# Bias Formulas for Violations of Proximal Identification Assumptions

Raluca Cobzaru, Roy Welsch, Stan Finkelstein, Kenney Ng, Zach Shahn

###### Abstract

Causal inference from observational data often rests on the unverifiable
assumption of no unmeasured confounding. Recently, Tchetgen Tchetgen and
colleagues have introduced proximal inference to leverage negative control
outcomes and exposures as proxies to adjust for bias from unmeasured
confounding [16]. However, some of the key assumptions that proximal inference
relies on are themselves empirically untestable. Additionally, the impact of
violations of proximal inference assumptions on the bias of effect estimates
is not well understood. In this paper, we derive bias formulas for proximal
inference estimators under a linear structural equation model data generating
process. These results are a first step toward sensitivity analysis and
quantitative bias analysis of proximal inference estimators. While limited to
a particular family of data generating processes, our results may offer some
more general insight into the behavior of proximal inference estimators.

##  1 Introduction

Causal inference using observational data often rests on the assumption of no
unmeasured confounding. This assumption is not empirically verifiable, but
sensitivity analysis methods (e.g., [2, 4, 10, 12]) are available to assess
robustness of results to possible unmeasured confounding. Alternatively,
investigators might turn to methods such as instrumental variable analysis or
difference-in-differences, which depend on different assumptions. Sensitivity
analyses for violations of the assumptions required by these alternative
methods are also available ([1, 9]).

There has been recent interest in the use of negative control methods to
detect and resolve confounder bias. A negative control outcome (NCO) is a
variable known not to be causally affected by the treatment of interest, while
a negative control exposure (NCE) is a variable known not to causally affect
the outcome of interest [15]. Tchetgen Tchetgen and colleagues have developed
a Proximal Inference framework ([3, 7, 16]) which uses NCE-NCO pairs sharing
the same unmeasured confounders as the treatment-outcome relationship of
interest as proxies to adjust for unmeasured confounding.

However, some of the assumptions that proximal inference relies on are
themselves empirically untestable [16] and bias resulting from violations of
proximal identification assumptions is not fully understood. In this paper, we
characterize bias from violations of proximal inference assumptions in a
linear structural equation data generating process. Our results build
understanding of the sensitivity of proximal inference to assumption
violations and serve as a first step toward sensitivity analysis and
quantitative bias analysis [6] tools for proximal inference.

The organization of the paper is as follows. In Section 2, we review proximal
inference. In Section 3, we describe the forms of bias that we will study. In
Sections 4 and 5, we derive bias formulas for the settings described in
Section 3. In Section 6, we present numerical experiments based on the bias
formulas from Sections 4 and 5 to explore their implications. In Section 7, we
conclude by discussing some potential insights into the sensitivity of
proximal inference estimators gained from our results.

##  2 Proximal Identification of the Average Treatment Effect

###  2.1 Review of Definitions and Assumptions

We use the potential outcome framework [13] to formally define causal effects.
Let $A$ denote the binary treatment of interest, $Y$ the observed post-
treatment outcome, and $Y(a),a=0,1$ the potential (counterfactual) outcome
that would have been observed had treatment $A$ been set to $a$. We implicitly
make the no-interference assumption that the potential outcome of each
individual does not depend on the treatments received by other individuals
[18]. We aim to estimate the average causal effect (ACE) of $A$ on $Y$,
defined as $\psi=\operatorname{\mathbb{E}}[Y(1)-Y(0)]$.

Let $L$ denote the set of measured covariates. We make the standard
assumptions of Consistency and Positivity, defined below.

######  Assumption 1.

(Consistency) $Y=Y(A)$

In other words, the observed value of $Y$ under treatment $A$ coincides with
the counterfactual outcome that would have been observed under the same
treatment value. Thus, we only observe the counterfactual outcome
corresponding to the treatment value that was actually administered in our
data.

######  Assumption 2.

(Positivity) $0<\mathbb{P}(A=a|L)<1$, for $a=0,1$

Assumption 2 states that both exposure levels are observed at all levels of
the observed covariates $L$.

Many analyses further make the assumption that there is no unobserved
confounding, i.e. that observed covariates block all non-direct (or
‘backdoor’) causal paths between treatment and outcome.

######  Assumption 3.

(Exchangeability) $Y(a)\perp\\!\\!\\!\perp A\ |\ L$, for $a=0,1$

Under Assumptions 1-3, the counterfactual mean
$\operatorname{\mathbb{E}}[Y(a)]$ is identified by the g-formula (introduced
in [5]):

$\operatorname{\mathbb{E}}[Y(a)]=\sum_{l}\operatorname{\mathbb{E}}[Y|A=a,L=l]\mathbb{P}(L=l).$
(1)

Exchangeability is a strong assumption that is empirically untestable. [8]
propose an alternative to Assumption 3 that allows us to identify the
counterfactual mean $\operatorname{\mathbb{E}}[Y(a)]$ despite the presence of
unobserved confounding. We review the alternative conditions developed by [8]
leading to the proximal g-formula, a counterpart to (1) allowing for some
unobserved confounding.

As in [3], we consider a (potentially multidimensional) variable $L$ that can
be partitioned into three types of variables $(X,Z,W)$, such that

  * 1)

$X$ includes observed variables that may be common causes of $A$ and $Y$
(observed confounders)

  * 2)

$Z$ includes treatment-inducing confounding proxies, i.e. $Z$ includes causes
of $A$ that share an unmeasured common cause $U_{Z}$ with $Y$

  * 3)

$W$ includes outcome-inducing confounding proxies, i.e. $W$ includes causes of
$Y$ that share an unmeasured common cause $U_{W}$ with $A$

Figure 1 contains DAGs representing each of the proxy types included in $L$.

Figure 1: DAGs representing the three types of variables $(X,Z,W)$
partitioning $L$

In [8], exchangeability is replaced with the assumptions:

######  Assumption 4.

(Treatment-inducing confounding proxy)

$Y(a,z)=Y(a),\ \ \textnormal{for all }a,z$ (2)

######  Assumption 5.

(Outcome-inducing confounding proxy)

$W(a,z)=W,\ \ \textnormal{for all }a,z$ (3)

######  Assumption 6.

(Latent unconfoundedness) If $U$ denotes the set of unobserved confounders,
then

$\displaystyle Z$ $\displaystyle\perp\\!\\!\\!\perp(Y(a),W)\ |\ U,X$ (4)
$\displaystyle W$ $\displaystyle\perp\\!\\!\\!\perp A\ |\ U,X$ (5)

Assumption 4 states that $Z$ does not have a direct effect on $Y$ upon
intervening on $A$, while Assumption 5 states that neither $A$ nor $Z$ have a
causal effect on $W$. Past works [15] refer to variables $Z$ satisfying (2)
and (4) as negative control exposure (NCE) variables, and to variables $W$
satisfying (3) and (5) as negative control outcome (NCO) variables. This
terminology is based on negative control methods employing variables that
share a confounding mechanism with the treatment-outcome relationship in view
to detect bias in epidemiological research. Although there is a subtle
distinction between the proxy and negative control nomenclature when
discussing the design of observational studies [16], for the theoretical
analysis employed in this paper we will be using treatment-inducing (outcome-
inducing) confounding proxies and NCE (NCO) variables interchangeably.

