Long-term outcomes of experimental evaluations are necessarily observed after
long delays. We develop semiparametric methods for combining the short-term
outcomes of experiments with observational measurements of short-term and
long-term outcomes, in order to estimate long-term treatment effects. We
characterize semiparametric efficiency bounds for various instances of this
problem. These calculations facilitate the construction of several estimators.
We analyze the finite-sample performance of these estimators with a simulation
calibrated to data from an evaluation of the long-term effects of a poverty
alleviation program.

Keywords: Long-Term Treatment Effects, Semiparametric Efficiency,
Experimentation
JEL Codes: C01, C13, C14, O10

##  1\. Introduction

Empirical researchers often aim to estimate the long-term effects of policies
or interventions. Randomized experimentation provides a simple and
interpretable approach to this problem (Fisher,, 1925; Duflo et al.,, 2007;
Athey and Imbens,, 2017). However, long-term outcomes of experimental
evaluations are necessarily observed after long and potentially costly delays.
Consequently, there is relatively limited experimental evidence on the long-
term effects of economic and social policies.111Bouguen et al., (2019) provide
a systematic review of randomized control trials in development economics and
report that only a small proportion evaluate long-term effects. Similarly, in
a review on experimental evaluations of the effects of early childhood
educational interventions, Tanner et al., (2015) identify only one randomized
evaluation that reports long-term employment and labor market effects (Gertler
et al.,, 2014).

In this paper, we develop methods for estimating long-term treatment effects
by combining short-term experimental and long-term observational data sets. We
consider two closely related models, proposed by Athey et al., 2020a  and
Athey et al., 2020b . In both cases, a researcher is interested in the average
effect of a binary treatment on a scalar long-term outcome. The researcher
observes two samples of data, an _experimental_ sample and an _observational_
sample. The experimental sample measures short-term outcomes of a randomized
evaluation of the treatment. the treatment The observational sample measures
both short-term and long-term outcomes, but may be subject to unmeasured
confounding. The two settings that we consider are distinguished by whether
treatment is observed in the observational sample. Similar identifying
assumptions are required in each case. We review these assumptions in Section
2.

Methods for combining experimental and observational data to estimate long-
term treatment effects are widely applicable in applied microeconomics and
online platform experimentation. The estimators proposed in Athey et al.,
2020b  have been used to estimate the long-term effects of free tuition on
college completion (Dynarski et al.,, 2021), of an agricultural monitoring
technology on farmer revenue in Paraguay (Dal Bó et al.,, 2021), and of
changes to Twitter’s platform on user engagement (Twitter Engineering,, 2021).
As a result, there is considerable interest in advancing a statistical and
methodological foundation for this problem (Gupta et al.,, 2019).

To that end, this paper offers two contributions. First, we develop
semiparametric theory for estimation of long-term average treatment effects in
Section 3. In particular, we derive the semiparametric efficiency bound, and
the corresponding efficient influence function, for estimating long-term
average treatment effects in each of the models that we consider. We then
demonstrate that the efficient influence function is the unique influence
function in each model, indicating that all regular and asymptotically linear
estimators have the same asymptotic variance and achieve the semiparametric
efficiency bound (Chen and Santos,, 2018; Newey,, 1994). In both cases, we
find that the efficient influence functions possess a “double-robust”
structure commonly found in causal estimation problems (Scharfstein et al.,,
1999; Kang and Schafer,, 2007).

These results are novel. In particular, our calculations correct statements
concerning efficient influence functions and semiparametric efficiency bounds
given in a working paper draft of Athey et al., 2020b .222The results given in
Athey et al., 2020b  were obtained through a non-rigorous calculation related
to standard heuristics involving a discretization of the sample space (see
e.g., Kennedy, (2022) and Ichimura and Newey, (2022) for discussion). Our
arguments are rigorous, following the method developed in Section 3.4 of
Bickel et al., (1993). Analogous results for the model considered in Athey et
al., 2020a  have not appeared before in the literature.

Second, in Section 4, we establish the consistency and asymptotic normality of
a suite of estimators. These estimators differ according to whether they are
based on moment conditions associated with the efficient influence functions
derived in Section 3. Moment conditions defined by efficient influence
functions are often referred to as Neyman orthogonal moment conditions, due to
their insensitivity to local perturbations of nuisance parameters (see e.g.,
Chernozhukov et al., 2018a, ; Foster and Syrgkanis,, 2019). The estimators
that we formulate can be viewed as instances of one-step estimators (Le Cam,,
1956; Bickel,, 1982; Newey,, 1994). Roughly, the conditional expectations that
compose an efficient influence function are estimated by augmenting standard
machine learning algorithms with cross-fitting. Estimates of long-term average
treatment effects are then formed by treating estimated efficient influence
functions as identifying moment functions. Estimators obtained from Neyman
orthogonal moment conditions admit a very general analysis, under high-level
sufficient conditions, due to Chernozhukov et al., 2018a . We adapt these
arguments to our setting.

We compare these estimators to a variety of alternative estimators based on
non-orthogonal moment conditions. In particular, we consider a set of
estimators that are analogous to standard inverse propensity score weighting
and outcome regression estimators for average treatment effects under
unconfoundedness (see e.g., Imbens, 2004 for a review). These estimators
include, but are not limited to, many of the estimators proposed in Athey et
al., 2020a  and Athey et al., 2020b , and applied by e.g., Dynarski et al.,
(2021). Here, to obtain theoretical guarantees, we restrict attention to
estimators that plug-in nuisance parameter estimates derived from the method
of sieves (Chen and Liao,, 2015; Chen et al.,, 2014). Relative to those for
estimators based on orthogonal moments, sufficient conditions for estimators
based on non-orthogonal moments appear more stringent.

Section 5 assesses the finite sample performance of these semiparametric
estimators with a simulation calibrated to data from a randomized evaluation
of the long-term effects of a poverty alleviation program originally analyzed
in Banerjee et al., (2015). We find that estimators based on orthogonal
moments are substantially more accurate than estimators based on non-
orthogonal moments. Section 6 concludes.

Proofs for all results stated in the main text are provided in Appendix A.
Appendices B, C, and D give additional results or details and will be
introduced at appropriate points throughout the paper. Code implementing the
estimators developed in this paper is available on GitHub at the link
https://github.com/DavidRitzwoller/longterm.

###  1.1. Related literature

Settings closely related to our own include Rosenman et al., (2018), Rosenman
et al., (2020), and Kallus and Mao, (2020). In Rosenman et al., (2018),
treatment assignment is unconfounded in both samples. In Rosenman et al.,
(2020), the outcome of interest is observed in both samples. In Kallus and
Mao, (2020), treatment is unconfounded in both samples considered jointly, but
not necessarily in either sample considered individually. Yang et al., (2020),
Gui, (2020), and Gechter and Meager, (2021) give complementary analyses of
related models. Hou et al., (2021) consider a model in which measurements of
long-term outcomes and treatments are missing completely at random. Imbens et
al., (2022) propose several alternative identification strategies for models
similar to the model that we consider. In contemporaneous work, Singh, (2021)
and Singh, (2022) propose estimators and confidence intervals similar to those
developed in Section 4 for the case in which treatment is not observed in the
observational data set.

This paper contributes to the literature on missing data models (Little and
Rubin,, 2019; Ridder and Moffitt,, 2007; Hotz et al.,, 2005), and more
specifically to the literature on semiparametric efficiency in missing data
models (Chen et al.,, 2008; Graham,, 2011; Muris,, 2020; Bia et al.,, 2020).
The models that we consider are related to the literatures on statistical
surrogacy (Prentice,, 1989; Begg and Leung,, 2000) and mediation analysis (van
der Laan and Petersen,, 2004; Imai et al.,, 2010).

###  1.2. Notation

Let $\lambda$ be a $\sigma$-finite measure on the measurable space
$\left(\Omega,\mathcal{F}\right)$, and let $\mathcal{M}_{\lambda}$ be the set
of all probability measures on $\left(\Omega,\mathcal{F}\right)$ that are
absolutely continuous with respect to $\lambda$.333The model developed in the
sequel does not depend on $\lambda$, as $\lambda$ can be chosen as the
population distribution of the data. For an arbitrary random variable $D$
defined on $\left(\Omega,\mathcal{F},Q\right)$ with
$Q\in\mathcal{M}_{\lambda}$, we let $\mathbb{E}_{Q}[D]$ denote its expected
value. The quantity $p(d\mid E)$ will denote the density of the random
variable $D$ at $d$ conditional on the event $E\in\Omega$ and $\ell(d\mid E)$
will denote the corresponding log-likelihood. This notation leaves the law of
the random variable $D$ implicit, but will cause no ambiguity. We let
$\|\cdot\|_{Q,q}$ denote the $L^{q}(Q)$ norm and $[n]$ denote the set
$\\{1,\ldots,n\\}$.

##  2\. Problem Formulation

Consider a researcher who conducts a randomized experiment aimed at assessing
the effects of a policy or intervention. For each individual in the
experiment, they measure a $q$-vector $X_{i}$ of pre-treatment covariates, a
binary variable $W_{i}$ denoting assignment to treatment, and a $d$-vector
$S_{i}$ of short-term post-treatment outcomes. The researcher is interested in
the effect of the treatment on a scalar, long-term, post-treatment outcome
$Y_{i}$ that is not measured in their experimental data. They are able to
obtain an auxiliary, observational data set containing measurements, for a
separate population of individuals, of the long-term outcome of interest, in
addition to records of the same pre-treatment covariates and short-term
outcomes that were measured in the experimental data set. This observational
data set may or may not record whether each individual was exposed to the
treatment of interest. In this paper, we develop methods for estimating the
effect of a treatment on long-term outcomes that combine experimental and
observational data sets with this structure.

To fix ideas, consider Dynarski et al., (2021), who randomize grants of free
college tuition to a population of high achieving, low income high school
students. They estimate the average effects of these free tuition grants on
college application and enrollment rates. The long-term effects of free
tuition grants on college completion rates may be of more direct interest to
policy-makers considering the expansion of college aid programs. However, it
will take several years before college completion is observed for the cohort
of students in the experimental sample. Given an observational data set that
records college enrollment and completion for a comparable population of high
school students, the methods developed in this paper may facilitate a more
timely quantification of the effect of free tuition grants on college
completion.

We begin this section by defining the data structure and estimands that we
consider. We then review, and restate in a common notation, two sets of
closely related sets of identifying assumptions proposed by Athey et al.,
2020a  and Athey et al., 2020b . We refer to these settings as the _Latent
Unconfounded Treatment_ and _Statistical Surrogacy_ Models, respectively.

###  2.1. Data

Consider the collection of random variables

$\\{{A}_{i}\\}_{i=1}^{n}=\\{(Y_{i}(0),Y_{i}(1),{S}_{i}(0),S_{i}(1),W_{i},G_{i},{X}_{i})\\}_{i=1}^{n}$

consisting of the potential outcomes and characteristics of a sample of
individuals drawn independently and identically from a distribution
$P_{\star}$. Here, $G_{i}$ is a binary indicator denoting whether the
observation was acquired in the observational sample ($G_{i}=1$) or the
experimental sample ($G_{i}=0$). The variables $Y_{i}(\cdot)$ are long-term
potential outcomes, $S_{i}(\cdot)$ are short-term potential outcomes, $W_{i}$
is a binary treatment indicator, and $X_{i}$ are covariates.

The data observable to the researcher are denoted by $\\{{B}_{i}\\}_{i=1}^{n}$
and are i.i.d. according to a distribution denoted by $P$. The _observed_
outcomes $S_{i}$ and $Y_{i}$ are given by
$S_{i}=W_{i}S_{i}(1)+(1-W_{i})S_{i}(0)$ and
$Y_{i}=W_{i}Y_{i}(1)+(1-W_{i})Y_{i}(0),$ respectively. The short-term outcomes
$S_{i}$ are observed in both the observational and experimental data sets. The
long-term outcome $Y_{i}$ is observed only in the observational data set.
Treatment $W_{i}$ may or may not be observed in the observational sample.
Thus, the observable data $B_{i}$ are given by
$(G_{i}Y_{i},S_{i},W_{i},G_{i},X_{i})$ if treatment is measured in the
observational data set and by $(G_{i}Y_{i},S_{i},(1-G_{i})W_{i},G_{i},X_{i})$
if treatment is not measured in the observational data set.

###  2.2. Estimands

In the main text, we consider estimation of the long-term average treatment
effect in the observational population, given by

$\tau_{1}=\mathbb{E}_{P_{\star}}\left[Y_{i}(1)-Y_{i}(0)\mid
G_{i}=1\right]\leavevmode\nobreak\ .$ (2.1)

Athey et al., 2020a  note that there is often reason to believe that features
of the observational population ($G_{i}=1$) are more “externally valid,” in
that they of greater interest to policymakers.

In Appendix B, we give results analogous to those presented in the main text
for the long-term average treatment effect in the experimental population

$\tau_{0}=\mathbb{E}_{P_{\star}}\left[Y_{i}(1)-Y_{i}(0)\mid
G_{i}=0\right]\leavevmode\nobreak\ ,$ (2.2)

which may be of interest in some contexts.444An alternative estimand is the
unconditional long-term average treatment effect
$\tau=\mathbb{E}_{P_{\star}}\left[Y_{i}(1)-Y_{i}(0)\right]$. The practical
interpretation of this parameter is somewhat nebulous, as it is unclear why it
would be desirable to weight the two samples in the definition of the
parameter according to their sizes. Athey et al., 2020a  and Athey et al.,
2020b  consider estimation of $\tau_{1}$ and $\tau_{0}$, respectively.

###  2.3. Identifying Assumptions

We consider two sets of assumptions, proposed in Athey et al., 2020a  and
Athey et al., 2020b . In both models, the long-term average treatment effect
$\tau_{1}$ is identified. The models differ according to whether they are
applicable to contexts in which treatment is or is not measured in the
observational data set. Both models assume that treatment assignment is
unconfounded in the experimental data set and that the probabilities of being
assigned treatment or of being measured in the observational data set satisfy
a strict overlap condition.

###### Assumption 2.1 (Experimental Unconfounded Treatment).

In the experimental data set, treatment is independent of short-term and long-
term potential outcomes conditional on pre-treatment covariates, in the sense
that

$W_{i}\indep(Y_{i}(0),S_{i}(0),Y_{i}(1),S_{i}(1))\mid{X}_{i},G_{i}=0\leavevmode\nobreak\
.$

###### Assumption 2.2 (Strict Overlap).

The probability of being assigned to treatment or of being measured in the
observational data set is strictly bounded away from zero and one, in the
sense that, for each $w$ and $g$ in $\\{0,1\\}$, the conditional probabilities

$P(W=w\mid S,X,G=g)\quad\text{and}\quad P(G=g\mid S,X,W=w)$

are bounded between $\varepsilon$ and $1-\varepsilon$, $\lambda$-almost
surely, for some fixed constant $0<\varepsilon<1/2$.

Unconfounded treatment and strict overlap are often satisfied in the
experimental sample by design. Assessing and accounting for violations of
overlap in observational data are important empirical and methodological
issues (Crump et al.,, 2009). In particular, strong overlap conditions can
place stringent restrictions on the data generating process when there are
many covariates or short-term outcomes (D’Amour et al.,, 2021). We view
systematic consideration of these issues in our context as an important area
for further research.

In addition, as our aim is to use the experimental sample to estimate a
feature of the observational population, we require an assumption limiting the
differences between the two populations.

###### Assumption 2.3 (Experimental Conditional External Validity).

The distribution of the potential outcomes is invariant to whether the data
belong to the experimental or observational data sets, in the sense that

$G_{i}\indep\left(Y_{i}(1),Y_{i}(0),S_{i}(1),S_{i}(0)\right)\mid{X}_{i}\leavevmode\nobreak\
.$

Assumption 2.3 implies that adjustments to the distribution of covariates in
the experimental data set are sufficient to obtain approximations to features
of the observational population, thereby ruling out unobserved systematic
differences between the two populations conditional on covariates.

####  2.3.1. Latent Unconfounded Treatment

If treatment is measured in the observational data set, then the key
identifying assumption is that treatment assignment is unconfounded with
respect to the long-term outcome if conditioned on the short-term potential
outcomes. We term this restriction “Latent Unconfounded Treatment” following
Athey et al., 2020a .

###### Assumption 2.4 (Observational Latent Unconfounded Treatment).

In the observational data set, treatment is independent of the long-term
potential outcomes conditional on the short-term potential outcomes and pre-
treatment covariates, in the sense that, for $w\in\\{0,1\\}$,

$W_{i}\indep Y_{i}(w)\mid S_{i}(w),{X}_{i},G_{i}=1\leavevmode\nobreak\ .$

Informally, Assumption 2.4 states that all unobserved confounding in the
observational sample is mediated through the short-term outcomes. Assumptions
2.1, 2.2, 2.3, and 2.4 are sufficient for identification of the long-term
treatment effect $\tau_{1}$. We summarize these assumptions with the following
shorthand.

######  Definition 2.1.

The collection of Assumptions 2.1, 2.2, 2.3, and 2.4, in a setting where
treatment is measured in the observational data set, is referred to as the
Latent Unconfounded Treatment Model.

Panel A of Figure 1 displays a causal Directed Acyclic Graph (DAG) that is
consistent with the restrictions that the Latent Unconfounded Treatment Model
place on the data generating process for the observational data set.555See
Pearl, (1995) for further discussion of the applications of manipulation of
graphical models to causal inference. We do not make use of do-calculus
methodology to establish identification. The following proposition is stated
as Theorem 1 in Athey et al., 2020a . We state and prove the result for
completeness.

###### Proposition 2.1 (Athey et al., 2020a ).

Under the Latent Unconfounded Treatment Model, the long-term average treatment
effect $\tau_{1}$ is point identified.

Figure 1. Causal DAGs Consistent with Assumptions on Observational Data

Panel A: Latent Unconfounded Treatment | Panel B: Statistical Surrogacy
---|---
WSYXX | WSYXXX

Notes: Panels A and B of Figure 1 display causal DAGs that describe the
restrictions on the data generating process for the observational data set
($G=1$) implied by the Latent Unconfounded Treatment and Statistical Surrogacy
Models, respectively. Light grey arrows denote the existence of an effect of
the tail variable on the head variable. Dark red arrows with x’s denote that
an effect of the the tail variable on the head variable is ruled out. Dashed
bidirectional arrows represent existence of some unobserved common causal
variable $U$, where we have a fork $\leftarrow U\rightarrow$.

