Agostino Capponi (Columbia): Matrix Completion Methods for Causal Inference under Simultaneous Irreversible Treatment

Agostino Capponi (Columbia): Matrix Completion Methods for Causal Inference under Simultaneous Irreversible Treatment 

Abstract: We consider the estimation of causal effects in panel data settings. During a given time period, one observes units of interest and stores the realized outcomes into a matrix. At a fixed point in time, a subset of the units is exposed to an irreversible treatment, i.e., the data matrix of treated units has a block structure. The objective is to design an estimator for the counterfactual outcomes of the block of treated units. For large sample sizes and under typical statistical settings, we show that the use of matrix completion (MC) estimators for counterfactual recovery yields phase transition (PT) phenomena, where it is possible to distinguish regions of the parameter space where a perfect estimation of the counterfactual is possible from those where it is not. We determine the separating line (the so-called phase transition (PT) curve) between the regions, and show that it admits a closed form expression that directly relates time series and cross-sectional heterogeneity among units to the number of untreated units and the initial time of the treatment. Our methodology is designed to handle settings where the starting time of the treatment and the number of control (untreated) units are not necessarily identical, i.e., where the block of counterfactuals in the matrix of control outcomes is a rectangular matrix. We support our theoretical analysis with numerical simulations, which further indicate that an exact counterfactual recovery is attainable even for fairly small sample sizes.