Learning Treatment Effects in Panels with General Intervention Patterns

Vivek Farias, Andrew A Li, Tianyi Peng

Keywords: Causal Inference, Treatment Effect, Low-Rank Matrix Estimation, Panel Data, Synthetic Control

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

Code: https://github.com/TianyiPeng/Causal-Inference-Code

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2106.02780/code)