# Instrumented Common Confounding

Christian Tien<EMAIL_ADDRESS>Faculty of Economics, University of Cambridge

Causal inference is difficult in the presence of unobserved confounders. We
introduce the _instrumented common confounding_ (ICC) approach to
(nonparametrically) identify average causal (structural) effects with
instruments, which are exogenous only conditional on some unobserved _common
confounders_. The ICC approach is most useful in rich observational data with
multiple sources of unobserved confounding, where instruments are at most
exogenous conditional on some unobserved common confounders. Suitable examples
of this setting are various identification problems in the social sciences,
dynamic panels, and problems with multiple endogenous confounders. The ICC
identifying assumptions are closely related to those in mixture models,
proximal learning and IV. Compared to mixture models (Bonhomme et al., 2016b),
we require less conditionally independent variables and do not need to model
the unobserved confounder. Compared to proximal learning (Cui et al., 2020),
we allow for non-common confounders, with respect to which the instruments are
conditionally exogenous. Compared to IV (Newey and Powell, 2003), we allow
instruments to be exogenous conditional on some unobserved common confounders,
for which a set of observed variables is complete. We prove point
identification with outcome model and alternatively first stage restrictions.
We provide a practical step-by-step guide to the ICC model assumptions and
present the causal effect of education on income as a motivating example.
Keywords:
Causal Inference, Unobserved Confounding, Instrumental Variables, proximal
learning, Proximal Learning

##  1 Introduction

Causal inference in observational data with unobserved confounders is
difficult. Researchers inevitably rely on some unverifiable assumptions, which
invite criticism. A popular approach towards identification is the use of
instrumental variables (IV). Famously, instruments need to satisfy a relevance
condition and exclusion restriction. Finding excluded instruments is often
difficult or impossible in applications. Our novel _instrumented common
confounding_ (ICC) approach identifies average causal (structural) effects
with instruments, which are excluded (and relevant) conditional on some
_unobserved_ confounders. We call these unobserved confounders _common_ ,
because we assume their association with other observed variables. In
economics, a well-known common confounder is ability (cognitive skill) in the
education production function. An alternative title for this paper would be
_IV with mismeasured confounders_ , where the measurements of the unobservable
confounders may be economically meaningful variables with their own effect on
the outcome.

The here introduced ICC approach links instrumental variable (IV) and proximal
learning (also negative control) methods. Compared to proximal learning (Cui
et al., 2020), we relax the conditional unconfoundedness assumption for
treatment $A$. Instead, we require conditional relevance of action-aligned
proxies $Z$ with respect to treatment $A$. With conditional relevance, the
assumptions imposed on action-aligned proxies $Z$ start to resemble those in
IV. The main difference compared to IV (Newey and Powell, 2003) is that
exclusion and relevance of instruments $Z$ are no longer required to hold
conditional only on observed confounders, but may hold conditional on
(observed and) unobserved common confounders $U$. Thus, the information
contained in $U$ differs from proximal learning to our common confounding
setting: In proximal learning, treatment $A$ is exogenous conditional on $U$.
In our setting, the instruments $Z$ are exogenous conditional on $U$. While in
proximal learning $U$ contains every unobserved source of variation that made
treatment $A$ endogenous, in our common confounding setting $U$ contains every
unobserved source of variation that made instruments $Z$ endogenous. This may
be a much more realistic identifying assumption, e.g. when treatment is chosen
by heterogeneous economic agents. The identifying assumptions of our ICC
method are also related to mixture models (Bonhomme et al., 2016b), compared
to which less than three conditionally independent measurements of the
unobservable are needed.

Our novel method, instrumented common confounding (ICC), for which we prove
and explain identification in detail, is not a panacea. It replaces some
strong, untestable identifying assumptions by other such assumptions. While
the relevance assumptions of ICC are testable, its conditional exclusion
restriction remains untestable, except for over-identifying restrictions tests
(as in IV). To shed light on these assumptions, estimation of the education
production function is thoroughly discussed as a motivating example of the ICC
approach.

In section 2, we briefly discuss the related IV and proximal learning
literature. Section 3 contains the model setup in detail. Our main
contribution with new identification results in the common confounding model
are in section 4 and 5. In section 4 the outcome model is linearly separable
in the disturbance, whereas in section 5 different first stage reduced form
monotonicity restrictions are considered. Some numerical examples are included
in section 6. We present a practical algorithm, and describe the returns to
education and a health treatment with individual choice as examples of ICC
models in section 7. We conclude in section 8 and provide proofs in the
appendix.

##  2 Related Literature

Instrumented common confounding (ICC) bridges the proximal learning (Cui et
al., 2020) and nonparametric IV (Newey and Powell, 2003; Imbens and Newey,
2009) methods. Recent proximal learning literature (Miao et al., 2018) extends
nonclassical measurement error models with mismeasured confounders (Mahajan,
2006; Hu, 2008; Kasahara and Shimotsu, 2009; Kuroki and Pearl, 2014). All
measurement error models are characterised by independence conditions between
some observed variables conditional on the unobserved variable. In this sense,
measurement error models are a specific application of mixture models, for
which identification results are similarly available (Hettmansperger and
Thomas, 2000; Hall et al., 2003; Allman et al., 2009; Bonhomme et al., 2016b).
These identification results in mixture models have one thing in common: They
assume the independence of three observed variables conditional on the
unobserved variable. Under this key assumptions, the entire model is
nonparametrically identified in conjunction with completeness conditions
(Bonhomme et al., 2016b), which impose richness requirements on the observed
variables relative to the unobserved variable.

The proximal learning literature focuses on the identification of average
causal effects (ATE, ATT) in the presence of a common, unobserved confounder
(Miao et al., 2018; Cui et al., 2020; Singh, 2020). Instead of identification
of the entire model, only a causal effect of a treatment on an outcome is
identified, using proxies to instrument for each other and thus account for
the unobserved confounder. This restricted focus enables the identification of
average causal effects with weaker conditional independence assumptions on the
proxies compared to traditional mixture models. Specifically, the model no
longer needs to contain three conditionally independent measurements of the
unobserved confounder. In instrumental common confounding, we retain the
conditional independence assumptions of proximal learning, but drop
unconfoundedness conditional on the unobservable in favour of a relevance
requirement for the excluded action-aligned proxies, which become our
instruments.

The resemblance of proximal learning and IV is noteworthy. In proximal
learning, proxies are used as instruments for each other to adjust for the
confounding effect of the unobservable common confounder. Contrary to
traditional nonparametric IV (Newey and Powell, 2003), the proximal learning
problem is not ill- but well-posed (Deaner, 2018). In proximal learning,
conditional moments of observed variables only are constructed to identify
bridge functions of observed variables. Non-unique bridge function are no
problem, because any valid bridge function can be used to point-identify the
average causal effect of interest (Kallus et al., 2021). To our knowledge, we
are the first authors to leverage the similarities in identifying assumptions
of IV and proximal learning for a novel identification approach. From the
perspective of proximal learning, we add a conditional relevance requirement
for action-aligned proxies $Z$ (our instruments) with respect to treatment
$A$. With this strengthened relevance requirement, we allow for conditional
confoundedness of the treatment $A$ due to non-common confounders. From the
perspective of nonparametric IV, we allow the instrument $Z$ to satisfy
exclusion conditional on some unobserved common confounder $U$. Then, we add
conditionally independent, observable outcome-inducing proxies $W$ to the
model, which are sufficiently relevant for the common confounder $U$. As their
name suggests, unlike usual proxies the outcome-inducing proxies $W$ may be
economically meaningful with their own direct effect on outcome $Y$.

##  3 Setup

A treatment (action) $A\in\mathcal{A}$ is discrete or continuous, with base
measure $\mu_{A}$ of $\mathcal{A}$. The counterfactual $Y(a)\in\mathbb{R}$
would be observed if we could set $a\in\mathcal{A}$. $Y=Y(A)$ is the outcome
of observed action $A$. For notational simplicity, conditioning on observed
covariates $X\in\mathcal{X}\subseteq\mathbb{R}^{d_{X}}$ is not made explicit,
but always possible. The causal effect of interest $J$ is a function of
counterfactuals with a contrast function $\pi\mathrel{\mathop{\mathchar
58\relax}}\mathcal{A}\to\mathbb{R}$.

$\displaystyle J$
$\displaystyle\coloneqq\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}Y(a)\pi(a)d\mu_{A}(a)\right].$
(3.1)

Due to unmeasured confounders, which may be discrete, continuous or any mix,
exchangeability is violated:
$Y(a)\centernot{\mathrel{\perp\mspace{-10.0mu}\perp}}A$.
$Z\in\mathcal{Z}\subseteq\mathbb{R}^{d_{Z}}$ is a vector of _conditionally
exogenous_ , relevant instruments for treatment $A$. Instruments $Z$, which
may be discrete, continuous or any mix, are exogenous conditional on a subset
of unobserved _common_ confounders $U\in\mathcal{U}$.

$\displaystyle Y(a)\centernot{\mathrel{\perp\mspace{-10.0mu}\perp}}Z\text{ but
}Y(a)\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$

In proximal learning, the equivalent of our instruments $Z$ is called action-
aligned proxies (Deaner, 2021). Just like the action-aligned proxies in
proximal learning, these instruments satisfy an exclusion restriction with
respect to the potential outcomes $Y(a)$ conditional on the common confounders
$U$ (3.1.2a). However, unlike in proximal learning where action-aligned
proxies may not directly affect treatment $A$ (Cui et al., 2020), we use
variation in $Z$ to instrument for treatment $A$. This instrumentation step
requires relevance of instruments $Z$ for treatment $A$ conditional on common
confounders $U$ (3.1.2b). Consequently, we prefer to call $Z$ conditionally
exogenous instruments rather than action-aligned proxies with a relevance
requirement, but either term would be equally valid.

Below, the set of conditional independence and relevance assumptions of the
simple common confounding model are listed. Noticeably, the below assumptions
only impose a stronger relevance requirement for instruments $Z$ (3.1.2b)
compared to standard proximal learning.

######  Assumption 3.1 (Simple Common Confounding Model).

  1. 1.

SUTVA: $Y=Y(A,Z)$ and $W=W(A,Z)$.

  2. 2.

Instruments

    1. (a)

Exclusion: $Y(a,z)=Y(a)\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U\
\forall a\in\mathcal{A}$.

    2. (b)

Relevance (completeness): For any $g(A,U)\in L_{2}(A,U)$,

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|Z\right]$
$\displaystyle=0\text{ only when }g(A,U)=0.$ (3.2)

  3. 3.

Outcome-inducing proxies

    1. (a)

Exclusion: $W(a,z)=W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U$.

    2. (b)

Relevance (bridge function): There exists some function $h_{0}\in L_{2}(A,W)$
such that

$\displaystyle\operatorname{\mathbb{E}}\left[h_{0}(A,W)|A,U\right]$
$\displaystyle=k_{0}(A,U)$ (3.3)

almost surely, where $k_{0}\in L_{2}(A,U)$ is a function of interest.

Assumption 3.1.1 is a stable unit treatment value assumption (Imbens and
Rubin, 2015). It implies no interference across units and is not the focus of
this work. Instruments $Z$ must be independent from the potential outcomes
$Y(a)$ conditional on common confounders $U$, including no direct effect on
outcomes other than through the treatment as stated in assumption 3.1.2a.
Hence, by definition the common confounders $U$ contain all unobservables
conditional on which the instruments would satisfy an exclusion restriction
with respect to the potential outcomes. Instrument relevance is formulated as
a completeness condition with respect to $(A,U)$ in assumption 3.1.2b. This
completeness requirement means that conditional on the common confounders, the
remaining variation in $Z$ must still be sufficiently relevant for treatment
$A$. In some sense, this assumption sounds very similar to the standard
relevance requirement in IV conditional on observed confounders: The exogenous
variation in the instruments must be sufficiently relevant for the treatment.

The outcome-inducing proxies $W\in\mathcal{W}\subseteq\mathbb{R}^{d_{W}}$ may
be discrete, continuous or any mix. outcome-inducing proxies $W$ may directly
affect $Y$, while independent from $(A,Z)$ conditional on $U$ (3.1.3a). Their
richness requirement 3.1.3b with respect to $U$ is stated as the existence of
a bridge function $h_{0}\in L_{2}(A,W)$, whose expectation conditional on
$(A,U)$ must equal a function of interest $k_{0}(A,U)$, which closely relates
to the causal effect of interest $J$ (see section 4 and 5). The function of
interest $k_{0}$ is either some average structural function, or defined by
moment restrictions as in assumption 4.1. Completeness of $W$ for $U$
(conditional on $A$) is sufficient for 3.1.3b and can thus be used
alternatively to ensure relevance of $W$ for $U$ without reference to a
specific function of interest $k_{0}$. The action bridge function $h$ will
generally not be unique whenever $W$ carries more information than $U$ (Kallus
et al., 2021).

Figure 1: Example DAG of an instrumented common confounding model

$U$$A$$Y$$\ \ \ Z\ \ \ $$W$$\tilde{U}$2a2b3b

  * $Y$:

Outcome

  * $A$:

Treatment

  * $Z$:

Instrument

  * $W$:

outcome-inducing proxy

  * $U$:

Common Confounder

  * $\tilde{U}$:

Non-Common Confounder

The directed acyclic graph (DAG) in figure 1 is one of many possible
representations of the conditional independences implied by assumption 3.1.
All unobserved variables and their direct effects are illustrated by dashed
nodes and edges. The unobserved confounders $U$ are common, as they affect all
observed variables $(Z,A,Y,W)$. At the same time, there are other unobserved
confounders $\tilde{U}$ for the effect of $A$ on $Y$, with respect to which
instruments $Z$ are exogenous (possibly conditional on $U$). The absence of
edges between $(Y,\tilde{U})$ and $Z$ captures the exclusion condition
satisfied by instruments $Z$ (3.1.2a): The instruments $Z$ satisfy exclusion
conditional on the unobservable common confounders $U$. The relevance
requirement for $Z$ is depicted with a thick directed edge from $Z$ to $A$
(3.1.2b). The outcome-inducing proxies $W$ have no direct edges to $(A,Z)$.
Any association between $W$ and $(A,Z)$ stems from $U$, conditional on which
they are independent, which reflects the exclusion restriction for $W$
(3.1.3a), even if it is not made as explicit in the graph as the exclusion
restrictions for instruments $Z$. The richness requirement for outcome-
inducing proxies $W$ with respect to $U$ (3.1.3b) is illustrated with a thick
arrow from $U$ to $W$. The confounder $\tilde{U}$ could be associated with
$U$, but this link is omitted in favour of tractability in the DAG in figure
1.

#### Notation

All results hold irrespective of whether we condition on covariates $X$, so
$X$ is dropped from notation for simplicity. $\mathbb{E}$ is the expectations
operator wrt $(Y,A,Z,W)$. $\mathbb{E}_{n}$ is the empirical average over $n$
observations of $(Y,A,Z,W)$. Let $L_{2}(O)$ be the space of square-integrable
functions of a variable $O$ measurable wrt $(Y,A,Z,W)$.

##  4 Identification with Outcome Model Restrictions

Identification of causal effect $J$ defined in 3.1 is the goal of this paper.
In section 3, we introduced the common confounding model. In this section, we
derive a main identification theorem, which relies on separability of the
outcome model in the disturbance in assumption 4.1 after introducing some
useful lemmas.

Using instruments for point identification of causal effects always requires
some parametric model assumptions. One common approach is to formulate
conditional moment restrictions for the outcome model. In assumption 4.1, we
formulate such conditional moment restrictions. Treatment effect estimation
with a continuous outcome fits into this framework.

