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The Proximal Surrogate Index: Long-Term Treatment Effects under Unobserved Confounding
\maketitle
\begin{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.
\end{abstract}
\section{Introduction}
\paragraph{Motivation}
Empirical researchers
are often interested in evaluating
the long-term effects
of interventions or treatments.
For example,
labor economists
are interested in evaluating
job training programs
on participants' long-term earnings and employment outcomes,
public economists
are interested in assessing
in-kind transfer programs
on recipients' long-term well-being,
and
health economists
study
medical treatments
on patients' survival and quality of life.
\paragraph{Challenges}
Although randomized controlled trials (RCTs)
are the gold standard
for causal inference,
they often face practical challenges
in measuring long-term outcomes.
These challenges include
limited funding,
high attrition rates,
and lengthy follow-up periods.
As a result,
researchers may only have access
to short-term outcomes
from experimental studies,
while long-term outcomes of interest
are only available
in observational datasets,
such as administrative records or surveys,
where the treatment assignment is unobserved.
\paragraph{Existing Approaches}
The challenge of identifying long-term effects
from combined experimental and observational data has
been addressed in prior literature.
The most related work is \cite{athey2025surrogate},
who propose a \emph{surrogate index} approach.
Their method leverages short-term surrogate outcomes
to predict long-term outcomes across datasets,
recovering the long-term treatment effect
under certain identification assumptions.
However,
these assumptions
require
the absence of unobserved confounders
that affect both the treatment assignment
and the long-term outcome,
or the surrogate outcomes
and the long-term outcome.
These assumptions may be violated in many practical settings.
For example, in job training program evaluations,
unobserved ability
may affect both participation and long-term earnings;
in healthcare studies, unobserved health consciousness
may confound the relationship between short-term behaviors
and long-term health.
These concerns
motivate our work.
\paragraph{Research Question}
In this paper,
we study the nonparametric identification
and estimation
of long-term treatment effects
under unobserved confounding,
by combining experimental and observational data,
allowing for
unobserved confounding through the use of proxy variables.
\paragraph{Our Contributions}
This paper contributes to the literature in three ways.
First, we study long-term treatment effects in a two-sample surrogate setting
where the experimental sample lacks the long-term outcome
and the observational sample lacks treatment status,
allowing for unobserved confounding.
We establish nonparametric identification by leveraging proxy variables
and deriving two complementary bridge-based strategies:
an outcome bridge that imputes the missing long-term outcome in the experimental sample,
and a surrogate bridge that reweights the observational sample to the experimental target.
Second, we combine these strategies to obtain a multiply robust identification formula
that remains valid under four alternative sets of correctly specified nuisance components,
and we characterize the corresponding efficient influence function (EIF)
and semiparametric local efficiency bound.
Third, we develop cross-fitted estimators based on
double/debiased machine learning (DML)
that accommodate flexible nuisance estimation,
and we provide consistency, asymptotic normality, and efficiency results.
In an application to Job Corps (JC), our approach closely matches experimental benchmarks in designs
where standard surrogate index methods are biased.
\paragraph{Related Literature}
Our paper is situated at the intersection of several strands of literature.
First, it is related to the literature on causal inference with surrogate outcomes
\citep{weir2006statistical, vanderweele2013surrogate, joffe2009related}.
These works explore the use of intermediate outcomes
as surrogates for long-term outcomes of interest.
Many criteria have been proposed
to ensure the validity of surrogates \citep{prentice1989surrogate,frangakis2002principal,lauritzen2004discussion}.
However, these criteria may produce ``surrogate paradoxes''
in the presence of unobserved confounding \citep{chen2007criteria}.
Our work addresses this limitation,
offering new strategies to use surrogates.
Second, this paper adds to the growing body of work on
combining different data sources
for causal inference \citep{colnet2024causal},
especially those that leverage short-term outcomes
to estimate long-term treatment effects.
Besides \cite{athey2025surrogate},
\cite{athey2025experimental},
\cite{chenSemiparametricEstimationLongterm2023},
\cite{ghassamiCombiningExperimentalObservational2022b},
and
\cite{imbensLongtermCausalInference2025a},
propose various methods
to combine experimental and observational samples
to estimate long-term effects,
assuming both samples
contain information on the treatment assignment,
which differs from our setting
where the treatment is unobserved
in the observational sample.
Finally, our work contributes to
the recent literature on proximal causal inference
\citep{miaoIdentifyingCausalEffects2018, tchetgentchetgenIntroductionProximalCausal2024, cuiSemiparametricProximalCausal2024},
which
has been applied
to various settings,
including
longitudinal studies
\citep{imbens2021controlling,deaner2023proxycontrolspaneldata,liu2024proximal,ying2023proximal,tchetgentchetgenIntroductionProximalCausal2024},
mediation analysis
\citep{dukes2023proximal,ghassami2025causal},
and optimal treatment regimes
\citep{bai2025proximal,gao2025multiple,shen2023optimal,qi2024proximal}.
But most existing works
focus on single-sample settings,
except for \cite{ghassamiCombiningExperimentalObservational2022b,imbensLongtermCausalInference2025a},
which also consider data combination settings but under different data structures and assumptions.
Our paper provides an extension
by adapting the proximal learning framework to a more challenging
data combination structure,
where key variables are missing across samples.
\paragraph{Organization of the Paper}
The rest of this paper is organized as follows.
In \cref{sec:setup-and-notation},
we introduce the setup and notation.
\cref{sec:athey-surrogate-index}
reviews the surrogate index approach
proposed by \cite{athey2025surrogate}.
In \cref{sec:identification},
three identification results
are presented.
The estimation and inference procedures
are discussed in \cref{sec:estimation-and-inference}.
\cref{sec:real-data-application}
presents the empirical application.
Finally,
\cref{sec:conclusion} concludes.
\section{Setup and Notation} \label{sec:setup-and-notation}
We consider a setting
where we have access to
an experimental dataset ($\text{E}$) and
an observational dataset ($\text{O}$),
with sample sizes $N_{\text{E}}$ and $N_{\text{O}}$ respectively.
It is convenient to view the pooled dataset
as consisting of $N = N_{\text{E}} + N_{\text{O}}$ units,
drawn from a superpopulation that mixes both experimental
and observational distributions,
where each unit belongs to either
the experimental sample ($G_i = \text{E}$) or
the observational sample ($G_i = \text{O}$).
We denote the marginal probability of being in the experimental sample
as $\pi_0 \equiv P(G_i = \text{E})$, with $0 < \pi_0 < 1$.
For each unit $i$, there is a binary treatment of interest,
$A_{i}\in\{0,1\}$,
a scalar primary outcome, $Y_{i}$,
a set of intermediate outcomes, $S_{i}$
(which we refer to as surrogates),
and a set of pre-treatment covariates, $X_{i}$.
The data structure is characterized
by differential availability across samples.
In the experimental sample ($\text{E}$),
the treatment $A_{i}$ is randomly assigned,
and we observe $(A_i, S_i, X_i)$.
Our goal is to recover the causal effect of $A_{i}$ on $Y_{i}$.
However, the primary long-term outcome $Y_{i}$
is not recorded in this sample.
To provide information on the long-term outcome,
the observational dataset ($\text{O}$) is collected,
which contains measurements of $(Y_i, S_i, X_i)$.
Nevertheless,
the treatment status $A_{i}$ is unobserved in the observational sample.
A key challenge in this surrogate setting,
which we will address in \cref{sec:athey-surrogate-index},
arises when the identification assumptions
required for the existing surrogate index approach
are violated due to unobserved confounders $U_i$ \citep{athey2025surrogate}.
To overcome this,
we draw insights from recent proximal inference literature
\citep{miaoIdentifyingCausalEffects2018, tchetgentchetgenIntroductionProximalCausal2024, cuiSemiparametricProximalCausal2024}.
We therefore assume that we have access to
two types of proxy variables:
a set of \emph{surrogate-aligned proxies} $Z_{i}$
and a set of \emph{outcome-aligned proxies} $W_{i}$.
While these proxies are not utilized in the
standard surrogate index approach reviewed in \cref{sec:athey-surrogate-index},
they are essential to our proposed identification strategy
in \cref{sec:identification}.
The list of variables and their availability across samples
is summarized in \cref{tab:variable-notation}.
\begin{table}[htbp]
\centering
\footnotesize
\caption{Variable notation and availability across samples.}
\label{tab:variable-notation}
\begin{tabular}{cccc}
\toprule
Variable & Description & Experimental Sample & Observational Sample \\
\midrule
$A$ & Treatment indicator & $\checkmark$ & ? \\
$Y$ & Primary outcome & ? & $\checkmark$ \\
$S$ & Surrogate outcomes & $\checkmark$ & $\checkmark$ \\
$X$ & Pre-treatment covariates & $\checkmark$ & $\checkmark$ \\
$Z$ & Surrogate-aligned proxies & ? & $\checkmark$ \\
$W$ & Outcome-aligned proxies & $\checkmark$ & $\checkmark$ \\
$U$ & Unobserved confounders & ? & ? \\
\bottomrule
\end{tabular}
\caption*{\textit{Note:}
The check marks ($\checkmark$)
indicate that the variable is observed
in the corresponding sample,
while question marks (?)
indicate that the variable is unobserved.
