  Marginal Structural Models (MSM) are the most popular models for causal
inference from time-series observational data. However, they have two main
drawbacks: (a) they do not capture subject heterogeneity, and (b) they only
consider fixed time intervals and do not scale gracefully with longer
intervals. In this work, we propose a new family of MSMs to address these two
concerns. We model the potential outcomes as a three-dimensional tensor of low
rank, where the three dimensions correspond to the agents, time periods and the
set of possible histories. Unlike the traditional MSM, we allow the dimensions
of the tensor to increase with the number of agents and time periods. We set up
a weighted tensor completion problem as our estimation procedure, and show that
the solution to this problem converges to the true model in an appropriate
sense. Then we show how to solve the estimation problem, providing conditions
under which we can approximately and efficiently solve the estimation problem.
Finally we propose an algorithm based on projected gradient descent, which is
easy to implement, and evaluate its performance on a simulated dataset.