####  2.3.2. Statistical Surrogacy

If treatment is not measured in the observational data set, an alternative
“Statistical Surrogacy” assumption, in the spirit of Prentice, (1989), is
required in the place of Assumption 2.4. Under this restriction, the short-
term outcomes can be interpreted as “proxies” or “surrogates” for the long-
term outcome.

###### Assumption 2.5 (Experimental Statistical Surrogacy).

In the experimental data set, treatment is independent of the long-term
observed outcomes conditional on the short-term observed outcomes and pre-
treatment covariates, in the sense that

$W_{i}\indep Y_{i}\mid S_{i},{X}_{i},G_{i}=0\leavevmode\nobreak\ .$

Informally, Assumption 2.5 additionally rules out a causal link from treatment
to the long-term outcome that is not mediated by the short-term
outcomes.666Athey et al., 2020b  provide a procedure for bounding the bias in
estimates of $\tau_{0}$ if Assumption 2.5 is violated under certain functional
form restrictions. We see the extension of these methods as a useful direction
for future research.

In addition to Assumption 2.3, a supplementary restriction is required to
ensure that the experimental and observational data sets are suitably
comparable, conditional on the observed outcomes.

###### Assumption 2.6 (Long-Term Outcome Comparability).

The distribution of the long-term outcome is invariant to whether the data
belong to the experimental or observational data sets conditional on the
short-term outcome and covariates, in the sense that

$G_{i}\indep Y_{i}\mid{X}_{i},S_{i}\leavevmode\nobreak\ .$

Assumption 2.3 is not strictly stronger than Assumption 2.6, as belonging to
the experimental or observational data sets is not necessarily independent of
treatment assignment. Statistical Surrogacy, in addition to Assumptions 2.1,
2.2, 2.6, and 2.3, is sufficient for identification of the long-term treatment
effect $\tau_{1}$ in settings where treatment is not measured in the
observational data set. Again, we summarize these conditions with the
following shorthand.

######  Definition 2.2.

The collection of Assumptions 2.1, 2.2, 2.6, 2.3, and 2.5, in a setting where
treatment is not measured in the observational data set, is referred to as the
Statistical Surrogacy Model.

Panel B of Figure 1 describes the restrictions that the Statistical Surrogacy
Model place on the data generating process for the observational data set. The
following proposition restates Theorem 1 of Athey et al., 2020b . Again, we
state and prove the result for completeness.777We note that Assumption 2.3 is
unnecessary for the identification of $\tau_{0}$ in the Statistical Surrogacy
Model.

###### Proposition 2.2 (Athey et al., 2020b ).

Under the Statistical Surrogacy Model, $\tau_{1}$ is point identified.

##  3\. Semiparametric Efficiency

In this section, we derive efficient influence functions and corresponding
semiparametric efficiency bounds for estimation of long-term average treatment
effects $\tau_{1}$ in observational populations. In Appendix B, we state
comparable results for long-term average treatment effects $\tau_{0}$ in
experimental populations.

###  3.1. Nuisance Functions

The efficient influence functions that we derive are expressed in terms of a
set of unknown, but identified, conditional expectations. We classify each of
these objects as being either a “long-term outcome mean” or a “propensity
score.” Each long-term outcome mean is an expectation of the long-term outcome
conditioned on other features of the data. Under the Latent Unconfounded
Treatment Model, where treatment is measured in the observational data set,
the long-term outcome means that appear in the efficient influence function
are given by

$\displaystyle\mu_{w}(s,x)$ $\displaystyle=\mathbb{E}_{P}[Y\mid
W=w,S=s,X=x,G=1]\quad\text{and}$ (3.1) $\displaystyle\overline{\mu}_{w}(x)$
$\displaystyle=\mathbb{E}_{P}[\mu_{w}(S,X)\mid
W=w,X=x,G=0]\leavevmode\nobreak\ .$ (3.2)

The function $\mu_{w}(s,x)$ is the mean of the long term outcomes in the
observational sample, conditioned on treatment $w$, the short-term outcomes
$s$, and the covariates $x$. The function $\overline{\mu}_{w}(x)$ is the
projection of $\mu_{w}(s,x)$ onto the experimental population conditional on
$w$ and $x$, integrated over $s$.

In turn, under the Statistical Surrogacy model, where treatment is not
measured in the observational data set, the analogous nuisance functions
appearing in the efficient influence function are given by

$\displaystyle\nu(s,x)$ $\displaystyle=\mathbb{E}_{P}[Y\mid
S=s,X=x,G=1]\quad\text{and}$ (3.3) $\displaystyle\overline{\nu}_{w}(x)$
$\displaystyle=\mathbb{E}_{P}[\nu(S,X)\mid W=w,X=x,G=0]\leavevmode\nobreak\ .$
(3.4)

The function $\nu(s,x)$ is similar to $\mu_{w}(s,x)$, but does not condition
on treatment, as treatment is not observed in the observational sample under
the Statistical Surrogacy Model. The function $\overline{\nu}_{w}(x)$ projects
$\nu(s,x)$ on the observational sample, conditional on $x$ and $w$, and
integrates over $s$.

It useful to note that the proofs of Propositions 2.1 and 2.2 operate by
establishing, in their respective models, that the equalities

$\mathbb{E}_{P_{*}}\left[Y(1)\mid
G=1\right]=\mathbb{E}_{P_{*}}\left[\overline{\mu}_{1}(X)\mid
G=1\right]\quad\text{and}\quad\mathbb{E}_{P_{*}}\left[Y(1)\mid
G=1\right]=\mathbb{E}_{P_{*}}\left[\overline{\nu}_{1}(X)\mid G=1\right]$

hold and that the objects (3.2) and (3.4) are identified from the observable
data. Section 4.2 presents a set of alternative moment conditions that
analogously identify $\tau_{1}$.

Each member of the second class of nuisance functions—propensity
scores—expresses either the probability of treatment or the probability of
inclusion in the observational sample conditioned on other features of the
data. In particular, let

$\displaystyle\rho_{w}(s,x)$ $\displaystyle=P_{\star}(W=w\mid
S(w)=s,X=x,G=1)\leavevmode\nobreak\ ,$ (3.5) $\displaystyle\varrho(s,x)$
$\displaystyle=P(W=1\mid S=s,X=x,G=0)\leavevmode\nobreak\ ,\quad\text{and}$
(3.6) $\displaystyle\varrho(x)$ $\displaystyle=P(W=1\mid X=x,G=0)$ (3.7)

denote the probabilities of treatment conditional on various features of the
data. Similarly, let

$\displaystyle\gamma(s,x)$ $\displaystyle=P(G=1\mid
S=s,X=x)\leavevmode\nobreak\ ,\gamma(x)=P(G=1\mid X=x)\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\pi=P(G=1)$ (3.8)

denote the probabilities of inclusion in the observational sample conditional
on various features of the data.888 Athey et al., 2020b  term $\nu(s,x)$ the
_surrogate index_ , $\varrho(s,x)$ the _surrogate score_ , and $1-\gamma(s,x)$
the _sampling score_. Each of these objects is bounded away from zero and one
$\lambda$-almost surely by strict overlap (Assumption 2.2). Although the
propensity score $\rho_{w}(s,x)$ includes the unobserved random variable
$S(w)$ in its conditioning set, we may write $\rho_{w}(s,x)$ in terms of
observables as

$\displaystyle\rho_{w}(s,x)$ $\displaystyle=\frac{P(G=1\mid
S=s,W=w,X=x)}{P(G=0\mid S=s,W=w,X=x)}$
$\displaystyle\quad\times\frac{P(G=0\mid W=w,X=x)}{P(G=1\mid
W=w,X=x)}P(W=w\mid X=x,G=1)$ (3.9)

by successive applications of Bayes’ rule, Assumption 2.1, and Assumption 2.3.

###  3.2. Influence Functions and Efficiency Bounds

We characterize efficient influence functions for $\tau_{1}$ with the approach
developed in Section 3.4 of Bickel et al., (1993). In particular, we
characterize the tangent spaces for the classes of distributions restricted by
the maintained assumptions. We then verify that the conjectured efficient
influence functions are pathwise derivatives of $\tau_{1}$ and are elements of
their respective tangent spaces. Recall that the Latent Unconfounded Treatment
Model and Statistical Surrogacy Model differ according to the assumptions they
impose on the data generating distribution $P_{\star}$ in addition to whether
treatment is assumed to have been measured in the observational sample.

###### Theorem 3.1.

Let $b=(y,s,w,g,x)$ denote a possible value for the observed data.

  1. (1)

Under the Latent Unconfounded Treatment Model, given in 2.1, the efficient
influence function for the parameter $\tau_{1}$ is given by

$\displaystyle\psi_{1}(b,\tau_{1},\eta)$
$\displaystyle=\frac{g}{\pi}\left(\frac{w(y-\mu_{1}(s,x))}{\rho_{1}(s,x)}-\frac{(1-w)(y-\mu_{0}(s,x))}{\rho_{0}(s,x)}+(\overline{\mu}_{1}(x)-\overline{\mu}_{0}(x))-\tau_{1}\right)$
(3.10)
$\displaystyle+\frac{1-g}{\pi}\left(\frac{\gamma(x)}{1-\gamma(x)}\left(\frac{w(\mu_{1}(s,x)-\overline{\mu}_{1}(x))}{\varrho(x)}-\frac{(1-w)(\mu_{0}(s,x)-\overline{\mu}_{0}(x))}{1-\varrho(x)}\right)\right),$

where the parameter $\eta$ collects the nuisance functions appearing in
(3.10).

  2. (2)

Under the Statistical Surrogacy Model, given in 2.2, the efficient influence
function for the parameter $\tau_{1}$ is given by

$\displaystyle\xi_{1}(b,\tau_{1},\varphi)=\frac{g}{\pi}\left(\frac{\gamma(x)}{\gamma(s,x)}\frac{1-\gamma(s,x)}{1-\gamma(x)}\frac{(\varrho(s,x)-\varrho(x))(y-\nu(s,x))}{\varrho(x)(1-\varrho(x))}+(\overline{\nu}_{1}(x)-\overline{\nu}_{0}(x))-\tau_{1}\right)$
$\displaystyle\quad\quad\quad+\frac{1-g}{\pi}\left(\frac{\gamma(x)}{1-\gamma(x)}\left(\frac{w(\nu(s,x)-\overline{\nu}_{1}(x))}{\varrho(x)}-\frac{(1-w)(\nu(s,x)-\overline{\nu}_{0}(x))}{1-\varrho(x)}\right)\right)\leavevmode\nobreak\
,$ (3.11)

where the parameter $\varphi$ collects nuisance functions appearing in
(3.11).999We thank Rahul Singh for noting a typo in the statement of
$\xi_{1}(\cdot)$ in a previous draft of this paper, which was missing the
factor $\gamma(x)/(1-\gamma(x))$ in the second term.

######  Remark 3.1.

The efficient influence functions $\psi_{1}(b,\tau_{1},\eta)$ and
$\xi_{1}(b,\tau_{1},\varphi)$ derived in Theorem 3.1 are additively separable
into two terms associated with the observational and experimental data sets,
respectively. The structure of the terms associated with the observational
data set resembles the structure of the efficient influence function for the
average treatment effect under unconfoundedness (Hahn,, 1998); we discuss the
relationship between these objects in Section 4.1.1.

######  Remark 3.2.

The efficient influence functions $\psi_{1}(b,\tau_{1},\eta)$ and
$\xi_{1}(b,\tau_{1},\varphi)$ possess a “double-robust” structure that is
prevalent in causal inference and missing data problems (see e.g., Kang and
Schafer,, 2007; Bang and Robins,, 2005). In particular, the mean-zero property
of $\psi_{1}(b,\tau_{1},\eta)$ is maintained even if some of the nuisance
functions are misspecified. That is, if we partition the nuisance function
$\eta$ into long-term outcome means, propensity scores, and $\pi$ by
$\eta=(\omega,\kappa,\pi)$ and let the arbitrary measurable functions
$\tilde{\omega}$ replace the long-term outcome means $\omega$, then
$\mathbb{E}_{P}\left[{\psi_{1}(B,\tau_{1},(\tilde{\omega},\kappa,\pi))}\right]=0.$
Similarly, if the arbitrary measurable functions $\tilde{\kappa}$ replace the
propensity scores $\kappa$, then we have that
$\mathbb{E}_{P}\left[{\psi_{1}(B,\tau_{1},(\omega,\tilde{\kappa},\pi))}\right]=0.$
The efficient influence function $\xi_{1}(b,\tau_{1},\varphi)$ is similarly
robust to misspecification of either the long-term outcome means or the
propensity scores. This result echoes an analogous double-robustness property
of the efficient influence function of the average treatment effect under
ignorability (Scharfstein et al.,, 1999), in which the efficient influence
function is mean zero under misspecification of either the conditional means
of the outcome variable or the propensity score. The form of the double-
robustness entailed here is slightly more general, as $\omega$ and $\kappa$
each collect several nuisance function.

Double robustness, in this form, has a useful implication for estimation. We
demonstrate in Section C.1 that, under appropriate regularity conditions,
estimators based on the efficient influence functions
$\psi_{1}(b,\tau_{1},\eta)$ or $\xi_{1}(b,\tau_{1},\varphi)$ are consistent
for $\tau_{1}$ if _either_ the outcome means $\omega$ or the propensity scores
$\kappa$ are estimated consistently. We analyze estimators of this form in
further detail in Section 4.

The population variance of the efficient influence function is the
semiparametric efficiency bound. The respective bounds are presented in 3.1.

###### Corollary 3.1.

Define the conditional variances

$\displaystyle\sigma_{w}^{2}(s,x)$
$\displaystyle=\mathbb{E}_{P_{\star}}[(Y(w)-\mu_{w}(S,X))^{2}\mid
S=s,X=x]\quad\text{and}$ $\displaystyle\sigma^{2}(s,x)$
$\displaystyle=\mathbb{E}_{P_{\star}}[(Y-\nu(S,X))^{2}\mid S=s,X=x]$

as well as the expressions

$\displaystyle\Gamma_{w,1}(s,x)$
$\displaystyle=\frac{\gamma(x)}{1-\gamma(x)}\frac{\left(\mu_{w}(s,x)-\overline{\mu}_{w}(x)\right)^{2}}{\varrho(x)^{w}(1-\varrho(x))^{1-w}}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\Lambda_{w,1}(s,x)=\frac{\gamma(x)}{1-\gamma(x)}\frac{\left(\nu(s,x)-\overline{\nu}_{w}(x)\right)^{2}}{\varrho(x)^{w}(1-\varrho(x))^{1-w}}\leavevmode\nobreak\
.$

  1. (1)

Under the Latent Unconfounded Treatment Model, given in 2.1, the
semiparametric efficiency bound for $\tau_{1}$ is given by

$\displaystyle V_{1}^{\star}$
$\displaystyle=\mathbb{E}_{P}\Bigg{[}\frac{\gamma(X)}{\pi^{2}}\Bigg{(}\frac{\sigma_{1}^{2}(S,X)}{\rho_{1}(S,X)}+\frac{\sigma_{0}^{2}(S,X)}{\rho_{0}(S,X)}$
$\displaystyle\quad\quad\quad\quad\quad\quad+(\overline{\mu}_{1}(X)-\overline{\mu}_{0}(X)-\tau_{1})^{2}+\Gamma_{0,1}(S,X)+\Gamma_{1,1}(S,X)\Bigg{)}\Bigg{]}\leavevmode\nobreak\
.$ (3.12)

  2. (2)

Under the Statistical Surrogacy Model, given in 2.2, the semiparametric
efficiency bound for $\tau_{1}$ is given by

$\displaystyle V_{1}^{\star\star}$
$\displaystyle=\mathbb{E}_{P}\Bigg{[}\frac{\gamma(X)}{\pi^{2}}\Bigg{(}\left(\frac{\gamma(X)}{\gamma(S,X)}\frac{1-\gamma(S,X)}{1-\gamma(X)}\frac{\varrho(S,X)-\varrho(X)}{\varrho(X)(1-\varrho(X))}\right)^{2}\sigma^{2}(S,X)$
$\displaystyle\quad\quad\quad\quad\quad\quad+(\overline{\nu}_{1}(X)-\overline{\nu}_{0}(X)-\tau_{1})^{2}+\Lambda_{0,1}(S,X)+\Lambda_{1,1}(S,X)\Bigg{)}\Bigg{]}\leavevmode\nobreak\
.$ (3.13)

######  Remark 3.3.

In Section C.2, we analyze how the semiparametric efficiency bounds derived in
3.1 change if different components of the nuisance parameters $\eta$ or
$\varphi$ are known a priori. In both models, if the classical propensity
score $\varrho(X)$, i.e., the probability of being assigned to treatment in
the experimental sample as a function of covariates, is known, then the
semiparametric efficiency bounds are unchanged.101010Invariance of the
semiparametric efficiency bounds to knowledge of the propensity score
$\varrho(X)$ would no longer hold if the estimands of interest were average
long-term effects for the treated population. This echoes an analogous
ancillarity result for estimation of average treatment effects under
unconfoundedness given in Hahn, (1998).

By contrast, both semiparametric efficiency bounds change if the propensity
score $\gamma(x)$, i.e., the probability of being assigned to the
observational sample as a function of covariates, is known. This result is
relevant for settings where the experimental sample is known to be drawn from
the same population as the observational sample and indicates that development
of estimators tailored to this setting may be fruitful.111111On the other
hand, in that context, consideration of the unconditional long-term treatment
effect $\tau=\mathbb{E}[Y(1)-Y(0)]$ is tenable and natural. We expect the
efficiency bound for this functional to be invariant to knowledge of the
propensity score $\gamma(x)$. In the Statistical Surrogacy Model, somewhat
curiously, the efficient influence function $\xi_{1}(b,\tau_{1},\varphi)$ is
additionally invariant to knowledge of the distribution, in the observational
sample, of the short-term outcomes conditional on covariates, i.e. the law
$S\mid X,G=1$.

Next, we demonstrate that the efficient influence functions
$\psi_{1}(b,\tau_{1},\eta)$ and $\xi_{1}(b,\tau_{1},\varphi)$ expressed in
Theorem 3.1 are, in fact, the unique influence functions for $\tau_{1}$ in
their respective models. We recall that an influence for $\tau_{1}$ is any
mean-zero and square integrable function function $\tilde{\psi}(b)$ that
satisfies the condition

$\displaystyle\tau_{1}^{\prime}=\mathbb{E}_{P}[\tilde{\psi}(B)\ell^{\prime}(B)]\leavevmode\nobreak\
,$

where $\tau_{1}^{\prime}$ is the pathwise derivative of $\tau_{1}$ along an
arbitrary parametric submodel evaluated at zero and $\ell^{\prime}(B)$ is the
score function of this submodel; see e.g., Chapter 25 of Van der Vaart, (1998)
for further discussion.