}
\end{table}
To formally define our causal estimand and identification assumptions,
we adopt the potential outcomes framework
\citep{rubin1974estimating, holland1986statistics, imbens2015causal}.
For any variable $V_{i}$ that can be affected
by the treatment $A_{i}$,
we denote $V_{i}(a)$ as its potential value
if $A_i$ were set to $a\in\{0,1\}$.
Similarly, we denote $V_{i}(s)$ as the potential value
if the surrogate were set to $s\in\mathcal{S}$,
and $V_{i}(a,s)$ if both were set.
We also use the nested potential outcomes notation $V_{i}(a,S_{i}(a'))$
introduced by
\cite{pearl2001direct} and \cite{robins1992identifiability}
to denote the potential value of $V_i$
when the treatment is set to $a$
and the surrogate is set to its potential value $S_i(a')$.
Throughout the paper,
we assume that all random variables
are independent and identically distributed (IID)
samples from a superpopulation,
although some variables are only observed
in one of the two samples.
We use $P(\cdot)$ and $\E[\cdot]$
to denote
the probability and expectation
with respect to this superpopulation,
and use $f(\cdot)$
to denote
the corresponding probability density/mass function
as appropriate.
We drop the subscript $i$
when there is no ambiguity.
\begin{assumption}[Random Sampling]
\label{asn:random-sampling}
We observe an IID sample
\begin{align*}
\mathcal{D}
= \bc{D_i}_{i = 1}^N
= \bc{\mathbf{1}_{\{G_i = \text{O}\}} Y_i, W_i, \mathbf{1}_{\{G_i = \text{O}\}} Z_i, S_i, \mathbf{1}_{\{G_i = \text{E}\}} A_i, X_i, G_i}_{i = 1}^N
\end{align*}
from the population
described in \cref{sec:setup-and-notation},
where $N = N_{\text{E}} + N_{\text{O}}$
is the total sample size,
and $\mathbf{1}_{\{G_i = g\}}$ indicates
whether unit $i$
belongs to sample $g \in \{\text{E}, \text{O}\}$.
We denote $\mathcal{D}_{\text{E}}$
and $\mathcal{D}_{\text{O}}$
as the subsets of $\mathcal{D}$
corresponding to the experimental
and observational samples, respectively.
\end{assumption}
We further maintain the stable unit treatment value assumption (SUTVA)
\citep{rubin1980discussion}:
for any variable $V$ affected by $A$ and/or $S$,
\begin{align*}
V = V(A) = V(S) = V(A, S) = V(A, S(A)).
\end{align*}
which links the observed variables
to the potential outcomes.
Our primary estimand of interest
is the average treatment effect (ATE)
on the primary outcome $Y$
in the experimental sample:
\begin{align}
\label{eq:ate}
\tau_0 = \E\bs{Y(a) - Y(a') \mid G = \text{E}},
\end{align}
where $a = 1$ and $a' = 0$.
\section{The Surrogate Index and Its Limitations} \label{sec:athey-surrogate-index}
In this section,
we briefly review the surrogate index approach proposed by \cite{athey2025surrogate}
and analyze its limitations
in the presence of unobserved confounding.
This discussion motivates our proximal approach.
\subsection{Identification via the Surrogate Index}
\cite{athey2025surrogate} rely on three key assumptions
to identify the long-term ATE.
First, the standard unconfoundedness assumption
in the experimental sample
($A \ind (Y(a), S(a)) \mid X, G = \text{E}$)
is assumed to hold.
Second, the \emph{surrogacy} assumption ($A \ind Y \mid S, X, G = \text{E}$),
originally proposed by \cite{prentice1989surrogate},
posits that
the treatment $A$
provides no additional information about the primary outcome $Y$
beyond what is already contained in the surrogates $S$
and pre-treatment variables $X$
in the experimental sample.
Third, the \emph{comparability} assumption
requires that the conditional outcome distribution
is stable across samples, i.e., $G \perp Y \mid S, X$.
Under these assumptions, the mean potential outcome is identified by
\begin{align}
\E\bs{Y(a) \mid G = \text{E}} =
\E\bs{\E\bs{\mu(S, X, \text{O}) \mid A = a, X, G = \text{E}} \mid G = \text{E}},
\label{eq:surrogate-index-identification-intro}
\end{align}
where $\mu(S, X, g) \equiv \E[Y \mid S, X, G = g]$
is called the \emph{surrogate index}.
\subsection{Limitations and Motivation}
While elegant,
the surrogate index approach
relies heavily on the validity of
the surrogacy and comparability assumptions.
\cref{fig:surrogate-model-violations}
illustrates three common scenarios
where these assumptions can be violated.
The first scenario (\cref{fig:surrogate-model-treatment-outcome-confounder})
involves an unobserved confounder $U$
that affects both the treatment $A$
and the primary outcome $Y$.
Structurally, this resembles the front-door model \citep{pearl1995causal}.
However, in our context, conditioning on $S$ opens the collider path $G \to A \leftarrow U \to Y$,
violating the comparability assumption.
The second scenario (\cref{fig:surrogate-model-surrogate-outcome-confounder})
involves an unobserved confounder $U$
that affects both the surrogates $S$
and the primary outcome $Y$,
resembling an instrumental variable (IV) model \citep{wrightTariffAnimalVegetable1928,imbensIdentificationEstimationLocal1994,imbensInstrumentalVariablesEconometricians2014}.
Here,
since $S$ is a collider,
conditioning on $S$ opens the collider path $A \to S \leftarrow U \to Y$,
violating the surrogacy and potentially leading to surrogate paradoxes \citep{chen2007criteria,frangakis2002principal}.
In addition,
the sample indicator $G$
becomes associated with $Y$
after conditioning on $S$,
violating the comparability assumption.
The third scenario (\cref{fig:surrogate-model-direct-effect})
involves a direct effect of $A$ on $Y$ that bypasses $S$,
as in mediation analysis \citep{pearl2001direct, robins1992identifiability, vanderweele2015explanation}.
Since $A$ and $Y$ are never observed together, this effect is fundamentally unidentified.
Like the standard surrogate index,
our proximal approach must assume the absence of direct effects (\cref{asn:no-direct-effect}).
We summarize the key features of these three scenarios
in comparison to the surrogate index setting
in \cref{tab:comparison-of-scenarios}.
\begin{figure}[htbp]
\begin{subfigure}[t]{0.3\textwidth}
\centering
\small
\begin{tikzpicture}[scale=1.1]
\node[name=a, cir]{$A$};
\node[name=s, right=5mm of a, cir]{$S$};
\node[name=y, right=5mm of s, cir]{$Y$};
\node[name=g, below=7mm of a, cir]{$G$};
\node[name=x, above=7mm of a, cir]{$X$};
\node[name=u, above=7mm of y, cir, draw=darkred]{$U$};
\draw[-Latex](a) to (s);
\draw[-Latex](s) to (y);
\draw[-Latex](g) to (a);
\draw[-Latex](g) to (s);
\draw[-Latex](x) to (a);
\draw[-Latex](x) to (s);
\draw[-Latex](x) to (y);
\draw[-Latex, bend right=45](x) to (g);
\draw[-Latex, darkred](u) to (a);
\draw[-Latex, darkred](u) to (y);
\end{tikzpicture}
\caption{The presence of unobserved treatment-outcome confounders.}
\label{fig:surrogate-model-treatment-outcome-confounder}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.3\textwidth}
\centering
\small
\begin{tikzpicture}[scale=1.1]
\node[name=a, cir]{$A$};
\node[name=s, right=5mm of a, cir]{$S$};
\node[name=y, right=5mm of s, cir]{$Y$};
\node[name=g, below=7mm of a, cir]{$G$};
\node[name=x, above=7mm of a, cir]{$X$};
\node[name=u, above=7mm of y, cir, draw=darkred]{$U$};
\draw[-Latex](a) to (s);
\draw[-Latex](s) to (y);
\draw[-Latex](g) to (a);
\draw[-Latex](g) to (s);
\draw[-Latex](x) to (a);
\draw[-Latex](x) to (s);
\draw[-Latex](x) to (y);
\draw[-Latex, bend right=45](x) to (g);
\draw[-Latex, darkred](u) to (s);
\draw[-Latex, darkred](u) to (y);
\end{tikzpicture}
\caption{The presence of unobserved surrogate-outcome confounders.}
\label{fig:surrogate-model-surrogate-outcome-confounder}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.3\textwidth}
\centering
\small
\begin{tikzpicture}[scale=1.1]
\node[name=a, cir]{$A$};
\node[name=s, right=5mm of a, cir]{$S$};
\node[name=y, right=5mm of s, cir]{$Y$};
\node[name=g, below=7mm of a, cir]{$G$};
\node[name=x, above=7mm of a, cir]{$X$};
\draw[-Latex](a) to (s);
\draw[-Latex, bend right=45, greenblue](a) to (y);
\draw[-Latex](s) to (y);
\draw[-Latex](g) to (a);
\draw[-Latex](g) to (s);
\draw[-Latex](x) to (a);
\draw[-Latex](x) to (s);
\draw[-Latex](x) to (y);
\draw[-Latex, bend right=45](x) to (g);
\end{tikzpicture}
\caption{The presence of direct effect of treatment on outcome.}
\label{fig:surrogate-model-direct-effect}
\end{subfigure}
\caption{Possible violations of the surrogacy or comparability assumptions.}
\label{fig:surrogate-model-violations}
\end{figure}
These limitations point to a clear strategy inspired by the front-door criterion.
