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Marginal Structural Models (MSM) are the most popular models for causalinference from time-series observational data. However, they have two maindrawbacks: (a) they do not capture subject heterogeneity, and (b) they onlyconsider fixed time intervals and do not scale gracefully with longerintervals. In this work, we propose a new family of MSMs to address these twoconcerns. We model the potential outcomes as a three-dimensional tensor of lowrank, where the three dimensions correspond to the agents, time periods and theset of possible histories. Unlike the traditional MSM, we allow the dimensionsof the tensor to increase with the number of agents and time periods. We set upa weighted tensor completion problem as our estimation procedure, and show thatthe solution to this problem converges to the true model in an appropriatesense. Then we show how to solve the estimation problem, providing conditionsunder which we can approximately and efficiently solve the estimation problem.Finally we propose an algorithm based on projected gradient descent, which iseasy to implement, and evaluate its performance on a simulated dataset.

Weighted Tensor Completion for Time-Series Causal Inference