In addition to Assumptions 1-6, [7] introduce the following completeness
conditions for the identification of $\operatorname{\mathbb{E}}[Y(a)]$:

######  Assumption 7.

(Completeness) For any $a$, $x$ and for any square-integrable function $g$:

  * (a)

If $\operatorname{\mathbb{E}}[g(U)|Z,A=a,X=x]=0$ almost surely, then $g(U)=0$
almost surely.

  * (b)

If $\operatorname{\mathbb{E}}[g(Z)|W,A=a,X=x]=0$ almost surely, then $g(Z)=0$
almost surely.

Assumption 7(a) can be interpreted as a requirement that the NCE $Z$ has
enough variability relative to the variability of $U$; similarly, assumption
7(b) requires the variability of $W$ to be large enough relative to the
variability of $Z$. Under conditions 7(a) and (b), we can essentially account
for $U$ in our ACE estimate without either measuring or modeling the
distribution of $U$. The role of completeness will be further explored in
Section 2.2, where we outline the analytical framework by which the ACE is
estimated using the proximal g-formula.

Completeness assumption 7(a) has a simple interpretation in the case where
confounders $U$ and the negative control pair $(Z,W)$ are all categorical. As
mentioned in [3], if $(U,Z,W)$ are categorical with respective number of
categories $(d_{u},d_{z},d_{w})$, then completeness 7(a) requires that

$\min(d_{z},d_{w})\geq d_{u}$ (6)

In other words, proximal inference can account for unmeasured confounding if
the number of categories of $U$ is less than that of either $Z$ or $W$. This
leads to the practical recommendation to measure a rich set of baseline
characteristics (which can be used as negative controls), such that the
proximal identification approach has a higher chance of mitigating unmeasured
confounder bias [3]. There is not such a straightforward method for expressing
the completeness condition in the case of continuous $U$ and negative controls
$(Z,W)$, though some intuition for nonparametric regression results from [3].
In Section 4, we investigate the behavior of proximal inference in LSEM setups
in which the completeness assumption 7(a) is violated.

Lastly, to be valid proxies the variables $(Z,W)$ must be $U-relevant$:

######  Assumption 8.

($U$-relevance)

$\displaystyle Z$ $\displaystyle\not\perp\\!\\!\\!\perp U\ |\ A,X$ (7)
$\displaystyle W$ $\displaystyle\not\perp\\!\\!\\!\perp U\ |\ X$ (8)

The $U$-relevance assumption (also known as $U$-comparability [15]) requires
the unmeasured confounders $U$ of the $A-Y$ relationship to be the same as the
unmeasured confounders of the $A-W$ and $Z-Y$ secondary treatment-outcome
associations. This is such that, by the negative control framework, any non-
null $A-W$ or $Z-Y$ association can be attributed to $U$ confounding the $A-Y$
relationship (while null associations imply no empirical evidence of
unmeasured confounding).

Throughout this paper, we suppress the observed confounders $X$ unless
otherwise stated. While we do not include $X$ in the sensitivity analysis
discussion of Section 6, the addition of $X$ is a straightforward extension of
our bias formulations.

###  2.2 Estimating the Proximal g-Formula via Moment Restriction

[8] introduce the notion of an outcome confounding bridge function, which
transforms the negative control outcome $W$ to match the confounding effect of
$U$ on $Y$. More precisely, an outcome confounding bridge function $h(W,A,X)$
is a function satisfying

$\operatorname{\mathbb{E}}[Y|U,A=a,X=x]=\operatorname{\mathbb{E}}[h(W,A,X)|U,A=a,X=x]$
(9)

for all values of $a,x$. In other words, if function $h(W,A,X)$ exists, then
the confounding effect of $U$ on the transformed variable $h(W,a,X)$ equals
the confounding effect of $U$ on $Y$ at exposure level $A=a$. Given
assumptions 1, 5, 6, and 8, [8] infer that

$\operatorname{\mathbb{E}}[Y(a)]=\operatorname{\mathbb{E}}[h(W,a,X)]\textnormal{
for all $a=0,1$}$ (10)

which means $\operatorname{\mathbb{E}}[Y(a)]$ can be estimated following the
identification of an outcome bridge function $h(W,A,X)$, if such a function is
assumed to exist.

[3, 7] established the following proximal identification result for the
outcome confounding bridge function that leverages the distribution of a NCE
$Z$:

######  Theorem 1.

Suppose there exists an outcome confounding bridge function $h(w,a,x)$ solving
the Fredholm integral equation

$\operatorname{\mathbb{E}}[Y|Z,A,X]=\int h(w,A,X)dF(w|Z,A,X)$ (11)

almost surely. Then, under Assumptions 1, 2, 4-6, and 7(a),

$\operatorname{\mathbb{E}}[Y|U,A,X]=\int h(w,A,X)dF(w|U,X)$ (12)

almost surely.

Under Assumption 6, we have
$\operatorname{\mathbb{E}}[Y(a)]=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[Y|U,A=a,X=x\right]\right]$
for all $a,x$. The counterfactual mean $\operatorname{\mathbb{E}}[Y(a)]$ can
then be computed as follows:

######  Corollary 1.1.

(Proximal g-formula) If (12) holds almost surely, then the counterfactual mean
$\operatorname{\mathbb{E}}[Y(a)]$, $a=0,1$ is nonparametrically identified by

$\operatorname{\mathbb{E}}[Y(a)]=\int_{\mathcal{X}}\int h(w,a,x)dF(w|x)dF(x)$
(13)

and the ACE is identified by

$\psi=\int_{\mathcal{X}}\int\left\\{h(w,1,x)-h(w,0,x)\right\\}dF(w|x)dF(x)$
(14)

######  Remark 1.

[3] establish a similar proximal identification result for the existence and
identification of a treatment confounding bridge function $q(Z,A,X)$ that
leverages the NCO variable $W$ (and an assumption analogous to completeness
Assumption 7) instead. Due to the higher complexity of
$\frac{1}{\mathbb{P}(A=a|U,X)}$ relative to
$\operatorname{\mathbb{E}}[Y|U,A,X]$ in our chosen LSEMs, we delegate
sensitivity analysis involving the treatment confounding bridge function to
future work.

