# Efficient adjustment for complex covariates:
Gaining efficiency with DOPE

Alexander Mangulad Christgau♯<EMAIL_ADDRESS>and  Niels Richard Hansen♯
<EMAIL_ADDRESS>♯Department of Mathematical Sciences, University of
Copenhagen
Universitetsparken 5, Copenhagen, 2100, Denmark

###### Abstract.

Covariate adjustment is a ubiquitous method used to estimate the average
treatment effect (ATE) from observational data. Assuming a known graphical
structure of the data generating model, recent results give graphical criteria
for optimal adjustment, which enables efficient estimation of the ATE.
However, graphical approaches are challenging for high-dimensional and complex
data, and it is not straightforward to specify a meaningful graphical model of
non-Euclidean data such as texts. We propose an general framework that
accommodates adjustment for any subset of information expressed by the
covariates. We generalize prior works and leverage these results to identify
the optimal covariate information for efficient adjustment. This information
is minimally sufficient for prediction of the outcome conditionally on
treatment.

Based on our theoretical results, we propose the Debiased Outcome-adapted
Propensity Estimator (DOPE) for efficient estimation of the ATE, and we
provide asymptotic results for the DOPE under general conditions. Compared to
the augmented inverse propensity weighted (AIPW) estimator, the DOPE can
retain its efficiency even when the covariates are highly predictive of
treatment. We illustrate this with a single-index model, and with an
implementation of the DOPE based on neural networks, we demonstrate its
performance on simulated and real data. Our results show that the DOPE
provides an efficient and robust methodology for ATE estimation in various
observational settings.

##  1\. Introduction

Estimating the population average treatment effect (ATE) of a treatment on an
outcome variable is a fundamental statistical task. A naive approach is to
contrast the mean outcome of a treated population with the mean outcome of an
untreated population. Using observational data this is, however, generally a
flawed approach due to confounding. If the underlying confounding mechanisms
are captured by a set of pre-treatment covariates $\mathbf{W}$, it is possible
to adjust for confounding by conditioning on $\mathbf{W}$ in a certain manner.
Given that multiple subsets of $\mathbf{W}$ may be valid for this adjustment,
it is natural to ask if there is an ‘optimal adjustment subset’ that enables
the most efficient estimation of the ATE.

Assuming a causal linear graphical model, Henckel et al. (2022) established
the existence of – and gave graphical criteria for – an _optimal adjustment
set_ for the OLS estimator. Rotnitzky & Smucler (2020) extended the results of
Henckel et al. (2022), by proving that the optimality was valid within general
causal graphical models and for all regular and asymptotically linear
estimators. Critically, this line of research assumes knowledge of the
underlying graphical structure.

To accommodate the assumption of no unmeasured confounding, observational data
is often collected with as many covariates as possible, which means that
$\mathbf{W}$ can be high-dimensional. In such cases, assumptions of a known
graph are unrealistic, and graphical estimation methods are statistically
unreliable (Uhler et al., 2013; Shah & Peters, 2020; Chickering et al., 2004).
Furthermore, for non-Euclidean data such as images or texts, it is not clear
how to impose any graphical structure pertaining to causal relations.
Nevertheless, we can in these cases still imagine that the information that
$\mathbf{W}$ represents can be separated into distinct components that affect
treatment and outcome directly, as illustrated in Figure 1.

$T$Treatment$Y$Outcome$\mathbf{W}$$\mathbf{C}_{1}$$\mathbf{C}_{2}$$\mathbf{C}_{3}$
Figure 1. The covariate $\mathbf{W}$ can have a complex data structure, even
if the information it represents is structured and can be categorized into
components that influence treatment and outcome separately.

In this paper, we formalize this idea and formulate a novel and general
adjustment theory with a focus on efficiency bounds for the estimation of the
average treatment effect. Based on the adjustment theory, we propose a general
estimation procedure and analyze its asymptotic behavior.

###  1.1. Setup

Throughout we consider a discrete treatment variable $T\in\mathbb{T}$, a
square-integrable outcome variable $Y\in$, and pre-treatment covariates
$\mathbf{W}\in\mathbb{W}$. For now we only require that $\mathbb{T}$ is a
finite set and that $\mathbb{W}$ is a measurable space. The joint distribution
of $(T,\mathbf{W},Y)$ is denoted by $P$ and it is assumed to belong to a
collection of probability measures $\mathcal{P}$. If we need to make the joint
distribution explicit, we denote expectations and probabilities with
$\mathbb{E}_{P}$ and $\mathbb{P}_{P}$, respectively, but usually the $P$ is
omitted for ease of notation.

Our model-free target parameters of interest are of the form

(1)
$\displaystyle\chi_{t}\coloneq\mathbb{E}[\mathbb{E}[Y\,|\,T=t,\mathbf{W}]]=\mathbb{E}\left[\frac{\mathbbl{1}(T=t)Y}{\mathbb{P}(T=t\,|\,\mathbf{W})}\right],\qquad
t\in\mathbb{T}.$

In words, these are treatment specific means of the outcome when adjusting for
the covariate $\mathbf{W}$. To ensure that this quantity is well-defined, we
assume the following condition, commonly known as positivity.

######  Assumption 1.1 (Positivity).

It holds that $0<\mathbb{P}(T=t\,|\,\mathbf{W})<1$ almost surely for each
$t\in\mathbb{T}$.

Under additional assumptions common in causal inference literature – which
informally entail that $\mathbf{W}$ captures all confounding – the target
parameter $\chi_{t}$ has the interpretation as the _interventional mean_ ,
which is expressed by $\mathbb{E}[Y\,|\,\mathrm{do}(T=t)]$ in do-notation or
by $\mathbb{E}[Y^{t}]$ using potential outcome notation (Peters et al., 2017;
van der Laan & Rose, 2011). Under such causal assumptions, the average
treatment effect is identified, and when $\mathbb{T}=\\{0,1\\}$ it is
typically expressed as the contrast $\chi_{1}-\chi_{0}$. The theory in this
paper is agnostic with regards to whether or not $\chi_{t}$ has this causal
interpretation, although it is the primary motivation for considering
$\chi_{t}$ as a target parameter.

Given $n$ i.i.d. observations of $(T,\mathbf{W},Y)$, one may proceed to
estimate $\chi_{t}$ by estimating either of the equivalent expressions for
$\chi_{t}$ in Equation (1). Within parametric models, the outcome regression
function

$g(t,\mathbf{w})=\mathbb{E}[Y\,|\,T=t,\mathbf{W}=\mathbf{w}]$

can typically be estimated with a $\sqrt{n}$-rate. In this case, the sample
mean of the estimated regression function yields a $\sqrt{n}$-consistent
estimator of $\chi_{t}$ under Donsker class conditions or sample splitting.
However, many contemporary datasets indicate that parametric model-based
regression methods get outperformed by nonparametric methods such as boosting
and neural networks (Bojer & Meldgaard, 2021).

Nonparametric estimators of the regression function typically converge at
rates slower than $\sqrt{n}$, and likewise for estimators of the propensity
score

$m(t\mid\mathbf{w})=\mathbb{P}(T=t\,|\,\mathbf{W}=\mathbf{w}).$

Even if both nonparametric estimators have rates slower than $\sqrt{n}$, it is
in some cases possible to achieve a $\sqrt{n}$-rate of $\chi_{t}$ by modeling
both $m$ and $g$, and then combining their estimates in a way that achieves
‘rate double robustness’ (Smucler et al., 2019). That is, an estimation error
of the same order as the product of the errors for $m$ and $g$. Two prominent
estimators that have this property are the Augmented Inverse Probability
Weighted estimator (AIPW) and the Targeted Minimum Loss-based Estimator (TMLE)
(Robins & Rotnitzky, 1995; Chernozhukov et al., 2018; van der Laan & Rose,
2011).

In what follows, the premise is that even with a $\sqrt{n}$-rate estimator of
$\chi_{t}$, it might still be intractable to model $m$ – and possibly also $g$
– as a function of $\mathbf{W}$ directly. This can happen for primarily two
reasons: (i) the sample space $\mathbb{W}$ is high-dimensional or has a
complex structure, or (ii) the covariate is highly predictive of treatment,
leading to unstable predictions of the inverse propensity score
$\mathbb{P}(T=t\,|\,\mathbf{W})^{-1}$.

In either of these cases, which are not exclusive, we can try to manage these
difficulties by instead working with a _representation_
$\mathbf{Z}=\varphi(\mathbf{W})$, given by a measurable mapping $\varphi$ from
$\mathbb{W}$ into a more tractable space such as d. In the first case above,
such a representation might be a pre-trained word embedding, e.g., the
celebrated BERT and its offsprings (Devlin et al., 2018). The second case has
been well-studied in the special case where $\mathbb{W}=^{k}$ and where
$\mathcal{P}$ contains the distributions that are consistent with respect to a
fixed DAG (or CPDAG). We develop a general theory that subsumes both cases,
and we discuss how to represent the original covariates to efficiently
estimate the adjusted mean $\chi_{t}$.

###  1.2. Relations to existing literature

Various studies have explored the adjustment for complex data structures by
utilizing a (deep) representation of the covariates, as demonstrated in works
such as Shi et al. (2019); Veitch et al. (2020). In a different research
direction, the Collaborative TMLE (van der Laan & Gruber, 2010) has emerged as
a robust method for estimating average treatment effects by collaboratively
learning the outcome regression and propensity score, particularly in
scenarios where covariates are highly predictive of treatment (Ju et al.,
2019). Our overall estimation approach shares similarities with the mentioned
strategies; for instance, our proof-of-concept estimator in the experimental
section employs neural networks with shared layers. However, unlike the cited
works, it incorporates the concept of efficiently tuning the representation
specifically for predicting outcomes, rather than treatment. Related to this
idea is another interesting line of research, which builds upon the outcome
adapted lasso proposed by Shortreed & Ertefaie (2017). Such works include Ju
et al. (2020); Benkeser et al. (2020); Greenewald et al. (2021); Baldé et al.
(2023). These works all share the common theme of proposing estimation
procedures that select covariates based on $L_{1}$-penalized regression onto
the outcome, and then subsequently estimate the propensity score based on the
selected covariates adapted to the outcome. The theory of this paper
generalizes the particular estimators proposed in the previous works, and also
allows for other feature selection methods than $L_{1}$-penalization.
Moreover, our generalization of (parts of) the efficient adjustment theory
from Rotnitzky & Smucler (2020) allows us to theoretically quantify the
efficiency gains from these estimation methods. Finally, our asymptotic theory
considers a novel regime, which, according to the simulations, seems more
adequate for describing the finite sample behavior than the asymptotic results
of Benkeser et al. (2020) and Ju et al. (2020).

Our general adjustment results in Section 3 draws on the vast literature on
classical adjustment and confounder selection, for example Rosenbaum & Rubin
(1983); Hahn (1998); Henckel et al. (2022); Rotnitzky & Smucler (2020); Guo et
al. (2022); Perković et al. (2018); Peters et al. (2017); Forré & Mooij
(2023). In particular, two of our results are direct extensions of results
from Rotnitzky & Smucler (2020); Henckel et al. (2022).

###  1.3. Organization of the paper

In Section 2 we discuss generalizations of classical adjustment concepts to
abstract conditioning on information. In Section 3 we discuss information
bounds in the framework of Section 2. In Section 4 we propose a novel method,
the DOPE, for efficient estimation of adjusted means, and we discuss the
asymptotic behavior of the resulting estimator. In Section 5 we implement the
DOPE and demonstrate its performance on synthetic and real data. The paper is
concluded by a discussion in Section 6.

##  2\. Generalized adjustment concepts

In this section we discuss generalizations of classical adjustment concepts.
These generalizations are motivated by the premise from the introduction: it
might be intractable to model the propensity score directly as a function of
$\mathbf{W}$, so instead we consider adjusting for a representation
$\mathbf{Z}=\varphi(\mathbf{W})$. This is, in theory, confined to statements
about conditioning on $\mathbf{Z}$, and is therefore equivalent to adjusting
for any representation of the form
$\widetilde{\mathbf{Z}}=\psi\circ\varphi(\mathbf{W})$, where $\psi$ is a
bijective and bimeasureable mapping. The equivalence class of such
representations is characterized by the $\sigma$-algebra generated by
$\mathbf{Z}$, denoted by $\sigma(\mathbf{Z})$, which informally describes the
information contained in $\mathbf{Z}$. In view of this, we define adjustment
with respect to sub $\sigma$-algebras contained in $\sigma(\mathbf{W})$.

######  Remark 2.1.

Conditional expectations and probabilities are, unless otherwise indicated,
defined conditionally on $\sigma$-algebras as in Kolmogoroff (1933), see also
Kallenberg (2021, Ch. 8). Equalities between conditional expectations are
understood to hold almost surely. When conditioning on the random variable $T$
and a $\sigma$-algebra $\mathcal{Z}$ we write ‘$\,|\,T,\mathcal{Z}$’ as a
shorthand for ‘$\,|\,\sigma(T)\vee\mathcal{Z}$’. Finally, we define
conditioning on both the event $(T=t)$ and on a $\sigma$-algebra
$\mathcal{Z}\subseteq\sigma(\mathbf{W})$ by
$\mathbb{E}[Y\,|\,T=t,\mathcal{Z}]\coloneq\frac{\mathbb{E}[Y\mathbbl{1}(T=t)\,|\,\mathcal{Z}]}{\mathbb{P}(T=t\,|\,\mathcal{Z})},$
which is well-defined under Assumption 1.1. ∎

######  Definition 2.2.

A sub $\sigma$-algebra $\mathcal{Z}\subseteq\sigma(\mathbf{W})$ is called a
_description_ of $\mathbf{W}$. For each $t\in\mathbb{T}$ and
$P\in\mathcal{P}$, and with $\mathcal{Z}$ a description of $\mathbf{W}$, we
define

$\displaystyle\pi_{t}(\mathcal{Z};P)$
$\displaystyle\coloneq\mathbb{P}_{P}(T=t\,|\,\mathcal{Z}),$ $\displaystyle
b_{t}(\mathcal{Z};P)$
$\displaystyle\coloneq\mathbb{E}_{P}[Y\,|\,T=t,\mathcal{Z}],$
$\displaystyle\chi_{t}(\mathcal{Z};P)$
$\displaystyle\coloneq\mathbb{E}_{P}[b_{t}(\mathcal{Z};P)]=\mathbb{E}_{P}[\pi_{t}(\mathcal{Z};P)^{-1}\mathbbl{1}(T=t)Y].$

If a description $\mathcal{Z}$ of $\mathbf{W}$ is given as
$\mathcal{Z}=\sigma(\mathbf{Z})$ for a representation
$\mathbf{Z}=\varphi(\mathbf{W})$, we may write $\chi_{t}(\mathbf{Z};P)$
instead of $\chi_{t}(\sigma(\mathbf{Z});P)$ etc.

We say that

$\mathcal{Z}$ is $P$-valid if:
$\displaystyle\qquad\chi_{t}(\mathcal{Z};P)=\chi_{t}(\mathbf{W};P),$
$\displaystyle\text{ for all }t\in\mathbb{T},$ $\mathcal{Z}$ is $P$-OMS if:
$\displaystyle\qquad b_{t}(\mathcal{Z};P)=b_{t}(\mathbf{W};P),$
$\displaystyle\text{ for all }t\in\mathbb{T},$ $\mathcal{Z}$ is $P$-ODS if:
$\displaystyle\qquad
Y\,{\perp\\!\\!\\!\perp}_{P}\,\mathbf{W}\,|\,T,\mathcal{Z}.$

Here OMS means _Outcome Mean Sufficient_ and ODS means _Outcome Distribution
Sufficient_. If $\mathcal{Z}$ is $P$-valid for all $P\in\mathcal{P}$, we say
that it is $\mathcal{P}$-valid. We define $\mathcal{P}$-OMS and
$\mathcal{P}$-ODS analogously. ∎

A few remarks are in order.

  * •

We have implicitly extended the shorthand notation described in Remark 2.1 to
the quantities in Definition 2.2.

  * •

The quantity $\chi_{t}(\mathcal{Z};P)$ is deterministic, whereas
$\pi_{t}(\mathcal{Z};P)$ and $b_{t}(\mathcal{Z};P)$ are
$\mathcal{Z}$-measurable real valued random variables. Thus, if $\mathcal{Z}$
is generated by a representation $\mathbf{Z}$, then by the Doob-Dynkin lemma
(Kallenberg, 2021, Lemma 1.14) these random variables can be expressed as
functions of $\mathbf{Z}$. This fact does not play a role until the discussion
of estimation in Section 4.

  * •

There are examples of descriptions $\mathcal{Z}\subseteq\sigma(\mathbf{W})$
that are not given by representations. Such descriptions might be of little
practical importance, but our results do not require $\mathcal{Z}$ to be given
by a representation. We view the $\sigma$-algebraic framework as a convenient
abstraction that generalizes equivalence classes of representations.

  * •

We have the following hierarchy of the properties in Definition 2.2:

$\displaystyle P\textnormal{-ODS}\Longrightarrow
P\textnormal{-OMS}\Longrightarrow P\textnormal{-valid},$

and the relations also hold if $P$ is replaced by $\mathcal{P}$.

  * •

The $P$-ODS condition relates to existing concepts in statistics. The
condition can be viewed as a $\sigma$-algebraic analogue of prognostic scores
(Hansen, 2008), and it holds that any prognostic score generates a $P$-ODS
description. Moreover, if a description $\mathcal{Z}=\sigma(\mathbf{Z})$ is
$P$-ODS, then its generator $\mathbf{Z}$ is $c$-equivalent to $\mathbf{W}$ in
the sense of Pearl (2009), see in particular the claim following his Equation
(11.8). In Remark 3.12 we discuss the relations between $\mathcal{P}$-ODS
descriptions and classical statistical sufficiency.

The notion of $\mathcal{P}$-valid descriptions can be viewed as a
generalization of valid adjustment sets, where subsets are replaced with sub
$\sigma$-algebras.