######  Assumption 4.1 (Outcome model linearly separable in disturbance).

There exists some function $k_{0}\in L_{2}(A,U)$ such that

$\displaystyle Y=Y(A)$ $\displaystyle=k_{0}(A,U)+\varepsilon,$
$\displaystyle\operatorname{\mathbb{E}}\left[\varepsilon|Z,U\right]$
$\displaystyle=0.$ (4.1)

In the model described by assumption 4.1, the outcome $Y$ is linearly
separable as a counterfactual mean function $k_{0}(A,U)$ and disturbance
$\varepsilon$. A conditional moment holds, which states the mean-independence
of disturbances $\varepsilon$ from instruments $Z$ given common confounders
$U$. The treatments $A$ may therefore be endogenous.

#### IV with Fully Exogenous Instruments

First, suppose the confounders $U$ were observed. Conditional on $U$, the
instruments $Z$ satisfy an exclusion restriction. Then, the counterfactual
mean function $k_{0}$ is identified by the completeness condition for $Z$ with
respect to $A$ conditional on $U$ (3.1.2b). $k_{0}$ can be estimated by a
variety of learning methods (Dikkala et al., 2020) from the conditional moment
restrictions

$\displaystyle\operatorname{\mathbb{E}}\left[\left.Y-k_{0}(A,U)\right|Z,U\right]$
$\displaystyle=0.$

The counterfactuals $Y(a)$ are closely related to the counterfactual mean
function $k_{0}$. Once $k_{0}$ is identified, the causal effect $J$ is also
identified. First, the contrast function $\pi$ is used to integrate out $A$ in
$k_{0}(A,U)$ while $U$ is held fixed, producing $\phi_{IV}(U;k_{0})$ as
defined in lemma 4.0.1. Then, the common confounders $U$ are integrated out
from $\phi_{IV}(U;k_{0})$ without dependence on treatment $A$ to obtain causal
effect $J$.

######  Lemma 4.0.1.

If assumption 4.1 holds, then

$\displaystyle J=$
$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{IV}(U;k_{0})\right]$ where
$\displaystyle\ \ \ \
\phi_{IV}(u;k_{0})\coloneqq\int_{\mathcal{A}}k_{0}(a^{\prime},u)\pi(a^{\prime})d\mu_{A}(a^{\prime})$

When the common confounders $U$ are unobserved, conditioning on them is of
course impossible. An alternative identification approach for $J$ is required.
That alternative identification approach relies on bridge functions. First, we
use that by assumption 3.1.3b there exists some bridge function $h_{0}\in
L_{2}(A,W)$, whose expectation conditional on $(A,U)$ equals the
counterfactual mean function $k_{0}(A,U)$. In this sense, $h$ bridges the
function spaces $L_{2}(A,U)$ and $L_{2}(A,W)$ for $k_{0}$. Let
$\mathbb{H}_{0}$ be the nonempty set of valid outcome bridge functions $h$
defined by

$\displaystyle\mathbb{H}_{0}$ $\displaystyle=\left\\{h\in
L_{2}(A,W)\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h(A,W)|A,U\right]=0\right\\}\neq\emptyset.$
(4.2)

A sufficient condition for the existence of the outcome bridge function $h$
(assumption 3.1.3b) is that $W$ is complete with respect to $U$ conditional on
$A$.

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|A,W\right]=0\text{ only
when }g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

Completeness can be understood as a nonparametric relevance requirement.
Intuitively, the richness of relevant variation in $W$ with respect to $U$
allows us to replicate any counterfactual mean $k_{0}(A,U)$ with the outcome
bridge function $h_{0}(A,W)$ by taking the expectation of the latter
conditional on $(A,U)$. Due to this replicability, the causal effect $J$ can
be retrieved similarly as in lemma 4.0.1. First, the contrast function $\pi$
is used to integrate out $A$ in a valid bridge function $h_{0}(A,W)$ while $W$
is held fixed, producing $\tilde{\phi}_{IV}(W;h_{0})$ as defined in lemma
4.0.2. Then, the outcome-inducing proxies $W$ are integrated out from
$\tilde{\phi}_{IV}(W;h_{0})$ without dependence on treatment $A$ to obtain
causal effect $J$. Lemma 4.0.2 describes this identification mechanism
formally.

######  Lemma 4.0.2.

Suppose 4.1 and 3.1.3 hold. For any $h_{0}\in\mathbb{H}_{0}$,

$\displaystyle J=$
$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IV}(w;h_{0})\right]$
where $\displaystyle\ \ \ \
\tilde{\phi}_{IV}(w;h_{0})\coloneqq\int_{\mathcal{A}}h_{0}(a,w)\pi(a)d\mu_{A}(a)$

While lemma 4.0.2 shifts the focus from identifying a function of partly
unobservable $k_{0}\in L_{2}(A,U)$ to a function of only observables $h_{0}\in
L_{2}(A,W)$, the moment conditions defining the valid set of outcome bridge
functions $\mathbb{H}_{0}$ are still conditional on partly unobservable
$(A,U)$. Hence, observed data cannot immediately identify any
$h_{0}\in\mathbb{H}_{0}$. The next step in the direction of identification is
lemma 4.0.3. The lemma states that any bridge function in the set
$\mathbb{H}_{0}$ also satisfies equation 4.3, which is a conditional moment
restriction involving only observed variables. In the conditional moment
restriction 4.3 we condition on the observable instruments $Z$ instead of
partly unobservable $(A,U)$.

######  Lemma 4.0.3.

Under assumptions 4.1 and 3.1.3, any $h_{0}\in\mathbb{H}_{0}$ satisfies that

$\displaystyle\operatorname{\mathbb{E}}\left[Y-h_{0}(A,W)|Z\right]$
$\displaystyle=0$ (4.3)

These conditional moment restrictions in lemma 4.0.3 resemble those of IV. A
main difference to IV with a conditional moment restriction of this type is
that the instruments $Z$ do not need to be relevant with respect to both $A$
and $W$, which would also imply the uniqueness of $h_{0}$. Instead, $h_{0}$
will often be non-unique (specifically if $W$ contains more information than
$U$), yet the causal effect $J$ is still point-identified. The conditional
moment restriction 4.3 defines a different set of observable bridge functions
$\mathbb{H}^{\text{obs}}_{0}$, defined below.

$\displaystyle\mathbb{H}^{\text{obs}}_{0}$ $\displaystyle=\\{h\in
L_{2}(A,W)\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[Y-h(A,W)|Z\right]=0\\}\neq\emptyset.$
(4.4)

To be able to use the observable bridge functions in the identification of
$J$, the remaining task is to relate $\mathbb{H}_{0}^{\text{obs}}$ and
$\mathbb{H}_{0}$. Only then, we can hope to identify $J$ with some
$h\in\mathbb{H}_{0}^{\text{obs}}$. Fortunately, the completeness condition
3.1.2b for $Z$ with respect to $A$ conditional on $U$ ensures the equivalence
of $\mathbb{H}_{0}^{\text{obs}}$ and $\mathbb{H}_{0}$. Lemma 4.0.4 states this
equivalence result formally.

######  Lemma 4.0.4.

Under assumption 3.1,

$\displaystyle\mathbb{H}_{0}=\mathbb{H}_{0}^{\text{obs}}.$ (4.5)

Now that the equivalence of the set of original bridge functions
$\mathbb{H}_{0}$ and observable bridge functions $\mathbb{H}_{0}^{\text{obs}}$
has been shown, our main theorem 4.1 in the outcome model restrictions case
follows straightforwardly. The causal effect $J$ is identifiable with any
observable bridge function $h_{0}\in\mathbb{H}_{0}^{\text{obs}}$. First, the
contrast function $\pi$ is used to integrate out $A$ in an observable and
valid bridge function $h_{0}(A,W)$, while $W$ is held fixed, producing
$\tilde{\phi}_{IV}(W;h_{0})$. Then, the outcome-inducing proxies $W$ are
integrated out from $\tilde{\phi}_{IV}(W;h_{0})$ without dependence on
treatment $A$ to obtain causal effect $J$.

######  Theorem 4.1.

Suppose 4.1 and 3.1 hold. For any $h_{0}\in\mathbb{H}_{0}^{\text{obs}}$,

$\displaystyle J=$
$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IV}(w;h_{0})\right]$
where $\displaystyle\ \ \ \
\tilde{\phi}_{IV}(w;h_{0})=\int_{\mathcal{A}}h_{0}(a,w)\pi(a)d\mu_{A}(a)$

#### Comment on Non-uniqueness

The non-uniqueness of $h_{0}$ and point identification of $J$ may at first
seem at odds. The observable bridge function
$h_{0}\in\mathbb{H}_{0}^{\text{obs}}$ will not be unique whenever $W$ contains
more information than necessary for completeness with respect to $U$. Hence,
the non-uniqueness of $h_{0}$ will only be reflected in its second argument,
the outcome-inducing proxies $W$. While the $\tilde{\phi}_{IV}(W;h_{0})$ may
be non-unique as a function of $W$, the causal effect $J$ is unique once we
take the expectation over $W$ in
$\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IV}(W;h_{0})\right]$ without
dependence on treatment $A$, similar to the arguments in standard proximal
learning (Kallus et al., 2021).

##  5 Identification with First Stage Restrictions

To obtain point identification in IV, first stage monotonicity assumptions
represent an alternative to linear separability in the outcome model
disturbance. The same logic applies to the common confounding model, but some
additional steps are needed, which we lay out in this section. Some form of
first stage strict monotonicity assumption could suit identification problems
with continuous treatments.

We distinguish three main cases of the control function approach: First, when
the common confounders $U$ are observed. Second, when the common confounders
$U$ do not enter the first stage reduced form. Third, when the common
confounders $U$ enter the first stage reduced form, but in a way that
restricts the complexity of interactions.

###  5.1 All Confounders Observed

The causal effect $J$ is identified by a simple control function strategy when
the common confounders $U$ are observed.

######  Assumption 5.1 (Strict Monotonicity with Observed Confounders).

$\displaystyle Y$ $\displaystyle=g(A,U,\varepsilon)$ (5.1) $\displaystyle A$
$\displaystyle=h(Z,U,\eta)$ (5.2)

  1. 1.

The reduced form $h(Z,U,r)$ is strictly monotonic in $r$ with probability 1,

  2. 2.

and $\eta$ is a continuously distributed scalar with a CDF that is strictly
increasing on the support of $\eta$ (conditional on $U$).

  3. 3.

$(\eta,\varepsilon)\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$.

In case of observed $U$, assumption 5.1.3 states that $\varepsilon$ and $Z$
are independent conditional on $U$. By theorem 1 in Imbens and Newey (2009),
$A$ and $\varepsilon$ are then independent conditional on the control variable

$\displaystyle V\coloneqq F_{A|Z,U}(A,Z,U)=F_{\eta|U}(\eta).$ (5.3)

$V$ is a one-to-one function of $\eta$ due to the strict monotonicity of $h$
in its third argument and the strictly increasing CDF $F_{\eta|U}$ (assumption
5.1.1/2). For this reason, conditioning on $(V,U)$ is equivalent to
conditioning on $(\eta,U)$. Then, as conditional on $(V,U)$ all variation in
treatment $A$ stems from the instruments $Z$, $A$ and $\varepsilon$ are
independent conditional on the control variable $(V,U)$ due to assumption
5.1.3.

Instrument relevance is expressed as a common support assumption 5.2 in this
model.

######  Assumption 5.2 (Common Support with Observed Confounders).

For all $A\in\mathcal{A}$ where $\pi(A)\neq 0$, the support of $V$ conditional
on $(A,U)$ equals the support of $V$ conditional on $U$.

A typical function of interest is the average structural function

$\displaystyle k_{0}(A,U)$
$\displaystyle=\int_{\mathcal{V}}g(A,U,e)\operatorname{d\\!}F_{\varepsilon|U}(e),$
(5.4)

which is identified by the standard nonparametric control function approach
when the common confounders $U$ are observed (Imbens and Newey, 2009).
However, moving forward $U$ will be unobserved, so we instead focus on
identification of the causal effect $J$ as defined in 3.1. While the average
structural function in 5.4 is a function of $U$, the causal effect $J$ in 3.1
averages out the common confounders $U$ without dependence on the treatment
$A$. $J$ is a less informative version of the average structural function
$k_{0}$, because in $J$ there is one more level of averaging compared to
$k_{0}$.

Let $k_{0,v_{(j)}}(a,v_{(j)},u)$ be the condtional expectation
$k_{0,v_{(j)}}(a,v_{(j)},u)\coloneqq\operatorname{\mathbb{E}}\left[Y|A=a,V_{(j)}=v_{(j)},U=u\right]$,
where $(j)$ may index different control variables $V_{(j)}$. The generalised
propensity score is the conditional density of $A$ given $V$ and $U$,
$f_{A|V,U}(a|v,u)$, relative to base measure $\mu_{A}$ (Hirano and Imbens,
2004). Lemma 5.0.1 below states that the IPW estimator $\phi_{IPW}$,
regression estimator $\phi_{REG}$ and doubly robust estimator $\phi_{DR}$
allow identification of the causal effect $J$ conditional on the control
variable $V$.

######  Lemma 5.0.1.

If assumptions 5.1 and 5.2 hold, then

$\displaystyle J=$
$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V,U;f_{A|V,U})\right]=\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0,v})\right]=\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V,U;f_{A|V,U},k_{0,v})\right]$
where $\displaystyle\ \ \ \
\phi_{IPW}(y,a,v,u;f_{A|V,U})=y\frac{\pi(a)}{f_{A|V,U}(a|v,u)}$
$\displaystyle\ \ \ \
\phi_{REG}(v,u;k_{0,v})=\int_{\mathcal{A}}k_{0,v}(a^{\prime},v,u)\pi(a^{\prime})d\mu_{A}(a^{\prime})$
$\displaystyle\ \ \ \
\phi_{DR}(y,a,v,u;f_{A|V,U},k_{0,v})=(y-k_{0,v}(a,v,u))\frac{\pi(a)}{f_{A|V,U}(a|v,u)}+\int_{\mathcal{A}}k_{0,v}(a^{\prime},v,u)\pi(a^{\prime})d\mu_{A}(a^{\prime})$

This baseline result is helpful, as we will find equivalents to it when the
common confounders $U$ are unobserved.

###  5.2 Common Confounders Do Not Affect the First Stage

If the common confounders $U$ are absent from the first stage reduced form,
including the independence of $Z$ and $\eta$ unconditional on $U$,
identification is again possible without further conditional independence
assumptions. This model is described in assumption 5.3.

######  Assumption 5.3 (Strict Monotonicity without Common Confounders in
First Stage).

$\displaystyle Y$ $\displaystyle=g(A,U,\varepsilon)$ (5.5) $\displaystyle A$
$\displaystyle=h(Z,\eta)$ (5.6)

  1. 1.

The reduced form $h(Z,r)$ is strictly monotonic in $r$ with probability 1,

  2. 2.

and $\eta$ is a continuously distributed scalar with a CDF that is strictly
increasing on the support of $\eta$.

  3. 3.

$\eta\mathrel{\perp\mspace{-10.0mu}\perp}Z$, and
$\varepsilon\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$.

Assumption 5.3 primarily restricts the complexity of the first stage reduced
form. The absence of $U$ from the reduced form does not imply that $U$ cannot
affect treatment $A$. Below we provide an example of a simple model in which
the common confounder $U$ is linearly associated with treatment $A$ and
instruments $Z$.