Recall that the front-door formula identifies the causal effect $A \to Y$
by composing the effect of $A$ the mediator $S$, and the effect of $S$ on $Y$.
In our setting, the first component ($A \to S$) is readily identified from the experimental sample due to randomization.
The challenge lies entirely in identifying the second component ($S \to Y$) from the observational sample,
where the relationship is confounded by $U$.
While the standard front-door approach assumes no unobserved $S$-$Y$ confounding,
we relax this by leveraging proxy variable ($W, Z$) to adjust for $U$.
Thus, our approach conceptually mirrors the front-door strategy
but adapts it to the data combination setting:
we combine the experimentally identified effect of $A$ on $S$
with the proximally identified effect of $S$ on $Y$ to recover the long-term ATE.
\begin{table}[htbp]
\centering
\footnotesize
\caption{Surrogacy, front-door, IV, and mediation.}
\label{tab:comparison-of-scenarios}
\begin{tabular}{lcccc}
\toprule
& Surrogacy & Front-door & IV & Mediation \\
\midrule
Role of $A$ & Treatment & Treatment & Instrument & Treatment \\
Role of $S$ & Surrogate & Mediator & Treatment & Mediator \\
Role of $Y$ & Outcome & Outcome & Outcome & Outcome \\
Parameter of Interest & $A \to Y$ & $A \to Y$ & $S \to Y$ & $A \to Y$ \\
& & & & $A \to S \to Y$ \\
\midrule
Feasible w/o joint $(A, Y)$ & Yes & No & No & No \\
Unobs.\ $A$-$Y$ confounders & No & Yes & No & No \\
Unobs.\ $S$-$Y$ confounders & No & No & Yes & No \\
Direct effect of $A$ on $Y$ & No & No & No & Yes \\
\bottomrule
\end{tabular}
\caption*{\textit{Note:}
We use ``surrogacy'' to refer to
\cite{athey2025surrogate}'s surrogate index setting.
$A \to Y$ and $S \to Y$ denote
the causal effect of $A$ on $Y$
and that of $S$ on $Y$ respectively.
$A \to S \to Y$ denotes
the indirect effect of $A$ on $Y$
mediated through $S$.
}
\end{table}
\section{Restoring Identification via Proxies} \label{sec:identification}
As discussed in \cref{sec:athey-surrogate-index},
identifying the long-term ATE requires addressing the
potential violations of surrogacy and comparability assumptions.
In this section, we present a proximal identification strategy that allows for unobserved confounders $U$.
To provide some intuition,
we adopt the single-world intervention graph (SWIG)
to visualize the causal structure
under a hypothetical intervention \citep{richardson2013single, heckman2015causal}.
We first maintain the following no direct effect assumption.
Similar assumptions
have been widely used
in the IV literature,
where it is often referred to as the exclusion restriction
\citep{imbensIdentificationEstimationLocal1994,imbensInstrumentalVariablesEconometricians2014}.
This assumption requires that $A$ has no direct causal link to $Y$ other than through $S$,
corresponding to the absence of an edge $a \to Y(a,s)$ in \cref{fig:proximal-surrogate-model-swig}.
\begin{assumption}[No Direct Effect]
\label{asn:no-direct-effect}
\begin{align}
Y(a, s) = Y(a', s) \quad \text{for all } a, a' \in \{0, 1\}, s \in \mathcal{S}.
\end{align}
\end{assumption}
Next, we characterize the confounding structure.
While unobserved confounders $U$ may bias the relationships in the observational sample,
we assume that $U$,
together with
the observed covariates $X$,
are the only sources of confounding.
In terms of \cref{fig:proximal-surrogate-model-swig},
this means that there are no other unmeasured common causes
between $A$ and $Y(a, s)$,
and between $S$ and $Y(s)$,
besides $U$.
\begin{assumption}[Unconfoundedness in the Observational Sample]
\label{asn:unconfoundedness-observational}
For all $a \in \{0, 1\}$ and $s \in \mathcal{S}$,
\begin{align}
A \ind \bp{\bp{Y(a)}_{a \in \{0, 1\}}, \bp{S(a)}_{a \in \{0, 1\}}} &\mid (X, U, G = \text{O}), \label{eq:unconfoundedness-observational-a} \\
S \ind \bp{Y(s)}_{s \in \mathcal{S}} &\mid (A, X, U, G = \text{O}), \label{eq:unconfoundedness-observational-s}
\end{align}
where $0 < P(A = 1 \mid X, U, G = \text{O}) < 1$ almost surely.
\end{assumption}
In addition,
we assume that
treatment assignment
is randomized
in the experimental sample,
as is standard in RCTs.
However, the surrogate $S$ is an intermediate outcome, not a randomized intervention.
Even within the experiment, individuals with the same treatment status
may have different surrogate values due to unobserved factors $U$ (e.g., motivation or ability),
which also affect the long-term outcome $Y$.
Similarly to \cref{asn:unconfoundedness-observational},
we assume that $U$ is the only source of unobserved confounding
in the experimental sample as well.
The following assumption formalizes these conditions.
\begin{assumption}[Unconfoundedness in the Experimental Sample]
\label{asn:unconfoundedness-experimental}
For all $a \in \{0, 1\}$ and $s \in \mathcal{S}$,
\begin{align}
A \ind \bp{\bp{Y(a)}_{a \in \{0, 1\}}, \bp{S(a)}_{a \in \{0, 1\}}, U, W} &\mid (X, G = \text{E}), \label{eq:unconfoundedness-experimental-a} \\
S \ind \bp{Y(s)}_{s \in \mathcal{S}} &\mid (A, X, U, G = \text{E}), \label{eq:unconfoundedness-experimental-s}
\end{align}
where $0 < e_0(X) < 1$ almost surely,
and $e_0(X) \equiv P(A = 1 \mid X, G = \text{E})$.
\end{assumption}
\begin{remark}
The SWIG in
\cref{fig:proximal-surrogate-model-swig},
is a pooled representation.
The randomization of $A$
in the experimental sample
is captured
by an directed edge from $G$ to $A$,
indicating that the treatment assignment mechanism
depends on the sample indicator $G$.
Although the pooled graph depicts a generic dependence $U \to A$,
the experimental design explicitly breaks this dependence
within the $G=\text{E}$ stratum.
\end{remark}
To address the unobserved confounding,
we need to leverage
proxy variables
that provide indirect information
about the unobserved confounders $U$.
We make the following assumption
regarding the availability of such proxies.
\begin{assumption}[Proxies Availability]
\label{asn:availability-of-proxies}
We observe $W$ in both samples and $Z$ only in the observational sample.
Moreover,
\begin{align}
Z \ind Y &\mid (S, X, U, G), \\
W \ind (A, S, Z) &\mid (X, U, G).
\end{align}
\end{assumption}
\cref{fig:proximal-surrogate-model-swig}
illustrates the roles of these proxies.
In this representation, $Z$ acts as a surrogate-aligned proxy.
While the figure depicts $Z$ as a post-treatment variable (e.g., a post-program survey),
the formal requirement is broader:
$Z$ must be conditionally independent of $Y$ given $S$, $X$, and $U$,
regardless of its temporal or causal positioning.
Conversely, $W$ acts as an outcome-aligned proxy.
Graphically, it behaves like a pre-treatment covariate that is a noisy measure of $U$.
The key requirement is that $W$ is sufficiently informative about $U$
but is structurally independent of the $(A, S, Z)$ given covariates.