Assuming the outcome confounding bridge function $h(W,A,X)$ exists and is
identifiable as a solution to (12), [16, 8] provide a practical approach for
estimating the proximal g-formula using the generalized method of moments
(GMM). Suppose one has access to $n$ i.i.d. samples
$D_{i}=(A_{i},Y_{i},L_{i})$, $L_{i}=(X_{i},Z_{i},W_{i})$ (where $Z$, $W$ are
assumed to be correctly classified as treatment- and outcome-inducing
confounding proxies, respectively). Moreover, suppose one has specified a
parametric model for the confounding bridge, $h(W,A,X)=h(W,A,X;b)$ (e.g.,
$h(W,A,X;b)$ is linear in $W$, $A$, $X$ with unknown parameter $b$). The true
model for $h(W,A,X)$ is unknown, but one may fit a fairly flexible model
(including, for instance, splines or interaction terms) to obtain a reasonable
estimation in practice.

We define the target parameter $\theta=(b,\psi)$ to encode the parameters $b$
of $h(W,A,X;b)$ and the ACE $\psi$, along with the moment restrictions

$h(D_{i};\theta)=\begin{pmatrix}\left\\{Y_{i}-h(W_{i},A_{i},X_{i})\right\\}\times\begin{pmatrix}1&A_{i}&Z_{i}&X_{i}&A_{i}X_{i}&A_{i}Z_{i}\end{pmatrix}^{T}\\\
\psi-\left\\{h(W_{i},V_{i},1;b)-h(W_{i},V_{i},0;b)\right\\}\end{pmatrix}$ (15)

Then, if $m_{n}(\theta)=\frac{1}{n}\sum_{i=1}^{n}h(D_{i};\theta)$, the GMM
solves

$\hat{\theta}=\operatorname*{arg\,min}_{\theta}m_{n}^{T}(\theta)m_{n}(\theta)$
(16)

which can be equivalently written as

$\hat{\theta}=\begin{pmatrix}\hat{b}\\\
\hat{\psi}\end{pmatrix}=\begin{pmatrix}\operatorname*{arg\,min}_{b}\operatorname{\mathbb{E}}\left[Y-h(W_{i},A_{i},X_{i})|Z_{i},A_{i},X_{i}\right]^{2}\\\
\operatorname*{arg\,min}_{\psi}\left(\psi-\left\\{h(W_{i},V_{i},1;b)-h(W_{i},V_{i},0;b)\right\\}\right)^{2}\end{pmatrix}$
(17)

The resulting parameter estimate $\hat{b}$ is unbiased by (9), while the ACE
estimate $\hat{\psi}$ is unbiased by (10).

##  3 Bias Settings

We have so far collected a series of untestable assumptions 4-8 that replace
exchangeability and account for the effect of unmeasured confounders $U$
without directly modeling or estimating $U$. The impact on the direction
and/or magnitude of bias resulting from violations of these assumptions has
not been explored. We trust the analyst to identify true negative control
exposures and outcomes in this work (Assumptions 4 and 5), as subject matter
knowledge should often be quite reliable on this point. Latent
unconfoundedness (Assumption 6) is not really an assumption since it
presumably holds for some sufficiently rich $U$. But the richer (or higher
dimensional) the $U$ required to satisfy latent unconfoundedness Assumption 6,
the less plausible it is that completeness (Assumption 7) or $U$-relevance
(Assumption 8) hold. If many components of $U$ are common causes of the
negative control exposures and outcomes, then completeness (Assumption 7) is
difficult to satisfy. And if many components of $U$ are required to block all
backdoor paths between A and Y, then they are less likely to all be associated
with both $Z$ and $W$, violating Assumption 8.

In Section 4, we characterize the proximal inference estimator bias in a LSEM
under scenarios in which each of $Z$ and $W$ are one-dimensional but $U$
(comprising common causes of any of $A$, $Y$, $Z$, and $W$) has two
independent components. We first consider the case where one component of $U$
is an ‘extra’ common cause of $Z$ and $W$ not associated with $A$ or $Y$
(which violates completeness (Assumption 7) and is illustrated in Figure 2),
then we consider the case where one component of $U$ is a common cause of $A$
and $Y$ but is not associated with either $Z$ or $W$ (which violates
$U$-relevance (Assumption 8) and is illustrated in Figure 3). We would argue
that it is difficult to guard against violations of Assumptions 7 and 8
arising in this way using subject matter knowledge, making sensitivity
analysis for violations of these types particularly valuable.

Figure 2: DAG encoding the causal relationships among variables in (24) in
which completeness 7(a) is violated Figure 3: DAG encoding causal
relationships among variables in (29) in which U-relevance is violated

Additionally, for the settings of Figures 2 and 3, we compare the bias of the
proximal estimator due to violations of Assumptions 7 and 8 to the bias of
alternative estimators of the ACE which the analyst might implement under an
incorrect unconfoundedness assumption. We consider

  * (1)

an outcome regression estimator (referred to as “OR”) which adjusts for
$(Z,W)$ via the g-formula (1) taking $L=\\{Z,W\\}$ and specifying outcome
regression model $E[Y|A,L]=\beta^{T}(A,Z,W,AZ,AW)$

  * (2)

an unadjusted estimator (referred to as “unadj”) which assumes no unobserved
confounding and estimates $E[Y(a)]$ as $\hat{E}[Y|A=a]$ via sample means.

In Section 5, we characterize the bias of the proximal estimator in a LSEM
where $Z$ and $W$ have the same (arbitrary) number of dimensions and $U$ has
at least as many components as either $Z$ or $W$, under the simplifying
assumption that the effect of $A$ on $Y$ is not modified by $U$ on the
additive scale. This simplifying assumption makes tractable calculations that
allow us to develop more general bias formulas for scenarios in which each
component of $U$ might have missing arrows into any of $A,Y,Z$, or $W$ in the
causal DAG.

##  4 Bias Formulas For Two Dimensional $U$

In this section, we will derive formulas for the bias resulting from applying
proximal inference under scenarios depicted in Figures 2 and 3 under a LSEM
data generating process. We base our LSEMs on the data generating process in
[7].