$\displaystyle A$ $\displaystyle=Z\zeta+U\gamma_{A}+\epsilon_{A},$
$\displaystyle U$ $\displaystyle=Z\tilde{\gamma}_{Z}+\epsilon_{U},$
$\displaystyle(\epsilon_{A},\epsilon_{U})$
$\displaystyle\mathrel{\perp\mspace{-10.0mu}\perp}Z$ $\displaystyle\implies A$
$\displaystyle=Z\left(\zeta+\tilde{\gamma}_{Z}\gamma_{A}\right)+\eta,$
$\displaystyle\eta$ $\displaystyle=\epsilon_{A}+\epsilon_{U}\gamma_{A},$
$\displaystyle\eta$ $\displaystyle\mathrel{\perp\mspace{-10.0mu}\perp}Z$

Assuming independence of instruments $Z$ from the structural disturbances
$(\epsilon_{A},\epsilon_{U})$ suffices to write a reduced form, in which the
instruments are independent from the disturbance $\eta$.

Due to the absence of $U$ from the first stage reduced form, the variation in
treatment $A$ can be separated into variation induced by instruments $Z$
versus disturbance $\eta$. Subject to assumption 5.3, the control function

$\displaystyle V_{\ref{a:strict-monotonicity-no-cf-fs}}$
$\displaystyle\coloneqq F_{A|Z}(A,Z)=F_{\eta}(\eta)$

is identifiable. $V_{\ref{a:strict-monotonicity-no-cf-fs}}$ is a one-to-one
function of $\eta$ due to the strict monotonicity of $h$ in its second
argument and the strictly increasing CDF $F_{\eta}$ (assumption 5.3.1/2). For
this reason, conditioning on $(V_{\ref{a:strict-monotonicity-no-cf-fs}},U)$ is
equivalent to conditioning on $(\eta,U)$. Then, as conditional on
$(V_{\ref{a:strict-monotonicity-no-cf-fs}},U)$ all variation in treatment $A$
stems from the instruments $Z$, $A$ and $\varepsilon$ are independent
conditional on the control variable $(V_{\ref{a:strict-monotonicity-no-cf-
fs}},U)$ due to assumption 5.3.3. A result of the simple one-to-one relation
of $V_{\ref{a:strict-monotonicity-no-cf-fs}}$ and $\eta$, the results of
section 5.4 apply regarding the identification of causal effect $J$ with
bridge functions.

###  5.3 Control Bridge Function

Another, more complicated, option to identify the causal effect $J$ relies on
identification of a surrogate control function, which would render treatment
$A$ exogenous if the common confounders $U$ were observed. First, this model
is formally described in assumption 5.4.

######  Assumption 5.4 (Strict Monotonicity).

$\displaystyle Y$ $\displaystyle=g(A,U,\varepsilon)$ (5.7) $\displaystyle A$
$\displaystyle=h(Z,m(U,\eta))$ (5.8)

  1. 1.

The reduced form $h(Z,m)$ is strictly monotonic in $m$ with probability 1,

  2. 2.

$m(U,t)$ for $m\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}\times\mathcal{H}\rightarrow\mathbb{R}$ is strictly
monotonic in $t$ with probability 1,

  3. 3.

and $\eta$ is a continuously distributed scalar with a CDF that is strictly
increasing on the support of $\eta$ (conditional on $U_{0}$).

  4. 4.

$(\eta,\varepsilon)\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$.

The above strict monotonicity assumption 5.4 reduces the amount of
interactions between $Z$ and $U$ in the reduced form for $A$ (5.4.2) on top of
imposing strict monotonicity in the scalar disturbance $\eta\in\mathcal{H}$
(5.4.1/2). The scalar $\eta$ is continuously distributed and independent from
$Z$ conditional on $U$, as is $\varepsilon$. The reduction in the dimension of
interactions in the first stage reduced form is needed for the identifiability
of some surrogate version of scalar $\eta$ when the common confounders are
unobserved.

To identify some surrogate version of $\eta$ in the first stage reduced form,
we need additional information in form of conditional independence
assumptions. The requirements formulated in assumption 5.5 enable
identification of the first stage to the degree that some useful control
function for the endogenous variation $\eta$ becomes obtainable. These
assumptions go beyond the basic common confounding model 3.1. This approach
can accommodate the fact that only a subset of common confounders
$U_{0}\in\mathcal{U}_{0}\subseteq\mathcal{U}$ may enter the first stage
reduced form in a complexity-restricted way. In fact, the relevance
requirements in assumption 5.5 then are with respect $U_{0}$ instead of the
more complex $U$. Our notation in this section sticks with $U$ throughout, but
a researcher may consider less complex $U_{0}$ in the first stage to motivate
the relevance requirements in assumption 5.5.

######  Assumption 5.5 (Additional First Stage Requirements).

  1. 1.

SUTVA: $A(Z,W_{0})=A$, $W_{1}(A,Z)=W_{1}$.

  2. 2.

First stage instrument-aligned proxy:
$A(z,w_{0})=A(z)\mathrel{\perp\mspace{-10.0mu}\perp}W_{0}\ |\ U$

  3. 3.

First stage action-inducing proxy:
$W_{1}(a,z)=W_{1}\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$

Assumption 5.5.2 and 5.5.3 are typical proximal learning conditional
independence assumptions applied to the first stage reduced form instead of
the outcome equation. $W_{0}$ and $W_{1}$ are variables, which satisfy
conditional independence assumptions with respect to $Z$ and $A$. For example,
$W_{0}$ and $W_{1}$ could be different parts of a vector-valued outcome-
inducing proxy $W$.

Neither the conditional mean $k_{0,v}(a,v,u)$ nor the propensity score
$f(a|v,u)$ are directly identifiable without $U$. We use variation in the
first stage proxies $W_{0}$ and $W_{1}$ to identify a _control bridge
function_. We define the control quantity of interest $V_{\ref{a:strict-
monotonicity}}$ as

$\displaystyle V_{\ref{a:strict-monotonicity}}$
$\displaystyle\coloneqq\int_{\mathcal{U}}F_{A|Z,U}(A|Z,u)\operatorname{d\\!}F(u).$
(5.9)

With the control quantity $V_{\ref{a:strict-monotonicity}}$ we can identify
the causal effect of interest conditional on $U$. In section 5.3.1, we prove
and discuss this claim. In section 5.3.2, we show how to identify the control
quantity from observed data.

####  5.3.1 Properties of the Control Bridge

The control quantity $V_{\ref{a:strict-monotonicity}}$ is useful, because it
captures information about $\eta$, which would allow the exact identification
of $\eta$ conditional on $U$. The strict monotonicity properties of the
reduced form for treatment $A$ (5.4) allow us to write the control quantity as
a function of $\eta$ and $U$. Let $m^{-1}(.,U)$ be the inverse of $m(U,t)$ in
$t$ and $h^{-1}(.,Z)$ the inverse of $h(Z,m)$ in $m$. Below, we show that the
control quantity is equal to the expected conditional density of $\eta$,
integrating out $U$ without dependence on instruments $Z$.

$\displaystyle V_{\ref{a:strict-monotonicity}}$
$\displaystyle=\int_{\mathcal{U}}F_{A|Z,U}(A,Z,u)\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}\Pr{\left(h(Z,m(u,\eta))\leq
A\right)}\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}\Pr{\left(\eta\leq
m^{-1}\left(h^{-1}(A,Z),u\right)\right)}\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}F_{\eta|U}\left(m^{-1}\left(h^{-1}(A,Z),u\right)\right)\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}F_{\eta|U}\left(\eta(u)\right)\operatorname{d\\!}F(u)$

The last line follows from monotonicity of the reduced form $h(Z,m(U,\eta))$
for treatment $A$ in $\eta$ and the continuous conditional distribution of
$\eta$ (assumption 5.4). The continuous conditional distribution of $\eta$
(5.4) implies that $\eta$ is exactly identified from $(A,Z,U)$, because it
corresponds to a unique $F_{A|Z,U}$. As $\eta$ is identified only up to $U$
when $U$ is unobserved, we write $\eta(U)\coloneqq
m^{-1}\left(h^{-1}(A,Z),U\right)$.

With the complexity restriction on the interactions of $Z$ and $(U,\eta)$
imposed by assumption 5.4, $(V_{\ref{a:strict-monotonicity}},U)$ contains the
same information as $(V,U)$ (and hence $(\eta,U)$), which is formulated in
lemma 5.0.2 using sigma algebras.

######  Lemma 5.0.2.

Suppose assumption 5.4 holds. Then, the sigma algebras associated with the
following three vectors of random variables are identical:

$(V_{\ref{a:strict-monotonicity}},U),\ (V,U),\ (\eta,U).$

We include graphical intuition for the result that $\eta$ is identified
exactly from the mean of its conditional density, $V_{\ref{a:strict-
monotonicity}}$, and $U$, in the proof of lemma 5.0.2 in the appendix section
of proofs 9. As a consequence of lemma 5.0.2, conditioning on the feasible
control quantity $V_{\ref{a:strict-monotonicity}}$ jointly with $U$ is as good
as conditioning on $V=F_{A|Z,U}(A|Z,U)$ (which corresponds to a unique $\eta$)
with $U$. Hence, if $U$ were observed, conditioning on $V_{\ref{a:strict-
monotonicity}}$ would still identify the causal effect $J$. The conditional
exogeneity of $Z$ renders $V_{\ref{a:strict-monotonicity}}$ a valid control
function conditional on $U$. We formulate this idea in lemma 5.1, which
formally states that conditioning the IPW, REG and DR estimators on
$V_{\ref{a:strict-monotonicity}}$ and $U$ instead of $V$ and $U$ leads to a
similarly unbiased estimator. Let the IPW, REG and DR estimators be defined as
in lemma 5.0.1.

######  Theorem 5.1.

Under assumptions 3.1, 5.4 and 5.5,

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V_{\ref{a:strict-
monotonicity}},U;f_{A|V_{\ref{a:strict-monotonicity}},U})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V,U;f_{A|V,U})\right],$
$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{REG}(V_{\ref{a:strict-
monotonicity}},U;k_{0,V_{\ref{a:strict-monotonicity}}})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0,v})\right],$
$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V_{\ref{a:strict-
monotonicity}},U;f_{A|V_{\ref{a:strict-
monotonicity}},U},k_{0,V_{\ref{a:strict-monotonicity}}})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V,U;f_{A|V,U},k_{0,v})\right].$

Theorem 5.1 is one of our main results, but not immediately useful. Neither
are $U$ ever observed, nor have we shown how to identify $V_{\ref{a:strict-
monotonicity}}$ from observable data. We address the identifiability of
control quantity $V_{\ref{a:strict-monotonicity}}$ in section 5.3.2 with
bridge functions.

####  5.3.2 Identification of the Control Quantity

We assume the existence of a control bridge function $\tau_{A,0}$ in
assumption 5.6. For its definition, we use variables $W_{0}$ and $W_{1}$,
where $W_{0}\in\mathcal{W}_{0}$ and $W_{1}\in\mathcal{W}_{1}$. $W_{0}$ and
$W_{1}$ need to have sufficient independent variation conditional on $U$, but
they must not be fully conditionally independent. That conditionally
independent part of variation in $W_{0}$ and $W_{1}$ must be sufficiently
relevant for $U$. $W_{0}$ and $W_{1}$ may be elements of the outcome-inducing
proxy $W\in\mathcal{W}$, or not.

######  Assumption 5.6.

Let $(W_{0},W_{1})\in\mathcal{W}_{0}\times\mathcal{W}_{1}$. There exists a
bridge function $\tau_{A,0}\in L_{2}(Z,W_{1})$ for all $A\in\mathcal{A}$ such
that

$\displaystyle\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,W_{0},U\right]$
$\displaystyle=F(A|Z,U).$ (5.10)

and $\kappa_{0}\in L_{2}(Z,W_{0})$ such that

$\displaystyle\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,W_{1},U\right]$
$\displaystyle=\frac{f(U)}{f(U|Z)}.$ (5.11)

Assumption 5.6 is a requirement on the amount of independent variation in
$W_{0}$ and $W_{1}$ conditional on $U$. $W_{0}$ and $W_{1}$ must not be fully
independent conditional on $U$. However, some independent variation must be
contained in $W_{0}$ and $W_{1}$, and this independent variation must be
sufficiently rich with respect to $U$.

Let $\mathbb{T}_{0}$ and $\mathbb{K}_{0}$ be the nonempty (assumption 5.6)
sets of valid control bridge functions defined by

$\displaystyle\mathbb{T}_{0}$ $\displaystyle=\left\\{\tau_{A}\in
L_{2}(Z,W_{1})\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\tau_{A}(Z,W_{1})|Z,U,W_{0}\right]=F(A|Z,U)\right\\}\neq\emptyset,$
(5.12) $\displaystyle\mathbb{K}_{0}$ $\displaystyle=\left\\{\kappa\in
L_{2}(Z,W_{0})\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\kappa(Z,W_{0})|Z,U,W_{1}\right]=\frac{f(U)}{f(U|Z)}\right\\}\neq\emptyset.$
(5.13)

So far, the existence of such bridge functions does not immediately help with
identification, because their defining conditional moments involve the
unobserved $U$. Using the same proximal learning logic as in section 4, lemma
5.1.1 states that all valid bridge functions above also satisfy a different
set of observed conditional moments.

######  Lemma 5.1.1.

Under assumption 5.5, any $\tau_{A,0}\in\mathbb{T}_{0}$ and
$\kappa_{0}\in\mathbb{K}_{0}$ satisfy that

$\displaystyle\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,W_{0}\right]$
$\displaystyle=F(A|Z,W_{0}).$ (5.14)
$\displaystyle\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,W_{1}\right]$
$\displaystyle=\frac{f(W_{1})}{f(W_{1}|Z)}.$ (5.15)

The above result suggests that we may learn the bridge function $\tau_{A,0}$
and $\kappa_{0}$ from the set of observed bridge functions
$\mathbb{T}_{0}^{\text{obs}}$ and $\mathbb{K}_{0}^{\text{obs}}$, defined
below.

$\displaystyle\mathbb{T}_{0}^{\text{obs}}$ $\displaystyle=\left\\{\tau_{A}\in
L_{2}(Z,W_{1})\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\tau_{A}(Z,W_{1})|Z,W_{0}\right]=F(A|Z,W_{0})\right\\}\neq\emptyset,$
(5.16) $\displaystyle\mathbb{K}_{0}^{\text{obs}}$
$\displaystyle=\left\\{\kappa\in L_{2}(Z,W_{0})\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\kappa(Z,W_{0})|Z,W_{1}\right]=\frac{f(W_{1})}{f(W_{1}|Z)}\right\\}\neq\emptyset.$
(5.17)

As no assumptions are imposed on the uniqueness of the observed bridge
functions, it remains uncertain how to identify the control quantity
$V_{\ref{a:strict-monotonicity}}$ from them. In this regard, lemma 5.1.2
states that any $\tau_{A}\in\mathbb{T}_{0}^{\text{obs}}$ can identify
$V_{\ref{a:strict-monotonicity}}$.

######  Lemma 5.1.2.

Suppose assumptions 5.5 and 5.6 hold. It follows that for any $\tau_{A}\in
L_{2}(Z,W_{1})$ and $\kappa_{0}\in\mathbb{K}_{0}^{\text{obs}}$,

$\displaystyle\int_{\mathcal{W}_{1}}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1})-V_{\ref{a:strict-
monotonicity}}$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\operatorname{\mathbb{E}}\left[\tau_{A}(Z,W_{1})-F(A|Z,W_{0})|Z,W_{0}\right]|Z\right].$

Hence, for any $\tau_{A}\in\mathbb{T}_{0}^{\text{obs}}$ as long as
$\mathbb{K}_{0}^{\text{obs}}\neq\emptyset$,

$\displaystyle V_{\ref{a:strict-monotonicity}}$
$\displaystyle=\int_{\mathcal{W}_{1}}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1}).$

The above result means that a control function exists and can be estimated
from $Z$, $W_{0}$ and $W_{1}$ under richness and (some, not full) independence
requirements for $W_{0}$ and $W_{1}$. Importantly, once the control function
is identified, the causal effect $J$ can be estimated with action and outcome
bridge functions similarly to standard proximal learning (Cui et al., 2020).