The proxies satisfying \cref{asn:availability-of-proxies}
are ubiquitous in practical applications.
For example,
in job training programs,
test scores collected before the training
can serve as outcome-aligned proxies $W$,
because they reflect
the participants' underlying abilities $U$,
but do not affect or are affected by
the training $A$ or the short-term outcomes $S$ directly.
Similarly,
post-training surveys about job search self-efficacy
or soft skills can serve as surrogate-aligned proxies $Z$.
These measures capture the participants' latent motivation $U$
and the immediate psychological impact of the training $A$,
but arguably do not directly determine long-term earnings $Y$
once the actual intermediate employment history $S$ is accounted for.
To generalize the relationships learned in the observational sample
to the experimental sample,
we require the mechanism underlying the data generation to be comparable across groups.
In the proximal learning framework,
this transportability must hold for
both the primary outcome and the outcome-aligned proxy.
\begin{assumption}[Transportability]
\label{asn:transportability}
\begin{align}
Y &\ind G \mid (S, X, U), \label{eq:transportability-outcome} \\
W &\ind G \mid (X, U), \label{eq:transportability-proxy}
\end{align}
and
\begin{align}
\frac{P(G = \text{E} \mid U, X)}{P(G = \text{O} \mid U, X)} < \infty \quad \text{almost surely}.
\end{align}
\end{assumption}
\cref{eq:transportability-outcome}
is analogous to
the comparability assumption
in \cite{athey2025surrogate},
but allows for unobserved confounders $U$.
It posits that given the surrogate
and the complete set of confounders (both observed and unobserved),
the distribution of the primary outcome $Y$
is stable across samples.
\cref{eq:transportability-proxy}
is a new requirement specific to the proximal approach.
It ensures that the outcome-aligned proxy $W$
relates to the latent confounder $U$ in an invariant manner across the
experimental and observational samples.
In our SWIG representation (\cref{fig:proximal-surrogate-model-swig}),
these assumptions correspond to the requirement that the group indicator $G$
has no directed edges on $Y$ or $W$.
These transportability assumptions are closely related to
the concept of S-admissibility in the data fusion literature
\citep{bareinboim2016causal, pearl2014external}.
Notably, our identification results do not require transportability for the surrogate-aligned proxy $Z$.
While $Z$ is crucial for identifying the bridge functions within the observational sample,
its distributional relationship with $U$
may vary across samples.
Thus, we allow for arbitrary dependence between $G$ and $Z$,
accommodating scenarios where the proxy measurement mechanism
for $Z$ differs across samples.
In the SWIG (\cref{fig:proximal-surrogate-model-swig}),
this is reflected by the presence of a directed edge from $G$ to $Z$.
\begin{figure}[htbp]
\centering
\small
\begin{tikzpicture}[scale=1.6]
\node[name=a, shape=swig vsplit]{
\nodepart{left}{$A$}
\nodepart{right}{$a$}
};
\node[name=s, shape=swig vsplit] at (2, 0) {
\nodepart{left}{$S(a)$}
\nodepart{right}{$s$}
};
\node[name=y, ell] at (4, 0) {$Y(a, s)$};
\node[name=g, cir] at (1, -1) {$G$};
\node[name=x, cir] at (3, -1) {$X$};
\node[name=u, cir, draw=darkred] at (3, 1) {$U$};
\node[name=z, ell] at (1, 1) {$Z(a)$};
\node[name=w, cir] at (5, 1) {$W$};
\draw[-Latex](a) to (s);
\draw[-Latex, in=180, out=110](a) to (z);
\draw[-Latex](s) to (y);
\draw[-Latex, in=250, out=180](g) to (a);
\draw[-Latex](g) to (s);
\draw[-Latex, in=240, out=120](g) to (z);
\draw[-Latex, in=270, out=150, looseness=0.5](x) to (a);
\draw[-Latex, in=270, out=150](x) to (s);
\draw[-Latex](x) to (y);
\draw[-Latex](x) to (g);
\draw[-Latex, bend left=45](x) to (z);
\draw[-Latex, bend right=45](x) to (w);
\draw[-Latex, in=90, out=210, darkred](u) to (s);
\draw[-Latex, darkred](u) to (z);
\draw[-Latex, darkred](u) to (y);
\draw[-Latex, darkred, in=90, out=210, looseness=0.5](u) to (a);
\draw[-Latex, darkred](u) to (x);
\draw[-Latex, darkred](u) to (w);
\draw[-Latex, darkred, in=90, out=180, looseness=1](u) to (g);
\draw[-Latex](z) to (s);
\draw[-Latex](w) to (y);
\end{tikzpicture}
\caption{The SWIG for the underlying causal model
in the proximal surrogate approach
when intervening on $A$ and $S$.}
\label{fig:proximal-surrogate-model-swig}
\end{figure}
\subsection{Identification via Outcome Bridge Function} \label{sec:identification-via-outcome-bridge-function}
We first introduce
the outcome bridge function,
which helps to impute
the missing long-term outcome $Y$
in the experimental sample.
\begin{assumption}[Outcome Bridge Function]
\label{asn:outcome-bridge-function}
There exists a square-integrable function
$h_0: \mathcal{W} \times \mathcal{S} \times \mathcal{X} \to \mathbb{R}$
for the observational sample $(G = \text{O})$
such that
\begin{align}
\E\bs{Y \mid Z, S, X, G = \text{O}}
= \E\bs{h_0 (W, S, X) \mid Z, S, X, G = \text{O}}.
\label{eq:outcome-bridge-function}
\end{align}
\end{assumption}
We further assume
that the surrogate-aligned proxy $Z$
is sufficiently informative
about the unobserved confounders $U$
to ensure that we can learn the unobserved confounding relationship
using $Z$.
\begin{assumption}[Completeness for Outcome Bridge Function]
\label{asn:completeness-outcome}
For any $g \in L_2$,
if $\E\bs{g(U) \mid Z = z, S = s, X = x} = 0$ for all $z, s, x$,
then $g(U) = 0$ almost surely.
\end{assumption}
This completeness condition
has been widely used
in the nonparametric IV models
\citep{neweyInstrumentalVariableEstimation2003,ai2003efficient,chernozhukov2005iv},
measurement error models \citep{hu2008instrumental,an2012well},
and the recent proximal inference literature
\citep{miaoIdentifyingCausalEffects2018,tchetgentchetgenIntroductionProximalCausal2024,cuiSemiparametricProximalCausal2024,miaoConfoundingBridgeApproach2024,ghassamiCombiningExperimentalObservational2022b,imbensLongtermCausalInference2025a}.
We establish the first identification result
using the outcome bridge function.
\begin{theorem}[Identification via Outcome Bridge Function]
\label{thm:outcome-bridge-identification}
\begin{enumerate}
\item[]
\item Under \cref{asn:outcome-bridge-function,asn:availability-of-proxies,asn:completeness-outcome},
the function $h_0$ in \cref{eq:outcome-bridge-function}
satisfies
\begin{align*}
\E\bs{Y \mid U, S, X, G = \text{O}}
= \E\bs{h_0(W, S, X) \mid U, X, G = \text{O}}.
\end{align*}
\item Suppose that the function $h_0$ satisfies \cref{eq:outcome-bridge-function}.
Under \cref{asn:unconfoundedness-observational,asn:unconfoundedness-experimental,asn:transportability},
the function $h_0$
also satisfies
\begin{align*}
\E\bs{Y \mid U, S, X, G = \text{E}}
&= \E\bs{h_0(W, S, X) \mid U, X, G = \text{E}}.
\end{align*}
That is,
the same function $h_0$ transports from the observational sample
to the experimental sample.
\item Under \cref{asn:unconfoundedness-observational,asn:unconfoundedness-experimental,asn:no-direct-effect,asn:availability-of-proxies,asn:outcome-bridge-function,asn:completeness-outcome,asn:transportability},
the mean potential outcome is identified by
\begin{align}
\E\bs{Y(a) \mid X, G = \text{E}}
= \E\bs{h_0(W, S, X) \mid A = a, X, G = \text{E}},
\label{eq:outcome-bridge-identification}
\end{align}
where $h_0$ satisfies \cref{eq:outcome-bridge-function},
and we call $h_0$ the \emph{proximal surrogate index}.
Thus, the ATE in the experimental sample is identified by
\begin{align*}
\tau_0
&= \E\bs{\bar{h}_0(1, X) - \bar{h}_0(0, X) \mid G = \text{E}} \\
&= \E\bs{\frac{A h_0(W, S, X)}{e_0(X)} - \frac{(1 - A) h_0(W, S, X)}{1 - e_0(X)} \;\middle|\; G = \text{E}},
\end{align*}
where for any $h: \mathcal{W} \times \mathcal{S} \times \mathcal{X} \to \mathbb{R}$,
we define
\begin{align*}
\bar{h}^{(h)}(A, X)
\equiv \E\bs{h(W, S, X) \mid A, X, G = \text{E}}.
