  We analyze the properties of the Synthetic Control (SC) and related
estimators when the pre-treatment fit is imperfect. In this framework, we show
that these estimators are generally biased if treatment assignment is
correlated with unobserved confounders, even when the number of pre-treatment
periods goes to infinity. Still, we show that a demeaned version of the SC
method can substantially improve in terms of bias and variance relative to the
difference-in-difference estimator. We also derive a specification test for the
demeaned SC estimator in this setting with imperfect pre-treatment fit. Given
our theoretical results, we provide practical guidance for applied researchers
on how to justify the use of such estimators in empirical applications.