###  5.4 Action and Outcome Bridge Functions

In subsections 5.2 and 5.3, we established that valid control functions exist
for the endogenous variation $\eta$ in treatment $A$ when the unobserved
common confounders $U$ are absent from the first stage (assumption 5.3), and
when $U$ enter the first-stage in a complexity-restricted way (assumptions
5.4, 5.5, 5.6). Identification of causal effect $J$ with unobserved common
confounders $U$ was not yet explained. In this section, we use ideas from
standard proximal learning (Cui et al., 2020) to address the remaining
confounding problem stemming from unobserved $U$ as a source of variation in
instruments $Z$. Properties of the action and outcome bridge function
properties are explained in section 5.4.1, their identification is discussed
in section 5.4.2.

####  5.4.1 Properties of the Action and Outcome Bridge Functions

First, we use assumption 5.7 to (re-)specify the exclusion and relevance for
$U$ of outcome-inducing proxies $W$. The outcome bridge 5.18 for the
regression $k_{0,\tilde{V}}(A,\tilde{V},U)$ sufficiently describes the
relevance requirement for $W$, and the action bridge 5.19 for $Z$ with the
generalised propensity score (with contrast function)
$\frac{\pi(A)}{f(A|\tilde{V},U)}$. The existence of the bridge functions is
the key identification assumption. For example, the action bridge 5.19
existence poses a stronger richness requirement on $Z$ compared to standard
proximal learning (Kallus et al., 2021). Intuitively, the instruments $Z$ must
be sufficiently relevant for $A$ and $U$, where the latter is satisfied by
definition when $U$ captures all unobserved variables conditional on which $Z$
would satisfy an exclusion restriction.

######  Assumption 5.7 (Bridge functions).

$W(a,z)=W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U$. Let
$\tilde{V}=V_{\ref{a:strict-monotonicity-no-cf-fs}}$ if 5.3 holds, and
$\tilde{V}=V_{\ref{a:strict-monotonicity}}$ if 5.4 holds. There exist some
function $h_{0}\in L_{2}(A,W)$ and $q_{0}\in\pi L\in L_{2}(A,Z)$ such that

$\displaystyle\operatorname{\mathbb{E}}\left[h_{0}(A,\tilde{V},W)|A,\tilde{V},U\right]$
$\displaystyle=k_{0,\tilde{V}}(A,\tilde{V},U),$ (5.18)
$\displaystyle\operatorname{\mathbb{E}}\left[\pi(A)q_{0}(A,\tilde{V},Z)|A,\tilde{V},U\right]$
$\displaystyle=\frac{\pi(A)}{f(A|\tilde{V},U)}$ (5.19)

almost surely.

Equation 5.18 is the outcome bridge function and 5.19 the action bridge
function. While their existence is assumed, their uniqueness is not.

Let $\mathbb{H}_{0}$ and $\mathbb{Q}_{0}$ be the nonempty (assumption 5.7)
sets of valid outcome and action bridge functions defined by

$\displaystyle\mathbb{H}_{0}$ $\displaystyle=\left\\{h\in
L_{2}(A,\tilde{V},W)\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[Y-h(A,\tilde{V},W)|A,\tilde{V},U\right]=0\right\\}\neq\emptyset,$
(5.20) $\displaystyle\mathbb{Q}_{0}$ $\displaystyle=\left\\{q\text{ with }\pi
q\in L_{2}(A,\tilde{V},Z)\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\pi(A)(q(A,\tilde{V},Z)-1/f(A|\tilde{V},U))|A,\tilde{V},U\right]=0\right\\}\neq\emptyset.$
(5.21)

Any valid bridge function can identify $J$. We adapt lemma 2 from Kallus et
al. (2021) to our common confounding model for lemma 5.1.3. The causal effect
$J$ is identified in the common confounding model with any valid bridge
function.

######  Lemma 5.1.3.

Let $\mathcal{T}\mathrel{\mathop{\mathchar
58\relax}}L_{2}(A,\tilde{V},W)\rightarrow L_{2}(\tilde{V},W)$ be the linear
operator defined by
$(\mathcal{T}h)(\tilde{v},w)=\int_{\mathcal{A}}h(a,\tilde{v},w)\pi(a)d\mu_{A}(a)$.
Suppose 5.2 and 5.7 hold. Suppose either 5.3 ($\tilde{V}=V_{\ref{a:strict-
monotonicity-no-cf-fs}}$), or (5.4, 5.5, 5.6) ($\tilde{V}=V_{\ref{a:strict-
monotonicity}}$) hold. Then, for any $h_{0}\in\mathbb{H}_{0}$ and
$q_{0}\in\mathbb{Q}_{0}$,

$\displaystyle J=$
$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IPW}(Y,A,\tilde{V},Z;q_{0})\right]=\operatorname{\mathbb{E}}\left[\tilde{\phi}_{REG}(\tilde{V},W;h_{0})\right]=\operatorname{\mathbb{E}}\left[\tilde{\phi}_{DR}(Y,A,\tilde{V},W;h_{0},q_{0})\right]$
where $\displaystyle\ \ \ \
\tilde{\phi}_{IPW}(y,a,\tilde{v},z;q_{0})=y\pi(a)q_{0}(a,\tilde{v},z)$
$\displaystyle\ \ \ \
\tilde{\phi}_{REG}(\tilde{v},w;h_{0})=(\mathcal{T}h_{0})(\tilde{v},w)$
$\displaystyle\ \ \ \
\tilde{\phi}_{DR}(y,a,\tilde{v},z,w;h_{0},q_{0})=(y-h_{0}(a,\tilde{v},w))\pi(a)q_{0}(a,\tilde{v},z)+(\mathcal{T}h_{0})(\tilde{v},w)$

Notably, all moment functions in lemma 5.1.3 are functions of observed
variables only. By simple replacement of the infeasible regression
$k_{0}(A,V,U)$ and generalised propensity score $f^{-1}(A|V,U)$ with their
feasible counterpart bridge functions $h_{0}(A,\tilde{V},W)$ and
$q_{0}(A,\tilde{V},Z)$, we can identify causal effect $J$. Again, this is only
useful if we can identify the bridge functions.

####  5.4.2 Identification of the Action and Outcome Bridge Functions

The existence of bridge functions as defined in assumption 5.7 is not
immediately useful, because these bridge functions are defined conditional on
the unobserved common confounders $U$. Fortunately, these bridge functions
also satisfy some conditional moment equations involving only observable data.
This result is stated in lemma 5.1.4 below, where the bridge functions defined
in assumption 5.7 are solutions to feasible conditional moment equations.

######  Lemma 5.1.4.

Suppose 5.2 and 5.7 hold ($\mathbb{H}_{0}\neq\emptyset$ and
$\mathbb{Q}_{0}\neq\emptyset$). Suppose either 5.3
($\tilde{V}=V_{\ref{a:strict-monotonicity-no-cf-fs}}$), or (5.4, 5.5, 5.6)
($\tilde{V}=V_{\ref{a:strict-monotonicity}}$) hold. Then, any
$h_{0}\in\mathbb{H}_{0}$ and $q_{0}\in\mathbb{Q}_{0}$ satisfy that

$\displaystyle\operatorname{\mathbb{E}}\left[\left.Y-h_{0}(A,\tilde{V},W)\right|A,\tilde{V},Z\right]$
$\displaystyle=0$ (5.22)
$\displaystyle\operatorname{\mathbb{E}}\left[\left.\pi(A)\left(q_{0}(A,\tilde{V},Z)-\frac{1}{f(A|\tilde{V},W)}\right)\right|A,\tilde{V},W\right]$
$\displaystyle=0$ (5.23)

With these conditional moments of observable data we may identify a different
set of observable bridge functions.

$\displaystyle\mathbb{H}^{\text{obs}}_{0}$ $\displaystyle=\\{h\in
L_{2}(A,\tilde{V},W)\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\left.Y-h(A,\tilde{V},W)\right|A,\tilde{V},Z\right]=0\\}\neq\emptyset$
(5.24) $\displaystyle\mathbb{Q}^{\text{obs}}_{0}$ $\displaystyle=\\{q\text{
with }\pi q\in L_{2}(A,\tilde{V},Z)\mathrel{\mathop{\mathchar
58\relax}}\operatorname{\mathbb{E}}\left[\left.\pi(A)(q(A,\tilde{V},Z)-1/f(A|\tilde{V},W))\right|A,\tilde{V},W\right]=0\\}\neq\emptyset$
(5.25)

From lemma 5.1.4 we know that all proper bridge functions according to
assumption 5.7, which are in the sets $\mathbb{H}_{0}$ and $\mathbb{Q}_{0}$,
are also in $\mathbb{H}^{\text{obs}}_{0}$ and $\mathbb{Q}^{\text{obs}}_{0}$:
$\mathbb{H}_{0}\subseteq\mathbb{H}^{\text{obs}}_{0}$ and
$\mathbb{Q}_{0}\subseteq\mathbb{Q}^{\text{obs}}_{0}$. Assumption 5.7 for the
existence of bridge functions ensures
$\mathbb{H}^{\text{obs}}_{0}\neq\emptyset$ and
$\mathbb{Q}^{\text{obs}}_{0}\neq\emptyset$. We follow the logic in Kallus et
al. (2021) for the common confounding model to show that any
$h_{0}\in\mathbb{H}^{\text{obs}}_{0}$ and
$q_{0}\in\mathbb{Q}^{\text{obs}}_{0}$, which satisfy the observed conditional
moments 5.22 and 5.23, also identify the causal effect $J$.

######  Lemma 5.1.5.

Suppose 5.2 holds, and $W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U$.
Suppose either 5.3 ($\tilde{V}=V_{\ref{a:strict-monotonicity-no-cf-fs}}$), or
(5.4, 5.5, 5.6) ($\tilde{V}=V_{\ref{a:strict-monotonicity}}$) hold. Take any
$h_{0}\in\mathbb{H}^{\text{obs}}_{0}$, assuming
$\mathbb{H}^{\text{obs}}_{0}\neq 0$ and $\mathbb{Q}_{0}\neq\emptyset$. Then,

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IPW}(Y,A,\tilde{V},Z;q)-J\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[h_{0}(A,\tilde{V},W)\operatorname{\mathbb{E}}\left[\pi(A)(q(A,\tilde{V},Z)-1/f(A|\tilde{V},W))|A,\tilde{V},W\right]\right].$
(5.26)

Take any $q_{0}\in\mathbb{Q}^{\text{obs}}_{0}$, assuming
$\mathbb{Q}^{\text{obs}}_{0}\neq 0$ and $\mathbb{H}_{0}\neq\emptyset$. Then,

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{REG}(\tilde{V},W;h)-J\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\pi(A)q_{0}(A,\tilde{V},Z)\operatorname{\mathbb{E}}\left[h(A,\tilde{V},W)-Y|A,\tilde{V},Z\right]\right].$
(5.27)

From lemma 5.1.5, it follows that for any
$h_{0}\in\mathbb{H}^{\text{obs}}_{0}$ and
$q_{0}\in\mathbb{Q}^{\text{obs}}_{0}$, all three estimators identify causal
effect $J$. We state this result formally in theorem 5.2.

######  Theorem 5.2.

Suppose 5.2 holds, and $W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U$.
Suppose either 5.3 ($\tilde{V}=V_{\ref{a:strict-monotonicity-no-cf-fs}}$), or
(5.4, 5.5, 5.6) ($\tilde{V}=V_{\ref{a:strict-monotonicity}}$) hold.

Suppose $\mathbb{Q}_{0}\neq\emptyset$ and
$\mathbb{H}_{0}^{\text{obs}}\neq\emptyset$. Then, for any
$q_{0}\in\mathbb{Q}_{0}^{\text{obs}}$

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IPW}(Y,A,\tilde{V},Z;q_{0})\right]$
$\displaystyle=J.$ (5.28)

Suppose $\mathbb{H}_{0}\neq\emptyset$ and
$\mathbb{Q}_{0}^{\text{obs}}\neq\emptyset$. Then, for any
$h_{0}\in\mathbb{H}_{0}^{\text{obs}}$

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{REG}(\tilde{V},W;h_{0})\right]$
$\displaystyle=J.$ (5.29)

Suppose either $\\{\mathbb{Q}_{0}\neq\emptyset$ and
$\mathbb{H}_{0}^{\text{obs}}\neq\emptyset\\}$ or
$\\{\mathbb{H}_{0}\neq\emptyset$ and
$\mathbb{Q}_{0}^{\text{obs}}\neq\emptyset\\}$. Then, for any
$h_{0}\in\mathbb{H}_{0}^{\text{obs}}$ and
$q_{0}\in\mathbb{Q}_{0}^{\text{obs}}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{DR}(Y,A,\tilde{V},Z,W;h_{0},q_{0})\right]$
$\displaystyle=J.$ (5.30)

This key result in theorem 5.2 establishes the identifiability of causal
effect $J$ from only observable data in the common confounding model with some
first stage reduced form monotonicity assumption.

As a side note, only a relaxed version of assumption 5.7 is needed, because
e.g. unbiasedness of the regression estimator $\tilde{\phi}_{REG}$ only
requires $\mathbb{Q}^{\text{obs}}_{0}\neq\emptyset$ instead of
$\mathbb{Q}_{0}\neq\emptyset$. With the doubly robust estimator, the
relaxation is even more explicit in theorem 5.2. Despite these technically
feasible relaxations, completeness assumptions on the structural form, i.e.
the completeness of $Z$ and $W$ with respect to $U$, likely are easier to
comprehend and already imply $\mathbb{H}_{0}\neq\emptyset$ and
$\mathbb{Q}_{0}\neq\emptyset$. Corollary 5.2.1 states a simplified version of
theorem 5.2 with these slightly stronger assumptions
($\mathbb{H}_{0}\neq\emptyset$ and $\mathbb{Q}_{0}\neq\emptyset$).

######  Corollary 5.2.1.

Suppose 5.2 and 5.7 hold ($\mathbb{H}_{0}\neq\emptyset$ and
$\mathbb{Q}_{0}\neq\emptyset$). Suppose either 5.3
($\tilde{V}=V_{\ref{a:strict-monotonicity-no-cf-fs}}$), or (5.4, 5.5, 5.6)
($\tilde{V}=V_{\ref{a:strict-monotonicity}}$) hold. Then, for any
$h_{0}\in\mathbb{H}_{0}^{\text{obs}}$ and
$q_{0}\in\mathbb{Q}_{0}^{\text{obs}}$,

$\displaystyle J=$
$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IPW}(Y,A,\tilde{V},Z;q_{0})\right]=\operatorname{\mathbb{E}}\left[\tilde{\phi}_{REG}(Y,A,\tilde{V},W;h_{0})\right]=\operatorname{\mathbb{E}}\left[\tilde{\phi}_{DR}(Y,A,\tilde{V},Z,W;h_{0},q_{0})\right].$
(5.31)

##  6 Numerical Examples

We provide examples of bridge functions in linear, discrete and nonparametric
models in this subsection.

