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## Treatment effects for endogenous treatments

### Highlights

• Relaxes conditional independence assumption
• Continuous, binary, count, fractional, and nonnegative outcomes
• Average treatment effects (ATEs)
• ATEs on the treated (ATETs)
• Potential-outcome means (POMs)

Treatment-effects estimators extract experimental-style causal effects from observational data. Conventional treatment-effects estimators require the conditional independence assumption. That is, we must assume that no unobserved variables affect both treatment assignment and the outcome.

If an unobserved variable affects which treatment a person gets and affects the outcome, we have an endogeneity problem and we cannot obtain accurate estimates of effects using conventional treatment-effects estimators. Endogenous treatment estimators address such cases.

Stata has three commands for endogenous treatment-effects estimation.

We can estimate endogenous treatment effects in the same potential-outcomes framework used by teffects—the parameters of interest are the treatment effects. It lets us model a wide range of outcomes: continuous, binary, count, fractional, and nonnegative outcomes.

We can also estimate a linear or Poisson regression model that includes an endogenous treatment by using either etregress or etpoisson. These commands are slightly different from eteffects. Because the methods implemented in these commands are not naturally in the potential-outcomes framework, we use margins to obtain treatment effects such as the ATE.

### Show me

We wish to measure the effect of college degree on wages, but we worry that unobserved ability will affect both the outcome and treatment and so confound our estimates.

First, we fit our model by using eteffects. We model wages as being determined by job tenure and age. We model college attainment by age and the number of parents who attended college.

. eteffects (wage tenure c.age##c.age) (college c.age##c.age i.pcollege)

Iteration 0:   EE criterion =  5.757e-21
Iteration 1:   EE criterion =  5.757e-21  (backed up)

Endogenous treatment-effects estimation         Number of obs     =      1,000
Outcome model  : linear
Treatment model: probit

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

ATE
college
(1 vs 0)     931.8384   42.18229    22.09   0.000     849.1627    1014.514

POmean
college
0     2160.561    31.4066    68.79   0.000     2099.005    2222.116


The resulting estimated ATE is $931.84 per month for college attainment. We could also fit this model by typing . etregress wage tenure c.age##c.age, treat(college=c.age##c.age i.pcollege)  The resulting estimated ATE is$903.40 per month.

Because we are using simulated data, we can tell you that the true ATE was $924. Both eteffects and etregress provide estimates that are close to the true value. Had we estimated the ATE ignoring the endogeneity, the estimate would have been$1,514. We would have been off substantially.

### Tell me more

For information about the eteffects estimator, check out the overview from the Stata News.

Read much more about endogenous treatment effects and see more examples in [TE] eteffects, [TE] etregress, and [TE] etpoisson, which can be found in Stata Treatment-Effects Reference Manual.