Let us consider i.i.d. data generated according to

$\begin{split}\begin{pmatrix}U\\\
X\end{pmatrix}\sim\mathcal{N}&\left(\begin{pmatrix}0\\\ 0\\\
0\end{pmatrix},\begin{pmatrix}1&0&\rho_{1}\\\ 0&1&\rho_{2}\\\
\rho_{1}&\rho_{2}&1\end{pmatrix}\right),\
\rho_{1},\rho_{2}\in\left(-1,1\right)\\\
\textnormal{logit}(\mathbb{P}(A=1|X,U))&=\alpha_{0}+\alpha_{x}X+\boldsymbol{\alpha}_{u}^{T}U\\\
Z&=\theta_{0}+\theta_{a}A+\theta_{x}X+\boldsymbol{\theta}_{u}^{T}U+\epsilon_{1}\\\
W&=\mu_{0}+\mu_{x}X+\boldsymbol{\mu}_{u}^{T}U+\epsilon_{2}\\\
Y(a)&=\gamma_{0}+\gamma_{a}a+\gamma_{x}X+\boldsymbol{\gamma}_{u}^{T}U+\gamma_{au1}aU_{1}+\epsilon_{3}\\\
\epsilon_{1},\epsilon_{2},\epsilon_{3}&\sim\mathcal{N}(0,1)\end{split}$ (18)

The causal DAG corresponding to this data generating process can be seen in
Figure 4. Parameter
$\boldsymbol{\alpha}_{u}=\begin{pmatrix}\alpha_{u1}&\alpha_{u2}\end{pmatrix}^{T}$
encodes the magnitude of confounding, while
$\boldsymbol{\theta}_{u}=\begin{pmatrix}\theta_{u1}&\theta_{u2}\end{pmatrix}^{T}$
and $\boldsymbol{\mu}_{u}=\begin{pmatrix}\mu_{u1}&\mu_{u2}\end{pmatrix}^{T}$
encode the association between confounder $U$ and the NCE/NCO, respectively.
We will explore the sensitivity of the proximal inference bias to particular
values of
$\left(\boldsymbol{\alpha}_{u},\boldsymbol{\theta}_{u},\boldsymbol{\mu}_{u}\right)$.

Figure 4: DAG encoding the causal relationships among variables in (18)

The NCE $Z$ is a post-treatment variable in this DGP. We note that DAGs other
than Figure 4 might also be compatible with proximal inference assumptions
(e.g., $Z\to A$ or no arrow between $A$ and $Z$, in the absence of other
changes). More examples of DAGs compatible with proximal inference assumptions
can be found in [15].

###  4.1 Base Case: No Violated Assumptions.

As a sample application of the proximal identification method, we identify the
confounding outcome bridge function $h(W,A,X)$ corresponding to the baseline
case of one-dimensional $U$ (that is,
$\alpha_{u2}=\theta_{u2}=\mu_{u2}=\gamma_{u2}=0$). For simplicity, we drop the
index denoting the first component of $U$. We have the following DGP:

$\begin{split}\begin{pmatrix}U\\\
X\end{pmatrix}&\sim\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}1&\rho\\\ \rho&1\end{pmatrix}\right),\
\rho\in\left(-1,1\right)\\\
\textnormal{logit}(\mathbb{P}(A=1|X,U))&=\alpha_{0}+\alpha_{x}X+\alpha_{u}U\\\
Z&=\theta_{0}+\theta_{a}A+\theta_{x}X+\theta_{u}U+\epsilon_{1}\\\
W&=\mu_{0}+\mu_{x}X+\mu_{u}U+\epsilon_{2}\\\
Y(a)&=\gamma_{0}+\gamma_{a}a+\gamma_{x}X+\gamma_{u}U+\gamma_{au}aU+\epsilon_{3}\\\
\epsilon_{1},\epsilon_{2},\epsilon_{3}&\sim\mathcal{N}(0,1)\end{split}$ (19)

The confounding bridge functions that solve both the outcome and treatment
bridge function equations take the form

$\displaystyle h(W,A,X;b)$
$\displaystyle=b_{0}+b_{a}A+b_{w}W+b_{x}X+b_{ax}AX+b_{aw}AW$ (20)
$\displaystyle q(Z,A,X;t)$
$\displaystyle=1+\exp\left\\{(-1)^{1-A}(t_{0}+t_{z}Z+t_{a}A+t_{x}X)\right\\}$
(21)

with fitted parameters

$\displaystyle(b_{0},b_{a},b_{x},b_{w},b_{ax},b_{aw})=\left(\gamma_{0}-\frac{\mu_{0}\gamma_{u}}{\mu_{u}},\gamma_{a}-\frac{\mu_{0}\gamma_{au}}{\mu_{u}},\gamma_{x}-\frac{\mu_{x}\gamma_{u}}{\mu_{u}},\frac{\gamma_{u}}{\mu_{u}},-\frac{\mu_{x}\gamma_{au}}{\mu_{u}},\frac{\gamma_{au}}{\mu_{u}}\right)$
(22)
$\displaystyle(t_{0},t_{a},t_{x},t_{z})=\left(-\alpha_{0}+\frac{\theta_{0}}{\theta_{u}}\alpha_{u}+\frac{0.5}{\theta_{u}^{2}}\alpha_{u}^{2},-\frac{1}{\theta_{u}^{2}}\alpha_{u}^{2}+\frac{\theta_{a}}{\theta_{u}}\alpha_{u},\frac{\theta_{x}}{\theta_{u}}\alpha_{u}-\alpha_{x},-\frac{\alpha_{u}}{\theta_{u}}\right)$
(23)

The proof for the correctness and uniqueness of the above bridge functions is
provided in Appendix A.1. The proximal g-formula using either bridge function
yields an unbiased estimate of the ACE.

###  4.2 Violations of Proximal Inference Assumptions

We examine setup (18) for two-dimensional $U$, which implies that at least one
of vectors $(\alpha_{u2},\gamma_{u2})$ and $(\theta_{u2},\mu_{u2})$ has all
nonzero entries.

In the case where $\boldsymbol{\theta}_{u}$ is nonzero (that is, there is a
nonzero association between the NCE $Z$ and at least one component of $U$),
the following theorem holds:

######  Theorem 2.

If $\boldsymbol{\theta}_{u}$ is nonzero (i.e., $Z$ is $U$-relevant), then the
LSEM (18) with Gaussian $(X,U)$ violates completeness assumption 7(a).

A proof of Theorem 2 which constructs a counterexample function $g(U)$ for
assumption 7(a) is provided in Appendix B.1. By Theorem 1, we know that
violating assumption 7(a) leads to a potentially biased ACE estimate as the
outcome confounding bridge function $\hat{h}(W,A,X)$ resulting from the GMM
procedure no longer satisfies (12). In the upcoming sections, we will derive
formulas for the resulting bias in the above LSEM when $Z$ is $U$-relevant,
for the particular cases

  * •

$\alpha_{u2}=\gamma_{u2}=0$ and $\theta_{u2},\mu_{u2}\neq 0$ (section 4.2.1)

  * •

$\theta_{u2}=\mu_{u2}=0$ and $\alpha_{u2},\gamma_{u2}\neq 0$ (section 4.2.2)

The two cases were treated separately for simplicity, but they may be combined
into a general sensitivity analysis in the context where either vector
$(\alpha_{u2},\gamma_{u2})$ or $(\theta_{u2},\mu_{u2})$ has all nonzero
entries (and the two cases are not mutually exclusive).

