Abstract

In this article, we study methods for estimating causal effects in settings with panel data, where some units are exposed to a treatment during some periods and the goal is estimating counterfactual (untreated) outcomes for the treated unit/period combinations. We propose a class of matrix completion estimators that uses the observed elements of the matrix of control outcomes corresponding to untreated unit/periods to impute the “missing” elements of the control outcome matrix, corresponding to treated units/periods. This leads to a matrix that well-approximates the original (incomplete) matrix, but has lower complexity according to the nuclear norm for matrices. We generalize results from the matrix completion literature by allowing the patterns of missing data to have a time series dependency structure that is common in social science applications. We present novel insights concerning the connections between the matrix completion literature, the literature on interactive fixed effects models and the literatures on program evaluation under unconfoundedness and synthetic control methods. We show that all these estimators can be viewed as focusing on the same objective function. They differ solely in the way they deal with identification, in some cases solely through regularization (our proposed nuclear norm matrix completion estimator) and in other cases primarily through imposing hard restrictions (the unconfoundedness and synthetic control approaches). The proposed method outperforms unconfoundedness-based or synthetic control estimators in simulations based on real data.

摘要

在本文中，我们研究了使用面板数据估计因果效应的方法，其中一些单位在某些时期接受治疗，目标是估计治疗单位/时期组合的反事实（未处理）结果。我们提出了一类矩阵完成估计器，它使用对应于未处理单元/周期的控制结果矩阵的观察元素来估算对应于处理单元/周期的控制结果矩阵的“缺失”元素。这导致矩阵很好地逼近原始（不完整）矩阵，但根据矩阵的核范数具有较低的复杂性。我们通过允许缺失数据的模式具有社会科学应用中常见的时间序列依赖结构来概括矩阵完成文献的结果。我们提出了关于矩阵完成文献、交互式固定效应模型文献和无混杂和综合控制方法下的程序评估文献之间联系的新见解。我们表明所有这些估计量都可以被视为关注相同的目标函数。它们的区别仅在于处理识别的方式，在某些情况下仅通过正则化（我们提出的核范数矩阵完成估计器），而在其他情况下主要通过强加硬限制（无混淆和合成控制方法）。