###### Theorem 3.2.

There are unique influence functions in each model:

  1. (1)

Under the Latent Unconfounded Treatment Model, given in 2.1,
$\psi_{1}(b,\tau_{1},\eta)$ is the unique influence function for $\tau_{1}$.

  2. (2)

Under the Statistical Surrogacy Model, given in 2.2,
$\xi_{1}(b,\tau_{1},\varphi)$ is the unique influence function for $\tau_{1}$.

######  Remark 3.4.

Let $\mathcal{P}\subset\mathcal{M}_{\lambda}$ denote the set of probability
distributions that satisfy either the Latent Unconfounded Treatment Model or
the Statistical Surrogacy Model. In the terminology of Chen and Santos,
(2018), Theorem 3.2, Part (1), demonstrates that $P$ is locally just-
identified by $\mathcal{P}$. As a result, by Theorem 3.1 of Chen and Santos,
(2018), all regular and asymptotically linear (RAL) estimators of $\tau_{1}$
are first-order equivalent under the maintained assumptions. In particular,
there are no RAL estimators of $\tau_{1}$ that have smaller asymptotic
variances than others.

Consequently, semiparametric efficiency is equivalent to regularity and
asymptotic linearity under the maintained assumptions. We note that if the
propensity score $\varrho(x)$ admits known restrictions, then the resultant
model would be semiparametrically over-identified. In this case, not all RAL
estimators are first-order equivalent. However, since the efficiency bound
does not change, the estimators we propose in the following section remain
efficient with known propensity score.121212The semiparametric literature on
average treatment effect and local average treatment effect estimation (e.g.
Hirano et al.,, 2003; Frölich,, 2007) discusses efficiency as well, despite
the fact that the models in question are similarly just-identified.

Moreover, again by Theorem 3.1 of Chen and Santos, (2018), the model
$\mathcal{P}$ does not have any locally testable restrictions in the sense
that there are no specification tests of the maintained identifying
assumptions with nontrivial local asymptotic power. Analogous statements
follow from Theorem 3.2, Part (2). In that sense, the maintained identifying
assumptions are minimal.

##  4\. Estimation

We now consider estimation of the long-term average treatment effect
$\tau_{1}$.131313In Appendix B, we provide an analogous treatment of
estimators of the long-term average treatment effect $\tau_{0}$ in the
experimental population. The estimators that we consider can each be viewed as
semiparametric $Z$-estimators associated with an identifying moment function.
That is, each estimator is premised on determining the value $\tau_{1}$ that
solves a sample analogue of a moment condition

$\mathbb{E}_{P}\left[g(B_{i},\tau_{1},\zeta)\right]=0\leavevmode\nobreak\ ,$

for some identifying moment function $g(\cdot)$, where $\zeta$ is an unknown
nuisance parameter, replaced with its estimated counterpart in practice.

Our treatment differs by whether the identifying moment function $g(\cdot)$ is
given by the efficient influence functions $\psi_{1}(\cdot)$ or
$\xi_{1}(\cdot)$, derived in Theorem 3.1 or given by some other moment
condition. Moment conditions defined by influence functions are often referred
to as Neyman orthogonal moment conditions. We adapt very general arguments
from Chernozhukov et al., 2018a  to establish that these estimators are
consistent and asymptotically normal.

Our consideration of estimators based on non-orthogonal moments is selective
and is more specialized. To obtain theoretical guarantees, we restrict
attention to estimators that plug-in nuisance parameter estimates derived from
the method of sieves (Chen and Liao,, 2015; Chen et al.,, 2014). Sufficient
conditions for estimators with this structure are generally available, but are
more delicate and difficult to verify. We state and verify these conditions
for one of the estimators that we consider.141414The general high-level
conditions in Chen and Liao, (2015) and Chen et al., (2014) apply to each of
estimators that we consider. Lower-level conditions, analogous to those
discussed in Section 4.2 will exist for these estimators as well. However, as
the derivation of these conditions is lengthy and cumbersome, we provide an
illustration of this argument for only one estimator.

Throughout, it is useful to keep in mind that the efficient influence
functions $\psi_{1}(\cdot)$ and $\xi_{1}(\cdot)$ are the only influence
functions in their respective models (i.e., Theorem 3.2). Consequently, all
regular and asymptotically linear estimators are first-order equivalent, that
is, their asymptotic variances are all equal to the semiparametric efficiency
bound.151515Moreover, as each of the semiparametric efficiency bounds are
invariant to knowledge of the propensity score $\varrho(X)$, all regular and
asymptotically linear estimators in each model will achieve their respective
semiparametric efficiency bounds irrespective of whether $\varrho(X)$ is
known. Thus, the asymptotic variances of different estimators are the same,
but the conditions under which they are asymptotically normal may be
different.

###  4.1. Orthogonal Moments

We begin by considering estimators that are built directly on the influence
functions $\psi_{1}(\cdot)$ or $\xi_{1}(\cdot)$, derived in Theorem 3.1 with
the “Double/Debiased Machine Learning” (DML) construction developed in
Chernozhukov et al., 2018a .

####  4.1.1. Construction

The DML construction proceeds in two steps. First, estimates of the nuisance
functions $\eta$ or $\varphi$, defined in Theorem 3.1, are computed with
cross-fitting. Second, estimates of $\tau_{1}$ are obtained by plugging the
estimated values of $\eta$ and $\varphi$ into their respective efficient
influence functions and solving for the values of $\tau_{1}$ that equate the
sample means of these estimates of the efficient influence functions with
zero.

######  Definition 4.1 (DML Estimators).

Let $\hat{\eta}(I)$ and $\hat{\varphi}(I)$ denote generic estimates of $\eta$
and $\varphi$ based on the data $\\{B_{i}\\}_{i\in I}$ for some subset
$I\subseteq[n]$. Let $\\{I_{\ell}\\}_{\ell=1}^{k}$ denote a random $k$-fold
partition of $[n]$ such that the size of each fold is $m=n/k$. The estimator
$\hat{\tau}_{1,\mathsf{DML}}$ is defined as the solution to

$\frac{1}{k}\sum_{\ell=1}^{k}\frac{1}{m}\sum_{i\in
I_{\ell}}\psi_{1}(B_{i},\hat{\tau}_{1,\mathsf{DML}},\hat{\eta}(I_{\ell}^{c}))=0\quad\text{or}\quad\frac{1}{k}\sum_{\ell=1}^{k}\frac{1}{m}\sum_{i\in
I_{\ell}}\xi_{1}(B_{i},\hat{\tau}_{1,\mathsf{DML}},\hat{\varphi}(I_{\ell}^{c}))=0$

for the Latent Unconfounded Treatment and Statistical Surrogacy Models,
respectively.

######  Remark 4.1.

The fundamental structures underlying standard estimators of average treatment
effects under unconfoundedness can be classified as being based on either
“inverse propensity score weighting” or “outcome regression” (Imbens,, 2004);
elements of each structure appear in the estimators formulated in
4.1.161616Imbens, (2004) also discusses estimators based on matching and
Bayesian calculations. We do not develop estimators with these structures in
this paper, and view their consideration as an interesting extension. Inverse
propensity score weighted (IPW) estimators, also referred to as Horvitz and
Thompson, (1952) estimators, are constructed by weighting the observed values
of outcomes by their inverse propensity scores; see e.g., Rosenbaum and Rubin,
(1983) and Hirano et al., (2003). By contrast, outcome regression estimators
are constructed by imputing unobserved potential outcomes with estimates of
their expectation conditioned on covariates.

The estimators formulated in 4.1 combine IPW and outcome regression components
with an error-correcting structure comparable to the augmented inverse
propensity weighted (AIPW) estimator of Robins et al., (1995). To illustrate,
observe that $\psi_{1}(b,\tau_{1},\eta)$ can be interpreted as first
approximating $\tau_{1}$ with the outcome regression
$\overline{\mu}_{1}(x)-\overline{\mu}_{0}(x)$ in the observational sample.
Heuristically, the biases in this approximation, e.g., induced by
regularization, are then corrected by applying IPW to the residuals of the
approximation of $\overline{\mu}_{w}(x)$ to $\mu_{w}(s,x)$ with

$\frac{w(\mu_{1}(s,x)-\overline{\mu}_{1}(x))}{\varrho(x)}-\frac{(1-w)(\mu_{0}(s,x)-\overline{\mu}_{0}(x))}{1-\varrho(x)}$

computed in the experimental sample and reweighed by $\gamma(x)/(1-\gamma(x))$
to represent an expectation over the observational sample. However, the
correction above may introduce additional biases through the estimation of
$\mu_{w}(s,x)$; these are in turn corrected by applying IPW to the residuals
of the approximation of $\mu_{w}(s,x)$ to $Y(w)$ with

$\frac{w(y-\mu_{1}(s,x))}{\rho_{1}(s,x)}-\frac{(1-w)(y-\mu_{0}(s,x))}{\rho_{0}(s,x)}$

computed in the experimental sample. An analogous interpretation can be
formulated for the structure of the efficient influence function
$\xi_{1}(b,\tau_{1},\varphi)$.171717Note that in the term corresponding to the
observational sample in $\xi_{1}(\cdot)$, the unobserved treatment indictor
$w$ is replaced by the probability of treatment conditional on short-term
outcomes and covariates. Further discussion, at varying levels of rigor, of
this “bias-correction” interpretation of the structure of estimators based on
efficient influence functions is given in Section 4 of Kennedy, (2023) and
Chapter 7 of Bickel et al., (1993).

######  Remark 4.2.

At a high-level, the cross-fitting construction used in 4.1 is implemented so
that the estimation errors, e.g.,
$\mu_{w}(S_{i},X_{i})-\hat{\mu}_{w}(S_{i},X_{i})$, and model errors, e.g.,
$Y(W_{i})-\mu_{w}(S_{i},X_{i})$, are unrelated for a given observation.
Association between these two forms of error may have particularly pernicious
effects in finite-samples if estimates of nuisance functions suffer from over-
fitting. More technically, cross-fitting allows us to avoid imposing Donsker-
type regularity conditions in our asymptotic analysis, which would exclude
estimators with non-negligible asymptotic regularization. Standard
implementations of popular machine learning algorithms may feature such
regularization; see Chernozhukov et al., (2016) for detailed discussion and
illustration of this point. Further discussion of cross-fitting methods in
semiparametric estimation is given in Klaassen, (1987) and Newey and Robins,
(2018).

######  Remark 4.3.

We require specialized approaches to estimate the nuisance functions
$\rho_{w}(s,x)$, $\overline{\mu}_{w}(x)$, and $\overline{\nu}_{w}(x)$. We
estimate $\rho_{w}(s,x)$ by combining separate estimates of each of the
objects displayed in Equation 3.9. We estimate $\overline{\mu}_{w}(x)$ and
$\overline{\nu}_{w}(x)$ by first computing estimates of $\mu_{w}(s,x)$ and
$\nu(s,x)$ in the observational sample, denoted by $\hat{\mu}_{w}(s,x)$ and
$\hat{\nu}(s,x)$, and then computing estimates of
$\mathbb{E}_{P}[\hat{\mu}_{w}(s,x)|W=w,X=x,G=0]$ and
$\mathbb{E}_{P}[\hat{\nu}(s,x)|W=w,X=x,G=0]$ in the experimental sample. In
Section C.3, we derive the rate of convergence for particular implementations
of estimators with this structure based on linear sieves.

####  4.1.2. Large-Sample Theory

We now study the asymptotic behavior of the estimators formulated in
LABEL:def:dml. First, in Section C.1, we demonstrate that, under weak
regularity conditions and under both models, the estimator
$\hat{\tau}_{1,\mathsf{DML}}$ is consistent for $\tau_{1}$ if either the long-
term outcome means or the propensity scores are estimated consistently.
Second, we establish asymptotic normality by providing conditions sufficient
for the application of Theorem 3.1 of Chernozhukov et al., 2018a . We impose a
set of standard bounds on moments of the data, and a set of conditions on the
uniform rates of convergence of nuisance parameter estimators. Throughout, for
a collection of scalar-valued nuisance parameters
$\theta=(\theta_{1},\ldots,\theta_{\ell})$, we let
$\|\theta\|_{P,q}=\max_{i\in[\ell]}\left({\mathbb{E}_{P}|\theta_{i}(B)|^{q}}\right)^{1/q}$.

###### Assumption 4.1 (Moment Bounds).

Let $C,c>0$ be constants. Under the Latent Unconfounded Treatment Model, the
moment bounds

$\displaystyle\|Y(w)\|_{P,q}\leq C,$
$\displaystyle\quad\mathbb{E}_{P}\left[\sigma^{2}_{w}(S,X)\mid X\right]\leq
C,$ $\displaystyle\mathbb{E}_{P}\left[(Y(w)-\mu_{w}(S,X))^{2}\right]\geq c,$
$\displaystyle\quad\text{ and }\quad
c\leq\mathbb{E}_{P}\left[(\mu_{w}(S,X)-\overline{\mu}_{w}(x))^{2}\mid
X\right]\leq C$

hold for each $w\in\\{0,1\\}$ and $\lambda$-almost every $X$. Analogous bounds
hold for the Statistical Surrogacy Model, where $\sigma(S,X)$, $\nu(S,X)$, and
$\overline{\nu}_{w}(x)$ replace $\sigma_{w}(S,X)$, $\mu_{w}(S,X)$, and
$\overline{\mu}_{w}(x)$, respectively.

###### Assumption 4.2 (Convergence Rates).

Let $\mathcal{P}\subset\mathcal{M}_{\lambda}$ be the set of all probability
distributions $P$ that satisfy the Latent Unconfounded Treatment Model stated
in 2.1. Partition the collection of nuisance functions $\eta$ appearing as an
argument in $\psi_{1}(B,\tau_{1},\eta)$ into long-term outcome means,
propensity scores, and $\pi$ by $\eta=(\omega,\kappa,\pi)$. Consider a
sequence of estimators
$\hat{\eta}_{n}(I_{n}^{c})=(\hat{\omega}_{n},\hat{\kappa}_{n},\hat{\pi}_{n})$
indexed by $n$, where $I_{n}\subset[n]$ is a random subset of size $m=n/k$ and
$\hat{\pi}_{n}=\frac{1}{n-m}\sum_{i\in I_{n}^{c}}G_{i}$. For some sequences
$\Delta_{n}\to 0$ and $\delta_{n}\to 0$ and constants $\varepsilon,C>0$ and
$q>2$, that do not depend on $P$, with $P$-probability at least
$1-\Delta_{n}$,

  1. 1\. (Consistency in 2-norm)

$n^{-1/2}\leq\lVert\hat{\eta}_{n}-\eta\rVert_{P,2}\leq\delta_{n}$,

  2. 2\. (Boundedness in $q$-norm)

$\lVert\hat{\eta}_{n}-\eta\rVert_{P,q}<C$,

  3. 3\. (Non-degeneracy)

$\varepsilon\leq\hat{\kappa}_{n}\leq 1-\varepsilon$, where the inequalities
apply entry-wise, and

  4. 4\. ($o(n^{-1/2})$ product rates)

$\|\hat{\omega}_{n}-\omega\|_{P,2}\cdot\|\hat{\kappa}_{n}-\kappa\|_{P,2}\leq\delta_{n}n^{-1/2}$.

Analogous conditions hold for the Statistical Surrogacy Model, where $\varphi$
replaces $\eta$.

######  Remark 4.4.

Assumption 4.2 imposes the restriction that the product of the estimation
errors for the long-term outcome means and propensity scores converges at the
rate $o(n^{-1/2})$.181818If the true propensity score $\varrho(x)$ is known,
then using this information when constructing $\hat{\tau}_{1,\mathsf{DML}}$
may produce an estimator that performs well in finite-samples. However,
plugging the true propensity score into an estimator based off of a non-
orthogonal moment would probably be inefficient. For example, in Hahn, (1998),
an IPW type estimator for the average treatment effect based off of the true
propensity score is shown to be inefficient. These rates can be achieved, even
if the dimensionality of the covariates or the short-term outcomes is
increasing with $n$, by many standard machine learning algorithms including
the Lasso and Dantzig selector (Bickel et al.,, 2009; Belloni et al.,, 2014),
boosting algorithms (Luo and Spindler,, 2017), regression trees and random
forests (Wager and Walther,, 2015), and neural networks (Chen and White,,
1999; Farrell et al.,, 2021) under appropriate conditions on the structure or
sparsity of the underlying model.

In Section C.3, we verify that estimates of $\overline{\mu}_{w}(x)$ and
$\overline{\nu}_{w}(x)$ based on linear sieves can achieve these rates under
sufficiently stringent restrictions on the smoothness of the long-term outcome
means. Analysis of more complicated estimators of nuisance parameters of this
form, e.g., featuring penalization or more complicated bases, is an
interesting direction for further research.

Theorem 4.1 establishes the asymptotic properties of the estimators formulated
in 4.1. For the sake of brevity, we state the result only for the Latent
Unconfounded Treatment Model. An analogous result holds for the Statistical
Surrogacy Model. We provide proofs for both results in Section A.7.

###### Theorem 4.1.

Let $\mathcal{P}\subset\mathcal{M}_{\lambda}$ be the set of all probability
distributions $P$ for $\\{B_{i}\\}_{i=1}^{n}$ that satisfy the Latent
Unconfounded Treatment Model stated in 2.1 in addition to Assumption 4.1. If
LABEL:as:rates holds for $\mathcal{P}$, then

$\displaystyle\sqrt{n}(\hat{\tau}_{1,\mathsf{DML}}-\tau_{1})\overset{d}{\to}\mathcal{N}(0,V_{1}^{\star})$
(4.1)

uniformly over $P\in\mathcal{P}$, where $\hat{\tau}_{1,\mathsf{DML}}$ is
defined in 4.1, $V_{1}^{\star}$ is defined in 3.1, and $\overset{d}{\to}$
denotes convergence in distribution. Moreover, we have that

$\displaystyle\hat{V}_{1}^{\star}=\frac{1}{k}\sum_{\ell=1}^{k}\frac{1}{m}\sum_{i\in
I_{\ell}}\left(\psi_{1}(B_{i},\hat{\tau}_{1,\mathsf{DML}},\hat{\eta}(I_{\ell}^{c}))\right)^{2}\overset{p}{\to}V_{1}^{\star}$
(4.2)

uniformly over $P\in\mathcal{P}$, where $\overset{p}{\to}$ denotes convergence
in probability. As a result, we obtain the uniform asymptotic validity of the
confidence intervals

$\displaystyle\lim_{n\to\infty}\sup_{P\in\mathcal{P}}\Big{|}P\left(\tau_{1}\in\left[\hat{\tau}_{1,\mathsf{DML}}\pm
z_{1-\alpha/2}\sqrt{\hat{V}_{1}^{\star}/n}\right]\right)-(1-\alpha)\Big{|}=0\leavevmode\nobreak\
,$ (4.3)

where $z_{1-\alpha/2}$ is the $1-\alpha/2$ quantile of the standard normal
distribution.