###  6.1 Linear model

Let $Y_{i}\in\mathbb{R}$, $A_{i}\in\mathbb{R}$, $U_{i}\in\mathbb{R}^{d_{U}}$,
$Z_{i}\in\mathbb{R}^{d_{Z}}$ and $W_{i}\in\mathbb{R}^{d_{W}}$, for observation
$i\in\\{1,2,\ldots,n\\}$. The parameter of interest is $\beta$, the linear
effect of treatment $A$ on outcome $Y$.

$\displaystyle Y_{i}$ $\displaystyle=A_{i}{\beta}+U_{i}\underset{d_{U}\times
1}{\gamma_{Y}}+W_{i}\underset{W\times 1}{\zeta}+\varepsilon_{Y,i}$
$\displaystyle\operatorname{\mathbb{E}}\left[\varepsilon_{Y,i}|Z_{i}\right]$
$\displaystyle=0,$ $\displaystyle A_{i}$
$\displaystyle=Z_{i}\underset{d_{Z}\times
d_{A}}{\pi}+U_{i}\underset{d_{U}\times d_{A}}{\gamma_{A}}+\varepsilon_{A,i}$
$\displaystyle\operatorname{\mathbb{E}}\left[\varepsilon_{A,i}|Z_{i}\right]$
$\displaystyle=0,\ \pi\neq 0,\ d_{Z}\geq(d_{A}+d_{U})$ $\displaystyle W_{i}$
$\displaystyle=U_{i}\underset{d_{U}\times
d_{W}}{\gamma_{W}}+\varepsilon_{W,i}$
$\displaystyle\operatorname{\mathbb{E}}\left[\varepsilon_{W,i}|Z_{i}\right]$
$\displaystyle=0$

Any bias in the IV estimator stems from changes in
$\operatorname{\mathbb{E}}\left[U_{i}|Z_{i}\right]$ as $Z_{i}$ changes.
Suppose the parameter matrix $\gamma_{W}$ has full rank. Then, if $d_{W}\geq
d_{U}$, holding $\operatorname{\mathbb{E}}\left[W_{i}|Z_{i}\right]$ constant
is equivalent to holding $\operatorname{\mathbb{E}}\left[U_{i}|Z_{i}\right]$
constant. Consequently, conditioning on
$\operatorname{\mathbb{E}}\left[W_{i}|Z_{i}\right]$, which itself has at most
column rank $d_{U}\leq d_{W}$, eliminates all confounding bias from the common
confounders $U$. We could write the parameter of interest, $\beta$, in the
ratio form

$\displaystyle\beta$
$\displaystyle=\frac{\operatorname{Cov}\left[Y_{i},Z_{i}\pi\left|\operatorname{\mathbb{E}}\left[W_{i}|Z_{i}\right]\right.\right]}{\operatorname{Cov}\left[A_{i},Z_{i}\pi\left|\operatorname{\mathbb{E}}\left[W_{i}|Z_{i}\right]\right.\right]}.$

From the above formula follows a consistent estimator $\hat{\beta}_{ICC}$,
which we can write in matrix notation. Let $(Y,A,Z,W,U)$ be the $n$-rowed
vectors and matrices of $n$ stacked sample observations. We let
$P_{Z}\coloneqq Z(Z^{\intercal}Z)^{-1}Z^{\intercal}$. Then, the estimator of
the $n\times d_{W}$ matrix $\operatorname{\mathbb{E}}\left[W|Z\right]$ is
$\hat{W}\coloneqq P_{Z}W$. For simplicity, suppose that $d_{W}=d_{U}$, which
implies that $\operatorname{\mathbb{E}}\left[W|Z\right]$ has full column rank.
If $d_{W}>d_{U}$, we would use exactly $d_{U}$ linearly independent columns of
the $\operatorname{\mathbb{E}}\left[W|Z\right]$ matrix. Let

$\displaystyle P_{\hat{W}}$
$\displaystyle\coloneqq\hat{W}\left(\hat{W}^{\intercal}\hat{W}\right)^{-1}\hat{W}^{\intercal}$
$\displaystyle=P_{Z}W\left(W^{\intercal}P_{Z}W\right)^{-1}W^{\intercal}P_{Z},$
$\displaystyle M_{\hat{W}}$ $\displaystyle\coloneqq
I_{n}-P_{\hat{W}}=I_{n}-P_{Z}W\left(W^{\intercal}P_{Z}W\right)^{-1}W^{\intercal}P_{Z}$

The reduced form of the outcome model and the ICC estimator can be written as

$\displaystyle Y$
$\displaystyle=A\beta+\operatorname{\mathbb{E}}\left[W|Z\right]\delta+\varepsilon_{ICC},\
\ \ \
\delta\coloneqq\gamma_{W}^{\intercal}\left(\gamma_{W}\gamma_{W}^{\intercal}\right)^{-1}\left(\gamma_{Y}+\gamma_{W}\zeta\right),$
$\displaystyle\varepsilon_{ICC}$
$\displaystyle\coloneqq\varepsilon_{Y}+\varepsilon_{W}\zeta+\left(U-\operatorname{\mathbb{E}}\left[U|Z\right]\right)\left(\gamma_{Y}+\gamma_{W}\zeta\right),$
$\displaystyle\hat{\beta}_{ICC}$
$\displaystyle=\left(A^{\intercal}M_{\hat{W}}P_{Z}A\right)^{-1}\left(A^{\intercal}M_{\hat{W}}P_{Z}Y\right).$

The asymptotics of the estimator then follow from a few simple steps of
algebra using a central limit theorem under standard assumptions.

$\displaystyle\sqrt{n}\left(\hat{\beta}_{ICC}-\beta\right)$
$\displaystyle=\left(\frac{1}{n}A^{\intercal}P_{Z}M_{\hat{W}}A\right)^{-1}\left(\frac{1}{\sqrt{n}}A^{\intercal}M_{\hat{W}}P_{Z}\varepsilon_{ICC}\right)$
$\displaystyle\overset{d}{\rightarrow}N\left(0,\left(A^{\intercal}P_{Z}M_{\hat{W}}A\right)^{-1}\left(A^{\intercal}M_{\hat{W}}P_{Z}\operatorname{\mathbb{E}}\left[\varepsilon_{ICC}\varepsilon_{ICC}^{\intercal}\right]P_{Z}M_{\hat{W}}A\right)\left(A^{\intercal}P_{Z}M_{\hat{W}}A\right)^{-1}\right).$

Another interpretation is to separate the estimator into stages, like 2SLS.
Here, we get the predicted values $\hat{A}\coloneqq P_{Z}A$ and
$\hat{W}=P_{Z}W$ in the first stage. Then we regress $Y$ on the predicted
values $\hat{A}$ and $\hat{W}$. The OLS estimates in the second stage are the
point estimates of the ICC estimator $\hat{\beta}_{ICC}$ and the auxiliary OLS
slope parameter $\hat{\delta}_{ICC}$. The correct standard errors are easy to
calculate from the above asymptotic distribution once we realise that
$\hat{\varepsilon}_{ICC}=Y-A\hat{\beta}_{ICC}-P_{Z}W\hat{\delta}_{ICC}$.
Estimating $\pi$ via OLS in the first stage conditional on $\hat{W}$, we could
have set up a numerically equivalent control function estimator using
$A-Z\hat{\pi}$ and $\hat{W}$.

Importantly, the variation in instruments $Z$ must still be relevant for
treatment $A$ after conditioning on the predicted values of $W$ given $Z$,
$P_{Z}W$. This requirement is generally asymptotically satisfied when
$d_{Z}\geq(d_{A}+d_{U})$ with individually relevant instruments for a
$d_{A}$-dimensional treatment $A$ (except for some numerical special cases
where $Z$ correlates with $A$ identically as with $U$). Despite this
asymptotic guarantee, sample variation implies that the estimator $P_{Z}W$
always has the largest possible column rank $\min\\{d_{W},d_{Z}\\}$, even when
the true column rank of its estimand
$\operatorname{\mathbb{E}}\left[W|Z\right]$ is $d_{U}<\min\\{d_{W},d_{Z}\\}$.
Thus, whenever $d_{W}\geq d_{Z}$, the rank of the estimator $P_{Z}W$ would be
$d_{Z}$, and therefore must be reduced to some rank $r$, which satisfies
$d_{U}\geq r<d_{Z}$. As $d_{U}$. This necessary in-sample rank reduction is
justified by the true rank $d_{U}$ of estimand
$\operatorname{\mathbb{E}}\left[W|Z\right]$, to which the estimator $P_{Z}W$
converges.

###  6.2 Discrete model

####  6.2.1 Discrete model with outcome model restrictions

In the case of outcome model restrictions, recall that

$\displaystyle Y=Y(A)$ $\displaystyle={k_{0}(A,U)}+\varepsilon,$
$\displaystyle\operatorname{\mathbb{E}}\left[\varepsilon|Z,U\right]$
$\displaystyle=0.$

Let $A,Z,W,U$ be discrete variables, which take values $a_{j_{a}}$,
$z_{j_{z}}$, $w_{j_{w}}$, $u_{j_{U}}$ for $j_{a}=1,\ldots,d_{A}$,
$j_{z}=1,\ldots,d_{Z}$, $j_{w}=1,\ldots,d_{W}$ and $j_{U}=1,\ldots,d_{U}$. To
define the control bridge function 3.3, we first define some useful matrices
of conditional probabilities:

$\displaystyle\underset{1\times d_{U}}{K_{0}(A=a_{j_{A}},\boldsymbol{U})}$
$\displaystyle\text{ has $(j_{U})$th element
}{k_{0}(A=a_{j_{A}},U=u_{j_{U}})},$ $\displaystyle\underset{d_{W}\times
d_{U}}{P\left(\boldsymbol{W}|\boldsymbol{U}\right)}$ $\displaystyle\text{ has
$(j_{W},j_{U})$th element
}\operatorname{\mathbb{P}}\left[W=w_{j_{W}}\left|U=u_{j_{U}}\right.\right]$
$\displaystyle\underset{(d_{A}\times d_{W})\times
d_{Z}}{P\left(\boldsymbol{A},\boldsymbol{W}|\boldsymbol{Z}\right)}$
$\displaystyle\text{ has $((j_{A}\times j_{W}),j_{U})$th element
}\operatorname{\mathbb{P}}\left[A=a_{j_{A}},W=w_{j_{W}}\left|U=u_{j_{U}}\right.\right],$
$\displaystyle\underset{1\times
d_{Z}}{\operatorname{\mathbb{E}}\left[Y|\boldsymbol{Z}\right]}$
$\displaystyle\text{ has $j_{Z}$th element
}\operatorname{\mathbb{E}}\left[Y|Z=z_{j_{Z}}\right].$

The control bridge function 3.3 here corresponds to the linear system

$\displaystyle\underset{1\times
d_{W}}{H_{0}\left(A=a_{j_{A}},\boldsymbol{W}\right)}\underset{d_{W}\times
d_{U}}{P\left(\boldsymbol{W}|\boldsymbol{U}\right)}$
$\displaystyle=\underset{1\times
d_{U}}{K_{0}\left(A=a_{j_{A}},\boldsymbol{U}\right)},$

Now if $P\left(\boldsymbol{W}|\boldsymbol{U}\right)$ has full column rank,
which requires $d_{W}\geq d_{U}$, the linear equations system above has a
solution. Unless $d_{W}=d_{U}$, there will be more than one solution and the
bridge function $H_{0}\left(A=a_{j_{A}},\boldsymbol{W}\right)$ will be non-
unique. As the common confounder $U$ is not observed, we can at best identify
some other bridge function which satisfies

$\displaystyle\underset{1\times(d_{A}\times
d_{W})}{H_{0}^{\text{obs}}(\boldsymbol{A},\boldsymbol{W})}\underset{(d_{A}\times
d_{W})\times
d_{Z}}{P\left(\boldsymbol{A},\boldsymbol{W}|\boldsymbol{Z}\right)}$
$\displaystyle=\underset{1\times
d_{Z}}{\operatorname{\mathbb{E}}\left[Y|\boldsymbol{Z}\right]}.$

The existence of this bridge function is guaranteed if $d_{W}\geq d_{U}$. Our
results imply that any solution
$H_{0}^{\text{obs}}(\boldsymbol{A},\boldsymbol{W})$ results in a unique

$H_{0}^{\text{obs}}(A=a_{j_{A}},\boldsymbol{W})P(\boldsymbol{W})=K_{0}\left(A=a_{j_{A}},\boldsymbol{U}\right)P\left(\boldsymbol{U}\right),$

which can be interpreted as an average structural function where the
confounders $U$ are integrated out without dependence on treatment $A$.
Required for this result is the sufficient relevance of instruments $Z$ for
treatment $A$ conditional on common confounders $U$, i.e.
$d_{Z}\geq(d_{U}\times d_{A})$.

####  6.2.2 Discrete model with first stage restrictions

Let $Z,W,U$ be discrete variables, which take values $z_{j_{z}}$, $w_{j_{w}}$,
$u_{j_{U}}$ for $j_{z}=1,\ldots,d_{Z}$, $j_{w}=1,\ldots,d_{W}$ and
$j_{U}=1,\ldots,d_{U}$. The matrices of conditional probabilities, vectors of
conditional expectations and densities are

$\displaystyle\underset{d_{W_{k}}\times
d_{U}}{P(\boldsymbol{W}_{k}|\boldsymbol{U},w_{i})}$ $\displaystyle\text{ has
$(j_{W_{k}},j_{U})$th element
}\operatorname{\mathbb{P}}\left[W=w_{j_{W_{k}}}|U=u_{j_{U}},W_{0}=w_{0}\right]\text{
for }k\neq i\in\\{0,1\\},$ $\displaystyle\underset{d_{W}\times
d_{U}}{P(\boldsymbol{W}|\boldsymbol{U},a,\tilde{v})}$ $\displaystyle\text{ has
$(j_{W},j_{U})$th element
}\operatorname{\mathbb{P}}\left[W=w_{j_{W}}|U=u_{j_{U}},A=a,\tilde{V}=v\right],$
$\displaystyle\underset{d_{Z}\times
d_{U}}{P(\boldsymbol{Z}|\boldsymbol{U},a,\tilde{v})}$ $\displaystyle\text{ has
$(j_{Z},j_{U})$th element
}\operatorname{\mathbb{P}}\left[Z=w_{j_{Z}}|U=u_{j_{U}},A=a,\tilde{V}=v\right],$
$\displaystyle\underset{1\times
d_{U}}{\operatorname{\mathbb{E}}\left[Y|\boldsymbol{U},a,\tilde{v}\right]}$
$\displaystyle\text{ has $j_{U}$th element
}\operatorname{\mathbb{E}}\left[Y|U=u_{j_{U}},A=a,\tilde{V}=\tilde{v}\right],$
$\displaystyle\underset{1\times d_{U}}{F^{*}(a|\boldsymbol{U},z)}$
$\displaystyle\text{ is a vector with $j_{U}$th entry }f(a|u_{j_{U}},z),$
$\displaystyle\underset{d_{U}\times 1}{F^{*}(a|\boldsymbol{U},\tilde{v})}$
$\displaystyle\text{ is a vector with $j_{U}$th entry
}f(a|u_{j_{U}},\tilde{v}),$ $\displaystyle\underset{1\times
d_{U}}{F(\boldsymbol{U},z)}$ $\displaystyle\text{ is a vector with $j_{U}$th
entry }\frac{f(u_{j_{U}})}{f(u_{j_{U}}|z)},$ $\displaystyle\text{and
}\underset{d_{U}\times 1}{\boldsymbol{e}}$ is a vector of ones.

