  In this paper 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.