######  Example 2.3 (Comparison with adjustment sets in causal DAGs).

Suppose $\mathbf{W}\in^{k}$ and let $\mathcal{D}$ be a DAG on the nodes
$\mathbf{V}=(T,\mathbf{W},Y)$. Let $\mathcal{P}=\mathcal{M}(\mathcal{D})$ be
the collection of continuous distributions (on k+2) that are Markovian with
respect to $\mathcal{D}$ and with $\mathbb{E}|Y|<\infty$.

Any subset $\mathbf{Z}\subseteq\mathbf{W}$ is a representation of $\mathbf{W}$
given by a coordinate projection, and the corresponding $\sigma$-algebra
$\sigma(\mathbf{Z})$ is a description of $\mathbf{W}$. In this framework, a
subset $\mathbf{Z}\subseteq\mathbf{W}$ is called a valid adjustment set for
$(T,Y)$ if for all $P\in\mathcal{P}$, $t\in\mathbb{T}$, and $y\in$

$\displaystyle\mathbb{E}_{P}\left[\frac{\mathbbl{1}(T=t)\mathbbl{1}(Y\leq
y)}{\mathbb{P}_{P}(T=t\,|\,\mathrm{pa}_{\mathcal{D}}(T))}\right]=\mathbb{E}_{P}[\mathbb{P}_{P}(Y\leq
y\,|\,T=t,\mathbf{Z})],$

see, e.g., Definition 2 in Rotnitzky & Smucler (2020). It turns out that
$\mathbf{Z}$ is a valid adjustment set if and only if

$\displaystyle\mathbb{E}_{P}\left[\frac{\mathbbl{1}(T=t)Y}{\mathbb{P}_{P}(T=t\,|\,\mathrm{pa}_{\mathcal{D}}(T))}\right]=\mathbb{E}_{P}[\mathbb{E}_{P}[Y\,|\,T=t,\mathbf{Z}]]=\chi_{t}(\mathcal{Z};P)$

for all $P\in\mathcal{P}$ and $t\in\mathbb{T}$. This follows111We thank
Leonard Henckel for pointing this out. from results of Perković et al. (2018),
which we discuss for completeness in Proposition A.1. Thus, if we assume that
$\mathbf{W}$ is a valid adjustment set, then any subset
$\mathbf{Z}\subseteq\mathbf{W}$ is a valid adjustment set if and only if the
corresponding description $\sigma(\mathbf{Z})$ is $\mathcal{P}$-valid. ∎

In general, $\mathcal{P}$-valid descriptions are not necessarily generated
from valid adjustment sets. This can happen if structural assumptions are
imposed on the conditional mean, which is illustrated in the following
example.

######  Example 2.4.

Suppose that the outcome regression is known to be invariant under rotations
of $\mathbf{W}\in^{k}$ such that for any $P\in\mathcal{P}$,

$\displaystyle\mathbb{E}_{P}[Y\,|\,T,\mathbf{W}]=\mathbb{E}_{P}[Y\,|\,T,\|\mathbf{W}\|\,]\eqcolon
g_{P}(T,\|\mathbf{W}\|).$

Without further graphical assumptions, we cannot deduce that any proper subset
of $\mathbf{W}$ is a valid adjustment set. In contrast, the magnitude
$\|\mathbf{W}\|$ generates a $\mathcal{P}$-OMS – and hence also
$\mathcal{P}$-valid – description of $\mathbf{W}$ by definition.

Suppose that there is a distribution $\tilde{P}\in\mathcal{P}$ for which
$g_{\tilde{P}}(t,\cdot)$ is bijective. Then $\sigma(\|\mathbf{W}\|)$ is also
the smallest $\mathcal{P}$-OMS description up to
$\mathbb{P}_{\tilde{P}}$-negligible sets: if $\mathcal{Z}$ is another
$\mathcal{P}$-OMS description, we see that
$b_{t}(\mathcal{Z};\tilde{P})=b_{t}(\mathbf{W};\tilde{P})=g_{\tilde{P}}(t,\|\mathbf{W}\|)$
almost surely. Hence

$\sigma(\|\mathbf{W}\|)=\sigma(g_{\tilde{P}}(t,\|\mathbf{W}\|))\subseteq\overline{\sigma(b_{t}(\mathcal{Z};\tilde{P}))}\subseteq\overline{\mathcal{Z}},$

where overline denotes the $\mathbb{P}_{\tilde{P}}$-completion of a
$\sigma$-algebra. That is, $\overline{\mathcal{Z}}$ is the smallest
$\sigma$-algebra containing $\mathcal{Z}$ and all its
$\mathbb{P}_{\tilde{P}}$-negligible sets. ∎

Even in the above example, where the regression function is known to depend on
a one-dimensional function of $\mathbf{W}$, it is generally not possible to
estimate the regression function at a $\sqrt{n}$-rate without restrictive
assumptions. Thus, modeling of the propensity score is required in order to
obtain a $\sqrt{n}$-rate estimator of the adjusted mean. If $\mathbf{W}$ is
highly predictive of treatment, then naively applying a doubly robust
estimator (AIPW, TMLE) can be statistically unstable due to large inverse
propensity weights. Alternatively, since $\sigma(\|\mathbf{W}\|)$ is a
$\mathcal{P}$-valid description, we could also base our estimator on the
pruned propensity $\mathbb{P}(T=t\,|\,\|\mathbf{W}\|)$. This approach should
intuitively provide more stable weights, as we expect $\|\mathbf{W}\|$ to be
less predictive of treatment. We proceed to analyze the difference between the
asymptotic efficiencies of the two approaches and we show that, under
reasonable conditions, the latter approach is never worse asymptotically.

##  3\. Efficiency bounds for adjusted means

We now discuss efficiency bounds based on the concepts introduced in Section
2. To this end, let $\mathbf{Z}=\varphi(\mathbf{W})$ be a representation of
$\mathbf{W}$. Under a sufficiently dense model $\mathcal{P}$, the influence
function for $\chi_{t}(\mathbf{Z};P)$ is given by

(2)
$\psi_{t}(\mathbf{Z};P)=b_{t}(\mathbf{Z};P)+\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathbf{Z};P)}(Y-b_{t}(\mathbf{Z};P))-\chi_{t}(\mathbf{Z};P).$

The condition that $\mathcal{P}$ is sufficiently dense entails that no
‘structural assumptions’ are imposed on the functional forms of $\pi_{t}$ and
$b_{t}$, see also Robins et al. (1994); Robins & Rotnitzky (1995); Hahn
(1998). Structural assumptions do not include smoothness conditions, but do
include, for example, parametric model assumptions on the outcome regression.

Using the formalism of Section 2, we define $\psi_{t}(\mathcal{Z};P)$
analogously to (2) for any description $\mathcal{Z}$ of $\mathbf{W}$. We
denote the variance of the influence function by

$\displaystyle\mathbb{V}_{t}(\mathcal{Z};P)$
$\displaystyle\coloneq\operatorname{Var}_{P}[\psi_{t}(\mathcal{Z};P)]$ (3)
$\displaystyle=\mathbb{E}_{P}\left[\frac{\operatorname{Var}_{P}(Y\,|\,T=t,\mathcal{Z})}{\pi_{t}(\mathcal{Z};P)}\right]+\operatorname{Var}_{P}(b_{t}(\mathcal{Z};P)).$

The importance of this variance was highlighted by Hahn (1998), who computed
$\mathbb{V}_{t}(\mathbf{W};P)$ as the semiparametric efficiency bound for
regular asymptotically linear (RAL) estimators of $\chi_{t}(\mathbf{W};P)$.
That is, for any $\mathcal{P}$-consistent RAL estimator
$\widehat{\chi}_{t}(\mathbf{W};P)$ based on $n$ i.i.d. observations, the
asymptotic variance of
$\sqrt{n}(\widehat{\chi}_{t}(\mathbf{W};P)-\chi_{t}(\mathbf{W};P))$ is bounded
below by $\mathbb{V}_{t}(\mathbf{W};P)$, provided that $\mathcal{P}$ does not
impose structural assumptions on $\pi_{t}(\mathbf{W})$ and
$b_{t}(\mathbf{W})$. Moreover, the asymptotic variance of both the TMLE and
AIPW estimators achieve this lower bound when the propensity score and outcome
regression can be estimated with sufficiently fast rates (Chernozhukov et al.,
2018; van der Laan & Rose, 2011).

Since the same result can be applied for the representation
$\mathbf{Z}=\varphi(\mathbf{W})$, we see that estimation of
$\chi_{t}(\mathbf{Z};P)$ has a semiparametric efficiency bound of
$\mathbb{V}_{t}(\mathbf{Z};P)$. Now suppose that $\sigma(\mathbf{Z})$ is a
$\mathcal{P}$-valid description of $\mathbf{W}$, which means that
$\chi_{t}\coloneq\chi_{t}(\mathbf{W};P)=\chi_{t}(\mathbf{Z};P)$ for all $P$.
It is then natural to ask if we should estimate $\chi_{t}$ based on
$\mathbf{W}$ or the representation $\mathbf{Z}$? We proceed to investigate
this question based on which of the efficiency bounds
$\mathbb{V}_{t}(\mathbf{Z};P)$ or $\mathbb{V}_{t}(\mathbf{W};P)$ is smallest.

If $\mathbb{V}_{t}(\mathbf{Z};P)$ is smaller than
$\mathbb{V}_{t}(\mathbf{W};P)$, then $\mathbb{V}_{t}(\mathbf{W};P)$ is not an
actual efficiency bound for $\chi_{t}(\mathbf{W};P)$. This does not contradict
the result of Hahn (1998), as we have assumed the existence of a non-trivial
$\mathcal{P}$-valid description of $\mathbf{W}$, which implicitly imposes
structural assumptions on the functional form of $b_{t}(\mathbf{W})$.
Nevertheless, it is sensible to compare the variances
$\mathbb{V}_{t}(\mathbf{Z};P)$ and $\mathbb{V}_{t}(\mathbf{W};P)$, as these
are the asymptotic variances of the AIPW when using either $\mathbf{Z}$ or
$\mathbf{W}$, respectively, as a basis for the nuisance functions.

We formulate our efficiency bounds in terms of the more general contrast
parameter

(4)
$\Delta=\Delta(\mathbf{W};P)=\sum_{t\in\mathbb{T}}c_{t}\chi_{t}(\mathbf{W};P),$

where $\mathbf{c}\coloneq(c_{t})_{t\in\mathbb{T}}$ are fixed real-valued
coefficients. The prototypical example, when $\mathbb{T}=\\{0,1\\}$, is
$\Delta=\chi_{1}-\chi_{0}$, which is the average treatment effect under causal
assumptions, cf. the discussion following Assumption 1.1. Note that the family
of $\Delta$-parameters includes the adjusted mean $\chi_{t}$ as a special
case.

To estimate $\Delta$, we consider estimators of the form

(5)
$\widehat{\Delta}(\mathcal{Z};P)=\sum_{t\in\mathbb{T}}c_{t}\widehat{\chi}_{t}(\mathcal{Z};P),$

where $\widehat{\chi}_{t}(\mathcal{Z};P)$ denotes a consistent RAL estimator
of $\chi_{t}(\mathcal{Z};P)$. Correspondingly, the efficiency bound for such
an estimator is

(6)
$\mathbb{V}_{\\!\Delta}(\mathcal{Z};P)\coloneq\operatorname{Var}_{P}\Big{(}\sum_{t\in\mathbb{T}}c_{t}\psi_{t}(\mathcal{Z};P)\Big{)}.$

It turns out that two central results by Rotnitzky & Smucler (2020),
specifically their Lemmas 4 and 5, can be generalized from covariate subsets
to descriptions. One conceptual difference is that there is a priori no
natural generalization of precision variables and overadjustment
(instrumental) variables. To wit, if $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$
are descriptions, there is no canonical way222equivalent conditions to the
existence of an independent complement are given in Proposition 4 in Émery &
Schachermayer (2001). to subtract $\mathcal{Z}_{2}$ from $\mathcal{Z}_{1}$ in
a way that maintains their join
$\mathcal{Z}_{1}\vee\mathcal{Z}_{2}\coloneq\sigma(\mathcal{Z}_{1},\mathcal{Z}_{2})$.
Apart from this technical detail, the proofs translate more or less directly.
The following lemma is a direct extension of Rotnitzky & Smucler (2020, Lemma
4).

######  Lemma 3.1 (Deletion of overadjustment).

Fix a distribution $P\in\mathcal{P}$ and let
$\mathcal{Z}_{1}\subseteq\mathcal{Z}_{2}$ be $\sigma$-algebras such that
$Y\,{\perp\\!\\!\\!\perp}_{P}\,\mathcal{Z}_{2}\,|\,T,\mathcal{Z}_{1}$. Then it
always holds that

$\mathbb{V}_{\\!\Delta}(\mathcal{Z}_{2};P)-\mathbb{V}_{\\!\Delta}(\mathcal{Z}_{1};P)=\sum_{t\in\mathbb{T}}c_{t}^{2}D_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2};P)\geq
0,$

where for each $t\in\mathbb{T}$,

$\displaystyle D_{t}$
$\displaystyle(\mathcal{Z}_{1},\mathcal{Z}_{2};P)\coloneq\mathbb{V}_{t}(\mathcal{Z}_{2};P)-\mathbb{V}_{t}(\mathcal{Z}_{1};P)$
$\displaystyle=\mathbb{E}_{P}\,\Big{[}\pi_{t}(\mathcal{Z}_{1};P)\operatorname{Var}(Y\,|\,T=t,\mathcal{Z}_{1})\operatorname{Var}\big{(}\pi_{t}(\mathcal{Z}_{2};P)^{-1}\big{|}\;T=t,\mathcal{Z}_{1}\big{)}\Big{]}.$

Moreover, if $\mathcal{Z}_{2}$ is a description of $\mathbf{W}$ then
$\mathcal{Z}_{1}$ is $P$-valid if and only if $\mathcal{Z}_{2}$ is $P$-valid.

The lemma quantifies the efficiency lost in adjustment when adding information
that is irrelevant for the outcome.

We proceed to apply this lemma to the minimal information in $\mathbf{W}$ that
is predictive of $Y$ conditionally on $T$. To define this information, we use
the regular conditional distribution function of $Y$ given
$T=t,\mathbf{W}=\mathbf{w}$, which we denote by

$\displaystyle F(y\,|\,t,\mathbf{w};P)\coloneq\mathbb{P}_{P}(Y\leq
y\,|\,T=t,\mathbf{W}=\mathbf{w}),\qquad
y\in,t\in\mathbb{T},\mathbf{w}\in\mathbb{W}.$

See Kallenberg (2021, Sec. 8) for a rigorous treatment of regular conditional
distributions. We will in the following, by convention, take

(7) $b_{t}(\mathbf{w};P)=\int y\,\mathrm{d}F(y\,|\,t,\mathbf{w};P),$

so that $b_{t}(\mathbf{W};P)$ is given in terms of the regular conditional
distribution.

######  Definition 3.2.

Define the $\sigma$-algebras

$\displaystyle\mathcal{Q}$
$\displaystyle\coloneq\bigvee_{P\in\mathcal{P}}\mathcal{Q}_{P},\qquad\mathcal{Q}_{P}\coloneq\sigma(F(y\,|\,t,\mathbf{W};P);\,y\in,t\in\mathbb{T}),$
$\displaystyle\mathcal{R}$
$\displaystyle\coloneq\bigvee_{P\in\mathcal{P}}\mathcal{R}_{P},\qquad\mathcal{R}_{P}\coloneq\sigma(b_{t}(\mathbf{W};P);\,t\in\mathbb{T}).$

∎

$T$Treatment$\mathcal{Q}$$\mathbf{W}$$Y$Outcome Figure 2. The $\sigma$-algebra
$\mathcal{Q}$ given in Definition 3.2 as a description of $\mathbf{W}$.

Note that $\mathcal{Q}_{P}$, $\mathcal{Q}$, $\mathcal{R}_{P}$ and
$\mathcal{R}$ are all descriptions of $\mathbf{W}$, see Figure 2 for a
depiction of the information contained in $\mathcal{Q}$. Note also that
$\mathcal{R}_{P}\subseteq\mathcal{Q}_{P}$ by the convention (7).

We now state one of the main results of this section.

######  Theorem 3.3.

Fix $P\in\mathcal{P}$. Then, under Assumption 1.1, it holds that

  1. (i)

If $\mathcal{Z}$ is a description of $\mathbf{W}$ and
$\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}$, then $\mathcal{Z}$ is
$P$-ODS. In particular, $\mathcal{Q}_{P}$ is a $P$-ODS description of
$\mathbf{W}$.

  2. (ii)

If $\mathcal{Z}$ is $P$-ODS description, then
$\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}$.

  3. (iii)

For any $P$-ODS description $\mathcal{Z}$ it holds that

(8)
$\mathbb{V}_{\\!\Delta}(\mathcal{Z};P)-\mathbb{V}_{\\!\Delta}(\mathcal{Q}_{P};P)=\sum_{t\in\mathbb{T}}c_{t}^{2}D_{t}(\mathcal{Q}_{P},\mathcal{Z};P)\geq
0,$

where $D_{t}$ is given as in Lemma 3.1.

Together parts $(i)$ and $(ii)$ state that a description of $\mathbf{W}$ is
$P$-ODS if and only if its $\mathbb{P}_{P}$-completion contains
$\mathcal{Q}_{P}$. Part $(iii)$ states that, under $P\in\mathcal{P}$,
$\mathcal{Q}_{P}$ leads to the optimal efficiency bound among all $P$-ODS
descriptions.

In the following corollary we use $\overline{\mathcal{Z}}^{P}$ to denote the
$\mathbb{P}_{P}$-completion of a $\sigma$-algebra $\mathcal{Z}$.

