CI?

Here's the code with output:

. webuse school;

. biprobit (private = years loginc vote) (vote = year loginc logptax ), robust;

Fitting comparison equation 1:

Iteration 0:   log pseudolikelihood = -31.967097
Iteration 1:   log pseudolikelihood = -30.915551
Iteration 2:   log pseudolikelihood = -30.890586
Iteration 3:   log pseudolikelihood = -30.890555
Iteration 4:   log pseudolikelihood = -30.890555

Fitting comparison equation 2:

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

Comparison:    log pseudolikelihood = -89.387844

Fitting full model:

Iteration 0:   log pseudolikelihood = -89.387844
Iteration 1:   log pseudolikelihood = -89.274953
Iteration 2:   log pseudolikelihood =  -89.21596
Iteration 3:   log pseudolikelihood = -89.209977
Iteration 4:   log pseudolikelihood = -89.209866
Iteration 5:   log pseudolikelihood = -89.209866

Seemingly unrelated bivariate probit              Number of obs   =         95
                                                  Wald chi2(6)    =      14.69
Log pseudolikelihood = -89.209866                 Prob > chi2     =     0.0228

------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
private      |
       years |  -.0082564   .0176725    -0.47   0.640    -.0428939    .0263811
      loginc |   .2227815   .7555846     0.29   0.768    -1.258137      1.7037
        vote |   .5056949    1.70658     0.30   0.767     -2.83914     3.85053
       _cons |   -3.64063   7.195005    -0.51   0.613    -17.74258    10.46132
-------------+----------------------------------------------------------------
vote         |
       years |  -.0175658   .0174327    -1.01   0.314    -.0517332    .0166017
      loginc |   .9860015   .4170414     2.36   0.018     .1686155    1.803388
     logptax |  -1.287585   .5158509    -2.50   0.013    -2.298634   -.2765358
       _cons |  -.4144151   4.354969    -0.10   0.924    -8.949997    8.121167
-------------+----------------------------------------------------------------
     /athrho |  -.6420293   1.542175    -0.42   0.677    -3.664636    2.380577
-------------+----------------------------------------------------------------
         rho |  -.5662797   1.047641                     -.9986888    .9830337
------------------------------------------------------------------------------
Wald test of rho=0:                 chi2(1) =  .173318    Prob > chi2 = 0.6772

. scoregof;

Murphy's score test for biprobit

             chi2(9) =    2.61
         Prob > chi2 =    0.9778

. bphltest;

Modified Hosmer-Lemeshow goodness-of-fit test for biprobit

           chi2( 21) =   27.63
         Prob > chi2 =    0.1510

. /* AME Margins Way */
> margins, dydx(vote) predict(pmarg1) force;
(note: prediction is a function of possibly stochastic quantities
other than e(b))

Average marginal effects                          Number of obs   =         95
Model VCE    : Robust

Expression   : Pr(private=1), predict(pmarg1)
dy/dx w.r.t. : vote

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        vote |    .103085   .4101601     0.25   0.802     -.700814     .906984
------------------------------------------------------------------------------

. /* AME by Hand # 1 */
> predict double xb2, xb2;

.  // linear index for equation 2
> ren vote Tvote;

. gen vote=0;

. predict double p0, pmarg1;

.  // success for equation 1 with vote == 0
> predict double xb0, xb1;

.    // index for equation 1 with vote == 0
>
> replace vote=1;
(95 real changes made)

. predict double p1, pmarg1;

.  // success for equation 1 with vote == 1
> predict double xb1, xb1;

.    // index for equation 1 with vote == 1
> gen double dp=p1-p0;

.        // calculate diff in
> sum dp;

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
          dp |        95    .0949194    .0151578   .0492599   .1230978

. reg dp;

      Source |       SS       df       MS              Number of obs =      95
-------------+------------------------------           F(  0,    94) =    0.00
       Model |           0     0           .           Prob > F      =       .
    Residual |   .02159723    94  .000229758           R-squared     =  0.0000
-------------+------------------------------           Adj R-squared =  0.0000
       Total |   .02159723    94  .000229758           Root MSE      =  .01516

------------------------------------------------------------------------------
          dp |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .0949194   .0015552    61.04   0.000     .0918316    .0980072
------------------------------------------------------------------------------

. /* AME by Hand # 2 */
> gen double pdx=(binormal(xb1,xb2,e(rho))-binormal(xb0,xb2,e(rho)))/normal(xb2) if Tvote==1;
(95 missing values generated)

. qui replace pdx=normal(xb1)-normal(xb0);

. su pdx;

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         pdx |        95    .0949194    .0151578   .0492599   .1230978

. loc ATE2=r(mean);

. reg pdx;

      Source |       SS       df       MS              Number of obs =      95
-------------+------------------------------           F(  0,    94) =    0.00
       Model |           0     0           .           Prob > F      =       .
    Residual |   .02159723    94  .000229758           R-squared     =  0.0000
-------------+------------------------------           Adj R-squared =  0.0000
       Total |   .02159723    94  .000229758           Root MSE      =  .01516

