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Suppose I run a RCT where the treatment is educating high school students. Outcome variable of interest $y$ is the wage the student will earn after graduating from high school. Suppose there is no non-compliance, and SUTVA is satisfied. I am interesting in obtaining Average Treatment Effect (ATE). However, wage is only observed for individuals who are employed, and suppose also that the treatment also increases the probability of individual being hired.

How should I estimate the treatment effect on individual wage? To be specific, is the following regression what I want?

$y = \alpha + \beta D + \varepsilon$, where $D$ is the dummy for treated group.

This regression will include individuals whose wage is 0 because it includes individuals who are not hired. My instinct tells me that I need to somehow instrument for the treatment effect on the probability of being hired, but I am unsure what I should do.

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I think you are bit overthinking it. Assuming wage does not affect treatment there is no need for IV because there is no endogeneity. In your case since you are using RCT this should be the case if you designed the RCT well.

Depending on what exactly is the purpose of the study I think either of the two is reasonable:

  1. While of course it is possible to work and have zero wage, and while this is certainly different from being unemployment, if the purpose of the study is to get effect of wages, I believe it is completely reasonable to include unemployed as people with zero wage. My justification for it would be that simply the treatment intervention (which presumably is supposed to increase value of employee for employer) failed to do it sufficiently to warrant employment even at minimum wage.

    This will likely lead to truncated sample, but you can deal with this issue using Tobit regression.

    You could also argue perhaps it is because some people are just less interested in getting employed, but then just control for their employment status, and once you control for this you will get rid of OVB due to this reason.

  2. You can treat people who are unemployed in similar fashion as people who drop out from randomized trial. I think the analogy makes somewhat sense, not participating in the labor market could be viewed as dropping out of trial, and there are medicines that make people more likely to drop out (due to nasty side effects) more often from treatment than from control.

    What is being done in medical research is that they typically try to impute missing values in such cases in one way or another. I believe you could just simply try to impute 'missing' wages based on background characteristics and work with that. Here is some paper discussing this approach in medicine.

    Also, an important caveat I am economist, I just know about this method from being part of multidisciplinary working group on causal inference at my university. I am not sure if the paper I recommended is necessarily the best, didn't try to apply it in my own work but it is something that makes sense to me in case like this, but I can be wrong, you should bounce this idea against other colleagues at your research institute.

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