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This paper addresses the problem of estimating heterogeneous treatment effects in panel data with non-uniform treatment assignments by framing it as a matrix completion problem. They develop a computationally efficient estimator that achieves a row-wise $\ell_2$ error of $\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})$ under standard low-rankness and regularity assumptions, without requiring knowledge of treatment propensities. This improves upon existing matrix completion guarantees, which are insufficient for obtaining meaningful per-row bounds necessary for estimating heterogeneous treatment effects.
Existing matrix completion methods are too blunt to estimate individual treatment effects, but this new estimator provides sharp, row-wise guarantees that unlock heterogeneous treatment effect estimation.
A central goal of modern causal inference is estimating heterogeneous treatment effects to answer questions like"how does an intervention affect each unit,"rather than only on average. We study this problem with panel-data where we observe $n$ units across $m$ times under unknown, non-uniform treatment assignments. The data in this setting is naturally represented as a matrix of all unit--time treatment effects. Estimating heterogeneous treatment effects can then be expressed as obtaining a good estimation of each row's average in this matrix. This allows us to formulate the problem as matrix completion, which can be solved under natural low-rankness assumptions. However, existing matrix-completion guarantees are not powerful enough to get meaningful bounds for the per-row guarantee required for estimating the heterogeneous treatment effect; roughly speaking, they are only useful for estimating average treatment effect bounds, as also illustrated in a recent line of work. We give a simple, computationally efficient estimator that, without knowledge of the propensities and under standard low-rankness and regularity assumptions, achieves a row-wise $\ell_2$ error of $\tilde{O}(\sqrt{\frac{1}{n} + \frac{n}{m^2}})$. Technically, our analysis establishes the first sharp row-wise $\ell_2$-perturbation bound for low-rank approximation, complementing existing spectral-, Frobenius-, and entrywise perturbation theory.
