Learning Treatment Effects in Panels with General Intervention Patterns
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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 O_ij := M^*_ij + E_ij + 𝒯_ij Z_ij where 𝒯_ij are unknown, heterogenous treatment effects. The problem requires we estimate the average treatment effect τ^* := ∑_ij𝒯_ij Z_ij / ∑_ij Z_ij. 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.
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