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

TL;DR

This work tackles the problem of identifying and estimating long-term treatment effects when unobserved confounding biases the link between short-term surrogates and outcomes, by combining experimental and observational data with proxy information. It develops a proximal framework built around two bridge functions—an outcome bridge $h_0$ and surrogate bridges $q_{a,0}$—together with completeness and transportability assumptions to restore identifiability in a two-sample setting. The authors establish four layers of multiply robust identification, derive the efficient influence function, and propose cross-fitted, doubly robust estimators that achieve consistency, asymptotic normality, and semiparametric efficiency under rate conditions. The methodology is validated on the Job Corps dataset, where it recovers experimental benchmarks and demonstrates substantial improvements over standard surrogate-index methods by leveraging outcome- and surrogate-aligned proxies. Overall, the paper extends proximal causal inference to a challenging data-fusion context, offering practical estimators that exploit proxies to mitigate unobserved confounding in long-term causal effect estimation.

Abstract

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.

Figures (2)

• Figure 1: Possible violations of the surrogacy or comparability assumptions.
• Figure 2: The SWIG for the underlying causal model in the proximal surrogate approach when intervening on $A$ and $S$.

Theorems & Definitions (25)

• Remark 1
• Theorem 1: Identification via Outcome Bridge Function
• proof
• Remark 2: Connection to athey2025surrogate
• Theorem 2: Identification via Surrogate Bridge Function
• proof
• Remark 3
• Remark 4: Connection to athey2025surrogate
• Theorem 3: Multiply Robust Identification
• proof