First, the control bridge functions 5.10 and 5.11 correspond to the linear
system

$\displaystyle\tau_{A,0}(z,\boldsymbol{W}_{1})\operatorname{\mathbb{P}}\left[\boldsymbol{W}_{1}|\boldsymbol{U},w_{0}\right]$
$\displaystyle=F^{*}(a|\boldsymbol{U},z)\text{ for all
}w_{0}\in\mathcal{W}_{0},$
$\displaystyle\kappa_{0}(z,\boldsymbol{W}_{0})\operatorname{\mathbb{P}}\left[\boldsymbol{W}_{0}|\boldsymbol{U},w_{1}\right]$
$\displaystyle=F(\boldsymbol{U},z)\text{ for all }w_{1}\in\mathcal{W}_{1}.$

A sufficient condition for the solutions to exist is that
$\operatorname{\mathbb{P}}\left[\boldsymbol{W}_{1}|\boldsymbol{U},w_{0}\right]$
and
$\operatorname{\mathbb{P}}\left[\boldsymbol{W}_{0}|\boldsymbol{U},w_{1}\right]$
have full rank for all $(w_{0},w_{1})\in\mathcal{W}$. This requires $W_{0}$
and $W_{1}$ to contain a minimum amount of conditionally independent
categories. Let $\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}}$ be the
union of all largest-possible non-overlapping subsets
$\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}}_{i}\in\mathcal{W}$ of
values for which
$(W_{0},W_{1})\in\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}}_{i}$ are
conditionally independent.
$i\in\\{1,2,\ldots,\mathbf{card}(\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}})\\}$
indexes each subset of
$\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}}_{i}\in\mathcal{W}$. A
sufficient condition for the existence of the bridge functions is that for
each subset $i$, there are at least as many categories $d_{W_{k},i}$ for both
$k\in\\{0,1\\}$ as categories of the common confounder $d_{U}$:
$d_{W_{k},i}\geq d_{U}$ for all subsets $i$ and both $k\in\\{0,1\\}$. If
$W_{1}\mathrel{\perp\mspace{-10.0mu}\perp}W_{0}$ everywhere, this condition is
satisfied if $d_{W_{k}}>d_{U}$ for both $k\in\\{0,1\\}$. However, such a full
conditional independence assumption often is too strong in observational data.

The bridge function equations for outcome 5.18 and action 5.19 correspond to
the linear system

$\displaystyle
h_{0}^{\intercal}(\boldsymbol{W},a,\tilde{v})P(\boldsymbol{W}|\boldsymbol{U},a,\tilde{v})$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y|\boldsymbol{U},a,\tilde{v}\right],$
$\displaystyle
q_{0}^{\intercal}(\boldsymbol{Z},a,\tilde{v})P(\boldsymbol{Z}|\boldsymbol{U},a,\tilde{v})F^{*}(a|\boldsymbol{U},\tilde{v})$
$\displaystyle=\boldsymbol{e}^{\intercal},$

in this model with discrete $Z,W,U$. Suppose that
$P(\boldsymbol{W}|\boldsymbol{U},a,\tilde{v})$ and
$P(\boldsymbol{Z}|\boldsymbol{U},a,\tilde{v})$ are full rank, which requires
$d_{W}\geq d_{U}$ and $d_{Z}\geq(d_{A}\times d_{U})$. Also suppose that
$f(a|u,\tilde{v})\neq 0$ for any $u\in\mathcal{U}$ and $\tilde{v}\in(0,1)$.
Then, the linear system has solutions. Bridge functions exist. The solutions
must not be unique unless $d_{W}=d_{U}$ and $d_{Z}=(d_{A}\times d_{U})$.

Nonsingularity of $P(\boldsymbol{Z}|\boldsymbol{U},a,\tilde{v})$ enforces a
strong richness requirement for instruments $Z$. A necessary condition is that
$\operatorname{\mathbb{P}}\left[z_{j_{Z}}|a,\tilde{v}\right]\in(0,1)$ for at
least $d_{U}$ different $z_{j_{Z}}\in\mathcal{Z}$ for each
$(a,\tilde{v})\in\mathcal{A}\times(0,1)$. In simple terms, there must be
sufficient relevant information in $Z$ for $A$ conditional on $U$.

###  6.3 Nonparametric model

####  6.3.1 Nonparametric model with outcome model restrictions

Once we assume some outcome model restrictions (4.1), identification of the
causal effect $J$ relies on richness requirements for $W$ with respect to $U$
(3.1.3b) and for $Z$ with respect to $U$ and $A$ (3.1.2b). The existence of
bridge function 3.3 ensures that outcome-inducing proxies $W$ are sufficiently
rich with respect to $U$ to obtain unbiased estimates of the causal effect
$J$. As this bridge function is a solution to a Fredholm integral equation of
the first kind, its existence is given by Picard’s theorem (Polyanin and
Manzhirov, 2008). Sufficient for the existence of this bridge function is a
completeness condition: For any $g\in L_{2}(A,U)$,

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|A,W\right]$
$\displaystyle=0\text{ only when }g(A,U)=0.$

This relevance requirement for the outcome-inducing proxies $W$ is identifical
to standard proximal learning (Kallus et al., 2021). Different is the
completeness condition on $Z$, which we included as one of our main
assumptions 3.1.3.2. For any $g\in L_{2}(A,U)$,

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|Z\right]$
$\displaystyle=0\text{ only when }g(A,U)=0.$

As $U$ captures all endogeneity in the instruments $Z$, we require all
exogenous variation in the instruments $Z$ to be complete with respect to
treatment $A$. In ICC, both conditions must hold simultaneously. The
instruments we can consider with these different idenifying assumptions are
significantly different from to traditional instruments. Our examples in
section 7 will provide further intuition.

####  6.3.2 Nonparametric model with first stage restrictions

Identification of the conditional moment equations for control (5.10, 5.11),
outcome (5.18) and action (5.19) bridge functions does not require parametric
assumptions other than the strict monotonicity assumption for the first stage
reduced form (5.4). In nonparametric models, richness requirements for $Z$,
$W$, and possibly ($W_{0}$, $W_{1}$) with respect to $U$ can be completeness
conditions. The bridge functions are Fredholm integral equations of the first
kind. The existence of solutions to these inverse learning problems are given
by Picard’s theorem (Polyanin and Manzhirov, 2008). Solutions to the control
bridge functions (5.10 and 5.11) exist if the below completeness conditions
are satisfied. For any $g\in L_{2}(U,Z)$,

$\displaystyle\operatorname{\mathbb{E}}\left[g(U,Z)|W_{1i},Z\right]$
$\displaystyle=0\text{ for }W_{1i}\in\mathcal{W}_{1i}\text{ and any
}i\in\\{1,2,\ldots,\mathbf{card}(\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}})\\}\text{
only when }g(U,Z)=0,$
$\displaystyle\operatorname{\mathbb{E}}\left[g(U,Z)|W_{0i},Z\right]$
$\displaystyle=0\text{ for }W_{0i}\in\mathcal{W}_{0i}\text{ and any
}i\in\\{1,2,\ldots,\mathbf{card}(\mathcal{W}^{\mathrel{\perp\mspace{-10.0mu}\perp}})\\}\text{
only when }g(U,Z)=0.$

These completeness conditions require that there is rich enough variation in
$W_{0}$ (and $W_{1}$) compared to $U$ after conditioning on $W_{1}$ (or
$W_{0}$).

Similarly, outcome (5.18) and action (5.19) bridge functions exist if the
below completeness conditions are satisfied. For any $g_{U,A,\tilde{V}}\in
L_{2}(U,A,\tilde{V})$,

$\displaystyle\operatorname{\mathbb{E}}\left[g_{U,A,\tilde{V}}(U,A,\tilde{V})|Z,A,\tilde{V}\right]$
$\displaystyle=0\text{ only when }g(U,A,\tilde{V})=0,$
$\displaystyle\operatorname{\mathbb{E}}\left[g_{U,A,\tilde{V}}(U,A,\tilde{V})|W,A,\tilde{V}\right]$
$\displaystyle=0\text{ only when }g(U,A,\tilde{V})=0.$

Only when $Z$ and $W$ vary in a sufficiently rich way with $U$ conditional on
control function $\tilde{V}$, the above conditions can hold. Intuitively, the
first completeness condition means that after using some variation in $Z$ for
the construction of $\tilde{V}$, the remaining variation in $Z$ varies with
$U$ in a rich manner. In the linear model above we saw this implies that the
dimension of the instrument $d_{Z}$ must be at least as large as the dimension
of treatment and unobserved confounder ($d_{A}+d_{U}$). The completeness
assumptions may be stronger than necessary for the existence of bridge
functions. In this paper, we follow the approach by Kallus et al. (2021) and
instead use the weaker, yet sufficient assumption of existence of bridge
functions as much as possible.

##  7 ICC in Practice

First, we present a practical algorithm to construct an ICC model. Then, the
algorithm is applied to an economic and a medical causal inference problem. In
the economic example, we provide evidence of the benefit of instrumented
common confounding in the estimation of returns to education. The medical
example attempts to identify the causal effect of a health treatment in
observational data.

###  7.1 Constructing an ICC Model

In comparison to IV, the ICC assumptions may seem daunting at first. The
validity of any identification approach ultimately relies on the theoretical
arguments about its untestable assumptions. To facilitate the theoretical
discussion about the validity of ICC assumptions, we provide a simple
algorithm to check whether an identification problem fits the ICC framework.

  1. 1.

$Z$ exogeneity (3.1.2a): Define common confounders $U$ such that

$\displaystyle Y(a,z)=Y(a)\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U\ \forall
a\in\mathcal{A}.$

  2. 2.

$W$ exogeneity (3.1.3a): Include in the common confounders $U$ any unobserved
variables necessary to justify

$\displaystyle W(a,z)=W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U.$

  3. 3.

$W$ relevance (3.1.3b): Check whether $W$ is complete with respect to $U$
given $A$:

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|A,W\right]=0\text{ only
when }g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

  4. 4.

$Z$ relevance (3.1.2b): Check whether $Z$ is complete with respect to $(A,U)$:

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|Z\right]=0\text{ only when
}g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

The above algorithm requires completeness of $W$ with respect to common
confounder $U$ (3.1.3b). This requirement can hold even if some variable in
$W$ directly causes $(Z,A)$, where such variables in $W$ would be included in
$U$, as long as $W$ remains complete for $U$. Vice-versa, some instruments $Z$
might be endogenous with a direct impact on outcome $Y$, where these
instruments would be included in $U$. Exogenous variation in the instruments
must remain complete with respect to the treatment $A$, and $W$ complete with
respect to the endogenous instruments included in $U$. Clearly, the
completeness assumptions are more likely to be satisfied the richer the
information in $Z$ and $W$. Put simply, the more measurements of the
unobserved confounders there are, the better we can justify the required
completeness conditions. In our subsequent examples, we hope to convince the
reader of the relevance of the common confounding assumptions for interesting
causal inference problems. Our logic will be based on the above algorithm.

###  7.2 Returns to Education

The education production function has been a key function of interest for
applied microeconometricians since the 1950s, with at least 1,120 estimates in
139 countries (Psacharopoulos and Patrinos, 2018). The formal modelling of
human capital formation is grounded in microeconomic theory, which nearly any
undergraduate Economics student will have come across (Schultz, 1960, 1961;
Mincer, 1958; Becker, 1964). Early contributions to this literature emphasised
the heterogeneity of returns to education across levels of education and
across individuals with the same level of education (Becker and Chiswick,
1966; Chiswick, 1974; Mincer, 1974). A nonlinear model naturally accommodates
this type of causal effect heterogeneity. A well-known confounder of the
effect of education on earnings is ability (Griliches, 1977). The dominant
solution to the ability bias problem is the use of instruments for education
(Heckman et al., 2006). Parental education and number of siblings (Willis and
Rosen, 1979; Taber, 2001) are poor instruments, as family background strongly
determines ability (Cunha et al., 2006). Another popular type of instrument is
geographic location, e.g. distance to college (Card, 1993; Kling, 2001;
Cameron and Taber, 2004). However, the correlation of distance to college and
an ability proxy casts doubt on its exogeneity (Carneiro and Heckman, 2002).
Tuition cost (Kane and Rouse, 1993) may be an invalid instrument due to its
correlation with college quality, which may have a direct effect on earnings
(Carneiro and Heckman, 2002). Local business cycle fluctuations were used as
instruments in other studies (Cameron and Heckman, 1998; Carneiro et al.,
2003; Cameron and Taber, 2004), but require that all permanent labour market
effects are conditioned out, which may be difficult. The well-known quarter-
of-birth instrument (Angrist and Krueger, 1991) is likely exogenous, but is
known to be weak (Staiger and Stock, 1997). Finding exogenous instruments is
hard. Once an exogenous instrument is found, it is often weak or affects the
treatment only in a subpopulation. In this case, the IV estimate can at best
be interpreted as a marginal treatment effect in a subpopulation (Heckman et
al., 2006).

Economic theory states that ability is a likely confounder of the effect of
education on earnings. Despite the previous long list of instruments, which
intend to circumvent ability bias, ability proxies tend to explain little
about the residuals in a regression of earnings on education and observable
characteristics (Cawley et al., 1995). Other unobserved characteristics like
(non-academic) attitude (Green et al., 1998) and communication skills (on the
Educational Quality of the Workforce, US) may be important determinants of
earnings. Better communication skills serve applicants in admissions
interviews and are highly valued in the labour market.

More generally, selection bias is inevitable when treatment is the result of
heterogeneous individuals’ utility-maximising choice. Education is at least in
part the result of individuals’ expected utility $u_{i}$ optimisation, where
the individuals’ information set is naturally larger than that of the
researcher.

$\displaystyle Y_{i}$ $\displaystyle=g(A_{i},U_{i},\varepsilon_{i})$ (7.1)
$\displaystyle A_{i}$ $\displaystyle=\max_{a\in\mathcal{A}\text{ s.t.
constraints}}\left\\{\operatorname{\mathbb{E}}\left[u_{i}(Y_{i},A_{i})|\text{
information set pre-college}\right]\right\\}=h(Z_{i},U_{i},\eta_{i})$ (7.2)

Mechanically, individuals’ optimising behaviour implies that in this model the
disturbances $\varepsilon_{i}$ and $\eta_{i}$ are closely related to each
other. Even conditional on the common confounder ability $U_{i}$, this
dependence persists. However, there may be some instruments $Z$ relevant for
education $A$ that satisfy an exclusion restriction conditional on ability
$U_{i}$.

To demonstrate how instrumented common confounding suits the returns to
education identification problems, we present a directed acyclic graph (DAG)
to describe the model’s conditional independence structures in figure 2. By
going through the ICC construction algorithm, we demonstrate how the returns
to education identification problem fits into the ICC framework. In terms of
conditional independence assumptions we attempt to be as conservative as
possible, which at worst leads to stronger than necessary richness
requirements for the instruments $Z$ and proximal learning outcomes $W$.

Figure 2: DAG of the Education Production Function

$U$$A$$Y$$\ \ \ Z\ \ \ $$W$$\tilde{U}$

  * $Y$:

Income 5 years post-college

  * $A$:

College GPA

  * $Z$:

GPA (high school), SAT, ACT

  * $U$:

Ability, attitude, communication skills, general health, addictive tendency

  * $\tilde{U}$:

General selection bias

  * $W$:

Pre-college physical and behavioural characteristics

  * $X$:

Family characteristics: parental education, income, occupation

  1. 1.

