Learning Treatment Effects in Panels with General Intervention Patterns

The problem of causal inference with panel data is a central econometric question. The following is a fundamental version of this problem: Let M^* be a low rank matrix and E be a zero-mean noise matrix. For a `treatment' matrix Z with entries in 0,1 we observe the matrix O with entries Oij := M^*ij + Eij + Tij Zij where Tij are unknown, heterogenous treatment effects. The problem requires we estimate the average treatment effect τ^* := sumij Tij Zij / sumij Zij. The synthetic control paradigm provides an approach to estimating τ^* when Z places support on a single row. This paper extends that framework to allow rate-optimal recovery of τ^* for general Z, thus broadly expanding its applicability. Our guarantees are the first of their type in this general setting. Computational experiments on synthetic and real-world data show a substantial advantage over competing estimators.