Evaluating Counterfactual Quantities Using Cross-World Factor Models

The evaluation of counterfactual queries is a central objective of causal inference. We develop an identification strategy for counterfactual queries using cross-world factor models, which provide a compact representation of causal relationships across $K$ treatment levels using $J$ scalar latent factors. First, we characterize when this representation is saturated, that is, when it can be adopted without loss of generality relative to an unrestricted structural causal model, and show that saturation requires at least as many latent factors as treatment levels ($J\ge K$). Next, under conditional mutual independence and additional regularity and loading-structure assumptions, we derive identification results for the joint distribution of the potential outcomes from their conditional marginal distributions. Our identification results require no more latent factors than treatment levels ($J\le K$), implying that a model satisfying both saturation and our identification conditions must use the same number of latent factors as treatment levels ($J=K$). We further develop a parametric maximum likelihood estimation procedure and demonstrate the proposed approach using a real-world randomized recommendation dataset.