\end{align*}
In particular, $\bar{h}_0 \equiv \bar{h}^{(h_0)}$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-outcome-bridge-identification}.
\end{proof}
\cref{thm:outcome-bridge-identification}
shows that
the missing long-term outcome $Y$
can be imputed
in the experimental sample
using the outcome bridge function $h$
learned from the observational sample.
The long-term ATE in the experimental sample
can then be identified
using standard methods,
such as outcome regression (OR)
or inverse probability weighting (IPW),
based on the imputed outcomes.
\begin{remark}[Connection to \cite{athey2025surrogate}]
\cref{thm:outcome-bridge-identification}
generalizes \cite{athey2025surrogate}'s surrogate index identification
by allowing for unobserved confounding
between treatment and outcome
in the observational sample.
See \cref{sec:connection-to-surrogate-index-identification}
for more details.
\end{remark}
\subsection{Identification via Surrogate Bridge Function} \label{sec:identification-via-surrogate-bridge-function}
An alternative identification strategy
leverages
the surrogate bridge function,
as introduced below.
\begin{assumption}[Surrogate Bridge Function]
\label{asn:surrogate-bridge-function}
For each treatment level $a \in \{0, 1\}$,
there exists a square-integrable function
$q_{a, 0}: \mathcal{Z} \times \mathcal{S} \times \mathcal{X} \to \mathbb{R}$
such that
\begin{align}
\E\bs{q_{a, 0}(Z, S, X) \mid W, S, X, G = \text{O}}
= \frac{f(W, S \mid A = a, X, G = \text{E}) f(X \mid G = \text{E})}{f(W, S, X \mid G = \text{O})}.
\label{eq:surrogate-bridge-function}
\end{align}
\end{assumption}
To ensure that
the surrogate bridge function (\cref{eq:surrogate-bridge-function})
can be used for reweighting,
we impose the following completeness condition.
\begin{assumption}[Completeness for Surrogate Bridge Function]
\label{asn:completeness-surrogate}
For any $g \in L_2$,
if $\E\bs{g(U) \mid W = w, S = s, X = x} = 0$ for all $w, s, x$,
then $g(U) = 0$ almost surely.
\end{assumption}
This completeness condition mirrors
that in \cref{asn:completeness-outcome},
guaranteeing
the outcome-aligned proxy $W$
is sufficiently informative
about the unobserved confounders $U$.
We now present
an alternative identification result
based on the surrogate bridge function.
\begin{theorem}[Identification via Surrogate Bridge Function]
\label{thm:surrogate-bridge-identification}
\begin{enumerate}
\item[]
\item Under \cref{asn:surrogate-bridge-function,asn:availability-of-proxies,asn:completeness-surrogate},
the function $q_a$ in \cref{eq:surrogate-bridge-function}
satisfies
\begin{align*}
\E\bs{q_{a, 0}(Z, S, X) \mid U, S, X, G = \text{O}}
= \frac{f(U, S \mid A = a, X, G = \text{E}) f(X \mid G = \text{E})}{f(U, S, X \mid G = \text{O})}.
\end{align*}
\item Under \cref{asn:unconfoundedness-observational,asn:unconfoundedness-experimental,asn:no-direct-effect,asn:availability-of-proxies,asn:surrogate-bridge-function,asn:completeness-surrogate,asn:transportability},
the mean potential outcome is identified by
\begin{align}
\E\bs{Y(a) \mid G = \text{E}}
= \E\bs{q_{a, 0}(Z, S, X) Y \mid G = \text{O}},
\label{eq:surrogate-bridge-identification}
\end{align}
where $q_{a, 0}$ satisfies \cref{eq:surrogate-bridge-function}.
Thus, the ATE in the experimental sample is identified
by
\begin{align*}
\tau_0 = \E\bs{q_{1, 0}(Z, S, X) Y \mid G = \text{O}} - \E\bs{q_{0, 0}(Z, S, X) Y \mid G = \text{O}}.
\end{align*}
\end{enumerate}
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-surrogate-bridge-identification}.
\end{proof}
\cref{thm:surrogate-bridge-identification}
shows that
the long-term ATE
in the experimental sample
can be identified
by reweighting
the observed outcomes $Y$
in the observational sample
using the surrogate bridge functions $q_{a, 0}$'s.
\begin{remark}
\cref{thm:surrogate-bridge-identification}
is new to the literature,
providing a novel form of bridge function (\cref{eq:surrogate-bridge-function})
for identifying the ATE
in the data combination setting.
\end{remark}
\begin{remark}[Connection to \cite{athey2025surrogate}]
\cref{thm:surrogate-bridge-identification}
generalizes
\cite{athey2025surrogate}'s surrogate score identification
by allowing for unobserved confounding.
See \cref{sec:connection-to-surrogate-score-identification}
for more details.
\end{remark}
\subsection{Multiply Robust Identification}
In
\cref{sec:identification-via-outcome-bridge-function,sec:identification-via-surrogate-bridge-function},
we present two distinct identification strategies
based on
the outcome bridge functions
and the surrogate bridge functions,
respectively.
Combining these two strategies,
we obtain
a multiply robust identification result
as follows.
\begin{theorem}[Multiply Robust Identification]
\label{thm:mutiply-robust-identification}
Let $\tau_0$ be the true ATE defined in \cref{eq:ate}.
Let $\eta = (e, h, \bar{h}, q_0, q_1)$
be a set of possibly misspecified nuisance functions.
Let $\eta_0 = (e_0, h_0, \bar{h}_0, q_{0,0}, q_{1,0})$
denote the true nuisance functions,
where $e_0$ is the true propensity score,
$h_0$ is the function satisfying \cref{eq:outcome-bridge-function},
$\bar{h}_0(A, X)$ is defined as $\E\bs{h_0(W, S, X) \mid A, X, G = \text{E}}$,
and $q_{a, 0}$'s are the functions satisfying \cref{eq:surrogate-bridge-function}.
If any of the following conditions on nuisance functions holds:
\begin{enumerate}
\item $h = h_0$, $\bar{h} = \bar{h}_0$;
\item $h = h_0$, $e = e_0$;
\item $q_a = q_{a, 0}$ for each $a \in \{0, 1\}$, $e = e_0$;
\item $q_a = q_{a, 0}$ for each $a \in \{0, 1\}$, $\bar{h} = \bar{h}^{(h)}$;
\end{enumerate}
then
\begin{align}
\label{eq:mutiply-robust-identification}
\tau_0 = \E\bs{\varphi(D; \eta)},
\end{align}
where $\varphi(D; \eta) \equiv \varphi_{\text{E}}(D; \eta) + \varphi_{\text{O}}(D; \eta)$
is defined as
\begin{align*}
\varphi_{\text{E}}(D; \eta)
&\equiv \frac{\mathbf{1}_{\bc{G = \text{E}}}}{\pi_0} \bp{\frac{(A - e(X))\bp{h(W, S, X) - \bar{h}(A, X)}}{e(X)(1 - e(X))} + \bar{h}(1, X) - \bar{h}(0, X)} \\
\varphi_{\text{O}}(D; \eta)
&\equiv \frac{\mathbf{1}_{\bc{G = \text{O}}}}{1 - \pi_0} \bp{\bp{q_1(Z, S, X) - q_0(Z, S, X)} (Y - h(W, S, X))}.
\end{align*}
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-mutiply-robust-identification}.
\end{proof}
\begin{remark}
\cref{thm:mutiply-robust-identification}
is multiply robust
in the sense that
it identifies the ATE $\tau_0$
if any one of the four sets of nuisance functions
is correctly specified.
This identification result
is motivated by
the EIF of $\tau_0$
under a certain nonparametric model (see \cref{sec:estimation-and-inference}),
extending the influence function-based identification
in \cite{athey2025surrogate}
to accommodate unobserved confounding.
See \cref{sec:connection-to-influence-function-identification}
for more details.
\end{remark}
\section{Estimation and Inference} \label{sec:estimation-and-inference}
\subsection{Cross-Fitted Estimators} \label{sec:cross-fitted-estimators}
In this section,
we propose a generic estimation procedure
for the ATE $\tau_0$.
This approach
builds upon the identification results in \cref{sec:identification}
and employs
the cross-fitting technique from
\cite{chernozhukov2018double}
to
reduce overfitting bias
when estimating nuisance parameters.
\begin{definition}[Cross-Fitted Estimators]
\label{def:cross-fitted-estimators}
Fix an integer $K \geq 2$.