###  4.2. Non-orthogonal Moments

The moment functions $\psi_{1}(\cdot)$ and $\xi_{1}(\cdot)$ considered in
Section 4.1 are quite complicated. Constructing the estimator
$\hat{\tau}_{1,\mathsf{DML}}$ requires estimating several propensity scores
and long-term outcome means. It is natural to ask whether it suffices to
consider simpler estimators based on moment conditions with fewer nuisance
functions.

In this section, we consider a suite of estimators that do not use the
efficient influence functions $\psi_{1}(\cdot)$ or $\xi_{1}(\cdot)$ as
identifying moment functions. We emphasize estimators that bear a similarity
to standard IPW or outcome regression estimators for average treatment
effects, several of which were initially proposed in Athey et al., 2020a  and
Athey et al., 2020b . We verify the asymptotic normality of one of these
estimators, when nuisance parameters are estimated with the method of sieves,
following ideas developed in (Chen and Liao,, 2015; Chen et al.,, 2014).

####  4.2.1. Construction

Each of the estimators that we consider can be viewed as a $Z$-estimator based
on a moment function $g(\cdot,\tau_{1},\zeta)$, taking as an argument an
unknown nuisance parameter $\zeta$.

######  Definition 4.2 (Non-orthogonal Moment Estimators).

The estimator $\hat{\tau}_{1}(g)$ is defined as the solution to the sample
moment condition

$\frac{1}{n}\sum_{i=1}^{n}g(B_{i},\hat{\tau}_{1}(g),\hat{\zeta})=0\leavevmode\nobreak\
,$

where $\hat{\zeta}$ is a generic estimate of $\zeta$ based on the data
$\\{B_{i}\\}_{i=1}^{n}$.

We consider two classes of moment functions, differing in whether or not the
estimators that they entail resemble IPW or outcome regression estimators for
average treatment effects. In the Latent Unconfounded Treatment Model, the
random variable

$\displaystyle g_{\mathsf{w}}(B_{i},\tau_{1},\zeta_{\mathsf{w}})$
$\displaystyle=\frac{G_{i}}{\pi}\left(\frac{W_{i}Y_{i}}{\rho_{1}(S_{i},X_{i})}-\frac{(1-W_{i})Y_{i}}{\rho_{0}(S_{i},X_{i})}\right)-\tau_{1}\leavevmode\nobreak\
,$ (4.4)

where $\zeta_{\mathsf{w}}=(\pi,\rho_{1},\rho_{0})$, has mean zero under $P$.
The estimator $\hat{\tau}_{1}(g_{\mathsf{w}})$, proposed originally by Athey
et al., 2020a , only requires estimation of the nuisance parameters in
$\zeta_{\mathsf{w}}$ and can be viewed as an analogue to the IPW estimator for
average treatment effects. In turn, the functions

$\displaystyle g_{\mathsf{or,1}}(B_{i},\tau_{1},\zeta_{\mathsf{or,1}})$
$\displaystyle=\frac{G_{i}}{\pi}\left(\overline{\mu}_{1}(X_{i})-\overline{\mu}_{0}(X_{i})\right)\quad\text{and}$
(4.5) $\displaystyle g_{\mathsf{or,0}}(B_{i},\tau_{1},\zeta_{\mathsf{or,0}})$
$\displaystyle=\frac{1-G_{i}}{\pi}\frac{\gamma(X_{i})}{1-\gamma(X_{i})}\left(\mu_{1}(S_{i},X_{i})-\mu_{0}(S_{i},X_{i})\right)\leavevmode\nobreak\
,$ (4.6)

where $\zeta_{\mathsf{or,1}}$ and $\zeta_{\mathsf{or,0}}$ collect nuisance
parameters, yield the outcome regression type estimators
$\hat{\tau}_{1}(g_{\mathsf{or,1}})$ and $\hat{\tau}_{1}(g_{\mathsf{or,0}})$.
The estimator $\hat{\tau}_{1}(g_{\mathsf{or,0}})$ was originally proposed by
Athey et al., 2020a .

Analogously, in the Statistical Surrogacy Model, the moment function

$\displaystyle h_{\mathsf{w}}(B_{i},\tau_{1},\varsigma_{\mathsf{w}})$
$\displaystyle=\frac{G_{i}Y_{i}}{\pi}\frac{\gamma(X_{i})}{1-\gamma(X_{i})}\frac{1-\gamma(S_{i},X_{i})}{\gamma(S_{i},X_{i})}\left({\frac{\varrho(S_{i},X_{i})}{\varrho(X_{i})}-\frac{1-\varrho(S_{i},X_{i})}{1-\varrho(X_{i})}}\right)-\tau_{1}\leavevmode\nobreak\
,$ (4.7)

where $\varsigma_{\mathsf{w}}$ collects nuisance parameters, yields an IPW-
type estimator $\hat{\tau}_{1}(h_{\mathsf{w}})$ that is similar to an
estimator proposed by Athey et al., 2020b . The moment functions

$\displaystyle h_{\mathsf{or,1}}(B_{i},\tau_{1},\varsigma_{\mathsf{or,1}})$
$\displaystyle=\frac{G_{i}}{\pi}\left(\overline{\nu}_{1}(X_{i})-\overline{\nu}_{0}(X_{i})\right)\quad\text{and}$
(4.8) $\displaystyle
h_{\mathsf{or,0}}(B_{i},\tau_{1},\varsigma_{\mathsf{or,0}})$
$\displaystyle=\frac{1-G_{i}}{\pi}\frac{\gamma(X_{i})}{1-\gamma(X_{i})}\left(\frac{W_{i}}{\varrho(X_{i})}-\frac{1-W_{i}}{1-\varrho(X_{i})}\right)\nu(S_{i},X_{i})\leavevmode\nobreak\
,$ (4.9)

result in the outcome regression type estimators
$\hat{\tau}_{1}(h_{\mathsf{or,1}})$ and $\hat{\tau}_{1}(h_{\mathsf{or,0}})$,
respectively. The estimator $\hat{\tau}_{1}(h_{\mathsf{or,0}})$ was originally
proposed by Athey et al., 2020b . Dynarski et al., (2021) use an estimator
closely related to $\hat{\tau}_{1}(h_{\mathsf{or,0}})$ with an estimate of
$\nu(s,x)$ based on linear regression, in their analysis of the effects of
college tuition grants on college complteion rates.

####  4.2.2. Large-Sample Theory

Theoretical analysis of the large-sample performance of estimators based on
non-orthogonal moment conditions requires a more specialized treatment.
Sufficient conditions for their asymptotic normality in the literature are
often more delicate and stronger than those for the DML estimators considered
in Section 4.1. We provide details of this analysis for the estimator
$\hat{\tau}_{1}(g_{\mathsf{w}})$ only, as stating and verifying sufficient
conditions for asymptotic linearity is cumbersome. Nevertheless, the basic
structure of the conditions that we pose, and the method of their
verification, is applicable to each of the estimators formulated above.

Constructing the estimator $\hat{\tau}_{1}(g_{\mathsf{w}})$ requires an
estimate of the nuisance parameter $\zeta_{\mathsf{w}}$. We restrict attention
to procedures that estimate the remaining components of $\zeta_{\mathsf{w}}$,
i.e., $\rho_{1}$ and $\rho_{0}$, with linear sieves. We detail this procedure
in Section A.8.

Theorem 4.2 establishes the asymptotic normality of the estimator
$\hat{\tau}_{1}(g_{\mathsf{w}})$ when the nuisance parameter
$\zeta_{\mathsf{w}}$ is estimated with the method of sieves. Stating the
precise sufficient conditions for this Theorem requires additional notation
and definitions, which we defer to Section A.8.

###### Theorem 4.2.

Under the Latent Unconfounded Treatment Model and Assumption A.3 stated in
Section A.8. we have that

$\sqrt{n}(\hat{\tau}_{1}(g_{\mathsf{w}})-\tau_{1})\overset{d}{\to}\mathcal{N}(0,V_{1}^{\star})\leavevmode\nobreak\
,$

where $V_{1}^{\star}$ is the semiparametric efficiency bound for $\tau_{1}$
defined in 3.1.

We again emphasize that, by Theorem 3.2, _any_ regular and asymptotically
linear estimator $\hat{\tau}_{1}$ for $\tau_{1}$ achieves the semiparametric
efficiency bound $V_{1}^{\star}$, if the assumptions that define the Latent
Unconfounded Treatment Model are the _only_ set of restrictions that are
imposed. Thus, the semiparametric efficiency of
$\hat{\tau}_{1}(g_{\mathsf{w}})$, per se, is to be expected. The substance of
Theorem 4.2 is the asymptotic linearity.

######  Remark 4.5.

Informally, Assumption A.3 requires that:

  1. (1)

$\hat{\zeta}_{\mathsf{w}}$ is $o(n^{-1/4})$-consistent;

  2. (2)

The parameter space containing $\zeta_{\mathsf{w}}$ is Donsker; and

  3. (3)

The sieve space chosen to approximate $\zeta_{\mathsf{w}}$ has limited
complexity.

Condition (1) is analogous to the product rate condition in Assumption 4.2. It
is weaker in the sense that no consistent estimators for the nuisance
parameters in $\eta$ that are not in $\zeta_{\mathsf{w}}$ are needed. On the
other hand, Condition (1) places more stringent conditions on the rate that
$\hat{\zeta}_{\mathsf{w}}$ estimates $\zeta_{\mathsf{w}}$. In particular,
Assumption 4.2 will still hold in situations where some elements of
$\zeta_{\mathsf{w}}$ are estimated at a rate slower than $o(n^{-1/4})$, so
long as the product of the errors in estimation of the long-term outcome means
and propensity scores is smaller than $o(n^{-1/2})$.191919Imbens, (2004)
outlines similar heuristics for average treatment effect estimation under
unconfoundedness. Condition (2) is imposed in order to ensure stochastic
equicontinuity for the moment condition $\zeta_{\mathsf{w}}\mapsto
g_{\mathsf{w}}(\cdot,\tau_{1},\zeta_{\mathsf{w}})$ treated as a process
indexed by $\zeta_{\mathsf{w}}$. This condition ensures that estimating
$\tau_{1}$ and $\zeta_{\mathsf{w}}$ using the same data does not induce errors
that are excessively large. Using sample-splitting would eliminate the need
for this condition. Condition (3) is specific to the sieve approach for
estimating nuisance parameters. It is not directly imposed in Assumption 4.2,
but may be needed to justify rate conditions when one estimates nuisance
parameters with sieves.

##  5\. Simulation

We now compare the estimators formulated in Section 4 with a simulation
calibrated to data from Banerjee et al., (2015). We find that the DML
estimators considered in Section 4.1 are more accurate than the estimators
based on non-orthogonal moments considered in Section 4.2, particularly if a
nonparametric approach is taken to nuisance parameter estimation.

###  5.1. Data, Calibration, and Design

Banerjee et al., (2015) study randomized evaluations of several similar
poverty-alleviation programs implemented by BRAC, a large non-governmental
organization. These programs allocated productive assets (typically livestock)
to participating households and measured both short-term and long-term
economic outcomes.

We restrict our attention to data from the evaluation of the program
implemented in Pakistan. For each of the $854$ households in our cleaned
sample, survey measurements of the consumption levels, food security, assets,
savings, and outstanding loans were taken prior to, as well as two and three
years after, treatment. We use the pre-treatment measurements as covariates
(i.e., $X_{i}$), the two-year post-treatment measurements as short-term
outcomes (i.e., $S_{i}$), and the three-year post-treatment measurements as
long-term outcomes (i.e., $Y_{i}$). There are $20$ pre-treatment covariates
and $21$ short-term outcomes. In the main text, the long-term outcome of
interest is total household assets; we give analogous results for total
household consumption in Section D.5. Section D.1 gives further information on
the construction and content of these data.

We calibrate a generative model to these data with a Generative Adversarial
Network (Goodfellow et al.,, 2014), following a method for simulation design
developed in Athey et al., (2021). The details of this calibration are given
in Section D.2.202020In Section D.2, we demonstrate that the joint
distribution of data drawn from this model matches the leading moments of
joint distribution of the data from Banerjee et al., (2015) remarkably
closely. Crucially, a sample drawn from this model consists of covariates and
both treated and untreated short-term and long-term potential outcomes for a
hypothetical household (the vector
$(X_{i},S_{i}(1),S_{i}(0),Y_{i}(1),Y_{i}(0))$ in our notation). That is, we
observe the true short-term and long-term treatment effects for each household
sampled from this model, and can measure true long-term average treatment
effects by averaging over many simulation draws.

With this calibrated model, we generate a collection of hypothetical data sets
that satisfy either the Latent Unconfounded Treatment Model or the Statistical
Surrogacy Model, as desired. The quality of various estimators is then
determined by measuring their average accuracy in recovering long-term average
treatment effects in a variety of metrics. To generate a hypothetical data
set, we draw a collection of samples from the generative mode of size $h\cdot
n$, where $h$ is some multiplier that we vary and $n$ is the sample size of
the Banerjee et al., (2015) data. Each observation is assigned to being either
“experimental” or “observational” with probability $1/2$.212121As a result,
there are no covariate differences in the observational or experimental
samples. Treatment for experimental samples is always assigned uniformly at
random. Treatment for observational samples is assigned with possible
confounding. Specifically, we determine treatment probabilities with an
increasing function, indexed by a parameter $\phi$, of each hypothetical
household’s true short-term treatment effects. Larger values of $\phi$
indicate more confounding. Details of this simulation design and
parameterization of confounding are given in Section D.3.

###  5.2. Comparison of Methods

We begin by comparing the estimators formulated in Section 4 with a simple
difference between the mean long-term treated and untreated outcomes in the
observational sample

$\hat{\tau}_{1,DM}=\frac{\sum_{i}G_{i}W_{i}Y_{i}}{\sum_{i}G_{i}W_{i}}-\frac{\sum_{i}G_{i}(1-W_{i})Y_{i}}{\sum_{i}G_{i}(1-W_{i})}\leavevmode\nobreak\
.$ (5.1)

This estimator is naive, making no adjustment for confounding in the
observational sample, and is infeasible if treatment is not observed in the
observational sample.

Figure 2 compares the absolute bias and root mean squared error of the
estimators formulated in Section 4 with the naive estimator
(5.1).222222Measurements of the variance of each estimator are displayed in
Figures D.9 and LABEL:fig:r3. We note that in the Latent Unconfounded
Treatment Model, one of the outcome regression estimators has very small
variance when nuisance parameters are estimated with linear regression.
However, the bias of this estimator is very large. We compare the use of the
Lasso (Tibshirani,, 1996), Generalized Random Forests (Athey et al.,, 2019),
and XGBoost (Chen and Guestrin,, 2016) for nuisance parameter estimation.
Details about the implementation of these estimators are given in Section D.4.
Panels A and B display results for the Latent Unconfounded Treatment and
Statistical Surrogacy Models, respectively. Columns within each panel vary the
sample size multiplier $h$. Each column displays results for the confounding
parameter $\phi$ set to zero, indicating no confounding in the observational
sample and labeled as “Baseline,” in addition to two parameterizations
indicating non-zero confounding.

Figure 2. Comparison of Estimators

Panel A: Latent Unconfounded Treatment
---
Panel B: Statistical Surrogacy

Notes: Figure 2 compares measurements of the absolute bias and root mean
squared error for the estimators formulated in Section 4, in addition to the
difference in means estimator defined in (5.1). The y-axes are displayed in
logs, base 10. The long-term outcome is total household assets. Panels A and B
display results for the Latent Unconfounded Treatment and Statistical
Surrogacy Models defined in 2.1 and 2.2, respectively. The columns of each
panel vary the sample size multiplier $h$. Each sub-panel displays results for
the baseline, unconfounded, case, as well as for the cases that the
confounding parameter $\phi$ has been set to $1/2$ and $2/3$. Results for each
estimator are displayed with dots of different colors. Results for different
nuisance parameter estimators are displayed with dots of different shapes.

In both the Latent Unconfounded Treatment Model and the Statistical Surrogacy
Model, the biases of the DML estimators considered in Section 4.1 tend to be
substantially smaller than the biases of the alternative outcome regression or
weighting estimators considered in Section 4.2, so long as the nuisance
parameters are not estimated by linear regression. The weighting estimator has
performance more comparable to the DML estimator in the Latent Unconfounded
Treatment Model. The performances of the weighting estimator and the outcome
regression estimators in the Statistical Surrogacy Model are more similar.

Figure 3 displays measurements of estimator quality for just the DML
estimators in a format analogous to Figure 2, with the addition of a third row
measuring one minus the coverage probability of confidence intervals
constructed around each estimator. We report these estimates of coverage
probabilities in Supplementary Section D.5. Confidence intervals constructed
with the intervals given in (A.41) around the estimators formulated in 4.1
have coverage probabilities that are reasonably close to the nominal level.

Figure 3. Finite-Sample Performance with Different Nuisance Parameter
Estimators

Panel A: Latent Unconfounded Treatment
---
Panel B: Statistical Surrogacy

Notes: Figure 3 displays measurements of the quality of the estimators
formulated in Section 4.1 implemented with several alternative choices of
nuisance parameter estimators. The long-term outcome is total household
assets. Panels A and B display results for the estimators defined in 4.1 for
the Latent Unconfounded Treatment and Statistical Surrogacy Models defined in
2.1 and 2.2, respectively. The columns of each panel vary the sample size
multiplier $h$. The rows of each panel display the absolute value of the bias,
one minus the coverage probability, and the root mean squared error of each
estimator, from top to bottom, respectively. A dotted line denoting one minus
the nominal coverage probability, 0.05, is displayed in each sub-panel in the
third row. Each sub-panel displays a bar graph comparing measurements of the
performance of the estimator defined in 4.1, constructed with three types of
nuisance parameter estimators, with the difference in means estimator (5.1).
Each sub-panel displays results for the baseline, unconfounded, case, as well
as for the cases that the confounding parameter $\phi$ has been set to $1/2$
and $2/3$.