####  4.2.1 Completeness Violation: Association between Negative Controls
through $U=(U_{1},U_{2})$ (as in Figure 2)

For simplicity, we exclude $X$ from these computations. Let us consider i.i.d.
data generated according to

$\begin{split}U&\sim\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\right)\\\
\textnormal{logit}(\mathbb{P}(A=1|U))&=\alpha_{0}+\alpha_{u1}U_{1}\\\
Z&=\theta_{0}+\theta_{a}A+\boldsymbol{\theta}_{u}^{T}U+\epsilon_{1}\\\
W&=\mu_{0}+\boldsymbol{\mu}_{u}^{T}U+\epsilon_{2}\\\
Y(a)&=\gamma_{0}+\gamma_{a}a+\gamma_{u1}U_{1}+\gamma_{au1}aU_{1}+2\epsilon_{3}\\\
\epsilon_{1},\epsilon_{2},\epsilon_{3}&\sim\mathcal{N}(0,1)\end{split}$ (24)

where $\boldsymbol{\theta}_{u},\boldsymbol{\mu}_{u}$ have all non-zero
entries.

From Theorem 2, we know that the above setup satisfies all assumptions except
7(a). Thus, proceeding to solve for the parameters $b$ of a linear outcome
bridge function (which is the functional form an investigator who was unaware
of $U_{2}$ would select) will lead to a biased estimate of the average
treatment effect, even if the linear bridge function is correctly specified.
The following theorem (see appendix B.1 for a proof) provides a formula for
this bias under a linear outcome bridge function:

######  Theorem 3.

If $(A,Y)\perp\\!\\!\\!\perp U_{2}\ |\ U_{1}$, then fitting a linear outcome
bridge function $h(W,A,X)=b_{0}+b_{a}A+b_{w}W+b_{x}X+b_{ax}AX+b_{aw}AW$ under
LSEM (18) yields a proximal outcome estimator bias equal to

$\begin{split}\delta_{POR}=&\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A](1-\operatorname{\mathbb{E}}[A])}\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}\cdot\\\
&\cdot\left[\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{au1}+\left(\frac{\operatorname{\mathbb{E}}[A]S_{1}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}\right]\end{split}$
(25)

where

$\displaystyle S_{1}$
$\displaystyle=\frac{(1-\operatorname{\mathbb{E}}[A])^{2}}{(1-\operatorname{\mathbb{E}}[A])(1-\operatorname{\mathbb{E}}[AU_{1}^{2}])-\operatorname{\mathbb{E}}[AU_{1}]^{2}}$
$\displaystyle S_{2}$
$\displaystyle=\frac{\operatorname{\mathbb{E}}[A]^{2}}{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]-\operatorname{\mathbb{E}}[AU_{1}]^{2}}$

In particular, for $\gamma_{au1}=0$, the bias can be written as

$\delta_{POR}=\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A](1-\operatorname{\mathbb{E}}[A])}\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}\cdot\left(\frac{\operatorname{\mathbb{E}}[A]S_{1}}{\mu_{u1}+S_{1}\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}$
(26)

In the remaining theoretical analysis of this case, we make the additional
simplifying assumption $\gamma_{au1}=0$. The general case $\gamma_{au1}\neq 0$
will be considered in the numerical experiments of Section 6, but we restrict
ourselves here for clarity.

##### Under setup $\gamma_{au1}=0$:

By comparison, the bias resulting from the non-proximal g-computation
estimator regressing $Y$ onto $(1,Z,W,A,AZ,AW)$ is

$\begin{split}\delta_{OR}=\frac{\left(\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}-\frac{1-\operatorname{\mathbb{E}}[A]}{S_{2}}\theta_{a}\theta_{u1}\right)\left(1+\mu_{u2}^{2}\right)+\left(\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}-\frac{1-\operatorname{\mathbb{E}}[A]}{S_{2}}\theta_{a}\frac{\mu_{u1}\mu_{u2}}{\theta_{u2}}\right)\theta_{u2}^{2}}{\left(1+\frac{1}{S_{2}}\theta_{u1}^{2}\right)\left(1+\mu_{u2}^{2}\right)+\left(1+\frac{1}{S_{2}}\mu_{u1}^{2}\right)\left(1+\theta_{u2}^{2}\right)-\left(1+\frac{2}{S_{2}}\theta_{u1}\mu_{u1}\theta_{u2}\mu_{u2}\right)}\gamma_{u1}\end{split}$
(27)

Additionally, the bias resulting from regressing $Y$ onto $A$ is

$\delta_{unadj}=\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A](1-\operatorname{\mathbb{E}}[A])}\gamma_{u1}$
(28)

The proofs for the non-proximal g-computation biases can be found in Appendix
C.3.

It turns out that, under certain setups $\left(\theta_{u},\mu_{u}\right)$
denoting the strengths of association between $(Z,W)$ and $U$, we are
guaranteed to obtain less bias from the proximal g-computation estimator than
from the unadjusted regression estimator. We formalize these setups in the
following theorem:

######  Theorem 4.

Under setup $\gamma_{au1}=0$, the proximal g-computation bias $\delta_{POR}$
and the unadjusted esimator bias $\delta_{unadj}$ can be compared as follows:

  * (i)

If $\theta_{u1}\mu_{u1}$ and $\theta_{u2}\mu_{u2}$ have the same sign (both
positive or both negative), then
$\left|\delta_{POR}\right|<\left|\delta_{unadj}\right|$.

  * (ii)

If $\theta_{u1}\mu_{u1}$ and $\theta_{u2}\mu_{u2}$ have different signs, then
$\begin{cases}\left|\delta_{POR}\right|>\left|\delta_{unadj}\right|&\textnormal{if
$\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}>-S_{1}(1-\operatorname{\mathbb{E}}[A])-S_{2}\operatorname{\mathbb{E}}[A]$}\\\
\left|\delta_{POR}\right|<\left|\hat{\delta}_{unadj}\right|&\textnormal{if
$\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}<-S_{1}(1-\operatorname{\mathbb{E}}[A])-S_{2}\operatorname{\mathbb{E}}[A]$}\end{cases}$

The proof of Theorem 4 can be found in Appendix C.5.

####  4.2.2 Partial $U$-relevance for Two-Dimensional Unobserved Confounder
$U$ (as in Figure 3)

For simplicity, we exclude $X$ from subsequent computations. Let us consider
i.i.d. data generated according to

$\begin{split}U&\sim\mathcal{N}\left(\begin{pmatrix}0\\\
0\end{pmatrix},\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\right)\\\
\textnormal{logit}(\mathbb{P}(A=1|U))&=\alpha_{0}+\boldsymbol{\alpha}_{u}^{T}U\\\
Z&=\theta_{0}+\theta_{a}A+\theta_{u1}U_{1}+\epsilon_{1}\\\
W&=\mu_{0}+\mu_{u1}U_{1}+\epsilon_{2}\\\
Y(a)&=\gamma_{0}+\gamma_{a}a+\boldsymbol{\gamma}_{u}^{T}U+\gamma_{au1}aU_{1}+\epsilon_{3}\\\
\epsilon_{1},\epsilon_{2},\epsilon_{3}&\sim\mathcal{N}(0,1)\end{split}$ (29)

where $\boldsymbol{\alpha}_{u},\boldsymbol{\gamma}_{u}$ have all non-zero
entries.