Let $\mathcal{I} \equiv \{1, \dots, N\}$ be the index set,
where $N$ is the total sample size.
Define $\mathcal{I}_{\text{E}} \equiv \{i \in \mathcal{I}: G_i = \text{E}\}$
and $\mathcal{I}_{\text{O}} \equiv \{i \in \mathcal{I}: G_i = \text{O}\}$
as the index sets for the experimental and observational samples, respectively.
Randomly partition $\mathcal{I}_{\text{E}}$ into $K$ disjoint subsets
$\{\mathcal{I}_{\text{E}, k}\}_{k = 1}^K$.
Similarly, randomly partition $\mathcal{I}_{\text{O}}$
into $K$ disjoint subsets $\{\mathcal{I}_{\text{O}, k}\}_{k = 1}^K$.
Define $\mathcal{D}_{k} \equiv \{D_i: i \in \mathcal{I}_{\text{E}, k} \cup \mathcal{I}_{\text{O}, k}\}$
as the evaluation set for fold $k$,
and $\mathcal{D}_{-k} \equiv \mathcal{D} \setminus \mathcal{D}_{k}$
as the training set for fold $k$.
For each fold $k$,
estimate the nuisance functions
$\hat{\eta}_{k} \equiv (\hat{e}_{k}, \hat{h}_{k}, \hat{\bar{h}}_{k}, \hat{q}_{0, k}, \hat{q}_{1, k})$
using the data excluding fold $k$, i.e., $\mathcal{D}_{-k}$,
where $\hat{\bar h}_k$ is fitted by regressing $\hat h_k(W,S,X)$ on $(A,X)$ using $\mathcal D_{\mathrm{E},-k}$.
The cross-fitted estimators are defined as
\begin{align*}
\hat{\tau}_{\text{OB-OR}}
&\equiv \frac{1}{N_{\text{E}}} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{E}, k}}
\bp{\hat{\bar{h}}_{k}(1, X_i) - \hat{\bar{h}}_{k}(0, X_i)}, \\
\hat{\tau}_{\text{OB-IPW}}
&\equiv \frac{1}{N_{\text{E}}} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{E}, k}}
\bp{\frac{A_i \hat{h}_{k}(W_i, S_i, X_i)}{\hat{e}_{k}(X_i)} - \frac{(1 - A_i) \hat{h}_{k}(W_i, S_i, X_i)}{1 - \hat{e}_{k}(X_i)}}, \\
\hat{\tau}_{\text{SB}}
&\equiv \frac{1}{N_{\text{O}}} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{O}, k}}
\bp{\hat{q}_{1, k}(Z_i, S_i, X_i) - \hat{q}_{0, k}(Z_i, S_i, X_i)} Y_i, \\
\hat{\tau}_{\text{MR}}
&\equiv \frac{1}{N_{\text{E}}} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{E}, k}}
\bp{\frac{(A_i - \hat{e}_{k}(X_i))(\hat{h}_{k}(W_i, S_i, X_i) - \hat{\bar{h}}_{k}(A_i, X_i))}{\hat{e}_{k}(X_i)(1 - \hat{e}_{k}(X_i))}} \\
&\quad + \frac{1}{N_{\text{E}}} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{E}, k}} \bp{\hat{\bar{h}}_{k}(1, X_i) - \hat{\bar{h}}_{k}(0, X_i)} \\
&\quad + \frac{1}{N_{\text{O}}} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{O}, k}}
\bp{(\hat{q}_{1, k}(Z_i, S_i, X_i) - \hat{q}_{0, k}(Z_i, S_i, X_i)) (Y_i - \hat{h}_{k}(W_i, S_i, X_i))}.
\end{align*}
\end{definition}
\subsection{Asymptotic Properties} \label{sec:asymptotic-properties}
Before stating the main theorems,
we introduce some additional definitions and assumptions.
First,
we define the linear operator $T: L_2(W, S, X) \to L_2(Z, S, X)$
as
\[
(T h)(Z, S, X) = \E\bs{h(W, S, X) \mid Z, S, X, G = \text{O}},
\]
and its adjoint operator
$T^*: L_2(Z, S, X) \to L_2(W, S, X)$
as
\[
(T^* q_a)(W, S, X) = \E\bs{q_a(Z, S, X) \mid W, S, X, G = \text{O}}.
\]
\subsubsection{Consistency}
Following the standard approach in \cite{imbensLongtermCausalInference2025a},
we impose the following convergence rate conditions
on the nuisance estimators to establish consistency.
\begin{assumption}[Convergence Rates]
\label{asn:convergence-rates}
Let $\| \cdot \|_{2}$ denote the $L_2$ norm.
For any square-integrable function $g(W, S, X)$,
let $\tilde{\bar{h}}^{(g)}(A, X)$ denote the population limit of the
estimator $\hat{\bar{h}}$ trained with pseudo-outcome $g(W, S, X)$ and predictors $(A, X)$
in the experimental sample.
For each fold $k$,
the nuisance estimators
satisfy the following convergence rates:
\begin{align*}
\|\hat{h}_{k} - \tilde{h}\|_{2} &= O_P(\delta_{h, N}), \quad
\|T(\hat{h}_{k} - \tilde{h})\|_{2} = O_P(\rho_{h, N}), \quad
\|\hat{\bar h}_{k} - \tilde{\bar{h}}^{(\hat{h}_k)}\|_2 = O_P(\delta_{\bar h,N}), \\
\|\hat{q}_{a, k} - \tilde{q}_{a}\|_{2} &= O_P(\delta_{q, N}), \quad
\|T^*(\hat{q}_{a, k} - \tilde{q}_{a})\|_{2} = O_P(\rho_{q, N}), \quad
\|\hat e_{k} - \tilde{e}\|_{2} = O_P(\delta_{e, N}),
\end{align*}
where $\delta_{h, N}$, $\rho_{h, N}$, $\delta_{\bar{h}, N}$, $\delta_{q, N}$, $\rho_{q, N}$, $\delta_{e, N}$
are sequences converging to zero as $N \to \infty$,
and the mapping $g \mapsto \tilde{\bar{h}}^{(g)}$ is Lipschitz continuous with respect to the $L_2$ norm.
\end{assumption}
\begin{theorem}[Consistency]
\label{thm:consistency}
\begin{enumerate}
\item[]
\item If the conditions in \cref{thm:outcome-bridge-identification},
\cref{asn:convergence-rates},
$\tilde{h} = h_0$ and $\tilde{\bar{h}}^{(h_0)} = \bar{h}_0$,
then $\hat{\tau}_{\text{OB-OR}}$ is consistent for $\tau_0$.
\item If the conditions in \cref{thm:outcome-bridge-identification},
\cref{asn:convergence-rates},
$\tilde{h} = h_0$ and $\tilde{e} = e_0$ hold,
then $\hat{\tau}_{\text{OB-IPW}}$ is consistent for $\tau_0$.
\item If the conditions in \cref{thm:surrogate-bridge-identification},
\cref{asn:convergence-rates},
$\tilde{q}_{a} = q_{a, 0}$ for each $a \in \{0, 1\}$
and $\tilde{e} = e_0$ hold,
then $\hat{\tau}_{\text{SB}}$ is consistent for $\tau_0$.
\item If \cref{asn:convergence-rates} holds, and any of the following sets of conditions is satisfied:
\begin{enumerate}
\item $\tilde{h} = h_0$ and $\tilde{\bar{h}}^{(h_0)} = \bar{h}_0$;
\item $\tilde{h} = h_0$ and $\tilde{e} = e_0$;
\item $\tilde{q}_{a} = q_{a, 0}$ for each $a \in \{0, 1\}$ and $\tilde{e} = e_0$;
\item $\tilde{q}_{a} = q_{a, 0}$ for each $a \in \{0, 1\}$ and $\tilde{\bar{h}}^{(\tilde{h})}(A, X) = \E[\tilde{h} \mid A, X, G=\text{E}]$,
\end{enumerate}
then $\hat{\tau}_{\text{MR}}$ is consistent for $\tau_0$.
\end{enumerate}
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-consistency}.
\end{proof}
\subsubsection{Asymptotic Normality}
While consistency requires only that the nuisance estimators converge to their true values,
establishing asymptotic normality requires stricter conditions on their convergence rates.
Specifically, we need to ensure that the second-order bias terms vanish faster than $N^{-1/2}$,
ensuring that the asymptotic distribution of $\hat{\tau}_{\text{MR}}$
is dominated by the first-order term.
We impose the following assumption on the product rates of the nuisance estimators.
\begin{assumption}[Product Rates of Convergence]
\label{asn:product-rates}
Suppose that \cref{asn:convergence-rates} holds.