######  Corollary 3.4.

Let $\mathcal{Z}$ be a description of $\mathbf{W}$. Then $\mathcal{Z}$ is
$\mathcal{P}$-ODS if and only if
$\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}^{P}$ for all
$P\in\mathcal{P}$. A sufficient condition for $\mathcal{Z}$ to be
$\mathcal{P}$-ODS is that $\mathcal{Q}\subseteq\overline{\mathcal{Z}}^{P}$ for
all $P\in\mathcal{P}$, in which case

(9)
$\mathbb{V}_{\\!\Delta}(\mathcal{Z};P)-\mathbb{V}_{\\!\Delta}(\mathcal{Q};P)=\sum_{t\in\mathbb{T}}c_{t}^{2}D_{t}(\mathcal{Q},\mathcal{Z};P)\geq
0,\qquad P\in\mathcal{P}.$

In particular, $\mathcal{Q}$ is a $\mathcal{P}$-ODS description of
$\mathbf{W}$, and (9) holds with $\mathcal{Z}=\sigma(\mathbf{W})$.

######  Remark 3.5.

It is a priori not obvious if $\mathcal{Q}$ is given by a representation,
i.e., if $\mathcal{Q}=\sigma(\varphi(\mathbf{W}))$ for some measurable mapping
$\varphi$. In Example 2.4 it is, since the arguments can be reformulated to
conclude that $\mathcal{Q}=\sigma(\|\mathbf{W}\|)$. ∎

Instead of working with the entire conditional distribution, it suffices to
work with the conditional mean when assuming, e.g., independent additive noise
on the outcome.

######  Proposition 3.6.

For fixed $P\in\mathcal{P}$, then $\mathcal{Q}_{P}=\mathcal{R}_{P}$ if any of
the following statements are true:

  * •

$F(y\,|\,t,\mathbf{W};P)$ is $\sigma(b_{t}(\mathbf{W};P))$-measurable for each
$t\in\mathbb{T},y\in$.

  * •

$Y$ is a binary outcome.

  * •

$Y$ has independent additive noise, i.e.,
$Y=b_{T}(\mathbf{W})+\varepsilon_{Y}$ with
$\varepsilon_{Y}\,{\perp\\!\\!\\!\perp}_{P}\,T,\mathbf{W}$.

If $\mathcal{Q}_{P}=\mathcal{R}_{P}$ holds, then (8) also holds for any
$P$-OMS description.

######  Remark 3.7.

When $\mathcal{Q}_{P}=\mathcal{R}_{P}$ for all $P\in\mathcal{P}$ we have
$\mathcal{Q}=\mathcal{R}$. There is thus the same information in the
$\sigma$-algebra $\mathcal{R}$ generated by the conditional means of the
outcome as there is in the $\sigma$-algebra $\mathcal{Q}$ generated by the
entire conditional distribution of the outcome. The three conditions in
Proposition 3.6 are sufficient but not necessary to ensure this. ∎

We also have a result analogous to Lemma 4 of Rotnitzky & Smucler (2020):

######  Lemma 3.8 (Supplementation with precision).

Fix $P\in\mathcal{P}$ and let $\mathcal{Z}_{1}\subseteq\mathcal{Z}_{2}$ be
descriptions of $\mathbf{W}$ such that
$T\,{\perp\\!\\!\\!\perp}_{P}\,\mathcal{Z}_{2}\,|\,\mathcal{Z}_{1}$. Then
$\mathcal{Z}_{1}$ is $P$-valid if and only if $\mathcal{Z}_{2}$ is $P$-valid.
Irrespectively, it always holds that

$\displaystyle\mathbb{V}_{\\!\Delta}(\mathcal{Z}_{1};P)-\mathbb{V}_{\\!\Delta}(\mathcal{Z}_{2};P)=\mathbf{c}^{\top}\operatorname{Var}_{P}[\mathbf{R}(\mathcal{Z}_{1},\mathcal{Z}_{2};P)]\mathbf{c}\geq
0$

where
$\mathbf{R}(\mathcal{Z}_{1},\mathcal{Z}_{2};P)\coloneq(R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2};P))_{t\in\mathbb{T}}$
with

$\displaystyle R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2};P)$
$\displaystyle\coloneq\left(\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{2};P)}-1\right)\left(b_{t}(\mathcal{Z}_{2};P)-b_{t}(\mathcal{Z}_{1};P)\right).$

Writing $R_{t}=R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2};P)$, the components of
the covariance matrix of $\mathbf{R}$ are given by

$\displaystyle\operatorname{Var}_{P}(R_{t})$
$\displaystyle=\mathbb{E}_{P}\left[\left(\frac{1}{\pi_{t}(\mathcal{Z}_{1};P)}-1\right)\operatorname{Var}_{P}[b_{t}(\mathcal{Z}_{2};P)\,|\,\mathcal{Z}_{1}]\right],$
$\displaystyle\operatorname{Cov}_{P}(R_{s},R_{t})$
$\displaystyle=-\mathbb{E}_{P}[\operatorname{Cov}_{P}(b_{s}(\mathcal{Z}_{2};P)b_{t}(\mathcal{Z}_{2};P)\,|\,\mathcal{Z}_{1})].$

As a consequence, we obtain the well-known fact that the propensity score is a
valid adjustment if $\mathbf{W}$ is, cf. Theorems 1–3 in Rosenbaum & Rubin
(1983).

######  Corollary 3.9.

Let $\Pi_{P}=\sigma(\pi_{t}(\mathbf{W};P)\colon\,t\in\mathbb{T})$. If
$\mathcal{Z}$ is a description of $\mathbf{W}$ containing $\Pi_{P}$, then
$\mathcal{Z}$ is $P$-valid and

$\mathbb{V}_{\\!\Delta}(\Pi_{P};P)-\mathbb{V}_{\\!\Delta}(\mathcal{Z};P)=\mathbf{c}^{\top}\operatorname{Var}_{P}[\mathbf{R}(\mathcal{R}_{P},\mathcal{Z};P)]\mathbf{c}\geq
0.$

The corollary asserts that while the information contained in the propensity
score is valid, it is asymptotically inefficient to adjust for in contrast to
all the information of $\mathbf{W}$. This is in similar spirit to Theorem 2 of
Hahn (1998), which states that the efficiency bound
$\mathbb{V}_{\\!\Delta}(\mathbf{W};P)$ remains unaltered if the propensity is
considered as known. However, the corollary also quantifies the difference of
the asymptotic efficiencies.

Corollary 3.4 asserts that $\mathcal{Q}$ is maximally efficient over all
$\mathcal{P}$-ODS descriptions $\mathcal{Z}$ satisfying
$\mathcal{Q}\subseteq\overline{\mathcal{Z}}^{P}$, and Proposition 3.6 asserts
that in special cases, $\mathcal{Q}$ reduces to $\mathcal{R}$. Since
$\mathcal{R}$ is $\mathcal{P}$-OMS, hence $\mathcal{P}$-valid, it is natural
to ask if $\mathcal{R}$ is generally more efficient than $\mathcal{Q}$. The
following example shows that their efficiency bounds may be incomparable
uniformly over $\mathcal{P}$.

######  Example 3.10.

Let $0<\delta<\frac{1}{2}$ be fixed and let $\mathcal{P}$ be the collection of
data generating distributions that satisfy:

  * •

$\mathbf{W}\in[\delta,1-\delta]$ with a symmetric distribution, i.e.,
$\mathbf{W}\stackrel{{\scriptstyle\mathcal{D}}}{{=}}1-\mathbf{W}$.

  * •

$T\in\\{0,1\\}$ with $\mathbb{E}(T\,|\,\mathbf{W})=\mathbf{W}$.

  * •

$Y=T+g(|\mathbf{W}-\frac{1}{2}|)+v(\mathbf{W})\varepsilon_{Y}$, where
$\varepsilon_{Y}\,\perp\\!\\!\\!\perp\,{}(T,\mathbf{W})$,
$\mathbb{E}[\varepsilon_{Y}^{2}]<\infty$, $\mathbb{E}[\varepsilon_{Y}]=0$, and
where $g\colon[0,\frac{1}{2}-\delta]\to$ and
$v\colon[\delta,1-\delta]\to[0,\infty)$ are continuous functions.

Letting $\mathbf{Z}=|\mathbf{W}-\frac{1}{2}|$, it is easy to verify directly
from Definition 3.2 that

$\mathcal{Q}=\sigma(\mathbf{W})\neq\sigma(\mathbf{Z})=\mathcal{R}.$

It follows that $\mathbf{Z}$ is $\mathcal{P}$-OMS but not $\mathcal{P}$-ODS.
However, $\mathbf{Z}$ is $\mathcal{P}_{1}$-ODS in the homoscedastic submodel
$\mathcal{P}_{1}=\\{P\in\mathcal{P}\,|\,v\equiv 1\\}$. In fact, it generates
the $\sigma$-algebra $\mathcal{Q}$ within this submodel, i.e.,
$\sigma(\mathbf{Z})=\vee_{P\in\mathcal{P}_{1}}\mathcal{Q}_{P}$. Thus
$\mathbb{V}_{\\!\Delta}(\mathbf{Z};P)\leq\mathbb{V}_{\\!\Delta}(\mathbf{W};P)$
for all $P\in\mathcal{P}_{1}$.

We refer to the supplementary Section A.6 for complete calculations of the
subsequent formulas.

From symmetry it follows that $\pi_{1}(\mathbf{Z})=0.5$ and hence we conclude
that $T\,\perp\\!\\!\\!\perp\,\mathbf{Z}$. By Lemma 3.8, it follows that $0$
(the trivial adjustment) is $\mathcal{P}$-valid, but with
$\mathbb{V}_{\\!\Delta}(\mathbf{Z};P)\leq\mathbb{V}_{\\!\Delta}(0;P)$ for all
$P\in\mathcal{P}$.

Alternatively, direct computation yields that

$\displaystyle\mathbb{V}_{t}(0)$
$\displaystyle=2\operatorname{Var}(g(\mathbf{Z}))+2\,\mathbb{E}[v(\mathbf{W})^{2}]\mathbb{E}[\varepsilon_{Y}^{2}]$
$\displaystyle\mathbb{V}_{t}(\mathbf{Z})$
$\displaystyle=\operatorname{Var}\left(g(\mathbf{Z})\right)+2\,\mathbb{E}[v(\mathbf{W})^{2}]\mathbb{E}[\varepsilon_{Y}^{2}]$
$\displaystyle\mathbb{V}_{t}(\mathbf{W})$
$\displaystyle=\operatorname{Var}\left(g(\mathbf{Z})\right)+\mathbb{E}\left[v(\mathbf{W})^{2}/\mathbf{W}\right]\mathbb{E}[\varepsilon_{Y}^{2}].$

With $\Delta=\chi_{t}$, the first two equalities confirm that
$\mathbb{V}_{\\!\Delta}(\mathbf{Z};P)\leq\mathbb{V}_{\\!\Delta}(0;P)$,
$P\in\mathcal{P}$, and the last two yield that indeed
$\mathbb{V}_{\\!\Delta}(\mathbf{Z};P)\leq\mathbb{V}_{\\!\Delta}(\mathbf{W};P)$
for $P\in\mathcal{P}_{1}$ by applying Jensen’s inequality. In fact, these are
strict inequalities whenever $g(\mathbf{Z})$ and $\varepsilon_{Y}$ are non-
degenerate.

Finally, we show that it is possible for
$\mathbb{V}_{\\!\Delta}(\mathbf{Z};P)>\mathbb{V}_{\\!\Delta}(\mathbf{W};P)$
for $P\notin\mathcal{P}_{1}$. Let $\tilde{P}\in\mathcal{P}$ be a data
generating distribution with $\mathbb{E}_{\tilde{P}}[\varepsilon_{Y}^{2}]>0$,
$v(\mathbf{W})=\mathbf{W}^{2}$, and with $\mathbf{W}$ uniformly distributed on
$[\delta,1-\delta]$. Then

$\displaystyle 2\,\mathbb{E}_{\tilde{P}}[v(\mathbf{W})^{2}]$
$\displaystyle=\,2\mathbb{E}_{\tilde{P}}[\mathbf{W}^{4}]=2\frac{(1-\delta)^{5}-\delta^{5}}{5(1-2\delta)}\xrightarrow{\delta\to
0}\frac{2}{5}$
$\displaystyle\mathbb{E}_{\tilde{P}}\left[\frac{v(\mathbf{W})^{2}}{\mathbf{W}}\right]$
$\displaystyle=\mathbb{E}_{\tilde{P}}[\mathbf{W}^{3}]=\frac{(1-\delta)^{4}-\delta^{4}}{4(1-2\delta)}\xrightarrow{\delta\to
0}\frac{1}{4}.$

So for sufficiently small $\delta$, it holds that
$\mathbb{V}_{\\!\Delta}(\mathbf{Z};\tilde{P})>\mathbb{V}_{\\!\Delta}(\mathbf{W};\tilde{P})$.
The example can also be modified to work for other $\delta>0$ by taking $v$ to
be a sufficiently large power of $\mathbf{W}$. ∎

######  Remark 3.11.

Following Theorem 1 of Benkeser et al. (2020), it is stated that the
asymptotic variance $\mathbb{V}_{\\!\Delta}(\mathcal{R}_{P};P)$ is generally
smaller than $\mathbb{V}_{\\!\Delta}(\mathbf{W};P)$. The example demonstrates
that this requires some assumptions on the outcome distribution, such as the
conditions in Proposition 3.6. ∎

######  Remark 3.12.

Suppose that $\mathcal{P}=\\{f_{\theta}\cdot\mu\colon\theta\in\Theta\\}$ is a
parametrized family of measures with densities
$\\{f_{\theta}\\}_{\theta\in\Theta}$ with respect to a $\sigma$-finite measure
$\mu$. Then informally, a _sufficient sub $\sigma$-algebra_ is any subset of
the observed information for which the remaining information is independent of
$\theta\in\Theta$, see Billingsley (2017) for a formal definition.
Superficially, this concept seems similar to that of a $\mathcal{P}$-ODS
description. In contrast however, the latter is a subset of the covariate
information $\sigma(\mathbf{W})$ rather than the observed information
$\sigma(T,\mathbf{W},Y)$, and it concerns sufficiency for the outcome
distribution rather than the entire data distribution. Moreover, the Rao-
Blackwell theorem asserts that conditioning an estimator of $\theta$ on a
sufficient sub $\sigma$-algebra leads to an estimator that is never worse.
Example 3.10 demonstrates that the situation is more delicate when considering
statistical efficiency for adjustment. ∎

##  4\. Estimation based on outcome-adapted representations

In this section we develop a general estimation method based on the insights
of Section 3, and we provide an asymptotic analysis of this methodology under
general conditions. Our method modifies the AIPW estimator, which we proceed
to discuss in more detail.

We have worked with the propensity score, $\pi_{t}$, and the outcome
regression, $b_{t}$, as random variables. We now consider, for each
$P\in\mathcal{P}$, their function counterparts obtained as regular conditional
expectations:

$\displaystyle m_{P}$ $\displaystyle\colon\mathbb{T}\times\mathbb{W}\to,$
$\displaystyle
m_{P}(t\,|\,\mathbf{w})\coloneq\mathbb{P}_{P}(T=t\,|\,\mathbf{W}=\mathbf{w}),$
$\displaystyle g_{P}$ $\displaystyle\colon\mathbb{T}\times\mathbb{W}\to,$
$\displaystyle
g_{P}(t,\mathbf{w})\coloneq\mathbb{E}_{P}[Y\,|\,T=t,\mathbf{W}=\mathbf{w}].$

To target the adjusted mean, $\chi_{t}$, the AIPW estimator utilizes the
influence function – given by (2) – as a score equation. To be more precise,
given estimates $(\widehat{m},\widehat{g})$ of the nuisance functions
$(m_{P},g_{P})$, the AIPW estimator of $\chi_{t}$ is given by

(10)
$\widehat{\chi}_{t}^{\mathrm{aipw}}(\widehat{m},\widehat{g})\coloneq\mathbb{P}_{n}\Big{[}\widehat{g}(t,\mathbf{W})+\frac{\mathbbl{1}(T=t)(Y-\widehat{g}(t,\mathbf{W}))}{\widehat{m}(t\,|\,\mathbf{W})}\Big{]},$

where $\mathbb{P}_{n}[\cdot]$ is defined as the empirical mean over $n$ i.i.d.
observations from $P$. The natural AIPW estimator of $\Delta$ given by
$\eqref{eq:hatDelta}$ is then
$\widehat{\Delta}^{\mathrm{aipw}}(\widehat{m},\widehat{g})=\sum_{t\in\mathbb{T}}c_{t}\widehat{\chi}_{t}^{\mathrm{aipw}}(\widehat{m},\widehat{g})$.

Roughly speaking, the AIPW estimator $\widehat{\chi}_{t}^{\mathrm{aipw}}$
converges to $\chi_{t}$ with a $\sqrt{n}$-rate if

(11)
$n\cdot\mathbb{P}_{n}\big{[}(\widehat{m}(t\,|\,\mathbf{W})-m_{P}(t\,|\,\mathbf{W}))^{2}\big{]}\cdot\mathbb{P}_{n}\big{[}(\widehat{g}(t\,|\,\mathbf{W})-g_{P}(t\,|\,\mathbf{W}))^{2}\big{]}\longrightarrow
0.$

If $(\widehat{m},\widehat{g})$ are estimated using the same data as
$\mathbb{P}_{n}$, then the above asymptotic result relies on Donsker class
conditions on the spaces containing $\widehat{m}$ and $\widehat{g}$. Among
others, Chernozhukov et al. (2018) propose circumventing Donsker class
conditions by sample splitting techniques such as $K$-fold cross-fitting.

###  4.1. Representation adjustment and the DOPE

Suppose that the outcome regression function factors through an intermediate
representation:

(12) $\displaystyle
g_{P}(t,\mathbf{w})=h_{P}(t,\varphi(\theta_{P},\mathbf{w})),\qquad
t\in\mathbb{T},\mathbf{w}\in\mathbb{W},$

where $\varphi\colon\Theta\times\mathbb{W}\to^{d}$ is a known measurable
mapping with unknown parameter $\theta\in\Theta$, and where
$h_{P}\colon\mathbb{T}\times^{d}\to$ is an unknown function. We use
$\mathbf{Z}_{\theta}=\varphi(\theta,\mathbf{W})$ to denote the corresponding
representation parametrized by $\theta\in\Theta$. If a particular covariate
value $\mathbf{w}\in\mathbb{W}$ is clear from the context, we also use the
implicit notation $\mathbf{z}_{\theta}=\varphi(\theta,\mathbf{w})$.

