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## Linear regression with endogenous treatment effects

Stata’s etregress allows you to estimate an average treatment effect (ATE) and the other parameters of a linear regression model augmented with an endogenous binary-treatment variable. You just specify the treatment variable and the treatment covariates in the treat() option. The average treatment effect on the treated (ATET) can also be estimated with etregress.

We estimate the ATE of being a union member on wages of women with etregress. Other outcome covariates include wage, school, grade, and tenure. Indicators for living in an SMSAâ€”standard metropolitan statistical area, being African American, and living in the southern region of the United States are also used as outcome covariates.

. webuse union3
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. etregress wage age grade smsa black tenure, treat(union = south black tenure)

Iteration 0:   log likelihood =  -3140.811
Iteration 1:   log likelihood = -3053.6629
Iteration 2:   log likelihood = -3051.5847
Iteration 3:   log likelihood =  -3051.575
Iteration 4:   log likelihood =  -3051.575

Linear regression with endogenous treatment     Number of obs     =      1,210
Estimator: maximum likelihood                   Wald chi2(6)      =     681.89
Log likelihood =  -3051.575                     Prob > chi2       =     0.0000

Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

wage

age     .1487409   .0193291     7.70   0.000     .1108566    .1866252

grade     .4205658   .0293577    14.33   0.000     .3630258    .4781058

smsa     .9117044   .1249041     7.30   0.000     .6668969    1.156512

black    -.7882471   .1367078    -5.77   0.000     -1.05619   -.5203048

tenure     .1524015   .0369596     4.12   0.000     .0799621    .2248409

1.union     2.945815   .2749621    10.71   0.000       2.4069    3.484731

_cons    -4.351572   .5283952    -8.24   0.000    -5.387208   -3.315936

union

south    -.5807419   .0851111    -6.82   0.000    -.7475566   -.4139271

black     .4557499   .0958042     4.76   0.000     .2679771    .6435226

tenure     .0871536   .0232483     3.75   0.000     .0415878    .1327195

_cons    -.8855758   .0724506   -12.22   0.000    -1.027576   -.7435753

/athrho    -.6544347   .0910314    -7.19   0.000     -.832853   -.4760164

/lnsigma     .7026769   .0293372    23.95   0.000      .645177    .7601767

rho    -.5746478    .060971                      -.682005   -.4430476

sigma     2.019151   .0592362                      1.906325    2.138654

lambda      -1.1603   .1495097                     -1.453334   -.8672668

LR test of indep. eqns. (rho = 0):   chi2(1) =    19.84   Prob > chi2 = 0.0000


The estimated ATE of being a union member is 2.95. The ATET is the same as the ATE in this case because the treatment indicator variable has not been interacted with any of the outcome covariates.

When there is a treatment variable and outcome covariate interaction, the parameter estimates from etregress can be used by margins to estimate the ATE. Now we use factor-variable notation to allow the tenure and black coefficients to vary based on union membership. We specify the vce(robust) because we need to specify vce(unconditional) when we use margins below.

. etregress wage age grade south i.union#c.(black tenure), treat(union = south black tenure) vce(robust)

Iteration 0:   log pseudolikelihood = -3093.9289
Iteration 1:   log pseudolikelihood = -3069.8014
Iteration 2:   log pseudolikelihood = -3069.0214
Iteration 3:   log pseudolikelihood = -3069.0106
Iteration 4:   log pseudolikelihood = -3069.0106

Linear regression with endogenous treatment     Number of obs     =      1,210
Estimator: maximum likelihood                   Wald chi2(8)      =     445.85
Log pseudolikelihood = -3069.0106               Prob > chi2       =     0.0000

Robust

Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]

wage

age     .1547605    .020634     7.50   0.000     .1143186    .1952025

grade     .4328724   .0372888    11.61   0.000     .3597876    .5059572

south    -.5060951   .2009611    -2.52   0.012    -.8999716   -.1122186

union#c.black

0     -.4695395   .2009365    -2.34   0.019    -.8633677   -.0757112

1     -.8580219   .2893336    -2.97   0.003    -1.425105   -.2909385

union#

c.tenure

0      .1802719   .0545018     3.31   0.001     .0734504    .2870934

1      .0848265   .0929442     0.91   0.361    -.0973408    .2669938

1.union     3.060777   .9504098     3.22   0.001     1.198008    4.923546

_cons    -3.847881   .6560055    -5.87   0.000    -5.133628   -2.562133

union

south    -.5041281   .0932344    -5.41   0.000    -.6868642    -.321392

black     .4506167   .0953425     4.73   0.000     .2637489    .6374845

tenure     .0917203   .0260037     3.53   0.000      .040754    .1426867

_cons    -.9325238   .0811249   -11.49   0.000    -1.091526   -.7735219

/athrho    -.5750886   .3420724    -1.68   0.093    -1.245538     .095361

/lnsigma     .6978439   .0973047     7.17   0.000     .5071302    .8885576

rho    -.5190865   .2499007                     -.8470277     .095073

sigma     2.009416   .1955256                      1.660519     2.43162

lambda    -1.043061   .5904939                     -2.200407    .1142862

Wald test of indep. eqns. (rho = 0): chi2(1) =     2.83   Prob > chi2 = 0.0927


The ATE of union membership is now estimated with margins. The “r.” notation tells margins to contrast the potential-outcome means for the treatment and control regimes.

. margins r.union, vce(unconditional) contrast(nowald)

Contrasts of predictive margins

Expression   : Linear prediction, predict()

Unconditional

Contrast   Std. Err.     [95% Conf. Interval]

union

(1 vs 0)       2.772612   .9382272      .9337209    4.611504



The estimate of the ATE is essentially the same as in the original model. Now we estimate the ATET of union membership with margins. We specify union in the subpop() option to restrict estimation to the treated subpopulation.

. margins r.union, vce(unconditional) contrast(nowald) subpop(union)

Contrasts of predictive margins

Expression   : Linear prediction, predict()

Unconditional

Contrast   Std. Err.     [95% Conf. Interval]

union

(1 vs 0)      2.704089   .9415909      .8586049    4.549573



The estimated ATET and ATE are close, indicating that the average predicted outcome for the treatment group is similar to the average predicted outcome for the whole population.