The convergence rates satisfy the following conditions:
\begin{align}
\delta_{e, N} \delta_{\bar{h}, N} = o_P(N^{-1/2}),
\label{eq:rate-cond-experimental} \\
\min \left\{ \delta_{q, N} \rho_{h, N}, \; \rho_{q, N} \delta_{h, N} \right\} = o_P(N^{-1/2}).
\label{eq:rate-cond-observational}
\end{align}
\end{assumption}
\begin{remark}
\cref{eq:rate-cond-experimental}
is standard in the DML literature
\citep{chernozhukov2018double},
requiring the product of the convergence rates of the propensity score
and the pseudo-outcome regression
that uses the estimated outcome bridge function as the outcome
to be $o_P(N^{-1/2})$.
\cref{eq:rate-cond-observational} is specific to the proximal inference setting.
Following \cite{imbensLongtermCausalInference2025a},
we distinguish between two types of convergence rates:
\emph{strong-metric errors} ($\delta_{h,N}$ and $\delta_{q,N}$)
measure $L_2$ deviation from the true bridge functions,
and \emph{weak-metric errors} ($\rho_{h,N}$ and $\rho_{q,N}$)
measure violations of the bridge equations via operators $T$ and $T^*$.
Because bridge-function estimation amounts to solving potentially ill-posed conditional moment equations,
an estimator can exhibit a slow strong-metric rate while still achieving a fast weak-metric rate
(e.g., via Tikhonov regularization or sieve methods).
\cref{eq:rate-cond-observational} exploits this asymmetry,
requiring only that the product of \emph{one} strong-metric error and \emph{one} weak-metric error
be $o_P(N^{-1/2})$,
a strictly weaker condition than requiring both strong rates
to satisfy $\delta_{h,N} \delta_{q,N} = o_P(N^{-1/2})$.
\end{remark}
The following theorem establishes the asymptotic normality of the
cross-fitted estimator $\hat{\tau}_{\text{MR}}$
when the bridge functions and propensity score
converge to their true values and the pseudo-outcome regression for $\hat{\bar h}_k$
is correctly specified.
\begin{theorem}[Asymptotic Normality]
\label{thm:asymptotic-normality}
Suppose \cref{asn:convergence-rates} holds with the correct limits
\[
\tilde{h} = h_0, \quad \tilde{e} = e_0, \quad \tilde{q}_a = q_{a,0} \text{ for } a \in \{0, 1\}, \quad \text{and} \quad
\tilde{\bar{h}}^{(h_0)} = \bar{h}_0.
\]
If \cref{asn:product-rates} also holds,
then the cross-fitted estimator $\hat{\tau}_{\text{MR}}$ is asymptotically normal:
\begin{align*}
\sqrt{N}(\hat{\tau}_{\text{MR}} - \tau_0) \xrightarrow{d} \mathcal{N}(0, V),
\end{align*}
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-asymptotic-normality}.
\end{proof}
An immediate consequence of \cref{thm:asymptotic-normality}
is that
we can construct
a consistent estimator for the asymptotic variance $V$
using the plug-in approach:
\begin{align*}
\hat{V}
&= \frac{N}{N_{\text{E}}^2} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{E}, k}}
\bp{\frac{(A_i - \hat{e}_{k}(X_i))(\hat{h}_{k}(W_i, S_i, X_i) - \hat{\bar{h}}_{k}(A_i, X_i))}{\hat{e}_{k}(X_i)(1 - \hat{e}_{k}(X_i))} + \hat{\bar{h}}_{k}(1, X_i) - \hat{\bar{h}}_{k}(0, X_i) - \hat{\tau}_{\text{MR}}}^2 \\
&\quad + \frac{N}{N_{\text{O}}^2} \sum_{k = 1}^K \sum_{i \in \mathcal{I}_{\text{O}, k}}
\bp{(\hat{q}_{1, k}(Z_i, S_i, X_i) - \hat{q}_{0, k}(Z_i, S_i, X_i)) (Y_i - \hat{h}_{k}(W_i, S_i, X_i))}^2.
\end{align*}
The following theorem establishes the consistency of this variance estimator
and the validity of the corresponding confidence interval (CI).
\begin{theorem}[Consistency of Variance Estimator]
\label{thm:consistency-of-variance-estimator}
Under the conditions of \cref{thm:asymptotic-normality},
the variance estimator $\hat{V}$ is consistent for $V$:
\begin{align*}
\hat{V} \xrightarrow{p} V.
\end{align*}
Consequently, the asymptotic $(1-\alpha)$ CI for $\tau_0$ is given by
\begin{align*}
\text{CI}_{1 - \alpha}
= \left( \hat{\tau}_{\text{MR}} - \Phi^{-1}(1 - \alpha/2) \sqrt{\hat{V}/N}, \quad
\hat{\tau}_{\text{MR}} + \Phi^{-1}(1 - \alpha/2) \sqrt{\hat{V}/N} \right),
\end{align*}
satisfying $\lim_{N \to \infty} P(\tau_0 \in \text{CI}_{1 - \alpha}) = 1 - \alpha$,
where $\Phi^{-1}(\cdot)$ is the quantile function of the standard normal distribution.
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-consistency-of-variance-estimator}.
\end{proof}
\subsubsection{Semiparametric Efficiency}
We now establish the local semiparametric efficiency of
our cross-fitted estimator $\hat{\tau}_{\text{MR}}$
under a statistical model $\mathcal{P}$
that imposes no restrictions
on the observed data law
other than the existence of an outcome bridge function $h_0$
(\cref{asn:outcome-bridge-function})
Following \cite{cuiSemiparametricProximalCausal2024} and \cite{imbensLongtermCausalInference2025a},
to ensure the EIF is well-defined,
we impose the following surjectivity condition,
which guarantees
the uniqueness of the bridge functions.
\begin{assumption}[Surjectivity]
\label{asn:surjectivity}
The linear operators $T$ and $T^*$
are surjective.
\end{assumption}
\begin{proposition}[Uniqueness of Bridge Functions]
\label{prp:surjectivity-uniqueness}
Under \cref{asn:surjectivity},
the outcome bridge function $h_0$
in \cref{eq:outcome-bridge-function}
and the surrogate bridge functions $q_{a, 0}$'s
in \cref{eq:surrogate-bridge-function}
are unique almost surely.
\end{proposition}
\begin{proof}
See \cref{sec:proof-of-surjectivity-uniqueness}.
\end{proof}
The following theorem characterizes the
EIF for the ATE $\tau_0$
and the corresponding semiparametric efficiency bound.
\begin{theorem}[EIF]
\label{thm:eif}
Consider the semiparametric model $\mathcal{P}$
evaluated at the submodel
where the \cref{asn:surrogate-bridge-function,asn:surjectivity} hold.
The EIF for the ATE $\tau_0$
is given by
\begin{align*}
&\mathrel{\phantom{=}} \IF(D) \\
&\equiv \frac{\mathbf{1}_{\bc{G = \text{E}}}}{\pi_0} \bp{\frac{(A - e_0(X))\bp{h_0(W, S, X) - \bar{h}_0(A, X)}}{e_0(X)(1 - e_0(X))} + \bar{h}_0(1, X) - \bar{h}_0(0, X) - \tau_0} \\
&\quad + \frac{\mathbf{1}_{\bc{G = \text{O}}}}{1 - \pi_0} \bp{\bp{q_{1, 0}(Z, S, X) - q_{0, 0}(Z, S, X)} (Y - h_0(W, S, X))}.
\end{align*}
Therefore,
the corresponding semiparametric local efficiency bound
for estimating $\tau_0$ is $V_\text{eff} = \E\bs{\IF(D)^2}$.
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-eif}.
\end{proof}
A direct consequence of \cref{thm:asymptotic-normality} and \cref{thm:eif}
is that
our cross-fitted estimator $\hat{\tau}_{\text{MR}}$
achieves the semiparametric local efficiency bound
as stated in the following theorem.
\begin{theorem}[Semiparametric Efficiency]
\label{thm:efficiency}
Suppose the conditions of \cref{thm:asymptotic-normality}
and \cref{asn:surjectivity} hold.
Then,
\begin{align*}
\sqrt{N} (\hat{\tau}_{\text{MR}} - \tau_0)
\xrightarrow{d} \mathcal{N}(0, V_\text{eff}),
\end{align*}
where $V_\text{eff}$ is the semiparametric efficiency bound
derived in \cref{thm:eif}.
\end{theorem}
\begin{proof}
See \cref{sec:proof-of-thm-efficiency}.
\end{proof}
\begin{comment}
\section{Numerical Studies} \label{sec:numerical-studies}
\subsection{A Simulation Study} \label{sec:simulation-study}
\end{comment}
\section{Real Data Application} \label{sec:real-data-application}
Following the seminal approach of \cite{lalonde1986evaluating},
we exploit the experimental data to
construct a ``ground truth'' benchmark
against which we evaluate
the performance of different methods.