######  Example 4.1 (Single-index model).

The (partial) single-index model applies to $\mathbf{W}\in^{k}$ and assumes
that (12) holds with $\varphi(\theta,\mathbf{w})=\mathbf{w}^{\top}\theta$,
where $\theta\in\Theta\subseteq^{k}$. In other words, it assumes that the
outcome regression factors through the linear predictor
$\mathbf{Z}_{\theta}=\mathbf{W}^{\top}\theta$ such that

(13) $\displaystyle\forall P\in\mathcal{P}:\qquad
Y=h_{P}(T,\mathbf{Z}_{\theta_{P}})+\varepsilon_{Y},\qquad\mathbb{E}_{P}[\varepsilon_{Y}\,|\,T,\mathbf{W}]=0.$

For each treatment $T=t$, the model extends the generalized linear model (GLM)
by assuming that the (inverse) link function $h_{P}(t,\cdot)$ is unknown.

The semiparametric least squares (SLS) estimator proposed by Ichimura (1993)
estimates $\theta_{P}$ and $h_{P}$ by performing a two-step regression
procedure. Alternative estimation procedures are given in Powell et al.
(1989); Delecroix et al. (2003). ∎

Given an estimator $\hat{\theta}$ of $\theta_{P}$, independent of
$(T,\mathbf{W},Y)$, we use the following notation for various functions and
estimators thereof:

$\displaystyle g_{\hat{\theta}}(t,\mathbf{w})$
$\displaystyle\coloneq\mathbb{E}_{P}[Y\,|\,T=t,\mathbf{Z}_{\hat{\theta}}=\mathbf{z}_{\hat{\theta}},\hat{\theta}\,],$
$\displaystyle m_{\hat{\theta}}(t\,|\,\mathbf{w})$
$\displaystyle\coloneq\mathbb{P}_{P}(T=t\,|\,\mathbf{Z}_{\hat{\theta}}=\mathbf{z}_{\hat{\theta}},\hat{\theta}\,),$
$\displaystyle\widehat{g}(t,\mathbf{w})$ $\displaystyle\colon\text{an
estimator of }g_{P}(t,\mathbf{w}),$ (14a)
$\displaystyle\widehat{g}_{\hat{\theta}}(t,\mathbf{w})$
$\displaystyle\colon\text{an estimator of
}g_{\hat{\theta}}(t,\mathbf{w})\text{ of the form
}\widehat{h}(t,\mathbf{z}_{\hat{\theta}}),$ (14b)
$\displaystyle\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{w})$
$\displaystyle\colon\text{an estimator of
}m_{\hat{\theta}}(t\,|\,\mathbf{w})\text{ of the form
}\widehat{f}(t,\mathbf{z}_{\hat{\theta}}).$

In other words, $g_{\hat{\theta}}$ and $m_{\hat{\theta}}$ are the theoretical
propensity score and outcome regression, respectively, when using the
_estimated_ representation
$\mathbf{Z}_{\hat{\theta}}=\varphi(\hat{\theta},\mathbf{W})$. Note that we
have suppressed $P$ from the notation on the left-hand side, as we will be
working under a fixed $P$ in this section.

Sufficient conditions are well known that ensure

$\displaystyle\sqrt{n}(\widehat{\chi}_{t}^{\mathrm{aipw}}(\widehat{m},\widehat{g})-\chi_{t})\xrightarrow{d}\mathrm{N}(0,\mathbb{V}_{t}(\mathbf{W};P)).$

Such conditions, e.g., Assumption 5.1 in Chernozhukov et al. (2018), primarily
entail the condition in (11). We will leverage (12) to derive more efficient
estimators under similar conditions.

Suppose for a moment that $\theta_{P}$ is known. Since
$\mathbf{Z}_{\theta_{P}}$ is $P$-OMS under (13), it holds that
$\chi_{t}(\mathbf{Z}_{\theta_{P}};P)=\chi_{t}$. Then under analogous
conditions for the estimators
$(\widehat{m}_{\theta_{P}},\widehat{g}_{\theta_{P}})$, we therefore have

$\displaystyle\sqrt{n}(\widehat{\chi}_{t}^{\mathrm{aipw}}(\widehat{m}_{\theta_{P}},\widehat{g}_{\theta_{P}})-\chi_{t})\xrightarrow{d}\mathrm{N}(0,\mathbb{V}_{t}(\mathbf{Z}_{\theta_{P}};P)).$

Under the conditions of Corollary 3.6, it holds that
$\mathbb{V}_{t}(\mathbf{Z}_{\theta_{P}};P)\leq\mathbb{V}_{t}(\mathbf{W};P)$.
In other words,
$\widehat{\chi}_{t}^{\mathrm{aipw}}(\widehat{m}_{\theta_{P}},\widehat{g}_{\theta_{P}})$
is asymptotically at least as efficient as
$\widehat{\chi}_{t}^{\mathrm{aipw}}(\widehat{m},\widehat{g})$.

In general, the parameter $\theta_{P}$ is, of course, unknown, and we
therefore consider adjusting for an estimated representation
$\mathbf{Z}_{\hat{\theta}}$. For simplicity, we present the special case
$\Delta=\chi_{t}$, but the results are easily extended to general contrasts
$\Delta$. Our generic estimation procedure is described in Algorithm 1, where

$\mathbf{Z}_{\hat{\theta},i}\coloneq\varphi(\hat{\theta},\mathbf{W}_{i})$

denotes the estimated representation of the $i$-th observed covariate. We
refer to the resulting estimator as the Debiased Outcome-adapted Propensity
Estimator (DOPE), and it is denoted by $\widehat{\chi}_{t}^{\mathrm{dope}}$.

1 input: observations $(T_{i},\mathbf{W}_{i},Y_{i})_{i\in[n]}$, index sets
$\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3}\subseteq[n]$;

2 options: method for computing $\hat{\theta}$, regression methods for
propensity score and outcome regression of the form (14);

3 begin

4 compute estimate $\hat{\theta}$ based on data
$(T_{i},\mathbf{W}_{i},Y_{i})_{i\in\mathcal{I}_{1}}$;

5

6 regress outcomes $(Y_{i})_{i\in\mathcal{I}_{2}}$ onto
$(T_{i},\mathbf{Z}_{\hat{\theta},i})_{i\in\mathcal{I}_{2}}$ to obtain
$\widehat{g}_{\hat{\theta}}(\cdot,\cdot)$;

7

8 regress treatments $(T_{i})_{i\in\mathcal{I}_{2}}$ onto
$(\mathbf{Z}_{\hat{\theta},i})_{i\in\mathcal{I}_{2}}$ to obtain
$\widehat{m}_{\hat{\theta}}(\cdot\,|\,\cdot)$;

9

10 compute AIPW based on data
$(T_{i},\mathbf{W}_{i},Y_{i})_{i\in\mathcal{I}_{3}}$ and nuisance estimates
$(\widehat{m}_{\hat{\theta}},\widehat{g}_{\hat{\theta}})$:
$\displaystyle\widehat{\chi}_{t}^{\mathrm{dope}}=\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})+\frac{\mathbbl{1}(T_{i}=t)(Y_{i}-\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i}))}{\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\right)$

11 end

12

return _DOPE: $\widehat{\chi}_{t}^{\mathrm{dope}}$._

Algorithm 1 Debiased Outcome-adapted Propensity Estimator

Algorithm 1 is formulated such that $\mathcal{I}_{1}$, $\mathcal{I}_{2}$, and
$\mathcal{I}_{3}$ can be arbitrary subsets of $[n]$, and for the asymptotic
theory we assume that they are disjoint. However, in practical applications it
might be reasonable to use the full sample for every estimation step, i.e.,
employing the algorithm with
$\mathcal{I}_{1}=\mathcal{I}_{2}=\mathcal{I}_{3}=[n]$. In this case, we also
imagine that line 4 and line 5 are run simultaneously given that
$\hat{\theta}$ may be derived from an outcome regression, cf. the SLS
estimator in the single-index model (Example 4.1). In the supplementary
Section C, we also describe a more advanced cross-fitting scheme for Algorithm
1

######  Remark 4.2.

Benkeser et al. (2020) use a similar idea as the DOPE, but their propensity
factors through the final outcome regression function instead of a general
intermediate representation. That our general formulation of Algorithm 1
contains their collaborative one-step estimator as a special case is seen as
follows. Suppose $T\in\\{0,1\\}$ is binary and let

$\Theta=\\{\theta=(g_{P}(0,\cdot),g_{P}(1,\cdot))\colon P\in\mathcal{P}\\}$

be the collection of outcome regression functions. Define the intermediate
representation as the canonical pairing
$\varphi(\theta,\mathbf{W})=(\theta_{1}(\mathbf{W}),\theta_{2}(\mathbf{W}))$.
Then, for any outcome regression estimate $\hat{\theta}$, we may set
$\hat{g}_{\hat{\theta}}(t,\mathbf{w})=(1-t)\hat{\theta}_{1}(\mathbf{w})+t\hat{\theta}_{2}(\mathbf{w})$,
and the propensity score $\hat{m}_{\hat{\theta}}(t\mid\mathbf{w})$ factors
through the outcome regressions. In this case and with
$\mathcal{I}_{1}=\mathcal{I}_{2}=\mathcal{I}_{3}$, Algorithm 1 yields the
collaborative one-step estimator of Benkeser et al. (2020, App. D).
Accordingly, we refer to this special case as DOPE-BCL (Benkeser, Cai and
Laan). ∎

###  4.2. Asymptotics of the DOPE

We proceed to discuss the asymptotics of the DOPE. For our theoretical
analysis, we assume that the index sets in Algorithm 1 are disjoint. That is,
the theoretical analysis relies on sample splitting.

######  Assumption 4.1.

The observations $(T_{i},\mathbf{W}_{i},Y_{i})_{i\in[n]}$ used to compute
$\widehat{\chi}_{t}^{\mathrm{dope}}$ are i.i.d. with the same distribution as
$(T,\mathbf{W},Y)$, and
$[n]=\mathcal{I}_{1}\cup\mathcal{I}_{2}\cup\mathcal{I}_{3}$ is a partition
such that $|\mathcal{I}_{3}|\to\infty$ as $n\to\infty$.

In our simulations, employing sample splitting did not seem to enhance
performance, and hence we regard Assumption 4.1 as a theoretical convenience
rather than a practical necessity in all cases. Our results can likely also be
established under alternative assumptions that avoid sample splitting, in
particular Donsker class conditions.

We also henceforth use the convention that each of the quantities
$\pi_{t},b_{t},\chi_{t}$, and $\mathbb{V}_{t}$ are defined conditionally on
$\hat{\theta}$, e.g.,

$\chi_{t}(\mathbf{Z}_{\hat{\theta}})=\mathbb{E}[b_{t}(\mathbf{Z}_{\hat{\theta}})\,|\,\hat{\theta}\,]=\mathbb{E}[\mathbb{E}[Y\,|\,T=t,\mathbf{Z}_{\hat{\theta}},\hat{\theta}\,]\,|\,\hat{\theta}\,].$

The error of the DOPE estimator can then be decomposed as

(15)
$\widehat{\chi}_{t}^{\mathrm{dope}}-\chi_{t}=(\widehat{\chi}_{t}^{\mathrm{dope}}-\chi_{t}(\mathbf{Z}_{\hat{\theta}}))+(\chi_{t}(\mathbf{Z}_{\hat{\theta}})-\chi_{t}).$

The first term is the error had our target been the adjusted mean when
adjusting for the estimated representation $\mathbf{Z}_{\hat{\theta}}$,
whereas the second term is the adjustment bias that arises from adjusting for
$\mathbf{Z}_{\hat{\theta}}$ rather than $\mathbf{W}$ (or
$\mathbf{Z}_{\theta}$).

####  4.2.1. Estimation error conditionally on representation

To describe the asymptotics of the first term of (15), we consider the
decomposition

(16) $\displaystyle\sqrt{|\mathcal{I}_{3}|}\widehat{\chi}_{t}^{\mathrm{dope}}$
$\displaystyle=U_{\hat{\theta}}^{(n)}+R_{1}+R_{2}+R_{3},$

where

$\displaystyle U_{\hat{\theta}}^{(n)}$
$\displaystyle\coloneq\frac{1}{\sqrt{|\mathcal{I}_{3}|}}\sum_{i\in\mathcal{I}_{3}}u_{i}(\hat{\theta}),$
$\displaystyle u_{i}(\hat{\theta})\coloneq
g_{\hat{\theta}}(t,\mathbf{W}_{i})+\frac{\mathbbl{1}(T_{i}=t)(Y_{i}-g_{\hat{\theta}}(t,\mathbf{W}_{i}))}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})},$
$\displaystyle R_{1}$
$\displaystyle\coloneq\frac{1}{\sqrt{|\mathcal{I}_{3}|}}\sum_{i\in\mathcal{I}_{3}}r_{i}^{1},$
$\displaystyle
r_{i}^{1}\coloneq\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)\left(1-\frac{\mathbbl{1}(T_{i}=t)}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\right),$
$\displaystyle R_{2}$
$\displaystyle\coloneq\frac{1}{\sqrt{|\mathcal{I}_{3}|}}\sum_{i\in\mathcal{I}_{3}}r_{i}^{2},$
$\displaystyle
r_{i}^{2}\coloneq(Y_{i}-g_{\hat{\theta}}(t,\mathbf{W}_{i}))\left(\frac{\mathbbl{1}(T_{i}=t)}{\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}-\frac{\mathbbl{1}(T_{i}=t)}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\right),$
$\displaystyle R_{3}$
$\displaystyle\coloneq\frac{1}{\sqrt{|\mathcal{I}_{3}|}}\sum_{i\in\mathcal{I}_{3}}r_{i}^{3},$
$\displaystyle
r_{i}^{3}\coloneq\big{(}\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\big{)}\left(\frac{\mathbbl{1}(T_{i}=t)}{\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}-\frac{\mathbbl{1}(T_{i}=t)}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\right).$

We show that the oracle term, $U_{\hat{\theta}}^{(n)}$, drives the asymptotic
limit, and that the terms $R_{1},R_{2},R_{3}$ are remainder terms, subject to
the conditions stated below.

######  Assumption 4.2.

For $(T,\mathbf{W},Y)\sim P$ satisfying the representation model (12), it
holds that:

  1. (i)

There exists $c>0$ such that
$\max\\{|\widehat{m}_{\hat{\theta}}-\frac{1}{2}|,|m_{\hat{\theta}}-\frac{1}{2}|\\}\leq\frac{1}{2}-c$.

  2. (ii)

There exists $C>0$ such that $\mathbb{E}[Y^{2}\,|\,\mathbf{W},T]\leq C$.

  3. (iii)

There exists $\delta>0$ such that
$\mathbb{E}\big{[}|Y|^{2+\delta}\big{]}<\infty$.

  4. (iv)

It holds that
$\mathcal{E}_{1,t}^{(n)}\coloneq\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})-m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i}))^{2}\xrightarrow{P}0$.

  5. (v)

It holds that
$\mathcal{E}_{2,t}^{(n)}\coloneq\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i}))^{2}\xrightarrow{P}0$.

  6. (vi)

It holds that
$|\mathcal{I}_{3}|\cdot\mathcal{E}_{1,t}^{(n)}\mathcal{E}_{2,t}^{(n)}\xrightarrow{P}0$.

Classical convergence results of the AIPW are proven under similar conditions,
but with $(vi)$ replaced by the stronger convergence in (11). We establish
conditional asymptotic results under conditions on the conditional errors
$\mathcal{E}_{1,t}^{(n)}$ and $\mathcal{E}_{2,t}^{(n)}$. To the best of our
knowledge, the most similar results that we are aware of are those of Benkeser
et al. (2020), and our proof techniques are most similar to those of
Chernozhukov et al. (2018); Lundborg & Pfister (2023).

We can now state our first asymptotic result for the DOPE.

######  Theorem 4.3.

Under Assumptions 4.1 and 4.2, it holds that

$\displaystyle\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})^{-1/2}\cdot
U_{\hat{\theta}}^{(n)}\xrightarrow{d}\mathrm{N}(0,1),\qquad\text{and}\qquad
R_{i}\xrightarrow{P}0,\quad i=1,2,3,$

as $n\to\infty$. As a consequence,

$\displaystyle\sqrt{|\mathcal{I}_{3}|}\cdot\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})^{-1/2}\left(\widehat{\chi}_{t}^{\mathrm{dope}}-\chi_{t}(\mathbf{Z}_{\hat{\theta}})\right)\xrightarrow{d}\mathrm{N}(0,1).$

In other words, we can expect the DOPE to have an asymptotic distribution,
conditionally on $\hat{\theta}$, approximated by

$\widehat{\chi}_{t}^{\mathrm{dope}}(\widehat{g}_{\hat{\theta}},\widehat{m}_{\hat{\theta}})\,|\,\hat{\theta}\quad\stackrel{{\scriptstyle
as.}}{{\sim}}\quad\mathrm{N}\Big{(}\chi_{t}(\mathbf{Z}_{\hat{\theta}}),\,\frac{1}{|\mathcal{I}_{3}|}\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})\Big{)}.$

Note that if $|\mathcal{I}_{3}|=\lfloor n/3\rfloor$, say, then the asymptotic
variance is $\frac{3}{n}\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})$. Our
simulation study indicates that the asymptotic approximation may be valid in
some cases without the use of sample splitting, in which case
$|\mathcal{I}_{3}|=n$ and the asymptotic variance is
$\frac{1}{n}\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})$. A direct
implementation of the sample splitting procedure thus comes with an efficiency
cost. In the supplementary Section C we discuss how the cross-fitting
procedure makes use of sample splitting without an efficiency cost.

