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A study with observational data has treatment and control group but the assignment is not randomised: some chose to be in the treament, some otherwise. But the choice had been made before the treatment was announced, so it is safe to assume that the groups had not been aware of the treatment and chosen the assignment by their own covariates.

After the treatment was annouced, some in control groups (presumely opportunistic) take up the treatment too (the treatment is a funding opportunity, so control group can change their behaviours to apply for the fund), and some in treatment group chose not to participate (for some reason). These take-ups are recorded in the follow-up survey.

I am interested in outcomes Y. For what I have read so far, Rubin causual model is a good candidate to deal with the setup.

Nonetheless, I still sense that somehow RCT literature can be used here. I am thinking to use difference-in-difference approach (for some outcomes with richer data, I want to use diff-in-diff-in-diff) AND use LATE effect to account for the imperfect compliance. The procedure is to run diff-in-diff 2SLS. I will estimate Logit/probit to see why they take up or not but return to 2SLS.

Could you please point out if I am wrong to think of complimenting Rubin causal model with diff-in-diff 2SLS?

(My aspiration is from Card and Krueger, 1994 for diff-in-diff when the treatment is not randomised, so the authors pick a neighbour state as control group. I am aware of the assumption that there must be a common trend between the groups but such thing is not allowed to be tested in short-term data.)

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This is a perfectly valid thing to do. You can instrument actual treatment status with initial assignment to treatment in a difference-in-differences setting which would give you the so-called intention-to-treat effect. A paper that uses a similar approach is Waldinger (2010) "Quality Matters: The Expulsion of Professors and the Consequences for PhD Student Outcomes in Nazi Germany" (link). A nice overview of the underlying idea is provided in these slides (p. 40 onwards). I'm sure there are other papers that also combine diff-in-diff with instrumental variables but this one came to mind first.

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  • $\begingroup$ Thanks Andy. I was reading the papers to see if I can apply in my case. I find another helpful handout from MIT page 12. Another question is whether DID works if I have data for 2 points of time (baseline survey in 2002, treatment from 2002-2006, and follow-up survey in 2006). From Waldinger (2010) and his note, one important assumption is common trend and he tests this by moving a placebo to pretreatment data. But does this require panel data (more than 2 points of time)? How could one test the common trend assumption? Thank you very much! $\endgroup$
    – Khan
    Jul 7, 2015 at 20:36
  • $\begingroup$ No you don't need panel data, it is sufficient if units are drawn from the same populations among the treatment and control group over time (there is a short explanation about this here: stats.stackexchange.com/questions/564/…). For the test of the parallel trends assumption, would you mind opening a new thread? There I can reply better with the maths and there is more space to write a proper answer. It would be a good question in its own right as well! $\endgroup$
    – Andy
    Jul 7, 2015 at 20:41
  • $\begingroup$ Thank you very much, I will open a new thread now but it takes me time to digest the explanation. This comment is to let you know my gratitude. Thank you! $\endgroup$
    – Khan
    Jul 7, 2015 at 20:44
  • $\begingroup$ I'm happy that you found my answers useful :-) $\endgroup$
    – Andy
    Jul 7, 2015 at 21:13

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