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## Endogenous treatment effects

### Highlights

• Relaxes conditional independence
• Correlated unobservables that affect treatment and outcome
• Selection on unobservables
• Estimators
• Linear regression
• Poisson regression
• Robust SEs to relax distributional assumptions
• Cluster–robust SEs to allow for correlated data
• Predictions
• Potential-outcome means
• Observed-outcome means
• Conditional treatment effects
• Probability of count and count-ranges for Poisson

### Show me

Treatment effects measure the causal effect of a treatment on an outcome. Sometimes, we do not have conditional independence; that is, unobserved variables affecting both treatment and outcome. Endogenous treatment estimators address such cases.

etregress handles continuous outcome variables.

etpoisson handles count outcomes.

Suppose we are studying acne vulgaris in adults. In the data, members of the treated group have a specific genetic marker. That marker is itself determined by covariates, say, x1, x2, and x4. The outcome variable we observe is the number of pimples, and it is determined by x1 and x3. We are also concerned that there are unobserved variables that could cause the presence of the marker and affect the number of pimples.

We fit an endogenous treatment-effects model by typing

. etpoisson pustules x1 x3, treat(marker = x1 x2 x4)

Poisson regression with endogenous treatment      Number of obs   =       1000
(24 quadrature points)                            Wald chi2(3)    =      81.50
Log likelihood = -2297.3045                       Prob > chi2     =     0.0000

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

pustules

x1     .0509081   .0442439     1.15   0.250    -.0358084    .1376246

x3     .1281427   .0313078     4.09   0.000     .0667806    .1895049

1.marker     1.329508   .1994485     6.67   0.000     .9385965     1.72042

_cons    -.6982827   .1139443    -6.13   0.000    -.9216094   -.4749559

marker

x1     .2638588   .0469415     5.62   0.000     .1718552    .3558624

x2    -.1921751    .048748    -3.94   0.000    -.2877194   -.0966308

x4     .4720434   .0465768    10.13   0.000     .3807546    .5633322

_cons    -.4946407   .1121999    -4.41   0.000    -.7145484   -.2747329

/athrho     .1350499   .1706788     0.79   0.429    -.1994743    .4695742

/lnsigma     -.285167   .0516515    -5.52   0.000     -.386402    -.183932

rho     .1342348   .1676033                     -.1968701    .4378552

sigma     .7518886   .0388361                      .6794973    .8319923

Wald test of indep. eqns. (rho = 0): chi2(1) =     0.63   Prob > chi2 = 0.4288


The output reveals no evidence of the correlation that would violate conditional independence; the point estimate of the correlation is 0.1342, and the test against 0 is insignificant.

We can obtain the average treatment effect (ATE) based on these estimates by typing

. margins r.marker

Contrasts of predictive margins
Model VCE    : OIM

Expression   : Potential-outcome mean, predict()

df        chi2     P>chi2

marker            1       28.30     0.0000

Delta-method

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

marker

(1 vs 0)      2.584824    .485849      1.632577     3.53707



The presence of the marker adds on average 2.58 pustules.

### Show me more

Read much more about endogenous treatment in the manuals at etregress and etpoisson.

See New in Stata 17 to learn about what was added in Stata 17.