  We propose a new method that uses tensor completion to estimate causal
effects with multivariate longitudinal data, data in which multiple outcomes
are observed for each unit and time period. Our motivation is to estimate the
number of COVID-19 fatalities prevented by government mandates such as travel
restrictions, mask-wearing directives, and vaccination requirements. In
addition to COVID-19 fatalities, we observe related outcomes such as the number
of fatalities from other diseases and injuries. The proposed method arranges
the data as a tensor with three dimensions (unit, time, and outcome) and uses
tensor completion to impute the missing counterfactual outcomes. We first prove
that under general conditions, combining multiple outcomes using the proposed
method improves the accuracy of counterfactual imputations. We then compare the
proposed method to other approaches commonly used to evaluate COVID-19
mandates. Our main finding is that other approaches overestimate the effect of
masking-wearing directives and that mask-wearing directives were not an
effective alternative to travel restrictions. We conclude that while the
proposed method can be applied whenever multivariate longitudinal data are
available, we believe it is particularly timely as governments increasingly
rely on longitudinal data to choose among policies such as mandates during
public health emergencies.
