  This paper studies inference on treatment effects in panel data settings with
unobserved confounding. We model outcome variables through a factor model with
random factors and loadings. Such factors and loadings may act as unobserved
confounders: when the treatment is implemented depends on time-varying factors,
and who receives the treatment depends on unit-level confounders. We study the
identification of treatment effects and illustrate the presence of a trade-off
between time and unit-level confounding. We provide asymptotic results for
inference for several Synthetic Control estimators and show that different
sources of randomness should be considered for inference, depending on the
nature of confounding. We conclude with a comparison of Synthetic Control
estimators with alternatives for factor models.
