Optimal Recovery for Causal Inference

Abstract: Problems in causal inference can be fruitfully addressed using signal processing techniques. As an example, it is crucial to successfully quantify the causal effects of an intervention to determine whether the intervention achieved desired outcomes. We present a new geometric signal processing approach to classical synthetic control called ellipsoidal optimal recovery (EOpR), for estimating the unobservable outcome of a treatment unit. EOpR provides policy evaluators with both worst-case and typical outcomes to help in decision making. It is an approximation-theoretic technique that relates to the theory of principal components, which recovers unknown observations given a learned signal class and a set of known observations. We show EOpR can improve pre-treatment fit and mitigate bias of the post-treatment estimate relative to other methods in causal inference. Beyond recovery of the unit of interest, an advantage of EOpR is that it produces worst-case limits over the estimates produced. We assess our approach on artificially-generated data, on datasets commonly used in the econometrics literature, and in the context of the COVID-19 pandemic, showing better performance than baseline techniques