##  6\. Conclusion

We study the estimation of long-term treatment effects through the combination
of short-term experimental and long-term observational data sets. We derive
efficient influence functions and calculate corresponding semiparametric
efficiency bounds for this problem. These calculations facilitate the
development of estimators that accommodate the applications of standard
machine learning algorithms for estimating nuisance parameters. We demonstrate
with simulation that these estimators are able to recover long-term treatment
effects in realistic settings.

Important unresolved practical issues remain. Methods for choosing valid and
informative short-term outcomes and assessing of the sensitivity of estimates
to violations of identifying assumptions would be valuable. Additional useful
extensions include the incorporation of instruments and continuous treatments
and the accommodation of settings with limited covariate overlap.

In the case of estimation of average treatment effects under unconfoundedness,
recent promising work (e.g., Athey et al.,, 2018; Bradic et al.,, 2019; Tan,,
2020) has developed estimators that are specifically optimized to handle high-
dimensional covariates and are able to attain various notions of optimality
under weak assumptions on, e.g., the sparsity of the outcome regression or
propensity score models. It is not immediately clear how to apply these ideas
to long-term average treatment effects. Further consideration of this problem
would be a potentially valuable extension, as the resultant estimators may be
particularly well-suited to the case where there are many short-term outcomes.
Some progress on a related problem (where, effectively, there is a single
short-term outcome) has been made by Viviano and Bradic, (2021).

## References

  *   * (1) Athey, S., Chetty, R., and Imbens, G. (2020a).  Combining experimental and observational data to estimate treatment effects on long term outcomes.  arXiv preprint arXiv:2006.09676.
  * (2) Athey, S., Chetty, R., Imbens, G., and Kang, H. (2020b).  Estimating treatment effects using multiple surrogates: The role of the surrogate score and the surrogate index.
  * Athey and Imbens, (2017) Athey, S. and Imbens, G. W. (2017).  The state of applied econometrics: Causality and policy evaluation.  Journal of Economic Perspectives, 31(2):3–32.
  * Athey et al., (2021) Athey, S., Imbens, G. W., Metzger, J., and Munro, E. (2021).  Using wasserstein generative adversarial networks for the design of monte carlo simulations.  Journal of Econometrics.
  * Athey et al., (2018) Athey, S., Imbens, G. W., and Wager, S. (2018).  Approximate residual balancing: debiased inference of average treatment effects in high dimensions.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(4):597–623.
  * Athey et al., (2019) Athey, S., Tibshirani, J., and Wager, S. (2019).  Generalized random forests.  The Annals of Statistics, 47(2):1148–1178.
  * Banerjee et al., (2015) Banerjee, A., Duflo, E., Goldberg, N., Karlan, D., Osei, R., Parienté, W., Shapiro, J., Thuysbaert, B., and Udry, C. (2015).  A multifaceted program causes lasting progress for the very poor: Evidence from six countries.  Science, 348(6236).
  * Bang and Robins, (2005) Bang, H. and Robins, J. M. (2005).  Doubly robust estimation in missing data and causal inference models.  Biometrics, 61(4):962–973.
  * Begg and Leung, (2000) Begg, C. B. and Leung, D. H. (2000).  On the use of surrogate end points in randomized trials.  Journal of the Royal Statistical Society: Series A (Statistics in Society), 163(1):15–28.
  * Belloni et al., (2014) Belloni, A., Chernozhukov, V., and Wang, L. (2014).  Pivotal estimation via square-root Lasso in nonparametric regression.  The Annals of Statistics, 42(2):757 – 788.
  * Bia et al., (2020) Bia, M., Huber, M., and Lafférs, L. (2020).  Double machine learning for sample selection models.  arXiv preprint arXiv:2012.00745.
  * Bickel, (1982) Bickel, P. J. (1982).  On adaptive estimation.  The Annals of Statistics, pages 647–671.
  * Bickel et al., (1993) Bickel, P. J., Klaassen, C. A., Ritov, Y., and Wellner, J. A. (1993).  Efficient and adaptive estimation for semiparametric models, volume 4.  Johns Hopkins University Press Baltimore.
  * Bickel et al., (2009) Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009).  Simultaneous analysis of lasso and dantzig selector.  The Annals of statistics, 37(4):1705–1732.
  * Bouguen et al., (2019) Bouguen, A., Huang, Y., Kremer, M., and Miguel, E. (2019).  Using randomized controlled trials to estimate long-run impacts in development economics.  Annual Review of Economics.
  * Bradic et al., (2019) Bradic, J., Wager, S., and Zhu, Y. (2019).  Sparsity double robust inference of average treatment e ects.
  * Chen and Guestrin, (2016) Chen, T. and Guestrin, C. (2016).  Xgboost: A scalable tree boosting system.  In Proceedings of the 22nd acm sigkdd international conference on knowledge discovery and data mining, pages 785–794.
  * Chen et al., (2015) Chen, T., He, T., Benesty, M., Khotilovich, V., Tang, Y., Cho, H., et al. (2015).  Xgboost: extreme gradient boosting.  R package version 0.4-2, 1(4):1–4.
  * Chen, (2007) Chen, X. (2007).  Large sample sieve estimation of semi-nonparametric models.  Handbook of econometrics, 6:5549–5632.
  * Chen et al., (2008) Chen, X., Hong, H., and Tarozzi, A. (2008).  Semiparametric efficiency in GMM models with auxiliary data.  The Annals of Statistics, 36(2):808 – 843.
  * Chen and Liao, (2015) Chen, X. and Liao, Z. (2015).  Sieve semiparametric two-step gmm under weak dependence.  Journal of Econometrics, 189(1):163–186.
  * Chen et al., (2014) Chen, X., Liao, Z., and Sun, Y. (2014).  Sieve inference on possibly misspecified semi-nonparametric time series models.  Journal of Econometrics, 178:639–658.
  * Chen and Santos, (2018) Chen, X. and Santos, A. (2018).  Overidentification in regular models.  Econometrica, 86(5):1771–1817.
  * Chen and Shen, (1998) Chen, X. and Shen, X. (1998).  Sieve extremum estimates for weakly dependent data.  Econometrica, pages 289–314.
  * Chen and White, (1999) Chen, X. and White, H. (1999).  Improved rates and asymptotic normality for nonparametric neural network estimators.  IEEE Transactions on Information Theory, 45(2):682–691.
  * (26) Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., and Robins, J. (2018a).  Double/debiased machine learning for treatment and structural parameters.  The Econometrics Journal, 21:C1–C68.
  * Chernozhukov et al., (2016) Chernozhukov, V., Escanciano, J. C., Ichimura, H., Newey, W. K., and Robins, J. M. (2016).  Locally robust semiparametric estimation.  arXiv preprint arXiv:1608.00033.
  * (28) Chernozhukov, V., Newey, W. K., and Singh, R. (2018b).  Learning l2 continuous regression functionals via regularized riesz representers.  arXiv preprint arXiv:1809.05224, 8.
  * Crump et al., (2009) Crump, R. K., Hotz, V. J., Imbens, G. W., and Mitnik, O. A. (2009).  Dealing with limited overlap in estimation of average treatment effects.  Biometrika, 96(1):187–199.
  * Dal Bó et al., (2021) Dal Bó, E., Finan, F., Li, N. Y., and Schechter, L. (2021).  Information technology and government decentralization: Experimental evidence from paraguay.  Econometrica, 89(2):677–701.
  * Duflo et al., (2007) Duflo, E., Glennerster, R., and Kremer, M. (2007).  Using randomization in development economics research: A toolkit.  Handbook of development economics, 4:3895–3962.
  * Dynarski et al., (2021) Dynarski, S., Libassi, C., Michelmore, K., and Owen, S. (2021).  Closing the gap: The effect of reducing complexity and uncertainty in college pricing on the choices of low-income students.  American Economic Review, 111(6):1721–56.
  * D’Amour et al., (2021) D’Amour, A., Ding, P., Feller, A., Lei, L., and Sekhon, J. (2021).  Overlap in observational studies with high-dimensional covariates.  Journal of Econometrics, 221(2):644–654.
  * Farrell et al., (2021) Farrell, M. H., Liang, T., and Misra, S. (2021).  Deep neural networks for estimation and inference.  Econometrica, 89(1):181–213.
  * Fisher, (1925) Fisher, R. A. (1925).  Statistical methods for research workers.  Statistical methods for research workers., (1st. ed).
  * Foster and Syrgkanis, (2019) Foster, D. J. and Syrgkanis, V. (2019).  Orthogonal statistical learning.  arXiv preprint arXiv:1901.09036.
  * Frölich, (2007) Frölich, M. (2007).  Nonparametric iv estimation of local average treatment effects with covariates.  Journal of Econometrics, 139(1):35–75.
  * Gechter and Meager, (2021) Gechter, M. and Meager, R. (2021).  Combining experimental and observational studies in meta-analysis: A mutual debiasing approach.”.  Technical report, Mimeo.
  * Gertler et al., (2014) Gertler, P., Heckman, J., Pinto, R., Zanolini, A., Vermeersch, C., Walker, S., Chang, S. M., and Grantham-McGregor, S. (2014).  Labor market returns to an early childhood stimulation intervention in jamaica.  Science, 344(6187):998–1001.
  * Goodfellow et al., (2014) Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. (2014).  Generative adversarial nets.  Advances in neural information processing systems, 27.
  * Graham, (2011) Graham, B. S. (2011).  Efficiency bounds for missing data models with semiparametric restrictions.  Econometrica, 79(2):437–452.
  * Gui, (2020) Gui, G. (2020).  Combining observational and experimental data using first-stage covariates.  arXiv preprint arXiv:2010.05117.
  * Gupta et al., (2019) Gupta, S., Kohavi, R., Tang, D., Xu, Y., Andersen, R., Bakshy, E., Cardin, N., Chandran, S., Chen, N., Coey, D., Curtis, M., Deng, A., Duan, W., Forbes, P., Frasca, B., Guy, T., Imbens, G. W., Saint Jacques, G., Kantawala, P., Katsev, I., Katzwer, M., Konutgan, M., Kunakova, E., Lee, M., Lee, M., Liu, J., McQueen, J., Najmi, A., Smith, B., Trehan, V., Vermeer, L., Walker, T., Wong, J., and Yashkov, I. (2019).  Top challenges from the first practical online controlled experiments summit.  ACM SIGKDD Explorations Newsletter, 21(1):20–35.
  * Hahn, (1998) Hahn, J. (1998).  On the role of the propensity score in efficient semiparametric estimation of average treatment effects.  Econometrica, pages 315–331.
  * Hastie and Qian, (2014) Hastie, T. and Qian, J. (2014).  Glmnet vignette.  Retrieved June, 9(2016):1–30.
  * Hirano et al., (2003) Hirano, K., Imbens, G. W., and Ridder, G. (2003).  Efficient estimation of average treatment effects using the estimated propensity score.  Econometrica, 71(4):1161–1189.
  * Horvitz and Thompson, (1952) Horvitz, D. G. and Thompson, D. J. (1952).  A generalization of sampling without replacement from a finite universe.  Journal of the American statistical Association, 47(260):663–685.
  * Hotz et al., (2005) Hotz, V. J., Imbens, G. W., and Mortimer, J. H. (2005).  Predicting the efficacy of future training programs using past experiences at other locations.  Journal of Econometrics, 125(1-2):241–270.
  * Hou et al., (2021) Hou, J., Mukherjee, R., and Cai, T. (2021).  Efficient and robust semi-supervised estimation of ate with partially annotated treatment and response.
  * Ichimura and Newey, (2022) Ichimura, H. and Newey, W. K. (2022).  The influence function of semiparametric estimators.  Quantitative Economics, 13(1):29–61.
  * Imai et al., (2010) Imai, K., Keele, L., and Tingley, D. (2010).  A general approach to causal mediation analysis.  Psychological methods, 15(4):309.
  * Imbens et al., (2022) Imbens, G., Kallus, N., Mao, X., and Wang, Y. (2022).  Long-term causal inference under persistent confounding via data combination.
  * Imbens, (2004) Imbens, G. W. (2004).  Nonparametric estimation of average treatment effects under exogeneity: A review.  Review of Economics and statistics, 86(1):4–29.
  * Kallus and Mao, (2020) Kallus, N. and Mao, X. (2020).  On the role of surrogates in the efficient estimation of treatment effects with limited outcome data.  arXiv preprint arXiv:2003.12408.
  * Kang and Schafer, (2007) Kang, J. D. and Schafer, J. L. (2007).  Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data.  Statistical science, pages 523–539.
  * Kennedy, (2022) Kennedy, E. H. (2022).  Semiparametric doubly robust targeted double machine learning: a review.  arXiv preprint arXiv:2203.06469.
  * Kennedy, (2023) Kennedy, E. H. (2023).  Semiparametric doubly robust targeted double machine learning: a review.
  * Klaassen, (1987) Klaassen, C. A. J. (1987).  Consistent Estimation of the Influence Function of Locally Asymptotically Linear Estimators.  The Annals of Statistics, 15(4):1548 – 1562.
  * Le Cam, (1956) Le Cam, L. (1956).  On the asymptotic theory of estimation and testing hypotheses.  In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, volume 3, pages 129–157. University of California Press.
  * Little and Rubin, (2019) Little, R. J. and Rubin, D. B. (2019).  Statistical analysis with missing data, volume 793.  John Wiley & Sons.
  * Luo and Spindler, (2017) Luo, Y. and Spindler, M. (2017).  $\ell_{2}$-boosting for economic applications.  American Economic Review, 107(5):270–73.
  * Muris, (2020) Muris, C. (2020).  Efficient gmm estimation with incomplete data.  Review of Economics and Statistics, 102(3):518–530.
  * Newey, (1994) Newey, W. K. (1994).  The asymptotic variance of semiparametric estimators.  Econometrica: Journal of the Econometric Society, pages 1349–1382.
  * Newey and McFadden, (1994) Newey, W. K. and McFadden, D. (1994).  Large sample estimation and hypothesis testing.  Handbook of econometrics, 4:2111–2245.
  * Newey and Robins, (2018) Newey, W. K. and Robins, J. R. (2018).  Cross-fitting and fast remainder rates for semiparametric estimation.
  * Pearl, (1995) Pearl, J. (1995).  Causal diagrams for empirical research.  Biometrika, 82(4):669–688.
  * Prentice, (1989) Prentice, R. L. (1989).  Surrogate endpoints in clinical trials: definition and operational criteria.  Statistics in medicine, 8(4):431–440.
  * Ridder and Moffitt, (2007) Ridder, G. and Moffitt, R. (2007).  The econometrics of data combination.  Handbook of econometrics, 6:5469–5547.
  * Robins et al., (1995) Robins, J. M., Rotnitzky, A., and Zhao, L. P. (1995).  Analysis of semiparametric regression models for repeated outcomes in the presence of missing data.  Journal of the american statistical association, 90(429):106–121.
  * Rosenbaum and Rubin, (1983) Rosenbaum, P. R. and Rubin, D. B. (1983).  The central role of the propensity score in observational studies for causal effects.  Biometrika, 70(1):41–55.
  * Rosenman et al., (2020) Rosenman, E., Basse, G., Owen, A., and Baiocchi, M. (2020).  Combining observational and experimental datasets using shrinkage estimators.  arXiv preprint arXiv:2002.06708.
  * Rosenman et al., (2018) Rosenman, E., Owen, A. B., Baiocchi, M., and Banack, H. (2018).  Propensity score methods for merging observational and experimental datasets.  arXiv preprint arXiv:1804.07863.
  * Scharfstein et al., (1999) Scharfstein, D. O., Rotnitzky, A., and Robins, J. M. (1999).  Adjusting for nonignorable drop-out using semiparametric nonresponse models.  Journal of the American Statistical Association, 94(448):1096–1120.
  * Singh, (2021) Singh, R. (2021).  A finite sample theorem for longitudinal causal inference with machine learning: Long term, dynamic, and mediated effects.
  * Singh, (2022) Singh, R. (2022).  Generalized kernel ridge regression for long term causal inference: Treatment effects, dose responses, and counterfactual distributions.
  * Tan, (2020) Tan, Z. (2020).  Model-assisted inference for treatment effects using regularized calibrated estimation with high-dimensional data.  The Annals of Statistics, 48(2):811–837.
  * Tanner et al., (2015) Tanner, J. C., Candland, T., and Odden, W. S. (2015).  Later impacts of early childhood interventions: a systematic review.  Washington: Independent Evaluation Group, World Bank Group.
  * Tibshirani, (1996) Tibshirani, R. (1996).  Regression shrinkage and selection via the lasso.  Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267–288.
  * Twitter Engineering, (2021) Twitter Engineering (2021).  “We built a causal estimation framework on the idea of statistical ‘surrogacy’ (Athey et al 2016) - when we can’t wait to observe long-run outcomes, we create a model based on intermediate data.”.  Tweet, @TwitterEng, 18 Oct, 12:18 P.M.
  * van der Laan and Petersen, (2004) van der Laan, M. J. and Petersen, M. L. (2004).  Estimation of direct and indirect causal effects in longitudinal studies.
  * Van der Vaart, (1998) Van der Vaart, A. W. (1998).  Asymptotic statistics.  Cambridge university press.
  * van der Vaart and Wellner, (1996) van der Vaart, A. W. and Wellner, J. A. (1996).  Weak convergence and empirical processes.  Springer.
  * Viviano and Bradic, (2021) Viviano, D. and Bradic, J. (2021).  Dynamic covariate balancing: estimating treatment effects over time.  arXiv preprint arXiv:2103.01280.
  * Wager and Walther, (2015) Wager, S. and Walther, G. (2015).  Adaptive concentration of regression trees, with application to random forests.  arXiv preprint arXiv:1503.06388.
  * Yang et al., (2020) Yang, J., Eckles, D., Dhillon, P., and Aral, S. (2020).  Targeting for long-term outcomes.  arXiv preprint arXiv:2010.15835.