From Theorem 2, we know that the above setup violates assumption 7(a). In
addition, we do not have a proof identifying the true outcome confounding
bridge function, so fitting a linear model might also be misspecified. The
following theorem (see Appendix C.1 for a proof) provides a formula for this
bias under a linear bridge function, which can be used in sensitivity
analysis.

######  Theorem 5.

If $(Z,W)\perp\\!\\!\\!\perp U_{2}\ |\ (A,U_{1})$, then fitting a linear
outcome bridge function
$h(W,A,X)=b_{0}+b_{a}A+b_{w}W+b_{x}X+b_{ax}AX+b_{aw}AW$ under LSEM (18) yields
a proximal outcome estimator bias equal to

$\delta_{POR}=-\frac{\left(\operatorname{\mathbb{E}}[AU_{1}^{2}]\left(1-\operatorname{\mathbb{E}}[AU_{1}^{2}]\right)-\operatorname{\mathbb{E}}[AU_{1}]^{2}\right)\operatorname{\mathbb{E}}[AU_{2}]}{\left(\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]-\operatorname{\mathbb{E}}[AU_{1}]^{2}\right)\left(\left(1-\operatorname{\mathbb{E}}[A]\right)\left(1-\operatorname{\mathbb{E}}[AU_{1}^{2}]\right)-\operatorname{\mathbb{E}}[AU_{1}]^{2}\right)}\gamma_{u2}$
(30)

By comparison, the bias resulting from the non-proximal g-computation
estimator regressing $Y$ onto $(1,Z,W,A,AZ,AW)$ can be obtained as in Section
4.2.1, but we omit the formula here due to space constraints and only include
this estimate in numerical experiments.

If we are not considering the proximal estimator, then we may not consider
adjusting for the post-exposure variables $Z$ and $W$. In this case, the bias
resulting from regressing $Y$ onto $A$ is

$\delta_{unadj}=\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)}\gamma_{u1}+\frac{\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]}\gamma_{au1}+\frac{\operatorname{\mathbb{E}}[AU_{2}]}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)}\gamma_{u2}$
(31)

######  Remark 2.

We note that, while the proximal estimator bias formula depends only on
$\gamma_{u2}$, $\alpha_{u2}$ (through $\operatorname{\mathbb{E}}[AU_{2}]$),
and $\alpha_{u1}$ (through $\operatorname{\mathbb{E}}[AU_{1}]$), the
unadjusted estimator bias depends additionally on parameters $\gamma_{u1}$ and
$\gamma_{au1}$ governing the strength of confounding introduced by $U_{1}$ in
the non-proximal case.

##  5 Bias Formulas in Arbitrary Dimension with No Confounder-Treatment
Interaction

We additionally look into a simplified case where $\gamma_{au}=0$ – that is,
the confounder is not an effect modifier. Moreover, we assume that the analyst
is aware of the lack of interaction between $A$ and $U$ in the true outcome
model, so we consider a simplified bridge function model
$h(W,A,X)=b_{0}+b_{a}A+b_{w}^{T}W+b_{x}^{T}X$ as input. This assumption allows
us to more easily obtain bias formulas in the general case of multi-
dimensional $Z,W,U,X$ with
$\left(dim(Z),dim(W),dim(U),dim(X)\right)=(m,n,p,q)$, for certain
relationships between $m,n,p,q$. For simplicity, we assume that the unobserved
and observed confounders $(U,X)$ jointly follow a multivariate normal
distribution with mean $\boldsymbol{0}_{p+q}$, $Var(U_{i})=Var(X_{j})=1$ for
all $i=1,\ldots,p$, $j=1,\ldots,q$ (under a potential transformation), and
some appropriate PSD covariance matrix such that
$Cov(U,X)=\rho\in(-1,1)^{p\times q}$.

Let us consider i.i.d. data generated according to

$\begin{split}\begin{pmatrix}U\\\
X\end{pmatrix}&\sim\mathcal{N}\left(\begin{pmatrix}\boldsymbol{0}_{p}\\\
\boldsymbol{0}_{q}\end{pmatrix},\begin{pmatrix}\boldsymbol{I}_{p}&\rho\\\
\rho^{T}&\boldsymbol{\Sigma}_{x}\end{pmatrix}\right),\
\rho\in\left(-1,1\right)^{p\times q}\\\
\textnormal{logit}(\mathbb{P}(A=1|U,X))&=\alpha_{0}+\alpha_{u}^{T}U+\alpha_{x}^{T}X\\\
Z&=\theta_{0}+\theta_{a}A+\theta_{u}^{T}U+\theta_{x}^{T}X+\epsilon_{1}\\\
W&=\mu_{0}+\mu_{u}^{T}U+\mu_{x}^{T}X+\epsilon_{2}\\\
Y(a)&=\gamma_{0}+\gamma_{a}a+\gamma_{u}^{T}U+\gamma_{x}X+\epsilon_{3}\\\
\epsilon_{1},\epsilon_{2},\epsilon_{3}&\sim\mathcal{N}(0,1)\end{split}$ (32)

The following theorem provides a formula for the proximal identification bias
under a linear bridge function.

######  Theorem 6.

Let
$\operatorname{\mathbb{E}}[AU]=\left(\operatorname{\mathbb{E}}[AU_{1}],\ldots,\operatorname{\mathbb{E}}[AU_{p}]\right)$,
$\operatorname{\mathbb{E}}[AX]=\left(\operatorname{\mathbb{E}}[AX_{1}],\ldots,\operatorname{\mathbb{E}}[AX_{p}]\right)$,
and

$B=\left(I_{p}-\rho\Sigma_{x}^{-1}\rho^{T}-\frac{\left(\operatorname{\mathbb{E}}[AU]-\rho\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]\right)\left(\operatorname{\mathbb{E}}[AU]^{T}-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\rho^{T}\right)}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]}\right)\theta_{u}.$

If $m=n<p$ and matrix $B^{T}\mu_{u}\in\operatorname{\mathbb{R}}^{m\times m}$
has full rank, then fitting a linear outcome bridge function
$h(W,A,X)=b_{0}+b_{a}A+b_{w}^{T}W+b_{x}^{T}X$ under LSEM (32) yields a
proximal outcome estimator bias equal to

$\delta=\frac{\operatorname{\mathbb{E}}[AU]^{T}-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\rho^{T}}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]}\left(I_{p}-\mu_{u}\left(B^{T}\mu_{u}\right)^{-1}B^{T}\right)\gamma_{u}$
(33)

A proof of Theorem 6 can be found in Appendix C.6.