Given nuisance estimates $(\widehat{g},\widehat{m})$ we consider the empirical
variance estimator given by

(17)
$\displaystyle\widehat{\mathcal{V}}_{t}(\widehat{g},\widehat{m})=\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\widehat{u}_{i}^{2}-\Big{(}\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\widehat{u}_{i}\Big{)}^{2}=\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\Big{(}\widehat{u}_{i}-\frac{1}{|\mathcal{I}_{3}|}\sum_{j\in\mathcal{I}_{3}}\widehat{u}_{j}\Big{)}^{2},$

where

$\displaystyle\widehat{u}_{i}=\widehat{u}_{i}(\widehat{g},\widehat{m})=\widehat{g}(t,\mathbf{W}_{i})+\frac{\mathbbl{1}(T=t)(Y_{i}-\widehat{g}(t,\mathbf{W}_{i}))}{\widehat{m}(t\,|\,\mathbf{W}_{i})}.$

The following theorem states that the variance estimator with nuisance
functions $(\widehat{g}_{\hat{\theta}},\widehat{m}_{\hat{\theta}})$ is
consistent for the asymptotic variance in Theorem 4.3.

######  Theorem 4.4.

Under Assumptions 4.1 and 4.2, it holds that

$\displaystyle\widehat{\mathcal{V}}_{t}(\widehat{g}_{\hat{\theta}},\widehat{m}_{\hat{\theta}})-\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})\xrightarrow{P}0$

as $n\to\infty$.

####  4.2.2. Asymptotics of representation induced error

We now turn to the discussion of the second term in (15), i.e., the difference
between the adjusted mean for the estimated representation
$\mathbf{Z}_{\hat{\theta}}$ and the adjusted mean for the full covariate
$\mathbf{W}$. Under sufficient regularity, the delta method (van der Vaart,
2000, Thm. 3.8) describes the distribution of this error:

######  Proposition 4.5.

Assume $(T,\mathbf{W},Y)\sim P$ satisfies the model (12) with
$\theta_{P}\in\Theta\subseteq^{p}$. Let $u\colon\Theta\to$ be the function
given by $u(\theta)=\chi_{t}(\varphi(\theta,\mathbf{W});P)$ and assume that
$u$ is differentiable in $\theta_{P}$. Suppose that $\hat{\theta}$ is an
estimator of $\theta_{P}$ with rate $r_{n}$ such that

$\displaystyle
r_{n}\cdot(\hat{\theta}-\theta_{P})\xrightarrow{d}\mathrm{N}(0,\Sigma).$

Then

$\displaystyle
r_{n}\cdot(\chi_{t}(\mathbf{Z}_{\hat{\theta}})-\chi_{t})\xrightarrow{d}\mathrm{N}(0,\nabla
u(\theta_{P})^{\top}\Sigma\nabla u(\theta_{P}))$

as $n\to\infty$.

The delta method requires that the adjusted mean,
$\chi_{t}=\chi_{t}(\varphi(\theta,\mathbf{W});P)$, is differentiable with
respect to $\theta\in\Theta$. The theorem below showcases that this is the
case for the single-index model in Example 4.1.

######  Theorem 4.6.

Let $(T,\mathbf{W},Y)\sim P$ be given by the single-index model in Example 4.1
with $h_{t}(\cdot)\coloneq h(t,\cdot)\in C^{1}(\mathbb{R})$. Assume that
$\mathbf{W}$ has a distribution with density $p_{\mathbf{W}}$ with respect to
Lebesgue measure on d and that $p_{\mathbf{W}}$ is continuous almost
everywhere with bounded support. Assume also that the propensity
$m(t\mid\mathbf{w})=\mathbb{P}(T=t\,|\,\mathbf{W}=\mathbf{w})$ is continuous
in $\mathbf{w}$.

Then $u\colon^{d}\to$, defined by
$u(\theta)=\chi_{t}(\mathbf{W}^{\top}\theta;P)$, is differentiable at
$\theta=\theta_{P}$ with

$\displaystyle\nabla
u(\theta_{P})=\mathbb{E}_{P}\Big{[}h_{t}^{\prime}(\mathbf{W}^{\top}\theta_{P})\Big{(}1-\frac{\mathbb{P}(T=t\,|\,\mathbf{W})}{\mathbb{P}(T=t\,|\,\mathbf{W}^{\top}\theta_{P})}\Big{)}\mathbf{W}\Big{]}.$

The theorem is stated with some restrictive assumptions that simplify the
proof, but these are likely not necessary. It should also be possible to
extend this result to more general models of the form (12), but we leave such
generalizations for future work. In fact, the proof technique of Theorem 4.6
has already found application in Gnecco et al. (2023) in a different context.

The convergences implied in Theorem 4.3 and Proposition 4.5, with
$r_{n}=\sqrt{|\mathcal{I}_{3}|}$, suggest that the MSE of the DOPE is of order

$\displaystyle\mathbb{E}[(\widehat{\chi}_{t}^{\mathrm{dope}}-\chi_{t})^{2}]$
$\displaystyle=\mathbb{E}[(\widehat{\chi}_{t}^{\mathrm{dope}}-\chi_{t}(\mathbf{Z}_{\hat{\theta}}))^{2}]+\mathbb{E}[(\chi_{t}(\mathbf{Z}_{\hat{\theta}})-\chi_{t})^{2}]$
$\displaystyle\qquad+2\,\mathbb{E}[(\widehat{\chi}_{t}^{\mathrm{dope}}-\chi_{t}(\mathbf{Z}_{\hat{\theta}}))(\chi_{t}(\mathbf{Z}_{\hat{\theta}})-\chi_{t})]$
(18)
$\displaystyle\approx\frac{1}{|\mathcal{I}_{3}|}\Big{(}\mathbb{E}[\mathbb{V}_{t}(\mathbf{Z}_{\hat{\theta}})]+\nabla
u(\theta_{P})^{\top}\Sigma\nabla
u(\theta_{P})\Big{)}+o\left(|\mathcal{I}_{3}|^{-1}\right)$

The informal approximation ‘$\approx$’ can be turned into an equality by
establishing (or simply assuming) uniform integrability, which enables the
distributional convergences to be lifted to convergences of moments.

######  Remark 4.7.

The expression in (4.2.2) suggests an approximate confidence interval of the
form

$\widehat{\chi}_{t}^{\mathrm{dope}}\pm\frac{z_{1-\alpha}}{\sqrt{|\mathcal{I}_{3}|}}(\widehat{\mathcal{V}}_{t}+\widehat{\nabla}^{\top}\widehat{\Sigma}\widehat{\nabla}),$

where $\widehat{\Sigma}$ is a consistent estimator of the asymptotic variance
of $\sqrt{|\mathcal{I}_{3}|}(\hat{\theta}-\theta_{P})$, and where
$\widehat{\nabla}$ is a consistent estimator of
$\nabla_{\theta}\chi_{t}(\varphi(\theta,\mathbf{W}))|_{\theta=\theta_{P}}$.
However, the requirement of constructing both $\widehat{\Sigma}$ and
$\widehat{\nabla}$ adds further to the complexity of the methodology, so it
might be preferable to rely on bootstrapping techniques in practice. In the
simulation study we return to the question of inference and examine the
coverage of a naive interval that does not include the term
$\widehat{\nabla}^{\top}\widehat{\Sigma}\widehat{\nabla}$. ∎

##  5\. Experiments

In this section, we showcase the performance of the DOPE on simulated and real
data. All code is publicly available on GitHub333
https://github.com/AlexanderChristgau/OutcomeAdaptedAdjustment.

###  5.1. Simulation study

We present results of a simulation study based on the single-index model from
Example 4.1. We demonstrate the performance of the DOPE from Algorithms 1 and
2, and compare with various alternative estimators.

####  5.1.1. Sampling scheme

We simulated datasets consisting of $n$ i.i.d. copies of $(T,\mathbf{W},Y)$
sampled according to the following scheme:

$\displaystyle\mathbf{W}=(W^{1},\ldots,W^{d})$
$\displaystyle\sim\mathrm{Unif}(0,1)^{\otimes d}$ $\displaystyle
T\,|\,\mathbf{W}$
$\displaystyle\sim\mathrm{Bern}(0.01+0.98\cdot\mathbbl{1}(W^{1}>0.5))$ (19)
$\displaystyle Y\,|\,T,\mathbf{W},\beta_{Y}$
$\displaystyle\sim\mathrm{N}(h(T,\mathbf{W}^{\top}\beta_{Y}),1)$

where $\beta_{Y}$ is sampled once for each dataset with

$\displaystyle\beta_{Y}$
$\displaystyle=(1,\tilde{\beta}_{Y}),\qquad\tilde{\beta}_{Y}\sim\mathrm{N}(0,\mathbf{I}_{d-1}),$

and where $n,d$, and $h$ are experimental parameters. The settings considered
were $n\in\\{300,900,2700\\}$, $d\in\\{4,12,36\\}$, and with $h$ being one of

(22)
$\displaystyle\begin{array}[]{ll}h_{\text{lin}}(t,z)=t+3z,&h_{\text{square}}(t,z)=z^{1+t},\\\
h_{\text{cbrt}}(t,z)=(2+t)z^{1/3},&h_{\text{sin}}(t,z)=(3+t)\sin(\pi
z).\end{array}$

For each setting, $N=300$ datasets were simulated.

Note that while $\mathbb{E}[T]=0.01+0.98\cdot\mathbb{P}(W^{1}>0.5)=0.5$, the
propensity score $m(t\mid\mathbf{w})=0.01+0.98\cdot\mathbbl{1}(w^{1}>0.5)$
takes the rather extreme values $\\{0.01,0.99\\}$. Even though it technically
satisfies (strict) positivity, these extreme values of the propensity makes
the adjustment for $\mathbf{W}$ a challenging task. For each dataset, the
adjusted mean $\chi_{1}$ (conditional on $\beta_{Y}$) was considered as the
target parameter, and the ground truth was numerically computed as the sample
mean of $10^{7}$ observations of $h(1,\mathbf{W}^{\top}\beta_{Y})$.

####  5.1.2. Simulation estimators

This section contains an overview of the estimators used in the simulation.
For a complete description see Section B in the supplementary material.

Two settings were considered for outcome regression (OR):

  * •

Linear: Ordinary Least Squares (OLS).

  * •

Neural network: A feedforward neural network with two hidden layers: a linear
layer with one neuron, followed by a fully connected ReLU-layer with 100
neurons. The first layer is a linear bottleneck that enforces the single-index
model, and we denote the weights by $\theta\in^{d}$. An illustration of the
architecture can be found in the supplementary Section B. For a further
discussion of leaning single-and multiple-index models with neural networks,
see Parkinson et al. (2023) and references therein.

For propensity score estimation, logistic regression was used across all
settings. A ReLU-network with one hidden layer with 100 neurons was also
considered for estimation of the propensity score, but it did not enhance the
performance of the resulting ATE estimators in this setting. Random forests
and other methods were also initially used for both outcome regression and
propensity score estimation. They were subsequently excluded because they did
not seem to yield any noteworthy insights beyond what was observed for the
methods above.

For each outcome regression, two implementations were explored: a stratified
regression where $Y$ is regressed onto $\mathbf{W}$ separately for each
stratum $T=1$ and $T=2$, and a joint regression where $Y$ is regressed onto
$(T,\mathbf{W})$ simultaneously. In the case of joint regression, the neural
network architecture represents the regression function as a single index
model, given by $Y=h(\alpha T+\theta^{\top}\mathbf{W})+\varepsilon_{Y}$. This
representation differs from the description of the regression function
specified in the sample scheme (5.1.1), which allows for a more complex
interaction between treatment and the index of the covariates. In this sense,
the joint regression is misspecified.

Based on these methods for nuisance estimation, we considered the following
estimators of the adjusted mean:

  * •

The regression estimator
$\widehat{\chi}_{1}^{\mathrm{reg}}\coloneq\mathbb{P}_{n}[\widehat{g}(1,\mathbf{W})]$.

  * •

The AIPW estimator given in Equation (10).

  * •

The DOPE from Algorithm 1, where
$\mathbf{Z}_{\hat{\theta}}=\mathbf{W}^{\top}\hat{\theta}$ and where
$\hat{\theta}$ contains the weights of the first layer of the neural network
designed for single-index regression. We refer to this estimator as the DOPE-
IDX.

  * •

The DOPE-BCL described in Remark 4.2, where the propensity score is based on
the final outcome regression. See also Benkeser et al. (2020, App. D).

We considered two versions of each DOPE estimator: one without sample
splitting and another using 4-fold cross-fitting444consistent with Rem. 3.1.
in Chernozhukov et al. (2018), which recommends using 4 or 5 folds for cross-
fitting.. For the latter, the final empirical mean is calculated based on the
hold-out fold, while the nuisance parameters are fitted using the remaining
three folds. For each fold $k=1,\ldots,4$, this means employing Algorithm 1
with $\mathcal{I}_{1}=\mathcal{I}_{2}=[n]\setminus J_{k}$ and
$\mathcal{I}_{3}=J_{k}$. The two versions showed comparable performance for
larger samples. In this section we only present the results for the DOPE
without sample splitting, which performed better overall in our simulations.
However, a comparison with the cross-fitted version can be found in Section B
in the supplement.

Numerical results for the IPW estimator
$\widehat{\chi}_{1}^{\mathrm{ipw}}\coloneq\mathbb{P}_{n}[\mathbbl{1}(T=t)Y/\widehat{m}(1\,|\,\mathbf{W})]$
were also gathered. It performed poorly in our settings, and we have omitted
the results for the sake of clarity in the presentation.

####  5.1.3. Empirical performance of estimators

Figure 3. Root mean square errors for various estimators of $\chi_{1}$ plotted
against sample size. Each data point is an average over 900 datasets, 300 for
each choice of $d\in\\{4,12,36\\}$. The bands around each line correspond to
$\pm 1$ standard error. For this plot, the outcome regression (OR) was fitted
separately for each stratum $T=1$ and $T=2$.

The results for stratified outcome regression are shown in Figure 3. Each
panel, corresponding to a link function in (22), displays the RMSE for various
estimators against sample size. Across different values of
$d\in\\{4,12,36\\}$, the results remain consistent, and thus the RMSE is
averaged over the $3\times 300$ datasets with varying $d$. In the upper left
panel, where the link is linear, the regression estimator
$\widehat{\chi}_{1}^{\mathrm{reg}}$ with OLS performs best as expected. The
remaining estimators exhibit similar performance, except for AIPW, which
consistently performs poorly across all settings. This can be attributed to
extreme values of the propensity score, resulting in a large efficiency bound
for AIPW. All estimators seem to maintain an approximate $\sqrt{n}$-consistent
RMSE for the linear link as anticipated.

For the nonlinear links, we observe that the OLS-based regression estimators
perform poorly and do not exhibit approximate $\sqrt{n}$-consistency. For the
sine link, the RMSEs for all OLS-based estimators are large, and they are not
shown for ease of visualization. For neural network outcome regression, the
AIPW still performs the worst across all nonlinear links. The regression
estimator and the DOPE estimators share similar performance when used with the
neural network, with DOPE-IDX being more accurate overall. Since the neural
network architecture is tailored to the single-index model, it is not
surprising that the outcome regression works well, and as a result there is
less need for debiasing. On the other hand, the debiasing introduced in the
DOPE does not hurt the accuracy, and in fact, improves it in this setting.

Figure 4. Root mean square errors for various estimators of $\chi_{1}$ plotted
against sample size. Each data point is an average over 900 datasets, 300 for
each choice of $d\in\\{4,12,36\\}$. The bands around each line correspond to
$\pm 1$ standard error. For this plot, the outcome regression (OR) was fitted
jointly onto $(T,\mathbf{W})$.

The results for joint regression of $Y$ on $(T,\mathbf{W})$ are shown in
Figure 4. The results for the OLS-based estimators provide similar insights as
previously discussed, so we focus on the results for the neural network based
estimators. The jointly trained neural network is, in a sense, misspecified
for the single-index model (except for the linear link), as discussed in
Subsection 5.1.2. Thus it is not surprising that the regression estimator
fails to adjust effectively for larger sample sizes. What is somewhat
unexpected, however, is that the precision of DOPE, especially the DOPE-IDX,
does not appear to be compromised by the misspecified outcome regression. In
fact, the DOPE-IDX even seems to perform better with joint outcome regression.
We suspect that this could be attributed to the joint regression producing
more robust predictions for the rare treated subjects with $W^{1}\leq 0.5$,
for which $m(1\,|\,\mathbf{W})=0.01$. The predictions are more robust since
the joint regression can leverage some of the information from the many
untreated subjects with $W^{1}\leq 0.5$ at the cost of introducing systematic
bias, which the DOPE-IDX deals with in the debiasing step. While this
phenomenon is interesting, a thorough exploration of its exact details, both
numerically and theoretically, is a task we believe is better suited for
future research.

In summary, the DOPE serves as a middle ground between the regression
estimator and the AIPW. It provides an additional safeguard against biased
outcome regression, all while avoiding the potential numerical instability
entailed by using standard inverse propensity weights.

####  5.1.4. Inference

We now consider approximate confidence intervals obtained from the empirical
variance estimator $\widehat{\mathcal{V}}$ defined in (17). Specifically, we
consider intervals of the form
$\widehat{\chi}_{1}\pm\frac{z_{0.975}}{\sqrt{n}}\widehat{\mathcal{V}}^{-1/2}$,
where $z_{0.975}$ is the $0.975$ quantile of the standard normal distribution.

Figure 5. Coverage of asymptotic confidence intervals of the adjusted mean
$\chi_{1}$, aggregated over $d\in\\{4,12,36\\}$ and with the true $\beta$
given in (5.1.1).

Figure 5 shows the empirical coverage of $\chi_{1}$ for the AIPW, DOPE-BCL and
DOPE-IDX, all based on neural network outcome regression fitted jointly. Based
on the number of repetitions, we can expect the estimated coverage
probabilities to be accurate up to $\pm 0.01$ after rounding.