Supplemental Appendix to: Semiparametric Estimation of
Long-Term Treatment Effects111Date:  Jiafeng Chen David M. Ritzwoller Harvard
Business School Stanford Graduate School of Business<EMAIL_ADDRESS><EMAIL_ADDRESS>

Contents
appendix.Asubsection.A.1subsection.A.2subsection.A.3subsection.A.4subsection.A.5subsection.A.6subsection.A.7subsection.A.8subsubsection.A.8.1subsubsection.A.8.2subsubsection.A.8.3appendix.Bsubsection.B.1subsection.B.2subsection.B.3appendix.Csubsection.C.1subsection.C.2subsubsection.C.2.1subsubsection.C.2.2subsubsection.C.2.3subsubsection.C.2.4subsubsection.C.2.5subsection.C.3subsubsection.C.3.1subsubsection.C.3.2subsubsection.C.3.3subsubsection.C.3.4appendix.Dsubsection.D.1subsection.D.2subsection.D.3subsection.D.4section*.51section*.52section*.53section*.54section*.55section*.56subsection.D.5

##  Appendix A Proofs for Results Presented in the Main Text

###  A.1. Proof of Proposition 2.1

Unless otherwise noted, we drop the dependence on the individual $i$. It will
suffice to consider the parameter

$\theta_{1,1}=\mathbb{E}_{P_{\star}}[Y(1)\mid G=1]\leavevmode\nobreak\ ,$

as the point-identification of $\theta_{1,0}=\mathbb{E}[Y(0)\mid G=1]$ will
follow by an analogous argument.

We consider Latent Unconfounded Treatment Model, specified in 2.1, and repeat
the argument provided in support of Theorem 1 of Athey et al., 2020a . Define
the functions

$\displaystyle\mu_{1}(s,x)$ $\displaystyle=\mathbb{E}_{P}[Y\mid
S=s,X=x,W=1,G=1]$ $\displaystyle=\mathbb{E}_{P_{\star}}[Y(1)\mid
S(1)=s,X=x,W=1,G=1]$

and

$\displaystyle\overline{\mu}_{1}(x)$
$\displaystyle=\mathbb{E}_{P}[\mu_{1}(S,X)\mid X=x,W=1,G=0]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\mu_{1}(S(1),X)\mid X=x,W=1,G=0]$

and observe that, by Assumption 2.2, $\mu_{1}(s,x)$ is identified from the
observational sample and $\overline{\mu}_{1}(x)$ is identified from the
experimental sample given the identification of $\mu_{1}(s,x)$. Observe that

$\displaystyle\mathbb{E}_{P_{\star}}[Y(1)\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
X,G=1]\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
X,G=0]\mid G=1]$ (Assumption 2.3)
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
S(1),X,G=1]\mid X,G=0]\mid G=1]$ (Assumption 2.3)
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
S(1),X,W=1,G=1]\mid X,G=0]\mid G=1]$ (Assumption 2.4)
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[\mu_{1}(S(1),X)]\mid
X,G=0]\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[\mu_{1}(S(1),X)]\mid
X,W=1,G=0]\mid G=1]$ (Assumption 2.1)
$\displaystyle=\mathbb{E}_{P_{\star}}[\overline{\mu}_{1}(X)\mid
G=1]\leavevmode\nobreak\ .$

Hence, $\tau_{1}$ is identified as $\overline{\mu}_{1}(x)$ is identified.

###  A.2. Proof of Proposition 2.2

We consider the Statistical Surrogacy Model, specified in 2.2. Define the
function

$\nu(s,x)=\mathbb{E}_{P}[Y\mid S=s,X=x,G=1]$

and observe that $\nu(s,x)$ is identified from the observational sample by
Assumption 2.2. Observe that

$\displaystyle\mathbb{E}_{P_{\star}}[Y(1)\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
X,G=1]\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
X,G=0]\mid G=1]$ (Assumption 2.3)
$\displaystyle=\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y(1)\mid
X,W=1,G=0]\mid G=1]\leavevmode\nobreak\ ,$ (Assumption 2.1)

and that

$\displaystyle\mathbb{E}_{P_{\star}}[Y(1)\mid X,W=1,G=0]$
$\displaystyle=\mathbb{E}_{P}[Y\mid X,W=1,G=0]$
$\displaystyle=\mathbb{E}_{P}[\mathbb{E}_{P}[Y\mid S,X,W=1,G=0]\mid
X,W=1,G=0]$ $\displaystyle=\mathbb{E}_{P}[\mathbb{E}_{P}[Y\mid S,X,G=0]\mid
X,W=1,G=0]$ (Assumption 2.5)
$\displaystyle=\mathbb{E}_{P}[\mathbb{E}_{P}[Y\mid S,X,G=1]\mid X,W=1,G=0]$
(Assumption 2.6) $\displaystyle=\mathbb{E}_{P}[\nu(S,X)\mid
X,W=1,G=0]\leavevmode\nobreak\ .$

It is clear that $\mathbb{E}_{P}[\nu(S,X)\mid X,W=1,G=0]$ is identified from
the experimental sample by Assumption 2.2, and that therefore $\tau_{1}$ is
identified.

###  A.3. Proof of Theorem 3.1. Part 1

Throughout, for a general functional $\varphi$ on $\mathcal{M}_{\lambda}$ and
an arbitrary random variable $D$ with distribution $Q$ in
$\mathcal{M}_{\lambda}$, we write $\varphi(Q)=\varphi(D)$ with a minor abuse
of notation.We let

$\varphi^{\prime}(D)=\varphi^{\prime}(D;0)=\frac{d}{d\varepsilon}\varphi(Q_{\varepsilon}){\small\bigg{\rvert}}_{\varepsilon=0}$

denote the pathwise derivative of $\varphi(\cdot)$ on some regular parametric
submodel $\mathcal{Q}=\\{Q_{\varepsilon}:\varepsilon\in[-1,1]\\}$ of
$\mathcal{M}_{\lambda}$ evaluated at $\varepsilon=0$.

It suffices to derive the efficient influence function for the parameter
$\theta_{1,1}$, defined more generally by
$\theta_{w,g}=\mathbb{E}[Y_{i}(w)\mid G_{i}=g]$, as an analogous argument will
hold for $\theta_{0,1}$. Unless otherwise noted, we drop the dependence on the
individual $i$. We omit the subscript $w$ on the nuisance functions
$\mu_{w}(s,x)$, $\overline{\mu}_{w}(x)$, and $\varrho_{w}(s,x)$, as only the
case $W=1$ will be relevant.

The presentation of the proof will be somewhat constructive, as we hope to
convey the rationale behind the form of the efficient influence function, for
which we have had to take a few educated guesses in forming a conjecture.
Nonetheless, we state that the efficient influence function $\psi_{1,1}(b)$
takes the form

$\psi_{1,1}(b)=\frac{gw(y-\mu(s,x))}{\pi\rho(s,x)}+\frac{(1-g)w}{(1-\gamma(x))\varrho(x)}\frac{\gamma(x)}{\pi}(\mu(s,x)-\overline{\mu}(x))+\frac{g}{\pi}(\overline{\mu}(x)-\theta_{1,1})$

The reader can of course verify Theorem 3.1 directly by using this stated form
of the influence function and checking a series of conditions that we lay out.

The argument proceeds as follows. First, we factorize the density of the
observed data and relate each factor to conditional densities of the complete
data distribution. Second, for an arbitrary smooth parametric submodel, we
characterize the tangent space of the distribution of the observed data as a
linear space of mean-zero square-integrable functions. Third, we conjecture a
functional form that explicitly resides in the tangent space, where we note
that the efficient influence function is the unique influence function for the
target statistical functional $\theta_{1,1}$ in the tangent space of
$\theta_{1,1}$ (Chapter 25 of Van der Vaart,, 1998). Fourth, we state the
conditions an influence function of the pathwise derivative of $\theta_{1,1}$
necessarily satisfies, and we verify that the conjectured efficient influence
function satisfies these conditions.

Density Factorization. Consider the random variable

$A_{1}=(Y(1),S(1),W,G,X)\equiv(Y^{*},S^{*},W,G,X)$

where we observe $B_{1}=(WGY^{*},WS^{*},W,G,X)$. Let $Y=WGY^{*}$ and
$S=WS^{*}$. Recall that the distribution of the complete data is denoted by
$P_{\star}$, with density $p_{\star}$ with respect to $\lambda$, and that the
distribution of the observed data is denoted by $P$, with density $p$ with
respect to $\lambda$. Observe that the densities under the complete data
distribution and the observed data distribution agree when data isn’t missing.
That is, for all $(y,s)\in\mathbb{R}\times\mathbb{R}^{d}$ and events
$E\in\mathcal{A}$, we have that

$p_{\star}(y\mid W=1,G=1,E)=p(y\mid W=1,G=1,E)$

and

$p_{\star}(s\mid W=1,E)=p(s\mid W=1,E)$

and that for all events $E$ measurable with respect to the $\sigma$-algebra
generated by $(W,G,X)$, we have that $P(E)=P_{\star}(E)$.

By Assumptions 2.1, 2.3, and 2.4, the density of the observed data $p$ admits
the factorization

$\displaystyle p(b_{1})=\,\,$ $\displaystyle p(y\mid
G=1,X=x,W=1,S=s)^{wg}p(s,W=1\mid G=1,X=x)^{wg}$ $\displaystyle p(s\mid
G=0,X=x,W=1)^{w(1-g)}$ $\displaystyle P(W=w\mid G=0,X=x)^{1-g}P(W=0\mid
G=1,X=x)^{(1-w)g}P(G=g\mid X=x)p(x)\leavevmode\nobreak\ .$

Each of the terms of this expression can be rewritten in terms of the density
of the complete data $p_{\star}$. In particular, we have that the first term
can be written

$\displaystyle p(y\mid G=1,X=x,W=1,S=s)$ $\displaystyle=p_{\star}(y\mid
G=1,X=x,W=1,S=s)$ $\displaystyle=p_{\star}(y\mid G=1,X=x,S=s)$ (Assumption
2.4) $\displaystyle=p_{\star}(y\mid x,s)\leavevmode\nobreak\ ,$ (Assumption
2.3, noting that $p_{\star}$ is the measure of the potential outcomes. )

the second and third terms can be written

$\displaystyle p(s,W=1\mid G=1,X=x)$ $\displaystyle=p_{\star}(S^{*}=s,W=1\mid
G=1,X=x)$ $\displaystyle=p_{\star}(W=1\mid
S^{*}=s,G=1,X=x)p_{\star}(S^{*}=s\mid G=1,X=x)$
$\displaystyle=\rho(s,x)p_{\star}(s\mid x)$ (Assumption 2.3)

and

$\displaystyle p(s\mid G=0,X=x,W=1)$ $\displaystyle=p_{\star}(S^{*}=s\mid
G=0,X=x)$ (Assumption 2.1) $\displaystyle=p_{\star}(s\mid
x)\leavevmode\nobreak\ ,$ (Assumption 2.3)

the fourth term can be written

$\displaystyle P(W=w\mid G=0,X=x)$
$\displaystyle=\varrho(x)^{w}(1-\varrho(x))^{1-w}\leavevmode\nobreak\ ,$
(Assumption 2.1)

and finally the fifth term can be written

$\displaystyle P(W=0\mid G=1,X=x)$ $\displaystyle=P_{\star}(W=0\mid G=1,X=x)$
$\displaystyle=\int P_{\star}(W=0\mid G=1,X=x,S=s)\,dP_{\star}(s\mid x)$
$\displaystyle=\int(1-\rho(s,x))p_{\star}(s\mid x)\,ds.$

As a result, we obtain a factorization of the density of the observed data in
terms of conditional distributions of the complete data

$\displaystyle p(b_{1})=\,\,$ $\displaystyle p_{\star}(y\mid
x,s)^{wg}p_{\star}(s\mid
x)^{w}\rho(s,x)^{wg}\left({\int(1-\rho(s,x))p_{\star}(s\mid
x)\,ds}\right)^{(1-w)g}$
$\displaystyle\gamma(x)^{g}(1-\gamma(x))^{1-g}(\varrho(x)^{w}(1-\varrho(x))^{1-w})^{1-g}p(x)\leavevmode\nobreak\
.$ (A.1)

Characterization of the Tangent Space. Let $\mathcal{P}$ be any regular
parametric submodel of $\mathcal{M}_{\lambda}$ indexed by
$\varepsilon\in\mathbb{R}$, with densities
$p_{\varepsilon}:=dP_{\varepsilon}/d\lambda$ with $p_{0}=dP/d\lambda$ and such
that Assumptions 2.1, 2.3, and 2.4 hold for each
$P_{\varepsilon}\in\mathcal{P}$. Let the subscript ⋆ distinguish log densities
of the complete data from log densities of the observed data. By the
factorization (A.1), we obtain the score

$\displaystyle\ell^{\prime}(b_{1})$
$\displaystyle=wg\cdot\ell^{\prime}_{*}(y\mid
x,s)+w\cdot\ell^{\prime}_{*}(s\mid x)+\ell^{\prime}_{*}(g,x)$
$\displaystyle-g(1-w)\left(\frac{\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)\mid G=0,W=1,X=x]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid
G=0,W=1,X=x]}\right)$
$\displaystyle+g\left(w\cdot\frac{\rho^{\prime}(s,x)}{\rho(s,x)}-(1-w)\frac{\mathbb{E}_{P_{\star}}[\rho^{\prime}(S^{*},X)\mid
G=0,W=1,X=x]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid G=0,W=1,X=x]}\right)$
$\displaystyle+(1-g)\varrho^{\prime}(x)\left(\frac{w-\varrho(x)}{\varrho(x)(1-\varrho(x))}\right)\leavevmode\nobreak\
.$ (A.2)

Thus, the tangent space $\mathcal{T}$ is given by mean-square closure of the
linear span of the functions

$\displaystyle s(b_{1})$ $\displaystyle=wg\cdot s_{1}(y\mid x,s)+w\cdot
s_{2}(s\mid x)+s_{3}(g,x)$
$\displaystyle-g(1-w)\left(\frac{\mathbb{E}_{P_{\star}}[\rho(S^{*},X)s_{2}(S^{*}\mid
X)\mid G=0,W=1,X=x]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid
G=0,W=1,X=x]}\right)$
$\displaystyle+g\left(w\cdot\frac{s_{4}(s,x)}{\rho(s,x)}-(1-w)\frac{\mathbb{E}_{P_{\star}}[s_{4}(S^{*},X)\mid
G=0,W=1,X=x]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid G=0,W=1,X=x]}\right)$
$\displaystyle+(1-g)\left(s_{5}(x)\frac{w-\varrho(x)}{\varrho(x)(1-\varrho(x))}\right)\leavevmode\nobreak\
,$

where the functions $s_{1}$ through $s_{5}$ range over the space of mean-zero
and square integrable functions that satisfy the restrictions

$\displaystyle\mathbb{E}_{P}[s_{1}(Y\mid S,X)\mid
W=1,G=1,S,X]=\mathbb{E}_{P_{\star}}[s_{1}(Y^{*}\mid S^{*},X)\mid S^{*}=s,X=x]$
$\displaystyle=0\leavevmode\nobreak\ ,$
$\displaystyle\mathbb{E}_{P}[s_{2}(S\mid X)\mid
W=1,G=0,X]=\mathbb{E}_{P_{\star}}[s_{2}(S^{*}\mid X)\mid X=x]$
$\displaystyle=0\leavevmode\nobreak\ \text{, and}$
$\displaystyle\mathbb{E}_{P}[s_{3}(G,X)]$ $\displaystyle=0\leavevmode\nobreak\
.$

Pathwise Differentiability Conditions. In the following calculations, it is
often easier to work with the complete data distribution. To that end, we note
that for an arbitrary measurable function $h$, we have that certain
conditional means of the observed distribution equal certain conditional means
of the complete data distribution:

$\displaystyle\mathbb{E}_{P}[h(Y,S,X)\mid W=1,G=1,X=x,S=s]$
$\displaystyle=\mathbb{E}_{P_{\star}}[h(Y^{*},S,X)\mid
G=1,X=x,S^{*}=s]\leavevmode\nobreak\ ,$
$\displaystyle\mathbb{E}_{P}[h(S,X)\mid G=0,W=1,X=x]$
$\displaystyle=\mathbb{E}_{P_{\star}}[h(S^{*},X)\mid G=0,X=x]$
$\displaystyle=\mathbb{E}_{P_{\star}}[h(S^{*},X)\mid
G=1,X=x]\leavevmode\nobreak\ ,$ $\displaystyle\mathbb{E}_{P}[h(X)\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{h(X)\mid
G=1}\right]\leavevmode\nobreak\ ,$

and, as a result, for $h$,

$\displaystyle\mathbb{E}_{P}[\mathbb{E}_{P}[\mathbb{E}_{P}[h(Y,S,X)\mid
W=1,G=1,X,S]\mid W=1,G=0,X]\mid G=1]$
$\displaystyle\quad=\mathbb{E}_{P_{\star}}\left[{h(Y^{*},S^{*},X)\mid
G=1}\right]=\mathbb{E}_{P_{\star}}\left[{\frac{G}{\pi}h(Y^{*},S^{*},X)}\right]\leavevmode\nobreak\
.$

Note that $\theta_{1,1}$ is such a functional with $h(y,s,x)=y$.

