We propose principled prediction intervals to quantify the uncertainty of a
large class of synthetic control predictions (or estimators) in settings with
staggered treatment adoption, offering precise non-asymptotic coverage
probability guarantees. From a methodological perspective, we provide a
detailed discussion of different causal quantities to be predicted, which we
call `causal predictands', allowing for multiple treated units with treatment
adoption at possibly different points in time. From a theoretical perspective,
our uncertainty quantification methods improve on prior literature by (i)
covering a large class of causal predictands in staggered adoption settings,
(ii) allowing for synthetic control methods with possibly nonlinear
constraints, (iii) proposing scalable robust conic optimization methods and
principled data-driven tuning parameter selection, and (iv) offering valid
uniform inference across post-treatment periods. We illustrate our methodology
with an empirical application studying the effects of economic liberalization
in the 1990s on GDP for emerging European countries. Companion general-purpose
software packages are provided in Python, R and Stata