Learning Treatment Effects in Panels with General Intervention Patterns

Abstract

The problem of causal inference with panel data is a central econometric question. The following is a fundamental version of this problem: Let $M^{\ast }$ 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 $O_{ij}:=M_{ij}^{\ast }+E_{ij}+{\mathcal{T}}_{ij}{Z}_{ij}$ where ${\mathcal{T}}_{ij}$ are unknown, heterogenous treatment effects. The problem requires we estimate the average treatment effect ${\tau }^{\ast }:={\sum }_{ij}{\mathcal{T}}_{ij}{Z}_{ij}/{\sum }_{ij}{Z}_{ij}$. The synthetic control paradigm provides an approach to estimating ${\tau }^{\ast }$ when $Z$ places support on a single row. This paper extends that framework to allow rate-optimal recovery of ${\tau }^{\ast }$ 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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