$Z$ exogeneity (3.1.2a): Define common confounders $U$ such that

$\displaystyle Y(a,z)=Y(a)\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U\ \forall
a\in\mathcal{A}.$

Having considered the forms of confounding from ability $U$ and general
selection bias $\tilde{U}$ in this model, the first task is to motivate the
exclusion restriction conditional on ability $U$ for instruments $Z$. As
instrument $Z$ we use pre-college test scores and GPAs. Clearly, pre-college
scores $Z$ are strongly associated with ability $U$. The main identifying
assumption is that pre-college test scores and GPA measures do not affect
post-college earnings through anything else than ability $U$ and college GPA
$A$. This is a relatively sensible assumption. Once college GPA $A$ is
determined, there is no reason why pre-college GPA $Z$ would matter regarding
an individual’s realised earnings post-college $Y$, other than through ability
or other (family) background characteristics, which we assume to observe.

  2. 2.

$W$ exogeneity (3.1.3a): Include in the common confounders $U$ any unobserved
variables necessary to justify

$\displaystyle W(a,z)=W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U.$

As outcome-inducing proxies $W$, we use a range of pre-college physical and
behavioural characteristics. These can include health status and behaviours,
height, fitness, and addictive behaviours like alcoholism and smoking. While
there may be some direct effects between pre-college health/behaviours and
test scores, we can let $U$ capture all relevant information to ensure the
required conditional independence.

  3. 3.

$W$ relevance (3.1.3b): Check whether $W$ is complete with respect to $U$
given $A$:

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|A,W\right]=0\text{ only
when }g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

There is substantial evidence for the relevance of the outcome-inducing
proxies $W$ including physical characteristics and behaviours for ability
(Gottfredson, 2004; Case and Paxson, 2008; Tabriz et al., 2015; Greengross and
Miller, 2011). Even allowing for some direct effects between academic
performance and health/behavioural characteristics, their relevance for
ability $U$ arguably remains likely. Ultimately, it depends on the richness of
available data whether completeness with respect to $U$ is satisfied. In some
datasets, like the NLS97 (Bureau of Labor Statistics, 2019), there are i.a.
rich observations of health and addictive behaviour.

  4. 4.

$Z$ relevance (3.1.2b): Check whether $Z$ is complete with respect to $(A,U)$:

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|Z\right]=0\text{ only when
}g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

Pre-college scores $Z$ are strong determinants of college GPA $A$, even once
ability $U$ is held fixed. Pre-college test scores can never measure ability
perfectly, and also reflect some exam performance skills, which are likely
carried on to college. The variation in $Z$ used to infer the effect of
college GPA $A$ on post-college earnings $Y$ would be the excluded, non-
ability determined variation in pre-college test scores. Again, satisfaction
of the relevance requirement for the instruments with respect to the treatment
relies on the available data. In some datasets, like the NLS97 (Bureau of
Labor Statistics, 2019), there are rich observations of pre-college test
scores $Z$.

As long as the observational data is sufficiently rich, ICC can
nonparametrically identify causal effects without too limiting exclusion
restrictions. Any argument against the model’s conditional exclusion
restriction can be countered by including the hypothesised unobserved
confounder into $U$. Clearly, at some point completeness of $Z$ with respect
to $(A,U)$ and of $W$ with respect to $U$ is no longer a reasonable assumption
as the number of hypothesised confounders keeps growing. In this sense, ICC
has to trade off relevance against exclusion requirements, similar to standard
IV. Unlike the rigid exclusion restriction in standard IV, ICC provides a way
to utilise rich observational data when instruments are at best excluded
conditional on some common confounders $U$. A good theoretical argument for
ICC will consist of a sensible set of hypothesised common confounders $U$ and
a convincing argument about the completeness of $Z$ and $W$ with respect to
those (and of $Z$ with respect to $A$ given this set $U$).

Ability or personality traits are optimal examples of such common confounders.
Hence, ICC is suitable for the identification of various causal effects
involving individuals’ choices in rich observational data. Returns to
education is, in our opinion, a good example of common confounding.

###  7.3 Causal Effect of a Health Treatment

Our second example involves the causal effect of a health treatment, which
could be a drug, on a health outcome. For drugs, randomised controlled trials
(RCTs) are the most common identification approach. Their advantages in
overcoming potential unobserved confounding are obvious. However, as RCTs are
costly, their sample size is mostly small and rather often effects of a drug
cannot be tested rigorously for different subgroups of the population. After
the introduction of a drug, the potential sample size hugely increases with
treated individuals. With the ICC approach, we could estimate the causal
effect of a treatment on a health outcome in observational data despite the
presence of unobserved confounders. Again, we provide a DAG and explain the
observed and unobserved variables we consider in this model in figure 3. Then,
we go through the ICC construction algorithm to explain how the health
treatment problem fits the ICC model.

Figure 3: DAG of a Health Treatment

$U$$A$$Y$$\ \ \ Z\ \ \ $$W$$\tilde{U}$

  * $Y$:

Post-treatment health outcome

  * $A$:

Health treatment

  * $Z$:

Health measures pre-treatment _with_ effect on treatment choice/assignment

  * $U$:

General health status

  * $\tilde{U}$:

Belief in treatment sucess (placebo effect)

  * $W$:

Health measures pre-treatment _without_ effect on treatment choice/assignment

  1. 1.

$Z$ exogeneity (3.1.2a): Define common confounders $U$ such that

$\displaystyle Y(a,z)=Y(a)\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U\ \forall
a\in\mathcal{A}.$

Instruments $Z$ are pre-treatment health measures with an effect on treatment
$A$ choice/assignment. Some of these instruments may even affect the post-
treatment health outcome $Y$ directly. These instruments would form part of
the unobserved common confounder $U$, which otherwise contains an individual’s
general health status. Individuals with generally better health status and
knowledge are more likely to choose treatment and might benefit more (or less)
from the treatment. E.g. if treatment is a protein supplement and muscle
growth the outcome variable, people with generally better health are likely to
exercise more and hence likely to experience more muscle growth than their
peers. Another obvious confounder is the placebo effect, where individuals
with greater belief in treatment success always benefit more from the
treatment. This inherent bias problem is captured by the unobserved confounder
$\tilde{U}$.

  2. 2.

$W$ exogeneity (3.1.3a): Include in the common confounders $U$ any unobserved
variables necessary to justify

$\displaystyle W(a,z)=W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U.$

$W$ contains pre-treatment health measures without effect on treatment
choice/assignment. $W$ will be associated with other pre-treatment health
measures via the general health status and some specific health issues which
are measures by similar variables in $W$ and $Z$, which are consequently
contained in $U$.

  3. 3.

$W$ relevance (3.1.3b): Check whether $W$ is complete with respect to $U$
given $A$:

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|A,W\right]=0\text{ only
when }g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

$W$ is complete with respect to $U$ as long as the information in its health
measures is sufficiently informative with respect to the general status of
health, as well as those specific pre-treatment health measures in $Z$ which
affect $Y$. When health records of each individual are rich, this may well be
the case.

  4. 4.

$Z$ relevance (3.1.2b): Check whether $Z$ is complete with respect to $(A,U)$:

$\displaystyle\operatorname{\mathbb{E}}\left[g(A,U)|Z\right]=0\text{ only when
}g(A,U)=0\text{ for any }g\in L_{2}(A,U).$

$Z$ is complete with respect to $U$ by construction. Conditional on $U$, these
pre-treatment health measures must be sufficiently relevant for treatment $A$.
This may be quite likely when health records of each individual are rich.

Importantly, the above example again shows that instruments $Z$ can affect
outcome $Y$ to a significant degree without violating the ICC assumptions,
simply by including those variables in $U$, and imposing richness requirements
on the outcome-inducing proxies $W$. For example, let us consider a
$J$-dimensional vector of relevant instruments $Z$ for the one-dimensional
treatment $A$. Then, $J-1$ instruments may have a direct effect on outcome
$Y$, as long as a $(J-1)$-dimensional vector of outcome-inducing proxies $W$
is complete for the $J-1$ endogenous instruments and excluded with respect to
the one exogenous instrument and treatment $A$ conditional on the $J-1$
endogenous instruments. Hence, ICC can be used to obtain unbiased estimates
even when some instruments are endogenous. It does not matter which
instruments these are exactly, but completeness of $W$ with respect to the
endogenous instruments is required, as well as the exclusion of $W$ with
respect to the treatment $A$ and any exogenous instruments conditional on the
endogenous ones. This flexibility of the ICC approach is one of its main
strengths in rich observational data.

##  8 Concluding Remarks

Instrumented common confounding (ICC) bridges identification theory between IV
and proximal learning. The approach is suitable for causal inference in the
social sciences, where ability or other character traits are common
confounders. However, ICC is versatile and can be a viable solution to
identification problems where instrument exclusion fails for a variety of
different reasons.

Our two examples, one economic and one medical, demonstrate how we may observe
instruments $Z$, which satisfy an exclusion restriction conditional on an
unobserved common confounder $U$, for which other relevant and conditionally
independent outcome-inducing proxies $W$ are available. In both examples, a
theoretical argument in favour of traditional IV exclusion is too difficult.
In ICC, we relax the exclusion restriction to exclusion conditional on an
unobserved common confounder $U$. The proposed ICC model construction
algorithm allows researchers to argue about the validity of ICC assumptions
straightforwardly. Exclusion conditional on an unobserved common confounder
$U$ is traded off with relevance of the observed variables $Z$ and $W$ for the
unobserved common confounder $U$ and treatment $A$.

While we are convinced that instrumented common confounding can be a useful
tool for causal inference, it is no panacea. It replaces some strong,
untestable identifying assumptions by other such assumptions, which are more
likely to be satisfied in rich observational data. The validity of its
assumptions must be justified rigorously by (economic) theory.

In this paper, we dealt only with the identification of ICC models, not their
estimation. While estimation is similar to standard IV, we intend to provide a
dedicated package for the simple estimation of ICC models using different
statistical learning tools and look forward to applying the ICC method to
various interesting empirical problems. We currently extend the ICC approach
to allow for the conditional dependence of outcome-aligned proxies $W$ and
treatment $A$.

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##  9 Proofs

###  9.1 Proofs with outcome model restrictions

###### Proof of lemma 4.0.1.

Let
$T_{A}(x)\coloneqq\int_{\mathcal{A}}x(a^{\prime})\pi(a^{\prime})\operatorname{d\\!}\mu_{A}(a^{\prime})$.

$\displaystyle J$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}Y(a^{\prime})\pi(a^{\prime})\operatorname{d\\!}\mu_{A}(a^{\prime})\right]=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\left(k_{0}(a^{\prime},U)+\varepsilon_{Y}\right)\pi(a^{\prime})\operatorname{d\\!}\mu_{A}(a^{\prime})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[T_{A}\left(k_{0}(a^{\prime},U)+\operatorname{\mathbb{E}}\left[\varepsilon_{Y}|A=a^{\prime},U\right]\right)\right]=\operatorname{\mathbb{E}}\left[T_{A}\left(k_{0}(a^{\prime},U)+\operatorname{\mathbb{E}}\left[\varepsilon_{Y}|U\right]\right)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[T_{A}\left(k_{0}(a^{\prime},U)+\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\varepsilon_{Y}|Z,U\right]|U\right]\right)\right]=\operatorname{\mathbb{E}}\left[T_{A}\left(k_{0}(a^{\prime},U)\right)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\left(\int_{\mathcal{A}}k_{0}(a^{\prime},U)\pi(a^{\prime})\operatorname{d\\!}\mu_{A}(a^{\prime})\right)\right]=\operatorname{\mathbb{E}}\left[\phi_{IV}(U;k_{0})\right]$

The second line follows as for any change of $a$ in $Y(a)$, $\epsilon_{Y}$ is
unchanged by definition. The third line follows as
$\operatorname{\mathbb{E}}\left[\varepsilon_{Y}|Z,U\right]=0$. ∎

###### Proof of lemma 4.0.2.

$\displaystyle\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}h_{0}(a,W)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}h_{0}(a,W)\pi(a)\operatorname{d\\!}\mu_{A}(a)|U\right]\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}h_{0}(a,W)\pi(a)\operatorname{d\\!}\mu_{A}(a)|A=a,U\right]\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[h_{0}(a,W)|A=a,U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0}(a,U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{IV}(U;k_{0})\right]=J$

From line one to two we used $W\mathrel{\perp\mspace{-10.0mu}\perp}A\ |\ U$
(A3.1.3a). From line three to four we used the definition of $\mathbb{H}_{0}$,
where $h_{0}\in\mathbb{H}_{0}$. On the last line we used lemma 4.0.1. ∎

###### Proof of lemma 4.0.3.

Any $h_{0}\in\mathbb{H}_{0}$ satisfies

$\displaystyle\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|A,U,Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|A,U\right]=0.$

The first equality holds by $W\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ (A,U)$
(assumption 3.1.3a). Consequently,

$\displaystyle\operatorname{\mathbb{E}}\left[Y-h_{0}(A,W)|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|U,Z\right]|Z\right]+\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\varepsilon|U,Z\right]|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|U,Z\right]|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|A,U,Z\right]|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|A,U\right]|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[0|Z\right]=0$

This proves that equation 4.3 of lemma 4.0.3 holds.

∎

###### Proof of lemma 4.0.4.

For any $h_{0}\in\mathbb{H}_{0}^{\text{obs}}$,

$\displaystyle\operatorname{\mathbb{E}}\left[Y-h_{0}(A,W)|Z\right]=\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|Z\right]=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|A,U\right]|Z\right]=0.$

Under completeness assumption 3.1.2b, the above can only be true if
$\operatorname{\mathbb{E}}\left[k_{0}(A,U)-h_{0}(A,W)|A,U\right]=0$. Hence,
any $h_{0}\in\mathbb{H}_{0}^{\text{obs}}$ also satisfies
$h_{0}\in\mathbb{H}_{0}$, which implies
$\mathbb{H}_{0}^{\text{obs}}\subseteq\mathbb{H}_{0}$. From lemma 4.0.3 it is
known that $\mathbb{H}_{0}\subseteq\mathbb{H}_{0}^{\text{obs}}$. Consequently,
$\mathbb{H}_{0}^{\text{obs}}=\mathbb{H}_{0}$. ∎

###### Proof of lemma 4.1.

Identification of $\tilde{\phi}_{IV}$. For any
$h_{0}\in\mathbb{H}_{0}^{\text{obs}}=\mathbb{H}_{0}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IV}(W;h_{0})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}h_{0}(a,W)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[h(a,W)|A=a,U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0}(a,U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{IV}(U;k_{0})\right]=J$

We move from the first to the second equation by assumption
$W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U$ of 3.1.3a. The step from
the second to the third line is by lemma 4.0.4 and 3.1.3b. The last line holds
by lemma 4.0.1. ∎

###  9.2 Proofs with first stage restrictions

###### Proof of lemma 5.0.1.

IPW. Identification of $\phi_{IPW}$.

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V,U;f_{A|V,U})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\frac{\pi(A)}{f_{A|V,U}(A|V,U)}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\frac{\pi(A)}{f_{A|\eta,U}(A|\eta,U)}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[g(A,U,\varepsilon)\frac{\pi(A)}{f_{A|\eta,U}(A|\eta,U)}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{E}}\int_{\mathcal{A}}g(a,U,e)\frac{\pi(a)}{f_{A|\eta,U}(a|\eta,U)}f_{A,\varepsilon|\eta,U}(a,e|\eta,U)\operatorname{d\\!}\mu_{A}(a)\operatorname{d\\!}\mu_{\varepsilon}(e)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{E}}\int_{\mathcal{A}}g(a,U,e)\frac{\pi(a)}{f_{A|\eta,U}(a|\eta,U)}f_{A|\eta,U}(a|\eta,U)\operatorname{d\\!}\mu_{A}(a)f_{\varepsilon|\eta,U}(e|\eta,U)\operatorname{d\\!}\mu_{\varepsilon}(e)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}g(a,U,\varepsilon)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}Y(a)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]=J$

On line one, we use assumption 5.2. Line two uses the one-to-one
correspondence of $V$ and $\eta$ conditional on $U$, which follows from
assumption 5.1. Line five then uses assumption 5.1.3, which implies the
independence of $A$ and $\varepsilon$ conditional on $(\eta,U)$.

