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# Endogenous treatment effects

## Highlights

• Endogenous treatments
• Continuous, fractional, binary, and count outcomes
• Average treatment effects (ATEs)
• ATEs on the treated (ATETs)
• Potential-outcome means (POMs)
• Control-function estimator
• Endogeneity Test

Treatment-effects estimators extract experimental-style causal effects from observational data.

Examples of treatment effects include effect of a drug regimen on blood pressure, effect of a surgical procedure on mobility, effect of a training program on employment, and effect of an ad campaign on sales.

If an unobserved variable affects the treatment and affects the outcome we have an endogeneity problem, the result of which is that we cannot obtain accurate estimates of effects using conventional treatment effect estimators.

For example, suppose we wish to know the effect on blood pressure of a antihypertensive drug. Further suppose that poor dietary habits affect blood pressure and individuals with poor dietary habits are more likely to be prescribed the antihypertensive drug. Suppose we do not observe dietary habits. We have an endogeneity problem.

Or, suppose we wish to know the effect on employment status of a job training program. Further suppose that motivation affects employment status and motivation affects participation. We do not observe motivation. We have an endogeneity problem.

Stata 14's new eteffects addresses the confounding effects of unobserved variables and allows us to test for endogeneity. In return, you must model both the treatment and the outcome.

If you didn't have concerns about endogeneity, you could use the standard teffects estimator and model either the treatment or the outcome, or you could model both and get the benefit of double robustness; only one of the models would have to be correct.

Nothing in this world is free, however. If you are concerned about endogeneity, you must model both the treatment and the outcome, and double robustnesss is out the window. Both models must be correct.

## Let's see it work

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.

We use eteffects rather than teffects. We type,

. 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


We modeled wages as being determined by job tenure and age.

We modeled college attainment by age and the number of parents who attended college.

The resulting estimated ATE is $931.84 per month for college attainment. The potential-outcome mean (the expected wage if no one attended college) is$2,160.56 per month.

eteffects also allow us to test if unobserved ability affects both outcome and treatment. This is a byproduct of the control-function approach, which estimates the correlation between the unobservables that affect treatment and outcome. If these correlations are zero, we have no endogeneity.

To test for endogeneity we type:

. estat endogenous

Test of endogeneity
Ho: treatment and outcome unobservables are uncorrelated

chi2(  2) =  305.38
Prob > chi2 =    0.0000


We reject the null hypothesis. There is evidence of endogeneity.

Had we not used eteffects to account for the endogeniety, we would have obtained incorrect estimates. Because we are using simulated data we can tell you that the true ATE was $924. The above estimate of$932 is close. Had we estimated the ATE ignoring the endogeneity, the estimate would have been \$1,514. We would have been off substantially. eteffects and its associated endogeneity test saved us from that fate.

## Tell me more

Read much more about endogenous treatment effects in the Stata Treatment-Effects Reference Manual; see [TE] eteffects.