######  Remark 3.

If $m=n=p$ and $B^{T}\mu_{u}$ has full rank, then $\delta=0$. If $p<m$ or
$p<n$, then we have a similar discussion as in [14] where we can either
consider the Moore-Penrose inverse of $B^{T}\mu_{u}$, or reduce the dimensions
of $Z$ and $W$ until they match the dimension of $U$.

######  Remark 4.

Theorem 6 enables sensitivity analysis. Note that the terms
$\operatorname{\mathbb{E}}[A]$ and $\operatorname{\mathbb{E}}[AX]$ in (33) are
straightforwardly estimated from data. Thus, to perform a sensitivity analysis
using the bias formula (33), it remains for the analyst to specify the
parameters $\operatorname{\mathbb{E}}[AU]$ (which is determined by
$\alpha_{u}$), $\mu_{u}$, $\gamma_{u}$, and $\rho$. An analyst could specify a
distribution over these parameters, which, via (33), would imply a
distribution over $\delta$ as each realization of the parameters drawn from
the distribution would correspond to a different bias $\delta$. The range of
magnitudes of the parameters governing the strength of association between $U$
and the other variables might be chosen based on the range of magnitudes of
associations between the observed variables. And probabilities of zero
components in $\theta_{u}$ and $\mu_{u}$ could determine the proportion of
components of $U$ that contribute to bias from $U-$irrelevance.

##  6 Numerical Experiments

In this section, we provide numerical examples to illustrate how the bias of
different estimators (proximal and non-proximal) varies with the strength and
direction of associations between unobserved $U$ and $(A,Y,Z,W)$. Due to the
relatively large number of parameters involved in the bias formulas, we fix
values $\alpha_{0}=\gamma_{0}=\theta_{0}=\mu_{0}=0$ ,
$\theta_{a}=\theta_{u1}=1$, $\mu_{u1}=0.5$, $\gamma_{u1}=1$,
$\gamma_{au1}=1.5$ similar to the simulation DGP in [8]. We then analyze bias
sensitivity to different values of
$(\alpha_{u2},\theta_{u2},\mu_{u2},\gamma_{u2})$, which encode how strongly
the proximal identification assumptions are violated in the presence of
$U_{2}$. The numerical results and plots in this discussion have been
outputted in Mathematica.

###  6.1 Completeness Violation: Association between Negative Controls through
$U=(U_{1},U_{2})$

####  6.1.1 Comparison of estimators for
$\theta_{u1},\theta_{u2},\mu_{u1},\mu_{u2}>0$

This is the case considered in Theorem 4(i), where it is shown that the bias
of the unadjusted estimator always exceeds that of the proximal estimator.
Figures 5-8 illustrate the change in absolute bias for each of the three
estimators. In all figures, we use the following notation:

  * (1)

The solid black curve (“PI”) corresponds to the (absolute) proximal estimator
bias

  * (2)

The dashed curve (“Unadj”) corresponds to the (absolute) unadjusted estimator
bias from regressing $Y$ on $A$

  * (3)

The dot-dashed curve (“OR”) corresponds to the (absolute) adjusted estimator
bias from regressing $Y$ on $(1,Z,W,A,AZ,AW)$

(a) $\mu_{u2}=0.5\theta_{u2}$

(b) $\mu_{u2}=\theta_{u2}$

(c) $\mu_{u2}=1.5\theta_{u2}$

Figure 5: Plots of the ACE estimate bias for $\alpha_{u1}=0.3$.

(a) $\mu_{u2}=0.5\theta_{u2}$

(b) $\mu_{u2}=\theta_{u2}$

(c) $\mu_{u2}=1.5\theta_{u2}$

Figure 6: Plots of the ACE estimate bias for $\alpha_{u1}=0.5$.

(a) $\mu_{u2}=0.5\theta_{u2}$

(b) $\mu_{u2}=\theta_{u2}$

(c) $\mu_{u2}=1.5\theta_{u2}$

Figure 7: Plots of the ACE estimate bias for $\alpha_{u1}=1$.

(a) $\mu_{u2}=0.5\theta_{u2}$

(b) $\mu_{u2}=\theta_{u2}$

(c) $\mu_{u2}=1.5\theta_{u2}$

Figure 8: Plots of the ACE estimate bias for $\alpha_{u1}=-0.3$.

Consistent with Theorem 4 (i), the proximal estimator always outperforms the
unadjusted estimator. However, we also note that there exist settings
$(\alpha_{u1},\theta_{u2},\mu_{u2})$ for which the non-proximal adjusted
estimate is (significantly) less biased than the proximal one. Although
standard criteria for variable adjustment do not usually include post-exposure
covariates in the adjustment set [17], the strong underlying association
between $U$ and the observed proxies $(Z,W)$ might actually help mitigate bias
through adjustment.

####  6.1.2 Comparison of Estimators for Different Directions of Association
Products $\theta_{u1}\mu_{u1}$, $\theta_{u2}\mu_{u2}$

##### Same sign of $\theta_{u1}\mu_{u1}$ and $\theta_{u2}\mu_{u2}$:

Figure 9 illustrates the change in absolute bias for each of the three
estimators relative to the value of $\theta_{u1}$, where it is assumed that
$\mu_{u2}=\theta_{u2}$ in all cases. We observe that the absolute unadjusted
bias is always greater than the proximal estimator bias, which is consistent
with the result in Theorem 4. Comparisons with the adjusted estimator are not
as straightforward, but there seem to exist threshold values of $\alpha_{u1}$
which determine whether the adjusted estimator bias ever exceeds the
unadjusted one (such as in Figures 9(a) and (d)). Moreover, when
$\alpha_{u1}>0$, there exists a threshold value of $|\theta_{u2}|$ which
determines whether the proximal estimator bias exceeds the adjusted bias.

The parameters used in these plots are:
$\alpha_{0}=\gamma_{0}=\theta_{0}=\mu_{0}=0$, $\theta_{a}=\theta_{u1}=1$,
$\gamma_{u1}=1$, $\gamma_{ua1}=1.5$, $\mu_{u1}=0.5$, $\gamma_{a}=0.5$.

(a) $\alpha_{u1}=0.3$, $\mu_{u2}=\theta_{u2}$

(b) $\alpha_{u1}=0.5$, $\mu_{u2}=\theta_{u2}$

(c) $\alpha_{u1}=1$, $\mu_{u2}=\theta_{u2}$

(d) $\alpha_{u1}=-0.5$, $\mu_{u2}=\theta_{u2}$

Figure 9: Plots of the ACE estimate bias for $\theta_{u1}=\mu_{u1}=0.5$.

