Can you use a count outcome with the synthetic control method? I am attempting to employ the synthetic control method (SCM) to analyze the causal impact of a UN peacekeeping operation (PKO) on post-conflict peace. I am operationalizing post-conflict peace as both the number of fatalities produced by terrorist violence and the count of terrorist attacks. In both cases, these outcome variables are count outcomes, rather than continuous outcomes. I have only ever seen the SCM used to estimate the causal effect of a treatment on a continuous outcome. So, I am not sure if using a count outcome is appropriate.
However, both of these count outcomes mimic continuous outcomes fairly well as both have a large degree of variation (ranging anywhere from 0 to well into the thousands). Further, for the case I am analyzing, there are no zeroes for the outcome variable. Given this, is moving forward with the count outcomes appropriate? Or should I log-transform these values?

Using counts of fatalities and attacks will not work well if your geographic units are of different sizes and those counts increase with size. You will be able to tell when that is the case if you cannot construct synthetic cohort(s) that resembles your treated unit(s) with low pre-treatment RMSPE (or just eyeballing the fit from the graph). It may still work if your treated units are medium-sized, so you can reweight large and small untreated to behave like them, though that sort of thing worries me a fair bit.
Here’s a brief outline of the argument in quotes from Abadie’s 2021 JEL paper (full citation at the bottom). Near the top right part of page 395, he writes:

Notice also that considering synthetic controls with weights that sum
to one may be warranted only if the variables in the data are rescaled
to correct for differences in size between units (e.g., per capita
income) or if such correction is not needed because the variables in
the data do not scale with size (e.g., prices).

On page 411, he adds:

If the unit affected by the intervention of interest is “extreme” in
the value of a particular variable, such a value may not be closely
approximated by a synthetic control.$^{16}$

Footnote 16 reads:

For example, Abadie, Diamond, and Hainmueller (2015) find that because
inflation levels were particularly low for West Germany before the
reunification, the value of this variable cannot be closely reproduced
by a synthetic control composed by other OECD countries.

I can’t recall any papers that directly model the outcome in levels when the units are geographic entities. Counts for GDP, STDs, and packs of cigarettes are put into per capita terms. The closest paper to your project that I know of is Donohue, Aneja, and Weber 2019 JELS on right-to-carry, where the outcomes are US state crime rates (violent crimes and homicides per 100K residents).
It is also reasonably straightforward to get back to levels after doing the per-capita analysis by multiplying by population, so you can still report that if you like or transform it to percentages.
To sum up, using counts is probably a bad idea. It might be doable if your units are all similar in size or if you use newer methods that allow for extrapolation rather than only interpolation that you get with constrained SC weights. But you should be able to tell if counts are not working pretty quickly. It may also be hard to get population data over time if you have many years of data.

Abadie, Alberto. 2021. “Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects.” Journal of Economic Literature, 59 (2): 391-425. DOI: 10.1257/jel.20191450
John J. Donohue, Abhay Aneja & Kyle D. Weber, Right-to-Carry Laws and Violent Crime: A Comprehensive Assessment Using Penal Data and a State-Level Synthetic Control Analysis, 16 Journal of Empirical Legal Studies 198 (2019) (available earlier as National Bureau of Economic Research, working paper no. 23510 (2017)).
