Estimating a causal effect with the synthetic control unit using a synthetic treated unit

I know that the synthetic control method is used to estimate a causal effect by finding the difference in outcome between a real treated unit and a synthetic non-treated unit. However, I was wondering if it is possible and mathematically sound to use the synthetic control method to estimate the opposite counter-factual.

Instead of using similar non-treated units to develop the synthetic unit, I imagine you could simply choose a non-treated unit as your real unit and then use similar treated units to develop the synthetic unit. Of course, if no event happened, this introduces a degree of subjectivity on the researcher's part to assign an artifical event. However, if said researcher has sound reasons to believe that an event could have (or almost did) occurred at a certain time (but it did not in the real world), I imagine you could artifically assign a date for when a hypothetical treatment could have occurred.

In theory, this sounds like it is feasible, but, given the fact that I haven't seen anyone else do this, I am cautious to explore this option further as there could be something inherent to the method that renders this inverse approach impractical.

• This sounds completely pointless. Isn't the point of research to characterize the effect of treatment? According to the Helsinki Declaration, clinical research is justified by equipoise: i.e. we don't the effect of treatment, and it could be better than (a suitable) control. Commented Dec 2, 2022 at 18:56
• I'm not sure what background you come from, but the counterfactual (Y | T1) is often equally as interesting as the counterfactual that the SCM estimates (Y | T0). Especially when the treatment of interest is a treatment that could have happpened but did not (i. e. had international intervention taken place in Country X, could genocide, war, etc. been prevented). In many areas, this counter-factual is the more important question. Commented Dec 2, 2022 at 19:28
• Retrospective analyses have their points- and that's precisely what you're describing. Causal modeling has a robust literature on estimating counterfactuals. What's not clear is how any prospective component factors in at all. In all the literature on synthetic controls, we deal with the problem of having a prospective study design where an experimental treatment is given in a non-random fashion, and we want to use historical data to model a control arm. Commented Dec 2, 2022 at 19:40
• Yeah, and that's where I've had my concerns with this approach because I haven't seen any other researcher attempt such a thing (which I do not take as a good sign). The setup of the SCM does almost beg the question to execute such a retrospective analysis because it is easy to 1) select a non-treated unit to observe over time, 2) develop a synthetic unit from treated units, and 3) artifically select a theoretically-informed time for the hypothetical treatment to occur. It can technically all be done, but I'm unsure if this works "under the hood" of the the method. Commented Dec 2, 2022 at 19:54
• In my opinion, what you're proposing is not well motivated, so whether the gap in literature is in fact problematic is a matter of opinion. Or one of misunderstanding - either yours or mine. To predict the response over time in a "treated" unit, you would need to simulate the effect of treatment according to some assumptions, and then could you really be surprised that your result is the same as you have assumed it to be? Commented Dec 2, 2022 at 20:09

I think this is an interesting idea and an important question to answer. But I can see some potential problems with using a SC approach.

Typically, you have one or a few treated units and many untreated ones. So you can usually construct a good synthetic cohort for the treated units since you have many potential donors to reweight. You don’t have that going in the other direction, which makes the job harder. This means you may not be able to model many peaceful countries. This becomes even trickier if units become treated at different times. I think this would also complicate inference for similar reasons.

Now suppose you give up on modeling all peaceful countries and pick just one.

For example, say you have Country A that has never had an intervention. You would like to say what would have happened to mortality had A had an intervention during 2010-2015. Say you can find k countries that always had intervention during those five years, but never before. You may not be able to reweight k countries to behave like A unless k is large.

You might then try to do this year by year. But then your counterfactual is always made up of fresh intervention countries. But this seems like a weird thought experiment if interventions have a life cycle (announcement, troop arrival, learning on the ground by commanders and locals, and eventual troop withdrawal, and adaptation to that).

Maybe you then freeze the pre model in 2009, but then add the new interventions in 2011 to those from 2010 to model the outcome in 2011. But this means you would have better pre fit in later periods since the choice of potential donors is better. I am also not sure how to adjust inference here to test cross-year hypotheses. It also makes it hard to do robustness checks where you drop adjacent countries from the analysis.

Maybe this can be easier with one of the new Diff in Diff methods that allow for staggered treatments and effects that depend on time.

I would also encourage you to define treatment more precisely and think about if SUTVA is a reasonable assumption given constraints on troops and spillovers across borders.