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.)