------------------------------------------------------------------------------
         pdx |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .0949194   .0015552    61.04   0.000     .0918316    .0980072
------------------------------------------------------------------------------

. replace vote=Tvote;
(36 real changes made)

.  // set vote back to normal again
>
> /* AME using ATT */
> bootstrap _b ate=r(ate), reps(500) saving("bs_ate.dta", replace): biprobittreat (private = years loginc vote) (vote = year loginc logptax), robust;
(running biprobittreat on estimation sample)

Bootstrap replications (500)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5
..................................................    50
..................................................   100
..................................................   150
..................................................   200
..................................................   250
..................................................   300
..................................................   350
.................................................x   400
..................................................   450
..................................................   500

Bootstrap results                               Number of obs      =        95
                                                Replications       =       499

      command:  biprobittreat (private = years loginc vote) (vote =
year loginc logptax), robust
    [_eq4]ate:  r(ate)

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
private      |
       years |  -.0082564   .0344714    -0.24   0.811     -.075819    .0593062
      loginc |   .2227815   .7495176     0.30   0.766    -1.246246    1.691809
        vote |   .5056949   1.331318     0.38   0.704     -2.10364    3.115029
       _cons |   -3.64063   7.297287    -0.50   0.618    -17.94305    10.66179
-------------+----------------------------------------------------------------
vote         |
       years |  -.0175658   .0257062    -0.68   0.494    -.0679491    .0328176
      loginc |   .9860015   .5231484     1.88   0.059    -.0393505    2.011354
     logptax |  -1.287585   .6197584    -2.08   0.038    -2.502289   -.0728808
       _cons |  -.4144151   5.665756    -0.07   0.942    -11.51909    10.69026
-------------+----------------------------------------------------------------
athrho       |
       _cons |  -.6420293   234.5964    -0.00   0.998    -460.4424    459.1584
-------------+----------------------------------------------------------------
_eq4         |
         ate |   .0949194   .3040172     0.31   0.755    -.5009434    .6907822
------------------------------------------------------------------------------
Note: one or more parameters could not be estimated in 1 bootstrap replicate;
      standard-error estimates include only complete replications.

. estat bootstrap, all

Bootstrap results                               Number of obs      =        95
                                                Replications       =       499

      command:  biprobittreat (private = years loginc vote) (vote =
year loginc logptax), robust
    [_eq4]ate:  r(ate)

------------------------------------------------------------------------------
             |    Observed               Bootstrap
             |       Coef.       Bias    Std. Err.  [95% Conf. Interval]
-------------+----------------------------------------------------------------
private      |
       years |  -.00825642  -.0080788   .03447135    -.075819   .0593062   (N)
             |                                       -.087569   .0253931   (P)
             |                                       -.082893   .0285328  (BC)
      loginc |   .22278152   .1409217   .74951755   -1.246246   1.691809   (N)
             |                                      -.8413433    1.90044   (P)
             |                                      -.9458166   1.668028  (BC)
        vote |   .50569492  -.5871846   1.3313176    -2.10364   3.115029   (N)
             |                                      -2.308104   1.569678   (P)
             |                                      -2.232462   1.608637  (BC)
       _cons |  -3.6406301  -.9503618   7.2972871   -17.94305   10.66179   (N)
             |                                      -20.36213   6.880115   (P)
             |                                      -20.31908   7.248422  (BC)
-------------+----------------------------------------------------------------
vote         |
       years |  -.01756575  -.0058208   .02570624   -.0679491   .0328176   (N)
             |                                      -.0884118    .013394   (P)
             |                                      -.0596505   .0210808  (BC)
      loginc |   .98600148   .0904138    .5231484   -.0393505   2.011354   (N)
             |                                       .0779691   2.152244   (P)
             |                                      -.1330201   2.017503  (BC)
     logptax |   -1.287585  -.1660917   .61975842   -2.502289  -.0728808   (N)
             |                                      -2.795679   -.418406   (P)
             |                                      -2.349517  -.0533561  (BC)
       _cons |  -.41441506   .3055246   5.6657558   -11.51909   10.69026   (N)
             |                                      -11.65625   10.40091   (P)
             |                                      -12.30915   9.867423  (BC)
-------------+----------------------------------------------------------------
athrho       |
       _cons |  -.64202932  -9.220272   234.59636   -460.4424   459.1584   (N)
             |                                      -16.54541    18.6691   (P)
             |                                      -17.94267    17.2223  (BC)
-------------+----------------------------------------------------------------
_eq4         |
         ate |   .09491939  -.1364229    .3040172   -.5009434   .6907822   (N)
             |                                      -.5932178   .3518947   (P)
             |                                      -.5716441   .3718321  (BC)
------------------------------------------------------------------------------
(N)    normal confidence interval
(P)    percentile confidence interval
(BC)   bias-corrected confidence interval
Note: one or more parameters could not be estimated in 1 bootstrap replicate;
      standard-error estimates include only complete replications.


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