Observe that the parameter of interest can be written

$\displaystyle\theta_{1,1}$ $\displaystyle=\mathbb{E}_{P_{\star}}[Y^{*}\mid
G=1]$ $\displaystyle=\mathbb{E}_{P}[\mathbb{E}_{P}[\mathbb{E}_{P}[Y\mid
W=1,G=1,X,S]\mid W=1,G=0,X]\mid G=1]$
$\displaystyle=\iiint\frac{y}{\pi}p_{\star}(y\mid s,x)p_{\star}(s\mid
x)p(x,G=1)\,\text{d}\lambda(y)\text{d}\lambda(s)\text{d}\lambda(x)\leavevmode\nobreak\
.$

The pathwise derivative of this parameter at 0 on $\mathcal{P}$ is given by

$\displaystyle\theta_{1,1}^{\prime}$
$\displaystyle=\mathbb{E}_{P_{\star}}[Y^{*}\ell^{\prime}(Y^{*}\mid
S^{*},X)\mid G=1]$
$\displaystyle+\mathbb{E}_{P_{\star}}[Y^{*}\ell^{\prime}(S^{*}\mid X)\mid
G=1]+\mathbb{E}_{P_{\star}}[Y^{*}\ell^{\prime}(X,G=1)\mid
G=1]\leavevmode\nobreak\ .$ (A.3)

Now, observe that

$\displaystyle G\cdot\ell^{\prime}_{*}(X,G=1)$
$\displaystyle=G\cdot\ell^{\prime}_{*}(X,G)-G\cdot\ell^{\prime}_{*}(G=1)\leavevmode\nobreak\
,$

and that

$\displaystyle\ell^{\prime}_{*}(G=1)$
$\displaystyle=\mathbb{E}_{P_{\star}}[\ell^{\prime}_{*}(X,G)\mid
G=1]\leavevmode\nobreak\ .$

Thus, each of the terms in (A.3) can be written in terms of conditional scores
by

$\displaystyle\mathbb{E}_{P_{\star}}[Y^{*}\ell^{\prime}(Y^{*}\mid S^{*},X)\mid
G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\pi^{-1}GY^{*}\ell^{\prime}_{*}(Y^{*}\mid
S^{*},X)]\leavevmode\nobreak\ ,$
$\displaystyle\mathbb{E}_{P_{\star}}[Y^{*}\ell^{\prime}(S^{*}\mid X)\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\pi^{-1}GY^{*}\ell^{\prime}_{*}(S^{*}\mid
X)]\leavevmode\nobreak\ \text{, and}$
$\displaystyle\mathbb{E}_{P_{\star}}[Y^{*}\ell^{\prime}(X,G=1)\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\pi^{-1}GY^{*}\ell^{\prime}_{*}(X,G=1)]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\pi^{-1}GY^{*}\ell^{\prime}_{*}(X,G)]-\mathbb{E}_{P_{\star}}[\pi^{-1}GY^{*}]\ell^{\prime}_{*}(G=1)$
$\displaystyle=\mathbb{E}_{P_{\star}}[\pi^{-1}GY^{*}\ell^{\prime}_{*}(X,G)]-\theta_{1,1}\cdot\mathbb{E}_{P_{\star}}[\ell^{\prime}_{*}(X,G)\mid
G=1]\leavevmode\nobreak\ ,$

respectively.

An influence function for $\theta_{1,1}$ is a mean-zero and square-integrable
function $\tilde{\psi}_{1,1}(B_{1})$ that satisfies the condition

$\theta_{1,1}^{\prime}=\mathbb{E}_{P}[\tilde{\psi}_{1,1}(B_{1})\ell^{\prime}(B_{1})]\leavevmode\nobreak\
.$ (A.4)

Hence, by (A.2) and (A.3), in order to establish that a mean-zero and square-
integrable function $\tilde{\psi}_{1,1}(B_{1})$ is an influence function for
$\theta_{1,1}$, it suffices to show that the terms in the right-hand side of
(A.4) match the terms in the right-hand side of (A.3):

$\displaystyle\frac{1}{\pi}\mathbb{E}_{P_{\star}}[GY^{*}\ell^{\prime}_{*}(Y^{*}\mid
S^{*},X)]$
$\displaystyle=\mathbb{E}_{P}\left[{\tilde{\psi}_{1,1}(B_{1})GW\ell^{\prime}_{*}(Y\mid
S,X)}\right]\leavevmode\nobreak\ ,$ (A.5)
$\displaystyle\frac{1}{\pi}\mathbb{E}_{P_{\star}}[GY^{*}\ell^{\prime}_{*}(S^{*}\mid
X)]$
$\displaystyle=\mathbb{E}_{P}\Bigg{[}\tilde{\psi}_{1,1}(B_{1})\Bigg{(}W\ell^{\prime}_{*}(S\mid
X)$
$\displaystyle\quad\quad\quad\quad-G(1-W)\frac{\int\rho(s,X)p^{\prime}_{*}(s\mid
X)\,\text{d}\lambda(s)}{1-\int\rho(s,X)p_{\star}(s\mid
X)\,\text{d}\lambda(s)}\Bigg{)}\Bigg{]}\leavevmode\nobreak\ ,$ (A.6)
$\displaystyle\mathbb{E}_{P}[\tilde{\psi}_{1,1}(B_{1})\ell^{\prime}_{*}(G,X)]$
$\displaystyle=\frac{1}{\pi}\mathbb{E}_{P_{\star}}[GY^{*}\ell^{\prime}_{*}(G,X)]-\theta_{1,1}\cdot\mathbb{E}_{P_{\star}}[\ell^{\prime}_{*}(G,X)\mid
G=1]\leavevmode\nobreak\ ,$ (A.7) $\displaystyle 0$
$\displaystyle=\mathbb{E}_{P}\Bigg{[}\tilde{\psi}_{1,1}(B_{1})G\Bigg{(}W\frac{\rho^{\prime}(S,X)}{\rho(S,X)}$
$\displaystyle\quad\quad\quad\quad-(1-W)\frac{\int\rho^{\prime}(s,X)p_{\star}(s\mid
X)\text{d}\lambda(s)}{1-\int\rho(s,X)p_{\star}(s\mid
X)\,\text{d}\lambda(s)}\Bigg{)}\Bigg{]}\leavevmode\nobreak\ ,$ (A.8)
$\displaystyle 0$
$\displaystyle=\mathbb{E}_{P}\left[{\tilde{\psi}_{1,1}(B_{1})(1-G)\frac{W-\varrho(X)}{\varrho(X)(1-\varrho(X))}\varrho^{\prime}(X)}\right]\leavevmode\nobreak\
.$ (A.9)

Conjectured Efficient Influence Function. We begin by considering an element
of the tangent space. For the unspecified functions $a(s,x)$ and $b(x)$,
consider the choices

$\displaystyle s_{1}(y,s,x)$
$\displaystyle=\frac{y-\mu(s,x)}{\pi\rho(s,x)}\leavevmode\nobreak\ ,$
$\displaystyle s_{2}(s,x)$
$\displaystyle=\frac{1}{\varrho(x)}\frac{a(s,x)}{(1-\gamma(x))}\leavevmode\nobreak\
,$ $\displaystyle s_{3}(g,x)$ $\displaystyle=g\cdot b(x),$ $\displaystyle
s_{4}(s,x)$ $\displaystyle=-\rho(s,x)s_{2}(s,x)\leavevmode\nobreak\ \text{,
and}$ $\displaystyle s_{5}(x)$ $\displaystyle=0\leavevmode\nobreak\ .$ (A.10)

These choices correspond to the following conjectured efficient influence
function

$\displaystyle\psi_{1,1}(b_{1})$ $\displaystyle=gw\cdot s_{1}(y,s,x)+w\cdot
s_{2}(s,x)-gw\cdot s_{2}(s,x)+gb(x)$ $\displaystyle=gw\cdot
s_{1}(y,s,x)+(1-g)w\cdot s_{2}(s,x)+g\cdot b(x)$
$\displaystyle=\frac{gw(y-\mu(s,x))}{\pi\rho(s,x)}+\frac{(1-g)w}{(1-\gamma(x))\varrho(x)}a(s,x)+g\cdot
b(x)\leavevmode\nobreak\ .$ (A.11)

In particular, $s_{4}$ is chosen so that certain terms cancel to construct the
term $(1-g)ws_{2}(s,x)$, which does not, prima facie, belong in the tangent
space (A.2).

In order to ensure that $\psi_{1,1}(b_{1})$ belongs to the tangent space, we
require

$\displaystyle\mathbb{E}_{P}[a(S,X)\mid W=1,G=0,X=x]$
$\displaystyle=\mathbb{E}_{P_{\star}}[a(S^{*},X)\mid
X=x]=0\leavevmode\nobreak\ \text{, and}$ (A.12)
$\displaystyle\mathbb{E}_{P}[G\cdot b(X)]$
$\displaystyle=0\leavevmode\nobreak\ .$ (A.13)

In the sequel, we verify that the choices

$\displaystyle a(s,x)$
$\displaystyle=\frac{\gamma(x)}{\pi}(\mu(s,x)-\overline{\mu}(x))\leavevmode\nobreak\
\text{, and}$ $\displaystyle b(x)$
$\displaystyle=\frac{\overline{\mu}(x)-\theta_{1,1}}{\pi}$

satisfy these conditions, and, moreover, give rise to the efficient influence
function by satisfying the conditions (A.5)–(A.9).

Verifying Pathwise Differentiability Conditions. We begin by stating and
proving a useful lemma for handling terms of the form $GWh(Y,S,X)$.

###### Lemma A.1.

Under the assumptions of Theorem 3.1, if $h(y,s,x)$ is any measurable
function, then

$\mathbb{E}_{P}\left[{\frac{GW}{\rho(S,X)}h(Y,S,X)}\right]=\mathbb{E}_{P_{\star}}\left[{Gh(Y^{*},S^{*},X)}\right]=\mathbb{E}_{P_{\star}}\left[{G\mathbb{E}[h(Y^{*},S^{*},X)\mid
S^{*},X]}\right]\leavevmode\nobreak\ .$

###### Proof.

Observe that

$\displaystyle\mathbb{E}_{P}\left[{\frac{GW}{\rho(S,X)}h(Y,S,X)}\right]$
$\displaystyle=\mathbb{E}_{P}\left[{\pi\frac{W}{\rho(S,X)}h(Y,S,X)\mid
G=1}\right]$
$\displaystyle=\pi\mathbb{E}_{P_{\star}}\left[{\frac{W}{\rho(S^{*},X)}h(Y^{*},S^{*},X)\mid
G=1}\right]$
$\displaystyle=\pi\mathbb{E}_{P_{\star}}\bigg{[}\rho^{-1}(S^{*},X)\mathbb{E}_{P_{\star}}[W\mid
S^{*},X,G=1]$
$\displaystyle\quad\quad\quad\mathbb{E}_{P_{\star}}[h(Y^{*},S^{*},X)\mid
S^{*},X,G=1]\mid G=1\bigg{]}$ (Assumption 2.4)
$\displaystyle=\mathbb{E}_{P_{\star}}[\pi h(Y^{*},S^{*},X)\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[Gh(Y^{*},S^{*},X)]\leavevmode\nobreak\
,$

giving the first equality. The second equality then follows from

$\displaystyle\mathbb{E}_{P_{\star}}[Gh(Y^{*},S^{*},X)]$
$\displaystyle=\pi\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[h(Y^{*},S^{*},X)\mid
S^{*},X,G=1]\mid G=1]$
$\displaystyle=\mathbb{E}_{P_{\star}}[G\mathbb{E}_{P_{\star}}[h(Y^{*},S^{*},X)\mid
S^{*},X,G=1]]$
$\displaystyle=\mathbb{E}_{P_{\star}}[G\mathbb{E}_{P_{\star}}[h(Y^{*},S^{*},X)\mid
S^{*},X]]\leavevmode\nobreak\ ,$

completing the proof.

It now suffices to verify the conditions (A.5)–(A.9) for choices of $a(S,X)$
and $b(X)$ that satisfy (A.12) and (A.13). The remainder of the proof verifies
these conditions sequentially.

Condition (A.5): Observe that

$\displaystyle\mathbb{E}_{P}\left[{\psi_{1,1}(B_{1})GW\ell^{\prime}_{*}(Y\mid
S,X)}\right]$
$\displaystyle=\mathbb{E}_{P}\left[{\frac{GWY}{\pi\rho(S,X)}\ell^{\prime}_{*}(Y\mid
S,X)}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\pi^{-1}GY^{*}\ell^{\prime}_{*}(Y^{*}\mid
S^{*},X)}\right]\leavevmode\nobreak\ ,$ (Lemma A.1)

as required.

Condition (A.6): We derive the restrictions on $a(s,x)$ implied by (A.6).
Observe that the inner product from (A.6) can be written

$\displaystyle\mathbb{E}_{P}\left[{\psi_{1,1}(B_{1})\left({W\ell^{\prime}_{*}(S\mid
X)-G(1-W)\frac{\mathbb{E}_{P_{\star}}[\rho(S,X)\ell^{\prime}_{*}(S\mid X)\mid
X]}{1-\mathbb{E}_{P_{\star}}[\rho(S,X)\mid X]}}\right)}\right]$
$\displaystyle\quad=\mathbb{E}_{P}\left[{\frac{GW(Y-\mu(S,X))\ell^{\prime}_{*}(S\mid
X)}{\pi\rho(S,X)}}\right]-\mathbb{E}_{P}\left[{\frac{b(X)G(1-W)\mathbb{E}_{P_{\star}}[\rho(S,X)\ell^{\prime}_{*}(S\mid
X)\mid X]}{1-\mathbb{E}_{P_{\star}}[\rho(S,X)\mid X]}}\right]$
$\displaystyle\quad+\mathbb{E}_{P}\left[{\frac{(1-G)Wa(S,X)}{(1-\gamma(X))\varrho(X)}\ell^{\prime}_{*}(S\mid
X)}\right]+\mathbb{E}_{P}\left[{GWb(X)\ell^{\prime}_{*}(S\mid
X)}\right]\leavevmode\nobreak\ .$

Note that the first term can be eliminated by Lemma A.1:

$\displaystyle\mathbb{E}_{P}\left[{\frac{GW(Y-\mu(S,X))\ell^{\prime}_{*}(S\mid
X)}{\pi\rho(S,X)}}\right]$ $\displaystyle=0\leavevmode\nobreak\ .$

Observe that

$\mathbb{E}_{P_{\star}}[G(1-W)\mid X]=P_{\star}(W\mid
G=1,X)\gamma(X)=(1-\mathbb{E}_{P_{\star}}[\rho(S,X)\mid X])\gamma(X),$

and thus the second term is given by

$\displaystyle\mathbb{E}_{P}\left[{\frac{b(X)G(1-W)\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)\mid X]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid X]}}\right]$
$\displaystyle\quad=\mathbb{E}_{P_{\star}}\left[{\gamma(X)b(X)\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)\mid X]}\right]$
$\displaystyle\quad=\mathbb{E}_{P_{\star}}[\gamma(X)b(X)\rho(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)]\leavevmode\nobreak\ .$

Next, observe that

$\mathbb{E}_{P_{\star}}[(1-G)W\mid S^{*},X]=(1-\gamma(X))P_{\star}(W=1\mid
G=0,S^{*},X)=(1-\gamma(X))\varrho(X),$

and therefore the third term can be written

$\mathbb{E}_{P}\left[{\frac{(1-G)W}{(1-\gamma(X))\varrho(x)}a(S,X)\ell^{\prime}_{*}(S\mid
X)}\right]=\mathbb{E}_{P_{\star}}\left[{a(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)}\right]\leavevmode\nobreak\ .$

The fourth term is given by

$\mathbb{E}_{P}[GWb(X)\ell^{\prime}_{*}(S\mid
X)]=\mathbb{E}_{P_{\star}}[\gamma(X)q(S^{*},X)b(X)\ell^{\prime}_{*}(S\mid
X)]\leavevmode\nobreak\ ,$

which cancels with the second term.

Thus, the condition (A.6) is equivalent to

$\mathbb{E}_{P_{\star}}[\ell^{\prime}_{*}(S^{*}\mid
X)a(S^{*},X)]=\mathbb{E}_{P_{\star}}\left[{\pi^{-1}GY^{*}\ell^{\prime}_{*}(S^{*}\mid
X))}\right].$ (A.14)

We proceed by verifying this condition for a choice of $a(S,X)$ that satisfies
(A.12), as required. Consider the function

$a(s,x)=\frac{\gamma(x)}{\pi}(\mu(s,x)-\overline{\mu}(x))\leavevmode\nobreak\
.$

Then $\mathbb{E}_{P_{\star}}[a(S^{*},X)\mid X]=0$, which satisfies (A.12).
Moreover,

$\displaystyle\mathbb{E}_{P_{\star}}[a(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{\gamma(X)}{\pi}\mu(S^{*},X)\ell^{\prime}_{*}(S^{*}\mid
X)}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{\mathbb{E}_{P_{\star}}[G\mid
S^{*},X]}{\pi}\mathbb{E}_{P_{\star}}\left[{Y^{*}\ell^{\prime}_{*}(S^{*}\mid
X)\mid S^{*},X}\right]}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\pi^{-1}GY^{*}\ell_{*}^{\prime}(S^{*}\mid
X)}\right]\leavevmode\nobreak\ ,$

where the second to last equality follows by Assumption 2.3, which verifies
(A.14) and thereby (A.6).

Condition (A.7): We derive the restrictions on $b(x)$ implied by (A.7). Note
that the conjectured influence function (A.11) can be written as

$\psi(b_{1})=gw\cdot\alpha(y,s,x)+(1-g)w\cdot\beta(s,x)+gb(x),$

where

$\displaystyle\mathbb{E}_{P}[\alpha(Y,S,X)\mid S=s,X=x,G=1,W=1]$
$\displaystyle=\mathbb{E}_{P}[\beta(S,X)\mid
G=0,W=1,X=x]=0\leavevmode\nobreak\ .$

We first show that the terms in (A.7) involving $\alpha$ and $\beta$ are zero.
Indeed, for terms involving $\alpha$, we have that

$\mathbb{E}_{P}[GW\cdot\alpha(Y,S,X)\ell^{\prime}_{*}(G,X)]=\pi\mathbb{E}_{P}[\alpha(Y,S,X)\ell^{\prime}_{*}(G,X)\mid
G=1,W=1]P(W=1\mid G=1)=0$

since $\mathbb{E}_{P}[\alpha(Y,S,X)\ell^{\prime}_{*}(G,X;0)\mid G=1,W=1]=0$.
Analogously, we have that

$\mathbb{E}_{P}[(1-G)W\beta(S,X)\ell^{\prime}_{*}(G,X;0)]=0.$

Therefore, (A.7) is equivalent to

$\frac{1}{\pi}\mathbb{E}_{P_{\star}}[GY^{*}\ell^{\prime}_{*}(G,X)]-\theta_{1,1}\mathbb{E}[\ell^{\prime}_{*}(G,X)\mid
G=1]=\mathbb{E}_{P}[Gb(X)\ell^{\prime}_{*}(G,X)].$ (A.15)

We proceed by verifying this condition for a choice of $b(X)$.