We analyze data from the JC program,
a large-scale randomized evaluation of job training programs
for disadvantaged youth in the United States
\citep{schochet2001national,schochet2008does}.
\paragraph{Design}
Our evaluation design proceeds in two steps.
First, we randomly split the experimental dataset
into two disjoint subsets: an ``Experimental Sample'' ($\text{E}$)
where the long-term outcome ($Y$) is masked,
and an ``Observational Sample'' ($\text{O}$)
where the treatment assignment ($A$) is masked.\footnote{
The dataset for this application
is publicly available at the \texttt{causalweight} R package
\citep{causalweight}.
}
This setup mimics the data combination problem
while ensuring, by construction,
that there is no selection bias between the two samples
(i.e., comparability or transportability assumptions hold).
This allows us to isolate
the performance of the estimators
in handling the failure of the surrogacy assumption
or the presence of unobserved confounding.
\paragraph{Variables}
We define the treatment ($A$)
as random assignment and
the long-term outcomes ($Y$)
as weekly earnings and proportion of weeks employed
in the fourth year.
The surrogate vector ($S$)
includes earnings and proportion of weeks employed
in years 2 and 3.
To address potential confounding,
we select possession of a general education development (GED) degree ($W$)
and English mother tongue ($Z$) as proxies.
These variables are determined pre-treatment
and likely provide information about unobserved confounders,
such as cognitive skills or long-term career orientation.
We also include baseline covariates ($X$),
including age, gender, race, education level, and pre-program earnings.
\paragraph{Methods Compared}
We compare four methods for estimating the ATE.
The first method
is the RCT benchmark,
which estimates the ATE
directly from the experimental sample
using a linear regression of $Y$ on $(1, A, X)$.
The second method
is the standard surrogate index estimator
\citep{athey2025surrogate},
which first fits a regression of $Y$ on $(1, S, X)$
in the observational sample,
then predicts $Y$ in the experimental sample,
and finally estimates the ATE
using a linear regression of the predicted $Y$ on $(1, A, X)$.
The third method
is similar to the second method
but includes the proxy variables ($W$ and $Z$)
as additional covariates
in both regressions.
The fourth method
is our proposed method,
which first estimates an linear outcome bridge function
$h(W, S, X; \beta)$
in the observational sample,
then predicts $Y$ in the experimental sample
using the fitted $h$,
and finally estimates the ATE
using a linear regression of the predicted $Y$ on $(1, A, X)$.
\paragraph{Results}
\begin{table}[ht]
\footnotesize
\centering
\caption{Estimates of the 4-year effects of the JC program on earnings and proportion of weeks employed.}
\label{tab:jc-ate}
\begin{tabular}{lccc}
\toprule
& Estimate & Standard Error & $p$-Value\\
\midrule
\addlinespace[0.3em]
\multicolumn{4}{l}{\textit{Panel A: Weekly Earnings}}\\
\hspace{1em}RCT & 15.30 & 5.49 & 0.005\\
\hspace{1em}Naive (1) & 7.05 & 3.70 & 0.057\\
\hspace{1em}Naive (2) & 7.30 & 3.69 & 0.048\\
\hspace{1em}Ours & 16.43 & 7.92 & 0.038\\
\addlinespace[0.3em]
\multicolumn{4}{l}{\textit{Panel B: Proportion of Weeks Employed}}\\
\hspace{1em}RCT & 2.87 & 1.14 & 0.012\\
\hspace{1em}Naive (1) & 0.80 & 0.67 & 0.233\\
\hspace{1em}Naive (2) & 0.84 & 0.67 & 0.210\\
\hspace{1em}Ours & 3.50 & 1.96 & 0.074\\
\bottomrule
\end{tabular}
\caption*{
\textit{Note:}
The standard errors
are computed using the plug-in sandwich estimator
based on the joint GMM framework.
The $p$-values
are based on two-sided Wald tests
using normal approximations.
}
\end{table}
\cref{tab:jc-ate}
presents
the estimates of the treatment effects
in the experimental sample.
While the true benchmark
shows a significant positive effect,
the standard surrogate index method
substantially underestimates
the effect for both outcomes,
and including the proxies
$W$ and $Z$
as additional covariates
does not improve the estimates.
In contrast,
our proposed method
that accounts for unobserved confounding
via the proxy variables
yields estimates
much closer to the benchmark results,
albeit with larger standard errors.
\paragraph{Discussion}
As found in \cite{schochet2008does},
program participation
temporarily suppresses short-term earnings and employment
because participants
spend time in training instead of working.
Theerefore, the substantial underestimation by the standard surrogate index
likely arises from unobserved confounding,
where latent ability $U$ positively affects both short-term ($S$) and long-term ($Y$) outcomes.
This confounding creates a positive bias that attenuates the true negative structural relationship between
$S$ and $Y$, where sacrificing short-term earnings represents an investment in future growth.
Consequently,
the standard method underestimates the true long-term effect of the program.
By leveraging proxies to adjust for $U$,
our method distinguishes between low earnings due to investment versus low ability,
thereby recovering the experimental benchmark.
\paragraph{Diagnostic Analysis}
We perform a diagnostic analysis
on the experimental sample
to investigate the validity
of the standard surrogacy assumption
and the role of unobserved confounding.
The results are summarized in \cref{tab:jc-diagnostic}.
First,
since we have access to the long-term outcome $Y$
in the experimental sample,
we can directly test the surrogacy assumption
by regressing $Y$ on $(1, A, S, X)$
and examining the coefficient of $A$.
The first rows from Panel A
and Panel B
of \cref{tab:jc-diagnostic}
show that
the coefficient of $A$
is statistically significant
for both outcomes,
indicating
a potential violation
of the surrogacy assumption.
Second,
to assess whether the proxies
$W$ and $Z$
help mitigate unobserved confounding,
we further fit an IV regression,
where the first stage regresses $W$ on $(1, Z, A, S, X)$
and the second stage regresses $Y$
on $(1, A, S, X, \hat W)$,
with $Z$ acting as the instrument for $W$.
In this setup,
$\hat W$
captures the variation coming from the unobserved confounders $U$,
so including it in the regression
helps adjust for unobserved confounding.
The second rows from Panel A
and Panel B
of \cref{tab:jc-diagnostic}
show that
the coefficient of $A$
becomes statistically insignificant
for both outcomes
after adjusting for $\hat W$,
suggesting
that
the proxies
$W$ and $Z$
effectively mitigate
unobserved confounding,
which results in
the improved performance
in \cref{tab:jc-ate}.
\begin{table}[ht]
\footnotesize
\centering
\caption{Diagnostic analysis on the experimental sample of the JC program.}
\label{tab:jc-diagnostic}
\begin{tabular}{lccc}
\toprule
& Coefficient on $A$ & Standard Error & $p$-Value\\
\midrule
\addlinespace[0.3em]
\multicolumn{4}{l}{\textit{Panel A: Weekly Earnings}}\\
\hspace{1em}OLS Regression & 8.603 & 4.286 & 0.045\\
\hspace{1em}IV Regression & -0.125 & 9.465 & 0.989\\
\addlinespace[0.3em]
\multicolumn{4}{l}{\textit{Panel B: Proportion of Weeks Employed}}\\
\hspace{1em}OLS Regression & 2.117 & 0.933 & 0.023\\
\hspace{1em}IV Regression & 0.546 & 1.848 & 0.768\\
\bottomrule
\end{tabular}
\caption*{
\textit{Note:}
The standard errors
are computed using heteroskedasticity-robust standard errors.
The $p$-values
are based on two-sided $t$-tests.
}
\end{table}
\section{Conclusion} \label{sec:conclusion}
We study the challenging problem
of identifying and estimating long-term treatment effects
by combining experimental and observational data
in the presence of unobserved confounding.
The standard surrogate index methods rely on
strong assumptions that rule out such confounding,
leading to potentially biased estimates in many empirical settings.
We develop new methods
for relaxing these stringent assumptions.
Our methods rely on proxy variables
that are informative about the unobserved confounders
and satisfy certain conditional independence properties.
We establish nonparametric identification results
based on bridge functions,
which generalize
existing identification strategies that assume no unobserved confounding.
We further propose
estimation and inference procedures
and establish their asymptotic properties,
including consistency, asymptotic normality, and semiparametric efficiency.
Applying our methods to
a real-world job training program evaluation,
we find that
the standard surrogate index approach
substantially underestimates
the true long-term effects,
while our proposed methods
that account for unobserved confounding
via proxy variables
yield estimates
much closer to the benchmark results
from the experimental data.
\phantomsection
\addcontentsline{toc}{section}{References}
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