  Quantifying cause and effect relationships is an important problem in many
domains. The gold standard solution is to conduct a randomised controlled
trial. However, in many situations such trials cannot be performed. In the
absence of such trials, many methods have been devised to quantify the causal
impact of an intervention from observational data given certain assumptions.
One widely used method are synthetic control models. While identifiability of
the causal estimand in such models has been obtained from a range of
assumptions, it is widely and implicitly assumed that the underlying
assumptions are satisfied for all time periods both pre- and post-intervention.
This is a strong assumption, as synthetic control models can only be learned in
pre-intervention period. In this paper we address this challenge, and prove
identifiability can be obtained without the need for this assumption, by
showing it follows from the principle of invariant causal mechanisms. Moreover,
for the first time, we formulate and study synthetic control models in Pearl's
structural causal model framework. Importantly, we provide a general framework
for sensitivity analysis of synthetic control causal inference to violations of
the assumptions underlying non-parametric identifiability. We end by providing
an empirical demonstration of our sensitivity analysis framework on simulated
and real data in the widely-used linear synthetic control framework.