Consider the function

$b(x)=\frac{1}{\pi}(\overline{\mu}(x)-\theta_{1,1})\leavevmode\nobreak\ ,$

which satisfies (A.13). We can compute

$\displaystyle\mathbb{E}_{P}[G\pi^{-1}\mu(X)\ell^{\prime}_{*}(G,X)]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{G}{\pi}\mathbb{E}_{P_{\star}}[\mu(S^{*},X)\mid
G=0,W=1,X]\ell^{\prime}_{*}(G,X;0)]}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{G}{\pi}\ell^{\prime}_{*}(G,X;0)\mathbb{E}_{P_{\star}}[\mathbb{E}_{P_{\star}}[Y\mid
W=1,G=1,S^{*},X]\mid X]}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{\gamma(X)}{\pi}\ell^{\prime}_{*}(G,X;0)\mathbb{E}_{P_{\star}}\left[{\mathbb{E}_{P_{\star}}\left[{Y^{*}\mid
S^{*},X}\right]\mid X}\right]}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{\gamma(X)}{\pi}\ell^{\prime}_{*}(G,X;0)Y^{*}}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{\mathbb{E}_{P_{\star}}[G\mid
Y^{*},X]}{\pi}\ell^{\prime}_{*}(G,X;0)Y^{*}}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{\frac{G}{\pi}Y^{*}\ell^{\prime}_{*}(G,X;0)}\right],$

and

$-\mathbb{E}_{P_{\star}}[G\pi^{-1}\theta_{1,1}\ell^{\prime}_{*}(G,X;0)]=-\theta_{1,1}\mathbb{E}_{P_{\star}}[\ell^{\prime}_{*}(1,X;0)\mid
G=1]$

which verifies (A.15) and thereby (A.7).

Condition (A.8): Since the first and second terms $\alpha(Y,S,X)$ and
$\beta(S,X)$ in the conjectured efficient influence function (A.11) are mean
zero conditional on $S^{*}$ and $X$ and $X$, respectively, the right-hand side
of (A.8) is equal to

$\displaystyle\mathbb{E}_{P}\left[{b(X)\left({GW\frac{\rho^{\prime}(S,X)}{\rho(S,X)}-G(1-W)\frac{\mathbb{E}_{P_{\star}}[\rho^{\prime}(S^{*},X)\mid
X]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid X]}}\right)}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{b(X)\gamma(X)\mathbb{E}_{P_{\star}}\left[{\rho(S^{*},X)\frac{\rho^{\prime}(S^{*},X)}{\rho(S^{*},X)}-(1-\rho(S^{*},X))\frac{\mathbb{E}_{P_{\star}}[\rho^{\prime}(S^{*},X)\mid
X]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid X]}\mid
G=1,S^{*},X}\right]}\right]$
$\displaystyle=\mathbb{E}_{P_{\star}}\left[{b(X)\gamma(X)\left({\rho^{\prime}(S^{*},X)-(1-\rho(S^{*},X))\frac{\mathbb{E}_{P_{\star}}[\rho^{\prime}(S^{*},X)\mid
X]}{1-\mathbb{E}_{P_{\star}}[\rho(S^{*},X)\mid X]}}\right)}\right]=0$

by iterated expectation conditional on $X$.

Condition (A.9): The only term on the right-hand side of (A.9) is

$\mathbb{E}_{P}[(1-G)W\cdot\beta(S,X)\cdot(W-\rho(X))\delta(X)]$

for some function $\delta(\cdot)$, where $(1-G)W\cdot\beta(S,X)$ is the second
term in the conjectured efficient influence function (A.11). We have:

$\displaystyle\mathbb{E}_{P}[(1-G)W\cdot\beta(S,X)\cdot(W-\varrho(X))\cdot\delta(X)]$
$\displaystyle=\mathbb{E}_{P}[\mathbb{E}_{P}[W\cdot\beta(S,X)\cdot(W-\varrho(X))\cdot\delta(X)\mid
G=0,X](1-\gamma(X))]$
$\displaystyle=\mathbb{E}_{P_{\star}}[\beta(S^{*},X)\varrho(X)(1-\varrho(X))\cdot\delta(X)]=0$

by the condition $\mathbb{E}_{P_{\star}}[\beta(S^{*},X)\mid X]=0$, completing
the proof.

###  A.4. Proof of Theorem 3.1. Part 2

We maintain the notation of the proof of Theorem 3.1, Part 1. In particular,
we omit the $w$ subscript in the nuisance function $\overline{\nu}_{w}(x)$ and
let

$A_{1}=(Y(1),S(1),W,G,X)=(Y^{*},S^{*},W,G,X)$

and $B_{1}=(Y,S,W,G,X)$ denote the complete and observed data, respectively.
The efficient influence function we construct is

$\displaystyle\xi_{1,1}(b)$
$\displaystyle=\frac{g}{\pi}\left(\frac{\gamma(x)}{\gamma(s,x)}\frac{1-\gamma(s,x)}{1-\gamma(x)}\frac{\varrho(s,x)(y-\nu(S,X))}{\varrho(x)}+(\overline{\nu}_{1}(x)-\theta_{1,1})\right)$
$\displaystyle+\frac{1-g}{\pi}\frac{\gamma(x)}{1-\gamma(x)}\left(\frac{w(\nu(S,X)-\overline{\nu}_{1}(x))}{\varrho(x)}\right)\leavevmode\nobreak\
.$

Density Factorization. By Assumptions 2.1, 2.6, 2.3, and 2.5, the density $p$
of the observed data distribution admits the factorization

$\displaystyle p(b_{1})=\,\,$ $\displaystyle
p(x)\gamma(x)^{g}(1-\gamma(x))^{1-g}$ $\displaystyle p(y\mid
G=1,s,x)^{g}p(s\mid G=1,x)^{g}$ $\displaystyle p(s\mid
G=0,W=1,x)^{w(1-g)}(\varrho(x)^{w}(1-\varrho(x))^{1-w})^{1-g}\leavevmode\nobreak\
.$ (A.16)

Characterization of the Tangent Space. Let $\mathcal{P}$ be any regular
parametric submodel of $\mathcal{M}_{\lambda}$ indexed by
$\varepsilon\in\mathbb{R}$, with densities
$p_{\varepsilon}:=dP_{\varepsilon}/d\lambda$ with $p_{0}=dP/d\lambda$ and such
that Assumptions 2.1, 2.6, 2.3, and 2.5 hold for each
$P_{\varepsilon}\in\mathcal{P}$. By the factorization (A.4), we obtain the
score

$\displaystyle\ell^{\prime}(b_{1})=\,\,$ $\displaystyle g\ell^{\prime}(y\mid
s,x,G=1)+g\ell^{\prime}(s\mid x,G=1)+w(1-g)\ell^{\prime}(s\mid
W=1,x,G=0)+\ell^{\prime}(x)$ $\displaystyle+$
$\displaystyle\frac{g-\gamma(x)}{\gamma(x)(1-\gamma(x))}\gamma^{\prime}(x)+(1-g)\frac{w-\varrho(x)}{\varrho(x)(1-\varrho(x))}\varrho^{\prime}(x).$
(A.17)

Thus, the tangent space $\mathcal{T}$ is given by the mean-square closure of
the linear span of the functions

$\displaystyle s(b_{1})=\,\,$ $\displaystyle g\cdot s_{1}(y\mid
s,x,G=1)+g\cdot s_{2}(s\mid x,G=1)+w(1-g)\cdot s_{3}(s\mid
W=1,x,G=0)+s_{4}(x)$ $\displaystyle+$
$\displaystyle\frac{g-\gamma(x)}{\gamma(x)(1-\gamma(x))}s_{5}(x)+(1-g)\frac{w-\varrho(x)}{\varrho(x)(1-\varrho(x))}s_{6}(x)\leavevmode\nobreak\
,$ (A.18)

where the functions $s_{1}$ through $s_{6}$ range over the space of mean-zero
and square integrable functions that satisfy the restrictions

$\displaystyle 0$ $\displaystyle=\mathbb{E}_{P}[s_{1}(Y\mid S,X,G=1)\mid
S,X,G=1]$ $\displaystyle=\mathbb{E}_{P}[s_{1}(Y\mid S,X,G=1)\mid
S,X]\leavevmode\nobreak\ ,$ (A.19) $\displaystyle 0$
$\displaystyle=\mathbb{E}_{P}[s_{2}(S\mid X,G=1)\mid
X,G=1]\leavevmode\nobreak\ ,$ (A.20) $\displaystyle 0$
$\displaystyle=\mathbb{E}_{P}[s_{3}(S\mid W=1,X,G=0)\mid W=1,X,G=0]$
$\displaystyle=\mathbb{E}_{P_{\star}}[s_{3}(S^{*}\mid W=1,X,G=0)\mid
X,G=0]\leavevmode\nobreak\ ,$ (A.21) $\displaystyle 0$
$\displaystyle=\mathbb{E}_{P}[s_{4}(X)]\leavevmode\nobreak\ ,$ (A.22)

and $s_{5},$ and $s_{6}$ are unconstrained, where (A.19) and (A.21) follow
from Assumption 2.6 and Assumption 2.1, respectively.

Pathwise Differentiability Representation. Recall from the proof of
Proposition 2.1 that, by Assumptions 2.1, 2.6, 2.3, and 2.5, the target
parameter $\theta_{1,1}$ can be written as

$\displaystyle\theta_{1,1}$ $\displaystyle=\mathbb{E}_{P_{\star}}[Y^{*}\mid
G=1]$
$\displaystyle=\mathbb{E}_{P}\left[{\mathbb{E}_{P}\left[{\mathbb{E}_{P}[Y\mid
S,X,G=1]\mid W=1,X,G=0}\right]\mid G=1}\right]$ $\displaystyle=\iiint yp(y\mid
s,x,G=1)p(s\mid x,W=1,G=0)p(x\mid
G=1)\,d\lambda(y)\,\,d\lambda(s)\,\,d\lambda(x)\leavevmode\nobreak\ .$

The pathwise derivative of this parameter is given by

$\displaystyle\theta_{1,1}^{\prime}=\mathbb{E}_{P}[\mathbb{E}_{P}[\mathbb{E}_{P}[YU(Y,S,X)\mid
S,X,G=1]\mid X,W=1,G=0]\mid G=1]$

with

$\displaystyle U(y,s,x)$ $\displaystyle=\ell^{\prime}(y\mid
s,x,G=1)+\ell^{\prime}(s\mid x,W=1,G=0)+\ell^{\prime}(x\mid G=1)$
$\displaystyle=\ell^{\prime}(y\mid s,x,G=1)+\ell^{\prime}(s\mid x,W=1,G=0)$
$\displaystyle+\ell^{\prime}(x)+\frac{\gamma^{\prime}(x)}{\gamma(x)}-\frac{\mathbb{E}_{P}[\gamma^{\prime}(X)+\gamma(X)\ell^{\prime}(X)]}{\pi}\leavevmode\nobreak\
,$

where final three terms follow from

$\displaystyle p^{\prime}(x\mid G=1)=\ell^{\prime}(x\mid G=1)p(x\mid G=1)$

and Bayes’ rule in the form of

$\ell(x\mid
G=1)=\log\gamma(x)+\ell(x)-\log\int\gamma(x)p(x)\,d\lambda(x)\leavevmode\nobreak\
.$

Observe that, using the definitions of $\nu(s,x)$ and $\overline{\nu}(x)$, we
may further simplify the pathwise derivative to give

$\displaystyle\theta_{1,1}^{\prime}=\,\,$
$\displaystyle\mathbb{E}_{P}[\mathbb{E}_{P}[\mathbb{E}_{P}[Y\ell^{\prime}(Y\mid
S,X,G=1)\mid S,X,G=1]\mid X,W=1,G=0]\mid G=1]$
$\displaystyle+\mathbb{E}_{P}[\mathbb{E}_{P}[\nu(S,X)\ell^{\prime}(S\mid
X,G=0,W=1)\mid X,W=1,G=0]\mid G=1]$
$\displaystyle+\pi^{-1}\mathbb{E}_{P}\left[{\overline{\nu}(X)\left({\gamma^{\prime}(X)+\gamma(X)\ell^{\prime}(X)}\right)}\right]-\pi^{-1}\mathbb{E}_{P}[\gamma^{\prime}(X)+\gamma(X)\ell^{\prime}(X)]\theta_{1,1}.$
(A.23)

An influence function for $\theta_{1,1}$ is a mean-zero and square-integrable
function $\tilde{\xi}_{1,1}(B_{1})$ that satisfies the condition

$\theta^{\prime}_{1,1}=\mathbb{E}[\tilde{\xi}(B_{1})\ell^{\prime}(B_{1})]\leavevmode\nobreak\
.$ (A.24)

We make a conjecture for such a function and verify that it satisfies this
condition and is an element of the tangent space $\mathcal{T}$, establishing
that it is the efficient influence function.

Conjectured Efficient Influence Function. We conjecture that the efficient
influence function takes the form

$\xi_{1,1}(b_{1})=g(y-\nu(s,x))\cdot
f_{1}(s,x)+w(1-g)(\nu(s,x)-\overline{\nu}(x))\cdot f_{2}(x)+g\cdot
f_{3}(x)\leavevmode\nobreak\ ,$ (A.25)

for functions $f_{1}(s,x)$, $f_{2}(x)$, and $f_{3}(x)$ to be specified, where
$f_{3}$ will be chosen to satisfy

$\mathbb{E}_{P}[f_{3}(X)\mid
G=1]=0=\mathbb{E}_{P}[f_{3}(X)\gamma(X)]\leavevmode\nobreak\ .$ (A.26)

Observe that for this choice, $\xi_{1,1}(b)$ will be an element of the tangent
space $\mathcal{T}$. This follows as

$s_{1}(y\mid s,x,G=1)=(y-\nu(s,x))f_{1}(s,x)$

satisfies (A.19), we make the choice $s_{2}(s\mid X,G=1)=0$ satisfying (A.20),

$s_{3}(s\mid W=1,x,G=0)=(\nu(s,x)-\overline{\nu}(x))f_{2}(x)$

satisfies (A.21), and that setting $s_{4}(x)=\gamma(x)f_{3}(x)$ and
$s_{5}(x)=\gamma(x)(1-\gamma(x))f_{3}(x)$ yields

$gf_{3}(x)=s_{4}(x)+\frac{g-\gamma(x)}{\gamma(x)(1-\gamma(x))}s_{5}(x)\leavevmode\nobreak\
,$

satisfying (A.22), the mean zero conditions for $s_{4}$ and $s_{5}$ by (A.26),
and implicitly satisfying (A.20) and the mean zero condition for $s_{6}$.

Verification of Pathwise Differentiability Representation. In the sequel, we
make choices of $f_{1}(s,x)$, $f_{2}(x)$, and $f_{3}(x)$ such that the
pathwise derivative of $\theta_{1,1}$ satisfies (A.24) and (A.26), completing
the proof.

Our conjecture for $\xi_{1,1}(b_{1})$ allows for some simplification of the
right-hand side of (A.24). Note that, for any measurable functions $h(S,X)$
and $k(X)$, we have

$\displaystyle\mathbb{E}_{P}[G(Y-\nu(S,X))f_{1}(S,X)h(S,X)]=0=\mathbb{E}_{P}[(1-G)W(\nu(S,X)-\overline{\nu}(X))f_{2}(X)k(X)]$

by iterated expectations. Thus, by additionally applying the mean-zero
property of the scores $\ell^{\prime}(Y\mid S,X,G=1)$, $\ell^{\prime}(S\mid
X,G=1)$, we can simplify the pathwise differentiability condition to give

$\displaystyle\mathbb{E}_{P}\left[{\xi_{1,1}(B_{1})\ell^{\prime}(B_{1})}\right]$
$\displaystyle=\mathbb{E}_{P}\left[{G(Y-\nu(S,X))f_{1}(S,X)\ell^{\prime}(Y\mid
S,X,G=1)}\right]$
$\displaystyle+\mathbb{E}_{P}\left[{W(1-G)(\nu(S,X)-\overline{\nu}(X))f_{2}(X)\ell^{\prime}(S\mid
W=1,X,G=0)}\right]$
$\displaystyle+\mathbb{E}_{P}\left[{\left({G\ell^{\prime}(X)+\gamma^{\prime}(X)}\right)f_{3}(X)}\right]\leavevmode\nobreak\
.$ (A.27)

Therefore, we obtain a set of sufficient conditions for (A.24) by matching
each term in (A.4) with (A.23), giving

$\displaystyle\mathbb{E}_{P}\left[{G(Y-\nu(S,X))f_{1}(S,X)\ell^{\prime}(Y\mid
S,X,G=1)}\right]$ (A.28)
$\displaystyle\quad\quad\quad=\mathbb{E}_{P}[\mathbb{E}_{P}[\mathbb{E}_{P}[Y\ell^{\prime}(Y\mid
S,X,G=1)\mid S,X,G=1]\mid X,W=1,G=0]\mid G=1]$
$\displaystyle\mathbb{E}_{P}\left[{W(1-G)(\nu(S,X)-\overline{\nu}(X))f_{2}(X)\ell^{\prime}(S\mid
W=1,G=0,X)}\right]$ (A.29)
$\displaystyle\quad\quad\quad=\mathbb{E}_{P}[\mathbb{E}_{P}[\mu(S,X)\ell^{\prime}(S\mid
X,W=1,G=0)\mid X,W=1,G=0]\mid G=1]$
$\displaystyle\mathbb{E}_{P}\left[{\left({G\ell^{\prime}(X)+\gamma^{\prime}(X)}\right)f_{3}(X)}\right]$
(A.30)
$\displaystyle\quad\quad\quad=\mathbb{E}_{P}\left[{\pi^{-1}\overline{\nu}(X)\left({\gamma^{\prime}(X)+\ell^{\prime}(X)\gamma(X)}\right)}\right]-\pi^{-1}\mathbb{E}_{P}[\gamma^{\prime}(X)+\gamma(X)\ell^{\prime}(X)]\theta_{1,1}.$

We complete the proof by considering each condition sequentially, making
appropriate choices of $f_{1}(s,x)$, $f_{2}(x)$, and $f_{3}(x)$, where we note
that each function appears in only one term.

Condition (A.28): Evaluating the left-hand side of (A.28) we find that

$\displaystyle\mathbb{E}_{P}\left[{G(Y-\nu(S,X))f_{1}(S,X)\ell^{\prime}(Y\mid
S,X,G=1)}\right]$
$\displaystyle=\mathbb{E}_{P}[GYf_{1}(S,X)\ell^{\prime}(Y\mid S,X,G=1)]$
$\displaystyle=\mathbb{E}_{P}[\gamma(X)\mathbb{E}_{P}[f_{1}(S,X)\mathbb{E}_{P}[Y\ell^{\prime}(Y\mid
S,X,G=1)\mid S,X,G=1]\mid X,G=1]].$

Thus, if we choose