The left column shows the results for no sample splitting, whereas the right
column shows the results for 4-fold cross-fitting. For all estimators, cross-
fitting seems to improve coverage slightly for large sample sizes. The AIPW is
a little anti-conservative, especially for the links $h_{\text{sin}}$ and
$h_{\text{cbrt}}$, but overall maintains approximate coverage. The slight
miscoverage is to be expected given the challenging nature with the extreme
propensity weights. For the DOPE-BCL and DOPE-IDX, the coverage probabilities
are worse, but with the DOPE-IDX having better coverage overall. According to
our asymptotic analysis, specifically Theorem 4.3, the DOPE-BCL and DOPE-IDX
intervals will only achieve asymptotic $95\%$ coverage of the _random_
parameters $\chi_{1}(\widehat{h}(t,\mathbf{Z}_{\hat{\theta}}))$ and
$\chi_{1}(\mathbf{Z}_{\hat{\theta}})$, respectively. Thus we do not anticipate
these intervals to asymptotically achieve the nominal $95\%$ coverage of
$\chi_{1}$. However, we show in Figure 9 in the supplementary Section B that
the average widths of the confidence intervals are significantly smaller for
the DOPE intervals than the AIPW interval. Hence it is possible that the DOPE
intervals can be corrected – as we have also discussed in Remark 4.7 – to gain
advantage over the AIPW interval, but we leave such explorations for future
work.

In summary, the lack of full coverage for the naively constructed intervals
are consistent with the asymptotic analysis conditionally on a representation
$\mathbf{Z}_{\hat{\theta}}$. In view of this, our asymptotic results might be
more realistic and pragmatic than the previously studied regime (Benkeser et
al., 2020; Ju et al., 2020), which assumes fast unconditional rates of the
nuisance parameters $(\widehat{m}_{\hat{\theta}},\widehat{g}_{\hat{\theta}})$.

###  5.2. Application to NHANES data

We consider the mortality dataset collected by the National Health and
Nutrition Examination Survey I Epidemiologic Followup Study (Cox et al.,
1997), henceforth referred to as the NHANES dataset. The dataset was initially
collected as in Lundberg et al. (2020)555 See
https://github.com/suinleelab/treeexplainer-study for their GitHub repository.
. The dataset contains several baseline covariates, and our outcome variable
is the indicator of death at the end of study. To deal with missing values, we
considered both a complete mean imputation as in Lundberg et al. (2020), or a
trimmed dataset where covariates with more than $50\%$ of their values missing
are dropped and the rest are mean imputed. The latter approach reduces the
number of covariates from $79$ to $65$. The final results were similar for the
two imputation methods, so we only report the results for the mean imputed
dataset here. In the supplementary Section B we show the results for the
trimmed dataset.

The primary aim of this section is to evaluate the different estimation
methodologies on a realistic and real data example. For this purpose we
consider a treatment variable based on pulse pressure and study its effect on
mortality. Pulse pressure is defined as the difference between systolic and
diastolic blood pressure (BP). High pulse pressure is not only used as an
indicator of disease but is also reported to increase the risk of
cardiovascular diseases (Franklin et al., 1999). We investigate the added
effect of high pulse pressure when adjusting for other baseline covariates, in
particular systolic BP and levels of white blood cells, hemoglobin,
hematocrit, and platelets. We do not adjust for diastolic BP, as it determines
the pulse pressure when combined with systolic BP, which would therefore lead
to a violation of positivity.

Figure 6. Distribution of estimated propensity scores based on the full
covariate set $\mathbf{W}$, the single-index $\mathbf{W}^{\top}\hat{\theta}$,
and the outcome predictions $\widehat{g}(1,\mathbf{W})$, respectively.

A pulse pressure of 40 mmHg is considered normal, and as the pressure
increases past 50 mmHg, the risk of cardiovascular diseases is reported to
increase. Some studies have used 60 mmHg as the threshold for high pulse
pressure (Homan et al., 2024), and thus we consider the following treatment
variable corresponding to high pulse pressure:

$\displaystyle T=\mathbbl{1}\left(\textrm{pulse pressure}>60\textrm{
mmHg}\right).$

For the binary outcome regression we consider logistic regression and a
variant of the neural network from Section 5.1.2 with an additional ‘sigmoid-
activation’ on the output layer. Based on 5-fold cross-validation, logistic
regression yields a log-loss of $0.50$, whereas the neural network yields a
log-loss of $0.48$ (smaller is better).

Figure 6 shows the distribution of the estimated propensity scores used in the
AIPW, DOPE-IDX, and DOPE-BCL, based on neural network regression. As expected,
we see that the full covariate set yields propensity scores that essentially
violate positivity, with this effect being less pronounced for DOPE-IDX and
even less so for DOPE-BCL, where the scores are comparatively less extreme.

Estimator | Estimate | BS se | BS CI
---|---|---|---
Regr. (NN) | 0.022 | 0.009 | (0.004, 0.037)
Naive contrast | 0.388 | 0.010 | (0.369, 0.407)
Regr. (Logistic) | 0.026 | 0.010 | (0.012, 0.049)
DOPE-BCL (Logistic) | 0.023 | 0.010 | (0.003, 0.040)
DOPE-BCL (NN) | 0.021 | 0.010 | (0.001, 0.039)
DOPE-IDX (NN) | 0.024 | 0.012 | (0.002, 0.048)
AIPW (NN) | 0.023 | 0.015 | (-0.009, 0.051)
AIPW (Logistic) | 0.026 | 0.015 | (-0.004, 0.056)
IPW (Logistic) | -0.040 | 0.025 | (-0.103, -0.006)
Table 1. Adjusted effect estimates of unhealthy pulse pressure on mortality
based on NHANES dataset. Estimators are sorted in increasing order after
bootstrap (BS) standard error.

Table 1 shows the estimated treatment effects
$\widehat{\Delta}=\widehat{\chi}_{1}-\widehat{\chi}_{0}$ for various
estimators together with $95\%$ bootstrap confidence intervals. The estimators
are sorted according to bootstrap variance based on $B=1000$ bootstraps.
Because we do not know the true effect, we cannot directly decide which
estimator is best. The naive contrast provides a substantially larger estimate
than all of the adjustment estimators, which indicates that the added effect
of high pulse pressure on mortality when adjusting is different from the
unadjusted effect. The IPW estimator, $\widehat{\chi}_{1}^{\mathrm{ipw}}$, is
the only estimator to yield a negative estimate of the adjusted effect, and
its bootstrap variance is also significantly larger than the other estimators.
Thus it is plausible, that the IPW estimator fails to adjust appropriately.
This is not surprising given extreme distribution of the propensity weights,
as shown in blue in Figure 6.

The remaining estimators yield comparable estimates of the adjusted effect,
with the logistic regression based estimators having a marginally larger
estimates than their neural network counterparts. The (bootstrap) standard
errors are comparable for the DOPE and regression estimators, but the AIPW
estimators have the slightly larger standard errors. As a result, the added
effect of high pulse pressure on mortality cannot be considered statistically
significant for the AIPW estimators, whereas it can for the other estimators.

In summary, the results indicate that the choice treatment effect estimator
impacts the final estimate and confidence interval. While it is uncertain
which estimator is superior for this particular application, the DOPE
estimators seem to offer a reasonable and stable estimate.

##  6\. Discussion

In this paper, we address challenges posed by complex data with unknown
underlying structure, an increasingly common scenario in observational
studies. Specifically, we have formulated a refined and general formalism for
studying efficiency of covariate adjustment. This formalism extends and builds
upon the efficiency principles derived from causal graphical models, in
particular results provided by Rotnitzky & Smucler (2020). Our theoretical
framework has led to the identification of the optimal covariate information
for adjustment. From this theoretical groundwork, we introduced the DOPE for
general estimation of the ATE with increased efficiency.

Several key areas within this research merit further investigation:

Extension to other targets and causal effects: Our focus has predominantly
been on the adjusted mean
$\chi_{t}=\mathbb{E}[\mathbb{E}[Y\,|\,T=t,\mathbf{W}]]$. However, the
extension of our methodologies to continuous treatments, instrumental
variables, or other causal effects, such as the average treatment effect among
the treated, is an area that warrants in-depth exploration. We suspect that
many of the fundamental ideas can be modified to prove analogous results for
such target parameters.

Beyond neural networks: While our examples and simulation study has applied
the DOPE with neural networks, its compatibility with other regression methods
such as kernel regression and gradient boosting opens new avenues for
alternative adjustment estimators. Investigating such estimators can provide
practical insights and broaden the applicability of our approach. The Outcome
Highly Adapted Lasso (Ju et al., 2020) is a similar method in this direction
that leverages the Highly Adaptive Lasso (Benkeser & van der Laan, 2016) for
robust treatment effect estimation.

Causal representation learning integration: Another possible direction for
future research is the integration of our efficiency analysis into the broader
concept of causal representation learning (Schölkopf et al., 2021). This line
of research, which is not concerned with specific downstream tasks, could
potentially benefit from some of the insights of our efficiency analysis and
the DOPE framework.

Implications of sample splitting: Our current asymptotic analysis relies on
the implementation of three distinct sample splits. Yet, our simulation
results suggest that sample splitting might not be necessary. Using the
notation of Algorithm 1, we hypothesize that the independence assumption
$\mathcal{I}_{3}\cap(\mathcal{I}_{1}\cup\mathcal{I}_{2})=\emptyset$ could be
replaced with Donsker class conditions. However, the theoretical
justifications for setting $\mathcal{I}_{1}$ identical to $\mathcal{I}_{2}$
are not as evident, thus meriting additional exploration into this potential
simplification.

Valid and efficient confidence intervals: The naive confidence intervals
explored in the simulation study, cf. Section 5.1.4, do not provide an
adequate coverage of the unconditional adjusted mean. As indicated in Remark
4.7, it would be interesting to explore methods for correcting the intervals,
for example by: (i) adding a bias correction, (ii) constructing a consistent
estimator of the unconditional variance, or (iii) using bootstrapping.
Appealing to bootstrapping techniques, however, might be computationally
expensive, in particular when combined with cross-fitting. To implement (i),
one approach could be to generalize Theorem 4.6 to establish general
differentiability of the adjusted mean with respect to general
representations, and then construct an estimator of the gradient. However, one
must asses whether this bias correction could negate the efficiency benefits
of the DOPE.

## Acknowledgments

We thank Leonard Henckel and Anton Rask Lundborg for helpful discussions. AMC
and NRH were supported by a research grant (NNF20OC0062897) from Novo Nordisk
Fonden.

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## Supplement to ‘Efficient adjustment for complex covariates: Gaining
efficiency with DOPE’

The supplementary material is organized as follows: Section A contains proofs
of the results in the main text and a few auxiliary results; Section B
contains details on the experiments in Section 5; and Section C contains a
description of the cross-fitting algorithm.

##  Appendix A Auxiliary results and proofs

The proposition below relaxes the criteria for being a valid adjustment set in
a causal graphical model. It follows from the results of Perković et al.
(2018), which was pointed out to the authors by Leonard Henckel, and is stated
here for completeness.

######  Proposition A.1.

Let $T,\mathbf{Z}$, and $Y$ be pairwise disjoint node sets in a DAG
$\mathcal{D}=(\mathbf{V},\mathbf{E})$, and let $\mathcal{M}(\mathcal{D})$
denote the collection of continuous distributions that are Markovian with
respect to $\mathcal{D}$ and with $\mathbb{E}|Y|<\infty$. Then $\mathbf{Z}$ is
an adjustment set relative to $(T,Y)$ if and only if for all
$P\in\mathcal{M}(\mathcal{D})$ and all $t$

$\displaystyle\mathbb{E}_{P}[Y\,|\,\mathrm{do}(T=t)]=\mathbb{E}_{P}\left[\frac{\mathbbl{1}(T=t)Y}{\mathbb{P}_{P}(T=t\,|\,\mathrm{pa}_{\mathcal{D}}(T))}\right]=\mathbb{E}_{P}[\mathbb{E}_{P}[Y\,|\,T=t,\mathbf{Z}]]$

###### Proof.

The ‘only if’ direction is straightforward if we use an alternative
characterization of $\mathbf{Z}$ being an adjustment set: for any density $p$
of a distribution $P\in\mathcal{P}$ it holds that

$p(y\,|\,\mathrm{do}(T=t))=\int
p(\mathbf{y}|\mathbf{z},t)p(\mathbf{z})\mathrm{d}\mathbf{z},$

where the ‘do-operator’ (and notation) is defined as in, e.g., Peters et al.
(2017). Dominated convergence then yields

$\displaystyle\mathbb{E}_{P}[Y\,|\,\mathrm{do}(T=t)]$
$\displaystyle=\mathbb{E}_{P}[\mathbb{E}_{P}[Y\,|\,T=t,\mathbf{Z}]],$

which is equivalent to the ‘only if’ part. On the contrary, assume that
$\mathbf{Z}$ is _not_ an adjustment set for $(T,Y)$. Then Theorem 56 and the
proof of Theorem 57 in Perković et al. (2018) imply the existence of a
Gaussian distribution $\tilde{P}\in\mathcal{P}$ for which

$\displaystyle\mathbb{E}_{\tilde{P}}[Y\,|\,\mathrm{do}(T=1)]\neq\mathbb{E}_{\tilde{P}}[\mathbb{E}_{\tilde{P}}[Y\,|\,T=1,\mathbf{Z}]]$

This implies the other direction. ∎

The following lemma is a generalization of Lemma 27 in Rotnitzky & Smucler
(2020).

######  Lemma A.2.

For any $\sigma$-algebras
$\mathcal{Z}_{1}\subseteq\mathcal{Z}_{2}\subseteq\sigma(\mathbf{W})$ it holds,
under Assumption 1.1, that

$\mathbb{E}_{P}\left[\pi_{t}(\mathcal{Z}_{2};P)^{-1}\,|\,T=t,\mathcal{Z}_{1}\right]=\pi_{t}(\mathcal{Z}_{1};P)^{-1}$

for all $P\in\mathcal{P}$.

###### Proof.

Direct computation yields

$\displaystyle\mathbb{E}_{P}\left[\pi_{t}(\mathcal{Z}_{2};P)^{-1}\,|\,T=t,\mathcal{Z}_{1}\right]\mathbb{P}_{P}(T=t\,|\,\mathcal{Z}_{1})$
$\displaystyle=\mathbb{E}_{P}\left[\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{2};P)}\,|\,\mathcal{Z}_{1}\right]=\mathbb{E}_{P}\left[\frac{\mathbb{E}_{P}[\mathbbl{1}(T=t)\,|\,\mathcal{Z}_{2}]}{\pi_{t}(\mathcal{Z}_{2};P)}\,|\,\mathcal{Z}_{1}\right]=1.$

∎

###  A.1. Proof of Lemma 3.1

Since $P\in\mathcal{P}$ is fixed, we suppress it from the notation in the
following computations.

From $Y\,\perp\\!\\!\\!\perp\,\mathcal{Z}_{2}\,|\,\mathcal{Z}_{1},T$ and
$\mathcal{Z}_{1}\subseteq\mathcal{Z}_{2}$, it follows that
$\mathbb{E}[Y\,|\,\mathcal{Z}_{1},T]=\mathbb{E}[Y\,|\,\mathcal{Z}_{2},T]$.
Hence, for each $t\in\mathbb{T}$,

(23)
$b_{t}(\mathcal{Z}_{1})=\mathbb{E}[Y\,|\,\mathcal{Z}_{1},T=t]=\mathbb{E}[Y\,|\,\mathcal{Z}_{2},T=t]=b_{t}(\mathcal{Z}_{2}).$

Therefore
$\chi_{t}(\mathcal{Z}_{1})=\mathbb{E}[b_{t}(\mathcal{Z}_{1})]=\mathbb{E}[b_{t}(\mathcal{Z}_{2})]=\chi_{t}(\mathcal{Z}_{2})$.
If $\mathcal{Z}_{2}$ is description of $\mathbf{W}$, this identity shows that
$\mathcal{Z}_{2}$ is $\mathcal{P}$-valid if and only if $\mathcal{Z}_{1}$ is
$\mathcal{P}$-valid.