REG. Identification of $\phi_{REG}$.

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0,v})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0,v}(a,V,U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0,\eta}(a,\eta,U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[g(A,U,\varepsilon)|A=a,\eta,U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\int_{\mathcal{E}}g(A,U,e)f_{\varepsilon|A,\eta,U}(e|a,\eta,U)\operatorname{d\\!}\mu_{\varepsilon}(e)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\int_{\mathcal{E}}g(A,U,e)f_{\varepsilon|\eta,U}(e|\eta,U)\operatorname{d\\!}\mu_{\varepsilon}(e)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{E}}\int_{\mathcal{A}}g(A,U,e)\operatorname{d\\!}\mu_{\varepsilon}(e)\pi(a)\operatorname{d\\!}\mu_{A}(a)f_{\varepsilon|\eta,U}(e|\eta,U)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}g(a,U,\varepsilon)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}Y(a)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]=J$

On line one, we use assumption 5.2. Line two uses the one-to-one
correspondence of $V$ and $\eta$ conditional on $U$, which follows from
assumption 5.1. Line five then uses assumption 5.1.3, which implies the
independence of $A$ and $\varepsilon$ conditional on $(\eta,U)$.

DR. Identification of $\phi_{DR}$.

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V,U;f_{A|V,U},k_{0,v})\right]=\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0,v})\right]=J$

∎

###### Proof of lemma 5.0.2.

Key to this proof is the strict monotonicity assumption 5.4. Let $h^{-1}(.,Z)$
be the inverse of $h(Z,m)$ in its second argument and $m^{-1}(.,U)$ be the
inverse of $m(U,\eta)$ in its second argument. Importantly, 5.4.2 implies that
$m^{-1}(m,U)$ is strictly monotonous in its first argument. As in the proof of
lemma 1 in Matzkin [2003], we write the control function as

$\displaystyle V_{\ref{a:strict-monotonicity}}$
$\displaystyle=\int_{\mathcal{U}}F_{A|Z,U}(A,Z,u)\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}\Pr{\left(h(Z,m(u,\eta))\leq
A\right)}\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}\Pr{\left(\eta\leq
m^{-1}\left(h^{-1}(A,Z),u\right)\right)}\operatorname{d\\!}F(u)$
$\displaystyle=\int_{\mathcal{U}}F_{\eta|U}\left(m^{-1}\left(h^{-1}(A,Z),u\right)\right)\operatorname{d\\!}F(u)$

Strict monotonocity of $m^{-1}(.,U)$ in its first argument,

$\displaystyle m^{-1}(m_{1},U)>m^{-1}(m_{0},U)\ \forall m_{1}>m_{0},\forall
U\in\mathcal{U},$

implies no-crossing:

$U$$\eta$$m_{0}$$m_{1}$$m_{2}$$m^{-1}(m_{2},\bar{u})$$m^{-1}(m_{1},\bar{u})$$m^{-1}(m_{0},\bar{u})$$\bar{u}$

Hence, the function $\eta(u)=m^{-1}\left(h^{-1}(a,z),u\right)$ is uniquely
identified by its mean

$\displaystyle\bar{M}=\int_{u}m^{-1}\left(h^{-1}(A,Z),u\right)dF(u).$

This implies the sigma algebras associated with the following three vectors of
random variables are identical: $(\bar{M},U),(\eta(U),U),(\eta,U)$.

$F_{\eta|U}(\eta)$ is strictly monotonous in $\eta$ on its support due to the
continuous conditional distribution of $\eta$ (assumption 5.4 3). In
combination with strict monotonicity of $m^{-1}(.,U)$ in its first argument,
this implies that $F_{\eta|U}\left(m^{-1}\left(h^{-1}(a,z),u\right)\right)$ is
also strictly monotonous in $h^{-1}(a,z)$.

$\bar{u}$$0$$1$$U$$F_{\eta|U}(\eta)$$m_{0}$$m_{1}$$m_{2}$$F_{\eta|U}\left(m^{-1}(m_{2},\bar{u})\right)$$F_{\eta|U}\left(m^{-1}(m_{1},\bar{u})\right)$$F_{\eta|U}\left(m^{-1}(m_{0},\bar{u})\right)$

Hence, the function
$F_{\eta|U}(\eta(u))=F_{\eta|U}\left(m^{-1}\left(h^{-1}(A,Z),u\right)\right)$
is uniquely identified by its mean

$\displaystyle\bar{V}=\int_{u}F_{\eta|U}\left(m^{-1}\left(h^{-1}(A,Z),u\right)\right)dF(u).$

This implies the sigma algebras associated with the following vectors of
random variables are identical:
$(\bar{V},U),(F_{\eta|U}(\eta(U)),U),(F_{\eta|U}(\eta),U),(\eta,U)$. The last
two associated sigma algebras’ equality follows from assumption 5.4.3
$F_{\eta|U}(\eta)$ is strictly monotonic on the support of $\eta$, the sigma
algebra of $(F_{\eta|U}(\eta),U)$ and $(\eta,U)$ are equal. ∎

###### Proof of lemma 5.1.

First, note that the main step of both proofs for IPW and REG estimators,
$\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V_{\ref{a:strict-
monotonicity}},U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V,U\right]\pi(a)\operatorname{d\\!}\mu(a)\right]$,
follows directly from lemma 5.0.2.

IPW. Want to show that
$\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V_{\ref{a:strict-
monotonicity}},U;f_{A|V_{\ref{a:strict-
monotonicity}},U})\right]=\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V,U;f_{A|V,U})\right]$.

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V_{\ref{a:strict-
monotonicity}},U;f_{A|V_{\ref{a:strict-monotonicity}},U})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\frac{\pi(A)}{f_{A|V_{\ref{a:strict-
monotonicity}},U}(A|V_{\ref{a:strict-monotonicity}},U)}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V_{\ref{a:strict-
monotonicity}},U\right]\frac{\pi(a)}{f_{A|V_{\ref{a:strict-
monotonicity}},U}(a|V_{\ref{a:strict-monotonicity}},U)}f_{A|V_{\ref{a:strict-
monotonicity}},U}(a|V_{\ref{a:strict-
monotonicity}},U)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V_{\ref{a:strict-
monotonicity}},U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V,U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{IPW}(Y,A,V,U;f_{A|V,U})\right]$

The penultimate line follows from lemma 5.0.2, and the last line from lemma
5.0.1.

REG. Want to show that
$\operatorname{\mathbb{E}}\left[\phi_{REG}(V_{\ref{a:strict-
monotonicity}},U;k_{0,V_{\ref{a:strict-
monotonicity}}})\right]=\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0,v})\right]$.

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{REG}(V_{\ref{a:strict-
monotonicity}},U;k_{0,V_{\ref{a:strict-monotonicity}}})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0}(a,V_{\ref{a:strict-
monotonicity}},U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V_{\ref{a:strict-
monotonicity}},U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[Y|A=a,V,U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0})\right]$

The last line follows from the definition of $\phi_{REG}(.;.)$ in lemma 5.0.1.

DR. Want to show that
$\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V_{\ref{a:strict-
monotonicity}},U;f,k_{0})\right]=\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V,U;f,k_{0})\right]$.

$\displaystyle\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V_{\ref{a:strict-
monotonicity}},U;f_{A|V_{\ref{a:strict-
monotonicity}},U},k_{0,V_{\ref{a:strict-monotonicity}}})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{REG}(V_{\ref{a:strict-
monotonicity}},U;k_{0,V_{\ref{a:strict-monotonicity}}})\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\phi_{REG}(V,U;k_{0,v})\right]=\operatorname{\mathbb{E}}\left[\phi_{DR}(Y,A,V,U;f_{A|V,U},k_{0,v})\right]$

The penultimate line follows from lemma 5.0.2, and the last line from lemma
5.0.1. ∎

###### Proof of lemma 5.1.1.

Proof of equation 5.14. For any $\tau_{A,0}\in\mathbb{T}_{0}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,U,W_{0}\right]$
$\displaystyle=F(A|Z,U)$
$\displaystyle\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,U,W_{0}\right]|Z,W_{0}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[F(A|Z,U)|Z,W_{0}\right]$
$\displaystyle\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,W_{0}\right]$
$\displaystyle=F(A|Z,W_{0}).$

We move from the second to third equation by assumption
$A\mathrel{\perp\mspace{-10.0mu}\perp}W_{0}\ |\ (Z,U)$ (assumption 5.5).

Proof of equation 5.15. For any $\kappa_{0}\in\mathbb{K}_{0}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,U,W_{1}\right]$
$\displaystyle=\frac{f(U)}{f(U|Z)}$
$\displaystyle\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,U,W_{1}\right]|Z,W_{1}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\frac{f(U)}{f(U|Z)}|Z,W_{1}\right]$
$\displaystyle\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,U,W_{1}\right]|Z,W_{1}\right]$
$\displaystyle=f(Z)\operatorname{\mathbb{E}}\left[\frac{1}{f(Z|U)}|Z,W_{1}\right]$
$\displaystyle\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,W_{1}\right]$
$\displaystyle=f(Z)\int_{\mathcal{U}}\frac{f(U|W_{1},Z)}{f(Z|U)}\operatorname{d\\!}\mu_{U}(U)$
$\displaystyle=f(Z)\int_{\mathcal{U}}\frac{f(W_{1},Z|U)f(U)}{f(Z|U)f(W_{1},Z)}\operatorname{d\\!}\mu_{U}(U)$
$\displaystyle=f(Z)\int_{\mathcal{U}}\frac{f(W_{1}|U)f(U)}{f(W_{1},Z)}\operatorname{d\\!}\mu_{U}(U)$
$\displaystyle=\frac{f(Z)}{f(Z|W_{1})}$
$\displaystyle=\frac{f(W_{1})}{f(W_{1}|Z)}$

We move from the fifth to the sixth equation by assumption
$W_{1}\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$ (assumption 5.5). ∎

###### Proof.

Proof of lemma 5.1.2 First, note that for any $\tau_{A,0}\in\mathbb{T}_{0}$,

$\displaystyle F(A|Z,U)$
$\displaystyle=\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,U,W_{0}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,U\right]$
$\displaystyle=\int_{\mathcal{W}_{1}}\tau_{A,0}(Z,W_{1})\operatorname{d\\!}F(W_{1}|U)$
$\displaystyle V_{\ref{a:strict-
monotonicity}}=\int_{\mathcal{U}}F(A|Z,U)\operatorname{d\\!}F(U)$
$\displaystyle=\int_{\mathcal{W}_{1}}\tau_{A,0}(Z,W_{1})\operatorname{d\\!}F(W_{1})$

We move from the second to the third line by assumption
$W_{1}\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ U$ (assumption 5.5).

Now, note that for any $\tau_{A}\in L_{2}(Z,W_{1})$ and
$\kappa_{0}\in\mathbb{K}_{0}^{\text{obs}}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\tau_{A}(Z,W_{1})|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\tau_{A}(Z,W_{1})|Z,W_{1}\right]|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})|Z,W_{1}\right]\tau_{A}(Z,W_{1})|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\frac{f(W_{1})}{f(W_{1}|Z)}\tau_{A}(Z,W_{1})|Z\right]$
$\displaystyle=\int_{\mathcal{W}_{1}}\frac{f(W_{1})}{f(W_{1}|Z)}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1}|Z)$
$\displaystyle=\int_{\mathcal{W}_{1}}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1})$

For any $\tau_{A}\in L_{2}(Z,W_{1})$ and
$\kappa_{0}\in\mathbb{K}_{0}^{\text{obs}}$, we write

$\displaystyle\int_{\mathcal{W}_{1}}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1})$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\tau_{A}(Z,W_{1})|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\operatorname{\mathbb{E}}\left[\tau_{A}(Z,W_{1})|Z,W_{0}\right]|Z\right].$

For any $\tau_{A,0}\in\mathbb{T}_{0}$ and
$\kappa_{0}\in\mathbb{K}_{0}^{\text{obs}}$, we write

$\displaystyle V_{\ref{a:strict-monotonicity}}$
$\displaystyle=\int_{\mathcal{W}_{1}}\tau_{A,0}(Z,W_{1})\operatorname{d\\!}F(W_{1})$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\tau_{A,0}(Z,W_{1})|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\operatorname{\mathbb{E}}\left[\tau_{A,0}(Z,W_{1})|Z,W_{0}\right]|Z\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})F(A|Z,W_{0})|Z\right].$

Now this implies that for any $\tau_{A}\in L_{2}(Z,W_{1})$,

$\displaystyle\int_{\mathcal{W}_{1}}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1})-V_{\ref{a:strict-
monotonicity}}$
$\displaystyle=\operatorname{\mathbb{E}}\left[\kappa_{0}(Z,W_{0})\operatorname{\mathbb{E}}\left[\tau_{A}(Z,W_{1})-F(A|Z,W_{0})|Z,W_{0}\right]|Z\right].$

Hence, for any $\tau_{A}\in\mathbb{T}_{0}^{\text{obs}}$ as long as
$\mathbb{K}_{0}^{\text{obs}}\neq\emptyset$,

$\displaystyle V_{\ref{a:strict-monotonicity}}$
$\displaystyle=\int_{\mathcal{W}_{1}}\tau_{A}(Z,W_{1})\operatorname{d\\!}F(W_{1}).$

∎

###### Proof of lemma 5.1.3.

IPW. Identification of $\tilde{\phi}_{IPW}$. For any $q_{0}\in\mathbb{Q}_{0}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{IPW}(Y,A,\tilde{V},Z)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\pi(A)q_{0}(A,\tilde{V},Z)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[Y\pi(A)q_{0}(A,\tilde{V},Z)|Y,A,\tilde{V},U\right]\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\operatorname{\mathbb{E}}\left[\pi(A)q_{0}(A,Z)|A,\tilde{V},U\right]\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\frac{\pi(A)}{f(A|\tilde{V},U)}\right]=\operatorname{\mathbb{E}}\left[{\phi}_{IPW}(Y,A,\tilde{V},U)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[Y\frac{\pi(A)}{f(A|\eta,U)}\right]=J$

We move from the second to third equation by assumption
$Y\mathrel{\perp\mspace{-10.0mu}\perp}Z\ |\ (A,U)$ of 5.5. If 5.4 holds, the
last line holds by theorem 5.1. Otherwise, the last line directly holds by the
steps in the proof of lemma 5.0.1, as 5.3 implies that
$\tilde{V}=V_{\ref{a:strict-monotonicity-no-cf-fs}}$ is a one-to-one
transformation of $\eta$.

REG. Identification of $\tilde{\phi}_{REG}$. For any $h_{0}\in\mathbb{H}_{0}$,

$\displaystyle\operatorname{\mathbb{E}}\left[\tilde{\phi}_{REG}(\tilde{V},W)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}h_{0}(a,\tilde{V},W)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}\operatorname{\mathbb{E}}\left[h(a,\tilde{V},W)|A=a,\tilde{V},U\right]\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0}(a,\tilde{V},U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]=\operatorname{\mathbb{E}}\left[\phi_{REG}(\tilde{V},U)\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\int_{\mathcal{A}}k_{0}(a,\eta,U)\pi(a)\operatorname{d\\!}\mu_{A}(a)\right]=J$

We move from the first to the second equation by assumption
$W\mathrel{\perp\mspace{-10.0mu}\perp}(A,Z)\ |\ U$ of 5.5. If 5.4 holds, the