The Proximal Surrogate Index: Long-Term Treatment Effects under Unobserved Confounding

We study the identification and estimation of long-term treatment effects under unobserved confounding by combining an experimental sample, where the long-term outcome is missing, with an observational sample, where the treatment assignment is unobserved. While standard surrogate index methods fail when unobserved confounders exist, we establish novel identification results by leveraging proxy variables for the unobserved confounders. We further develop multiply robust estimation and inference procedures based on these results. Applying our method to the Job Corps program, we demonstrate its ability to recover experimental benchmarks even when unobserved confounders bias standard surrogate index estimates.