To compare the asymptotic efficiency bounds for $\mathcal{Z}_{2}$ with
$\mathcal{Z}_{1}$ we use the law of total variance. Note first that

$\displaystyle\mathbb{E}[\psi_{t}(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1}]$
$\displaystyle=\mathbb{E}\Big{[}\mathbbl{1}(T=t)(Y-b_{t}(\mathcal{Z}_{2}))\frac{1}{\pi_{t}(\mathcal{Z}_{2})}+b_{t}(\mathcal{Z}_{2})-\chi_{t}\,|\,T,Y,\mathcal{Z}_{1}\Big{]}$
$\displaystyle=\mathbbl{1}(T=t)(Y-b_{t}(\mathcal{Z}_{1}))\,\mathbb{E}\Big{[}\frac{1}{\pi_{t}(\mathcal{Z}_{2})}\,|\,T=t,Y,\mathcal{Z}_{1}\Big{]}+b_{t}(\mathcal{Z}_{1})-\chi_{t}$
$\displaystyle=\mathbbl{1}(T=t)(Y-b_{t}(\mathcal{Z}_{1}))\mathbb{E}\left[\frac{1}{\pi_{t}(\mathcal{Z}_{2})}\,|\,T=t,\mathcal{Z}_{1}\right]+b_{t}(\mathcal{Z}_{1})-\chi_{t}$
$\displaystyle=\mathbbl{1}(T=t)(Y-b_{t}(\mathcal{Z}_{1}))\frac{1}{\pi_{t}(\mathcal{Z}_{1})}+b_{t}(\mathcal{Z}_{1})-\chi_{t}=\psi_{t}(\mathcal{Z}_{1}).$

Second equality is due to $\mathcal{Z}_{1}$-measurability and (23), whereas
the third equality follows from
$Y\,\perp\\!\\!\\!\perp\,\mathcal{Z}_{2}\,|\,\mathcal{Z}_{1},T$ and the last
inequality is due to Lemma A.2. On the other hand,

$\displaystyle\phantom{=}\mathbb{E}\big{[}\operatorname{Var}(\psi_{t}(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1})\big{]}$
$\displaystyle=\mathbb{E}\Big{[}\operatorname{Var}\Big{(}\mathbbl{1}(T=t)(Y-b_{t}(\mathcal{Z}_{1}))\frac{1}{\pi_{t}(\mathcal{Z}_{2})}+b_{t}(\mathcal{Z}_{1})-\chi_{t}(P)\;\big{|}\;T,Y,\mathcal{Z}_{1}\Big{)}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\mathbbl{1}(T=t)(Y-b_{t}(\mathcal{Z}_{1}))^{2}\operatorname{Var}\Big{(}\frac{1}{\pi_{t}(\mathcal{Z}_{2})}\;\big{|}\;T=t,Y,\mathcal{Z}_{1}\Big{)}\Big{]}$
$\displaystyle=\mathbb{E}\Big{[}\pi_{t}(\mathcal{Z}_{1})\operatorname{Var}(Y\,|\,T=t,\mathcal{Z}_{1})\operatorname{Var}\Big{(}\frac{1}{\pi_{t}(\mathcal{Z}_{2})}\;\big{|}\;T=t,\mathcal{Z}_{1}\Big{)}\Big{]}.$

Combined we have that

$\displaystyle\mathbb{V}_{t}(\mathcal{Z}_{2})$
$\displaystyle=\operatorname{Var}(\psi_{t}(\mathcal{Z}_{2}))=\operatorname{Var}(\mathbb{E}[\psi_{t}(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1}])+\mathbb{E}[\operatorname{Var}[\psi_{t}(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1}]]$
$\displaystyle=\mathbb{V}_{t}(\mathcal{Z}_{1})+\mathbb{E}\Big{[}\pi_{t}(\mathcal{Z}_{1})\operatorname{Var}(Y\,|\,T=t,\mathcal{Z}_{1})\operatorname{Var}\Big{(}\frac{1}{\pi_{t}(\mathcal{Z}_{2})}\Big{|}\;T=t,Y,\mathcal{Z}_{1}\Big{)}\Big{]}.$

To prove the last part of the lemma, we first note that
$\psi_{t}(\mathcal{Z}_{2})$ and $\psi_{t^{\prime}}(\mathcal{Z}_{2})$ are
conditionally uncorrelated for $t\neq t^{\prime}$. This follows from (23),
from which the conditional covariance reduces to

$\displaystyle\operatorname{Cov}(\psi_{t}(\mathcal{Z}_{2}),\psi_{t^{\prime}}(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1})=\mathbbl{1}(T=t)\mathbbl{1}(T=t^{\prime})\operatorname{Cov}(*,*\,|\,T,Y,\mathcal{Z}_{1})=0.$

Thus, letting
$\psi(\mathcal{Z}_{2})=(\psi_{t}(\mathcal{Z}_{2}))_{t\in\mathbb{T}}$ and using
the computation for $\mathbb{V}_{t}$, we obtain that:

$\displaystyle\mathbb{V}_{\Delta}(\mathcal{Z}_{2})=\operatorname{Var}(\mathbf{c}^{\top}\psi(\mathcal{Z}_{2}))$
$\displaystyle=\operatorname{Var}(\mathbb{E}[\mathbf{c}^{\top}\psi(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1}])+\mathbb{E}[\operatorname{Var}[\mathbf{c}^{\top}\psi(\mathcal{Z}_{2})\,|\,T,Y,\mathcal{Z}_{1}]]$
$\displaystyle=\mathbb{V}_{\Delta}(\mathcal{Z}_{1})+\sum_{t\in\mathbb{T}}c_{t}^{2}\,\mathbb{E}\Big{[}\pi_{t}(\mathcal{Z}_{1})\operatorname{Var}(Y\,|\,T=t,\mathcal{Z}_{1})\operatorname{Var}\Big{(}\frac{1}{\pi_{t}(\mathcal{Z}_{2})}\Big{|}\;T=t,Y,\mathcal{Z}_{1}\Big{)}\Big{]}.$

This concludes the proof. ∎

###  A.2. Proof of Theorem 3.3

(i) Assume that $\mathcal{Z}\subseteq\sigma(\mathbf{W})$ is a description of
$\mathbf{W}$ such that $\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}$.
Observe that almost surely,

$\mathbb{P}_{P}(Y\leq
y\,|\,T,\mathbf{W})=\sum_{t\in\mathbb{T}}\mathbbl{1}(T=t)F(y\mid
t,\mathbf{W};P)$

is $\sigma(T,\mathcal{Q}_{P})$-measurable and hence also
$\sigma(T,\overline{\mathcal{Z}})$-measurable. It follows that
$\mathbb{P}_{P}(Y\leq y\,|\,T,\mathbf{W})$ is also a version of
$\mathbb{P}_{P}(Y\leq y\,|\,T,\overline{\mathcal{Z}})$. As a consequence,
$\mathbb{P}_{P}(Y\leq y\,|\,T,\mathbf{W})=\mathbb{P}_{P}(Y\leq
y\,|\,T,\mathcal{Z})$ almost surely. From Doob’s characterization of
conditional independence (Kallenberg 2021, Theorem 8.9), we conclude that
$Y\,{\perp\\!\\!\\!\perp}_{P}\,\mathbf{W}\mid T,\mathcal{Z}$, or equivalently
that $\mathcal{Z}$ is $P$-ODS. The ‘in particular’ follows from setting
$\mathcal{Z}=\mathcal{Q}_{P}$.

(ii) Assume that $\mathcal{Z}\subseteq\sigma(\mathbf{W})$ is $P$-ODS. Using
Doob’s characterization of conditional independence again,
$\mathbb{P}_{P}(Y\leq y\,|\,T,\mathbf{W})=\mathbb{P}_{P}(Y\leq
y\,|\,T,\mathcal{Z})$ almost surely. Under Assumption 1.1, this entails that
$F(y\,|\,T=t,\mathbf{W};P)=\mathbb{P}_{P}(Y\leq y\,|\,T=t,\mathcal{Z})$ almost
surely, and hence $F(y\,|\,T=t,\mathbf{W};P)$ must be
$\overline{\mathcal{Z}}$-measurable. As the generators of $\mathcal{Q}_{P}$
are $\overline{\mathcal{Z}}$-measurable, we conclude that
$\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}$.

(iii) Let $\mathcal{Z}$ be a $P$-ODS description. From (i) and (ii) it follows
that $\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}$, and since
$\mathcal{Z}\subseteq\sigma(\mathbf{W})$ it also holds that
$Y\,{\perp\\!\\!\\!\perp}_{P}\,\overline{\mathcal{Z}}\mid T,\mathcal{Q}_{P}$.
Since
$\mathbb{V}_{\Delta}(\mathcal{Z};P)=\mathbb{V}_{\Delta}(\overline{\mathcal{Z}};P)$,
Lemma 3.1 gives the desired conclusion. ∎

###  A.3. Proof of Corollary 3.4

Theorem 3.3 (i,ii) implies that $\mathcal{Z}$ is $\mathcal{P}$-ODS if and only
if $\overline{\mathcal{Z}}^{P}$ contains $\mathcal{Q}_{P}$ for all
$P\in\mathcal{P}$.

If $\mathcal{Q}\subseteq\overline{\mathcal{Z}}^{P}$ for all $P\in\mathcal{P}$
then $\mathcal{Q}_{P}\subseteq\overline{\mathcal{Z}}^{P}$ for all
$P\in\mathcal{P}$ by definition, and $\mathcal{Z}$ is $\mathcal{P}$-ODS.
Equation (9) follows from the same argument as Theorem 3.3 $(iii)$. That is,
$\mathcal{Q}\subseteq\overline{\mathcal{Z}}^{P}$, and since $\mathcal{Q}$ is
$P$-ODS and $\mathcal{Z}\subseteq\sigma(\mathbf{W})$ we have
$Y\,{\perp\\!\\!\\!\perp}_{P}\,\overline{\mathcal{Z}}^{P}\mid T,\mathcal{Q}$.
Lemma 3.1 can be applied for each $P\in\mathcal{P}$ to obtain the desired
conclusion. ∎

###  A.4. Proof of Proposition 3.6

Since $\mathcal{R}_{P}\subseteq\mathcal{Q}_{P}$ always holds, so it suffices
to prove that $\mathcal{Q}_{P}\subseteq\mathcal{R}_{P}$. If
$F(y\,|\,t,\mathbf{W};P)$ is $\sigma(b_{t}(\mathbf{W};P))$-measurable, then
$\mathcal{Q}_{P}\subseteq\mathcal{R}_{P}$ follows directly from definition. If
$Y$ is a binary outcome, then the conditional distribution is determined by
the conditional mean, i.e., $F(y\,|\,t,\mathbf{W};P)$ is
$\sigma(b_{t}(\mathbf{W};P))$-measurable. If
$Y=b_{T}(\mathbf{W})+\varepsilon_{Y}$ with
$\varepsilon_{Y}\,\perp\\!\\!\\!\perp\,T,\mathbf{W}$, then
$F(y\,|\,t,\mathbf{w};P)=F_{P}(y-b_{t}(\mathbf{w}))$ where $F_{P}$ is the
distribution function for the distribution of $\varepsilon_{Y}$. Again,
$F(y\,|\,t,\mathbf{W};P)$ becomes $\sigma(b_{t}(\mathbf{W};P))$-measurable.

To prove the last part, let $\mathcal{Z}$ be a $P$-OMS description. Since
$b_{t}(\mathbf{W};P)=b_{t}(\mathcal{Z};P)$ is $\mathcal{Z}$-measurable for
every $t\in\mathbb{T}$, it follows that
$\mathcal{R}_{P}\subseteq\overline{\mathcal{Z}}$. Hence the argument of
Theorem 3.3 $(iii)$ establishes Equation (8) when
$\mathcal{Q}_{P}=\mathcal{R}_{P}$. ∎

###  A.5. Proof of Lemma 3.8

Since $P\in\mathcal{P}$ is fixed throughout this proof, we suppress it from
notation. From
$\mathcal{Z}_{1}\subseteq\mathcal{Z}_{2}\subseteq\sigma(\mathbf{W})$ and
$T\,{\perp\\!\\!\\!\perp}_{P}\,\mathcal{Z}_{2}\,|\,\mathcal{Z}_{1}$, it
follows that $\pi_{t}(\mathcal{Z}_{1})=\pi_{t}(\mathcal{Z}_{1})$, and hence

$\chi_{t}(\mathcal{Z}_{1})=\mathbb{E}[\pi_{t}(\mathcal{Z}_{1})^{-1}\mathbbl{1}(T=t)Y]=\mathbb{E}[\pi_{t}(\mathcal{Z}_{2})^{-1}\mathbbl{1}(T=t)Y]=\chi_{t}(\mathcal{Z}_{2}).$

This establishes the first part. For the second part, we use that
$\pi_{t}(\mathcal{Z}_{1})=\pi_{t}(\mathcal{Z}_{2})$ and
$\chi_{t}(\mathcal{Z}_{1})=\chi_{t}(\mathcal{Z}_{2})$ to see that

$\displaystyle\psi_{t}(\mathcal{Z}_{1})-\psi_{t}(\mathcal{Z}_{2})$
$\displaystyle=b_{t}(\mathcal{Z}_{1})-b_{t}(\mathcal{Z}_{2})+\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{2})}(b_{t}(\mathcal{Z}_{2})-b_{t}(\mathcal{Z}_{1}))$
$\displaystyle=\Big{(}\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{2})}-1\Big{)}(b_{t}(\mathcal{Z}_{2})-b_{t}(\mathcal{Z}_{1}))\eqcolon
R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2}).$

Since
$\mathbb{E}[R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})\,|\,\mathcal{Z}_{2}]=0$ we
have that
$\mathbb{E}[\psi_{t}(\mathcal{Z}_{2})R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})]=0$,
which means that $\psi_{t}(\mathcal{Z}_{2})$ and
$R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})$ are uncorrelated. Note that from
$\pi_{t}(\mathcal{Z}_{1})=\pi_{t}(\mathcal{Z}_{2})$ it also follows that

$\displaystyle
b_{t}(\mathcal{Z}_{1})=\mathbb{E}\Big{[}\frac{Y\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{1})}\,|\,\mathcal{Z}_{1}\Big{]}=\mathbb{E}\Big{[}\mathbb{E}\Big{[}\frac{Y\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{2})}\,|\,\mathcal{Z}_{2}\Big{]}\,|\,\mathcal{Z}_{1}\Big{]}=\mathbb{E}[b_{t}(\mathcal{Z}_{2})\,|\,\mathcal{Z}_{1}]$

This means that
$\mathbb{E}[(b_{t}(\mathcal{Z}_{2})-b_{t}(\mathcal{Z}_{1}))^{2}\,|\,\mathcal{Z}_{1}]=\operatorname{Var}(b_{t}(\mathcal{Z}_{2})\,|\,\mathcal{Z}_{1})$.
These observations let us compute that

$\displaystyle\mathbb{V}_{t}(\mathcal{Z}_{1})$
$\displaystyle=\operatorname{Var}(\psi_{t}(\mathcal{Z}_{1}))$
$\displaystyle=\operatorname{Var}(\psi_{t}(\mathcal{Z}_{2}))+\operatorname{Var}\left(R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})\right)$
$\displaystyle=\operatorname{Var}(\psi_{t}(\mathcal{Z}_{2}))+\mathbb{E}[\mathbb{E}[R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})^{2}\,|\,\mathcal{Z}_{2}]]$
$\displaystyle=\mathbb{V}_{t}(\mathcal{Z}_{2})+\mathbb{E}\Big{[}(b_{t}(\mathcal{Z}_{2})-b_{t}(\mathcal{Z}_{1}))^{2}\mathbb{E}\Big{[}\frac{\mathbbl{1}(T=t)^{2}}{\pi_{t}(\mathcal{Z}_{1})^{2}}-\frac{2\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{1})}+1\,|\,\mathcal{Z}_{2}\Big{]}\Big{]}$
(24)
$\displaystyle=\mathbb{V}_{t}(\mathcal{Z}_{2})+\mathbb{E}\Big{[}\operatorname{Var}(b_{t}(\mathcal{Z}_{2})\,|\,\mathcal{Z}_{1})\Big{(}\frac{1}{\pi_{t}(\mathcal{Z}_{1})}-1\Big{)}\Big{]}.$

This establishes the formula in the case $\Delta=\chi_{t}$.

For general $\Delta$, note first that $\mathbf{R}$ and
$(\psi_{t}(\mathcal{Z}_{2}))_{t\in\mathbb{T}}$ are uncorrelated since
$\mathbb{E}[\mathbf{R}\,|\,\mathcal{Z}_{2}]=0$. Therefore we have

$\displaystyle\mathbb{V}_{\\!\Delta}(\mathcal{Z}_{1})$
$\displaystyle=\operatorname{Var}\Big{(}\sum_{t\in\mathbb{T}}c_{t}\psi_{t}(\mathcal{Z}_{1})\Big{)}$
$\displaystyle=\operatorname{Var}\Big{(}\sum_{t\in\mathbb{T}}c_{t}\psi_{t}(\mathcal{Z}_{2})\Big{)}+\operatorname{Var}\Big{(}\sum_{t\in\mathbb{T}}c_{t}R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})\Big{)}$
$\displaystyle=\mathbb{V}_{\\!\Delta}(\mathcal{Z}_{2})+\mathbf{c}^{\top}\operatorname{Var}\Big{(}\mathbf{R}\Big{)}\mathbf{c}.$

It now remains to establish the covariance expressions for the last term,
since the expression of
$\operatorname{Var}\left(R_{t}(\mathcal{Z}_{1},\mathcal{Z}_{2})\right)$ was
found in (A.5). To this end, note first that for any $s,t\in\mathbb{T}$ with
$s\neq t$,

$\displaystyle\mathbb{E}\left[\Big{(}\frac{\mathbbl{1}(T=s)}{\pi_{s}(\mathcal{Z}_{2})}-1\Big{)}\Big{(}\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathcal{Z}_{2})}-1\Big{)}\,|\,\mathcal{Z}_{2}\right]=-1.$

Thus we finally conclude that

$\displaystyle\operatorname{Cov}(R_{s},R_{t})$
$\displaystyle=-\mathbb{E}\left[(b_{s}(\mathcal{Z}_{2})-b_{s}(\mathcal{Z}_{1}))(b_{t}(\mathcal{Z}_{2})-b_{t}(\mathcal{Z}_{1}))\right]$
$\displaystyle=-\mathbb{E}\left[\operatorname{Cov}(b_{s}(\mathcal{Z}_{2}),b_{t}(\mathcal{Z}_{2})\,|\,\mathcal{Z}_{1})\right],$

as desired. ∎

###  A.6. Computations for Example 3.10

Using symmetry, we see that

$\displaystyle\pi_{1}(\mathbf{Z})$
$\displaystyle=\mathbb{E}[T\mid\mathbf{Z}]=\mathbb{E}[\mathbf{W}\mid\mathbf{Z}]$
$\displaystyle=(0.5-\mathbf{Z})\mathbb{P}(\mathbf{W}\leq
0.5\,|\,\mathbf{Z})+(0.5+\mathbf{Z})\mathbb{P}(\mathbf{W}>0.5\,|\,\mathbf{Z})\
=\frac{1}{2}.$

From $T\,\perp\\!\\!\\!\perp\,\mathbf{Z}$ we observe that

$\displaystyle b_{t}(\mathbf{W})$
$\displaystyle=t+\mathbb{E}[g(\mathbf{Z})\,|\,T=t,\mathbf{W}]=t+g(\mathbf{Z}),$
$\displaystyle b_{t}(\mathbf{Z})$
$\displaystyle=t+\mathbb{E}[g(\mathbf{Z})\,|\,T=t,\mathbf{Z}]=t+g(\mathbf{Z}),$
$\displaystyle b_{t}(0)$
$\displaystyle=t+\mathbb{E}[g(\mathbf{Z})\,|\,T=t]=t+\mathbb{E}[g(\mathbf{Z})].$

Plugging these expressions into the asymptotic variance yields:

$\displaystyle\mathbb{V}_{t}(0)$
$\displaystyle=\operatorname{Var}\left(b_{t}(0)+\frac{\mathbbl{1}(T=t)}{\pi_{t}(0)}(Y-b_{t}(0))\right)$
$\displaystyle=\pi_{t}(0)^{-2}\operatorname{Var}\left(\mathbbl{1}(T=t)(Y-b_{t}(0))\right)$
$\displaystyle=4\mathbb{E}[\mathbbl{1}(T=t)(g(\mathbf{Z})-\mathbb{E}[g(\mathbf{Z})]+v(\mathbf{W})\varepsilon_{Y})^{2}]$
$\displaystyle=4\mathbb{E}[\mathbf{W}(g(\mathbf{Z})-\mathbb{E}[g(\mathbf{Z})])^{2}]+2\mathbb{E}[v(\mathbf{W})\varepsilon_{Y}^{2}]$
$\displaystyle=2\operatorname{Var}(g(\mathbf{Z}))+2\mathbb{E}[v(\mathbf{W})^{2}]\mathbb{E}[\varepsilon_{Y}^{2}]$
$\displaystyle\mathbb{V}_{t}(\mathbf{Z})$
$\displaystyle=\operatorname{Var}\left(b_{t}(\mathbf{Z})+\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathbf{Z})}(Y-b_{t}(\mathbf{Z}))\right)$
$\displaystyle=\operatorname{Var}\left(g(\mathbf{Z})+2\mathbbl{1}(T=t)v(\mathbf{W})\varepsilon_{Y}\right)$
$\displaystyle=\operatorname{Var}\left(g(\mathbf{Z})\right)+2\mathbb{E}[v(\mathbf{W})^{2}]\mathbb{E}[\varepsilon_{Y}^{2}]$
$\displaystyle\mathbb{V}_{t}(\mathbf{W})$
$\displaystyle=\operatorname{Var}\left(b_{t}(\mathbf{W})+\frac{\mathbbl{1}(T=t)}{\pi_{t}(\mathbf{W})}(Y-b_{t}(\mathbf{W}))\right)$
$\displaystyle=\operatorname{Var}\left(g(\mathbf{Z})+\frac{\mathbbl{1}(T=t)v(\mathbf{W})\varepsilon_{Y}}{\mathbf{W}}\right)$
$\displaystyle=\operatorname{Var}\left(g(\mathbf{Z})\right)+\mathbb{E}\left[\frac{v(\mathbf{W})^{2}}{\mathbf{W}}\right]\mathbb{E}[\varepsilon_{Y}^{2}].$

∎

### Proof of Theorem 4.3

We first prove that $R_{i}\xrightarrow{P}0$ for $i=1,2,3$. The third term is
taken care of by combining Assumption 4.2 (i,iv) with Cauchy-Schwarz:

$\displaystyle|R_{3}|$
$\displaystyle\leq\sqrt{|\mathcal{I}_{3}|}\sqrt{\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\big{(}\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})^{-1}-m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})^{-1}\big{)}^{2}}\sqrt{\mathcal{E}_{2,t}^{(n)}}$
$\displaystyle\leq\ell_{c}\sqrt{|\mathcal{I}_{3}|\mathcal{E}_{1,t}^{(n)}\mathcal{E}_{2,t}^{(n)}}\xrightarrow{P}0,$

where $\ell_{c}>0$ is the Lipschitz constant of $x\mapsto x^{-1}$ on
$[c,1-c]$.

For the first term, $R_{1}$, note that conditionally on $\hat{\theta}$ and the
observed estimated representations
$\mathbf{Z}_{\hat{\bullet}}\coloneq(\mathbf{Z}_{\hat{\theta},i})_{i\in\mathcal{I}_{3}}=(\varphi(\hat{\theta},\mathbf{W}_{i}))_{i\in\mathcal{I}_{3}}$,
the summands are conditionally i.i.d. due to Assumption 4.1. They are also
conditionally mean zero since

$\displaystyle\mathbb{E}[r_{i}^{1}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\,]$
$\displaystyle=\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)\Big{(}1-\frac{\mathbb{E}[\mathbbl{1}(T_{i}=t)\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\,]}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\Big{)}$
$\displaystyle=\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)\Big{(}1-\frac{\mathbb{E}[\mathbbl{1}(T_{i}=t)\,|\,\mathbf{Z}_{\hat{\theta},i},\hat{\theta}]}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\Big{)}=0,$

for each $i\in\mathcal{I}_{3}$. Using Assumption 4.2 (i), we can bound the
conditional variance by

$\displaystyle\operatorname{Var}\left(r_{i}^{1}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\,\right)$
$\displaystyle=\frac{\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)^{2}}{m_{\hat{\theta}}(t\,|\,\mathbf{W})^{2}}\operatorname{Var}\left(\mathbbl{1}(T=t)\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\right)$
$\displaystyle=\frac{\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)^{2}}{m_{\hat{\theta}}(t\,|\,\mathbf{W})}(1-m_{\hat{\theta}}(t\,|\,\mathbf{W}))$
$\displaystyle\leq\frac{(1-c)}{c}\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)^{2}$

Now it follows from Assumption 4.2 v and the conditional Chebyshev inequality
that

$\displaystyle\mathbb{P}\left(|R_{1}|>\epsilon\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\,\right)$
$\displaystyle\leq\epsilon^{-2}\operatorname{Var}(R_{1}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta})\leq\frac{1-c}{\epsilon^{2}c}\mathcal{E}_{2,t}^{(n)}\xrightarrow{P}0,$

from which we conclude that $R_{1}\xrightarrow{P}0$.

The analysis of $R_{2}$ is similar. Observe that the summands are
conditionally i.i.d. given $\mathbf{Z}_{\hat{\bullet}},\hat{\theta}$, and
$T_{\bullet}\coloneq(T_{i})_{i\in\mathcal{I}_{3}}$. They are also
conditionally mean zero since

$\displaystyle\mathbb{E}\left[r_{i}^{2}\,|\,\mathbf{Z}_{\hat{\bullet}},T_{\bullet},\hat{\theta}\right]=\left(\mathbb{E}[Y_{i}\,|\,\mathbf{Z}_{\hat{\bullet}},T_{\bullet},\hat{\theta}]-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)\left(\frac{\mathbbl{1}(T_{i}=t)}{\widehat{m}_{\hat{\theta}}(\mathbf{W}_{i})}-\frac{\mathbbl{1}(T_{i}=t)}{m_{\hat{\theta}}(\mathbf{W}_{i})}\right)=0,$

where we use that on the event $(T_{i}=t)$,

$\mathbb{E}[Y_{i}\,|\,\mathbf{Z}_{\hat{\theta},i},T_{i},\hat{\theta}\,]=\mathbb{E}[Y_{i}\,|\,\mathbf{Z}_{\hat{\theta},i},T_{i}=t,\hat{\theta}\,]=g_{\hat{\theta}}(t,\mathbf{W}_{i}).$

To bound the conditional variance, note first that Assumption 4.1 and
Assumption 4.2 ii imply

$\displaystyle\operatorname{Var}\big{(}Y_{i}\,|\,\mathbf{Z}_{\hat{\theta},i},T_{i},\hat{\theta}\,\big{)}\leq\mathbb{E}\big{[}Y_{i}^{2}\,|\,\mathbf{Z}_{\hat{\theta},i},T_{i},\hat{\theta}\,\big{]}=\mathbb{E}[\mathbb{E}\big{[}Y_{i}^{2}\,|\,\mathbf{W}_{i},T_{i},\hat{\theta}\,\big{]}\,|\,\mathbf{Z}_{\hat{\theta},i},T_{i},\hat{\theta}\,\big{]}\leq
C,$

which in return implies that

$\displaystyle\operatorname{Var}\left(r_{i}^{2}\,|\,\mathbf{Z}_{\hat{\bullet}},T_{\bullet},\hat{\theta}\right)$
$\displaystyle=\mathbbl{1}(T_{i}=t)\left(\frac{1}{\widehat{m}_{\hat{\theta}}(\mathbf{W}_{i})}-\frac{1}{m_{\hat{\theta}}(\mathbf{W}_{i})}\right)^{2}\operatorname{Var}\left(Y_{i}\,|\,\mathbf{Z}_{\hat{\theta},i},T_{i},\hat{\theta}\,\right)$
$\displaystyle\leq
C\left(\frac{1}{\widehat{m}_{\hat{\theta}}(\mathbf{W}_{i})}-\frac{1}{m_{\hat{\theta}}(\mathbf{W}_{i})}\right)^{2}.$

It now follows from the conditional Chebyshev’s inequality that

$\displaystyle\mathbb{P}(|R_{2}|>\epsilon\,|\,\mathbf{Z}_{\hat{\bullet}},T_{\bullet},\hat{\theta})$
$\displaystyle\leq\epsilon^{-2}\operatorname{Var}(R_{2}\,|\,\mathbf{Z}_{\hat{\bullet}},T_{\bullet},\hat{\theta})$
$\displaystyle\leq\frac{C}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\left(\frac{1}{\widehat{m}_{\hat{\theta}}(\mathbf{W}_{i})}-\frac{1}{m_{\hat{\theta}}(\mathbf{W}_{i})}\right)^{2}\leq
C\ell_{c}^{2}\mathcal{E}_{1,t}^{(n)}.$

Assumptions 4.2 iv implies convergence to zero in probability, and from this
we conclude that also $R_{2}\xrightarrow{P}0$.

We now turn to the discussion of the oracle term $U_{\hat{\theta}}^{(n)}$.
From Assumption 4.1 it follows that
$(T_{i},\mathbf{W}_{i},Y_{i})_{i\in\mathcal{I}_{3}}\,\perp\\!\\!\\!\perp\,\hat{\theta}$,
and hence the conditional distribution
$U_{\hat{\theta}}^{(n)}\,|\,\hat{\theta}=\theta$ is the same as
$U_{\theta}^{(n)}$. We show that this distribution is asymptotically Gaussian
uniformly over $\theta\in\Theta$. To see this, note that for each
$\theta\in\Theta$, the terms of $U_{\theta}^{(n)}$ are i.i.d. with mean
$\chi_{t}(\mathbf{Z}_{\theta})$ and variance
$\mathbb{V}_{t}(\mathbf{Z}_{\theta})$. The conditional Jensen’s inequality and
Assumption i imply that

$\displaystyle\mathbb{E}[|u_{i}(\theta)|^{2+\delta}]$ $\displaystyle\leq
C_{2+\delta}(\mathbb{E}[|g_{\theta}(t,\mathbf{W})|^{2+\delta}]+c^{-(1+\delta)}\mathbb{E}[|Y-g_{\theta}(t,\mathbf{W})|^{2+\delta}\,])$
$\displaystyle\leq C^{\prime}\mathbb{E}[|Y|^{2+\delta}],$

where $C^{\prime}>0$ is a constant depending on $c$ and $\delta$. Thus we can
apply the Lindeberg-Feller CLT, as stated in Shah & Peters (2020, Lem. 18), to
conclude that

$S_{n}(\theta)\coloneq\sqrt{|\mathcal{I}_{3}|}\mathbb{V}_{t}(\mathbf{Z}_{\theta})^{-1/2}(U_{\theta}^{(n)}-\chi_{t}(\mathbf{Z}_{\theta}))\xrightarrow{d/\Theta}\mathrm{N}(0,1),$

where the convergence holds uniformly over $\theta\in\Theta$. With
$\Phi(\cdot)$ denoting the CDF of a standard normal, this implies that

$\displaystyle\sup_{t\in}|\mathbb{P}(S_{n}(\hat{\theta})\leq t)-\Phi(t)|$
$\displaystyle=\sup_{t\in}\left\lvert\int(\mathbb{P}(S_{n}(\hat{\theta})\leq
t\,|\,\hat{\theta}=\theta)-\Phi(t))P_{\hat{\theta}}(\mathrm{d}\theta)\right\rvert$
(25)
$\displaystyle\leq\sup_{\theta\in\Theta}\sup_{t\in}|\mathbb{P}(S_{n}(\theta)\leq
t)-\Phi(t)|\longrightarrow 0.$

This shows that $S_{n}(\hat{\theta})\xrightarrow{d}\mathrm{N}(0,1)$ as
desired. The last part of the theorem is a simple consequence of Slutsky’s
theorem. ∎

### Proof of Theorem 4.4

For each $i\in\mathcal{I}_{3}$, let
$\widehat{u}_{i}=u_{i}(\hat{\theta})+r_{i}^{1}+r_{i}^{2}+r_{i}^{3}$ be the
decomposition from (16). For the squared sum, it is immediate from Theorem 4.3
that

$\displaystyle\left(\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\widehat{u}_{i}\right)^{2}=\left(\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}(\hat{\theta})\right)^{2}+o_{P}(1).$

For the sum of squares, we first expand the squares as

(26)
$\displaystyle\widehat{u}_{i}^{2}=u_{i}(\hat{\theta})^{2}+2(r_{i}^{1}+r_{i}^{2}+r_{i}^{3})u_{i}(\hat{\theta})+(r_{i}^{1}+r_{i}^{2}+r_{i}^{3})^{2}.$

We show that the last two terms are convergent to zero in probability. For the
cross-term, we note by Cauchy-Schwarz that

$\displaystyle\Big{|}\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1}+r_{i}^{2}+r_{i}^{3})u_{i}(\hat{\theta})\Big{|}\leq\left(\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1}+r_{i}^{2}+r_{i}^{3})^{2}\right)^{1/2}\left(\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}(\hat{\theta})^{2}\right)^{1/2}.$

We show later that the sum
$\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}^{2}$ is convergent
in probability. Thus, to show that the last two terms of (26) converge to
zero, it suffices to show that
$\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1}+r_{i}^{2}+r_{i}^{3})^{2}\xrightarrow{P}0$.
To this end, we observe that

$\displaystyle\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1}+r_{i}^{2}+r_{i}^{3})^{2}\leq\frac{3}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1})^{2}+(r_{i}^{2})^{2}+(r_{i}^{3})^{2}$

The last term is handled similarly to $R_{3}$ in the proof of Theorem 4.3,

$\displaystyle\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{3})^{2}$
$\displaystyle=\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)^{2}\left(\frac{\mathbbl{1}(T_{i}=t)}{\widehat{m}_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}-\frac{\mathbbl{1}(T_{i}=t)}{m_{\hat{\theta}}(t\,|\,\mathbf{W}_{i})}\right)^{2}$
$\displaystyle\leq\ell_{c}^{2}n\mathcal{E}_{1}^{(n)}\mathcal{E}_{2,t}^{(n)}\xrightarrow{P}0,$

where we have used the naive inequality $\sum a_{i}^{2}b_{i}^{2}\leq(\sum
a_{i}^{2})(\sum b_{i}^{2})$ rather than Cauchy-Schwarz. From the proof of
Theorem 4.3, we have that

$\displaystyle\mathbb{E}[(r_{i}^{1})^{2}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}]=\operatorname{Var}[r_{i}^{1}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}]\leq\frac{1-c}{c}\left(\widehat{g}_{\hat{\theta}}(t,\mathbf{W}_{i})-g_{\hat{\theta}}(t,\mathbf{W}_{i})\right)^{2},$

and hence the conditional Markov’s inequality yields

$\displaystyle\mathbb{P}\left(\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1})^{2}>\epsilon\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\right)$
$\displaystyle\leq\epsilon^{-1}\mathbb{E}\left[\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1})^{2}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\right]$
$\displaystyle=\epsilon^{-1}\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[(r_{i}^{1})^{2}\,|\,\mathbf{Z}_{\hat{\bullet}},\hat{\theta}\right]$
$\displaystyle\leq\frac{1-c}{\epsilon
c}\mathcal{E}_{2,t}^{(n)}\xrightarrow{P}0.$

Thus we also conclude that
$\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{1})^{2}\xrightarrow{P}0$.
Analogously, the final remainder
$\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}(r_{i}^{2})^{2}$ can be
shown to converge to zero in probability by leveraging the argument for
$R_{2}\xrightarrow{P}0$ in the proof of Theorem 4.3.

Combining the arguments above we conclude that

$\displaystyle\widehat{\mathcal{V}}_{t}(\widehat{g},\widehat{m})=\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}(\hat{\theta})^{2}-\Big{(}\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}(\hat{\theta})\Big{)}^{2}+o_{P}(1)$

As noted in the proof of Theorem 4.3, for each $\theta\in\Theta$ the terms
$\\{u_{i}(\theta)\\}_{i\in\mathcal{I}_{3}}$ are i.i.d with
$\mathbb{E}[|u_{i}(\theta)|^{2+\delta}]\leq
C^{\prime}\mathbb{E}[|Y|^{2+\delta}]<\infty$. Hence, the uniform law of large
numbers, as stated in Shah & Peters (2020, Lem. 19), implies that

$\displaystyle\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}(\theta)-\mathbb{E}[u_{1}(\theta)])\xrightarrow{P/\Theta}0,\qquad\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}^{2}(\theta)-\mathbb{E}[u_{1}(\theta)^{2}]\xrightarrow{P/\Theta}0,$

where the convergence in probability holds uniformly over $\theta\in\Theta$.
Since
$\operatorname{Var}(u_{1}(\theta))=\operatorname{Var}(\psi_{t}(\mathbf{Z}_{\theta}))=\mathbb{V}_{t}(\mathbf{Z}_{\theta})$,
this lets us conclude that

$\displaystyle
S_{n}(\theta)\coloneq\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}{u}_{i}(\theta)^{2}-\Big{(}\frac{1}{|\mathcal{I}_{3}|}\sum_{i\in\mathcal{I}_{3}}u_{i}(\theta)\Big{)}^{2}-\mathbb{V}_{t}(\mathbf{Z}_{\theta})\xrightarrow{P/\Theta}0.$

We now use that convergence in distribution is equivalent to convergence in
probability for deterministic limit variables, see Christgau et al. (2023,
Cor. B.4.) for a general uniform version of the statement. Since Assumption
4.1 implies that the conditional distribution
$S_{n}(\hat{\theta})\,|\,\hat{\theta}=\theta$ is the same as $S_{n}(\theta)$,
the computation in (A) with $\Phi(\cdot)=\mathbbl{1}(\cdot\geq 0)$ yields that
$S_{n}(\hat{\theta})\xrightarrow{d}0$. This lets us conclude that

$\widehat{\mathcal{V}}_{t}(\widehat{g},\widehat{m})=S_{n}(\hat{\theta})+o_{P}(1)\xrightarrow{P}0,$

which finishes the proof. ∎

###  A.7. Proof of Theorem 4.6

We fix $P\in\mathcal{P}$ and suppress it from notation throughout the proof.
Given $\theta\in^{d}$, the single-index model assumption (13) implies that

$\displaystyle