Entrywise Inference for Causal Panel Data: A Simple and Instance-Optimal Approach

In causal inference with panel data under staggered adoption, the goal is to estimate and derive confidence intervals for potential outcomes and treatment effects. We propose a computationally efficient procedure, involving only simple matrix algebra and singular value decomposition. We derive non-asymptotic bounds on the entrywise error, establishing its proximity to a suitably scaled Gaussian variable. Despite its simplicity, our procedure turns out to be instance-optimal, in that our theoretical scaling matches a local instance-wise lower bound derived via a Bayesian Cram\'{e}r-Rao argument. Using our insights, we develop a data-driven procedure for constructing entrywise confidence intervals with pre-specified coverage guarantees. Our analysis is based on a general inferential toolbox for the SVD algorithm applied to the matrix denoising model, which might be of independent interest.