##### Different signs of $\theta_{u1}\mu_{u1}$ and $\theta_{u2}\mu_{u2}$:

Figure 10 illustrates the change in absolute bias for each of the three
estimators relative to the value of $\theta_{u1}$, where it is assumed that
$\mu_{u2}=\theta_{u2}$ in all cases. We observe that, beyond a certain
threshold in the value of $|\theta_{u2}|$, the proximal estimation bias
exceeds that of the unadjusted estimator (and even the adjusted estimator
bias, for $\alpha_{u1}>0$), which is consistent with Theorem 4.

The parameters used in these plots are:
$\alpha_{0}=\gamma_{0}=\theta_{0}=\mu_{0}=0$, $\theta_{a}=\theta_{u1}=1$,
$\gamma_{u1}=1$, $\gamma_{ua1}=1.5$, $\mu_{u1}=-0.5$, $\gamma_{a}=0.5$.

(a) $\alpha_{u1}=0.3$, $\mu_{u2}=\theta_{u2}$

(b) $\alpha_{u1}=0.5$, $\mu_{u2}=\theta_{u2}$

(c) $\alpha_{u1}=1$, $\mu_{u2}=\theta_{u2}$

(d) $\alpha_{u1}=-0.5$, $\mu_{u2}=\theta_{u2}$

Figure 10: Plots of the ACE estimate bias for $\theta_{u1}=\mu_{u1}=-0.5$.

###  6.2 Partial $U$-relevance for Two-Dimensional Unobserved Confounder $U$

##### Same directions of association $\gamma_{u1},\gamma_{u2}$:

Figure 11 illustrates the change in absolute bias for the proximal and
unadjusted estimators relative to the value of $\alpha_{u2}$.

The parameters used in these plots are:
$\alpha_{0}=\gamma_{0}=\theta_{0}=\mu_{0}=0$, $\theta_{a}=\theta_{u1}=1$,
$\mu_{u1}=1$, $\gamma_{au1}=1$, $\gamma_{u1}=1.5$, $\gamma_{u2}=1$,
$\gamma_{a}=0.5$.

(a) $\alpha_{u1}=0.3$

(b) $\alpha_{u1}=0.5$

(c) $\alpha_{u1}=1$

(d) $\alpha_{u1}=-0.5$

Figure 11: Plots of the ACE estimate bias for $\gamma_{u1}=1.5,\gamma_{u2}>0$.

##### Opposite directions of association $\gamma_{u1},\gamma_{u2}$:

Figure 12 illustrates the change in absolute bias for the proximal and
unadjusted estimators relative to the value of $\alpha_{u2}$.

The parameters used in these plots are:
$\alpha_{0}=\gamma_{0}=\theta_{0}=\mu_{0}=0$, $\theta_{a}=\theta_{u1}=1$,
$\mu_{u1}=1$, $\gamma_{ua1}=1$, $\gamma_{u1}=-1.5$, $\gamma_{u2}=1$,
$\gamma_{a}=0.5$.

(a) $\alpha_{u1}=0.3$

(b) $\alpha_{u1}=0.5$

(c) $\alpha_{u1}=1$

(d) $\alpha_{u1}=-0.5$

Figure 12: Plots of the ACE estimate bias for
$\gamma_{u1}=-1.5,\gamma_{u2}>0$.

The distributions of bias appear almost shifted by translation. We observe a
reversal in which estimator has less bias compared to the case of
$\gamma_{u1},\gamma_{u2}>0$.

##  7 Discussion

By deriving bias formulas for proximal inference estimators under violations
of completeness and $U$-relevance under a LSEM, we begin to gain some insight
into the sensitivity of the proximal inference estimator to these sources of
bias. For example, under some settings, it is possible for completeness
violations alone without any failure of $U$-relevance (i.e. too many common
causes of the negative control exposure and negative control outcome) to lead
the proximal inference estimator to be arbitrarily more biased than an
unadjusted estimator completely subject to unobserved confounding (see Figure
10). However, under a LSEM, if the unobserved confounder leads to a positive
association between the negative control exposure and negative control
outcome, then under a completeness violation with full $U$-relevance the
proximal inference estimator is guaranteed to perform better than an
unadjusted one, no matter how strong the completeness violation (as shown in
Theorem 4). A tentative rule of thumb for the design of proximal inference
studies, pending additional evidence from other data generating processes,
should be to select negative control exposures and outcomes that are
positively associated.

Additionally, as discussed in Remark 4, we can use our bias formula results
(in particular, (33)) to devise schemes for sensitivity analyses of proximal
inference studies. While (33) was derived under the strong assumptions that
the data generating process is a LSEM and $U$ is not an effect modifier, an
analyst might reasonably conduct a sensitivity analysis using (33) as we
described even if they did not believe their data were generated by a LSEM
(and did not construct their proximal inference estimators under that
assumption), and even if they did not believe that $U$ is not an effect
modifier. There is a long history of unrealistic simplifying assumptions in
sensitivity analysis. For example, [11] and [18] both assume a one-dimensional
binary confounder for tractable sensitivity analysis of no unobserved
confounding. Later, [4] developed an approach that made far fewer
restrictions. We are in the early stages of proximal inference, so we
currently need to settle for sensitivity analyses that make strong simplifying
assumptions.

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Appendices

##  Appendix A Bridge Function Parameters for Post-Treatment NCE

###  A.1 Bridge functions derivation for one-dimensional unobserved $U$ – case
of no violations

We identify coefficients $(b_{0},b_{a},b_{x},b_{w},b_{ax},b_{aw})$ and
$(t_{0},t_{a},t_{x},t_{z})$ such that

$\displaystyle\operatorname{\mathbb{E}}[Y|U,a,X]$ $\displaystyle=\int
h(w,a,X)dF(w|U,X),a=0,1$ (34) $\displaystyle\frac{1}{\mathbb{P}[A=a|U,X]}$
$\displaystyle=\int q(z,a,X)dF(z|U,a,X),a=0,1$ (35)

##### Coefficients of $h$:

We have that
$\operatorname{\mathbb{E}}[Y|U,A,X]=\gamma_{0}+\gamma_{a}A+\gamma_{x}X+\gamma_{u}U+\gamma_{au}AU$,
so (34) implies

$\displaystyle\gamma_{0}+\gamma_{a}A+\gamma_{x}X+\gamma_{u}U+\gamma_{au}AU=b_{0}+b_{a}A+b_{x}X+b_{ax}AX+\int(b_{w}+b_{aw}A)w\cdot
dF(w|U,X)\iff$
$\displaystyle\gamma_{0}+\gamma_{a}A+\gamma_{x}X+\gamma_{u}U+\gamma_{au}AU=b_{0}+b_{a}A+b_{x}X+b_{ax}AX+(b_{w}+b_{aw}A)\operatorname{\mathbb{E}}[W|U,X]$

Since $W|U,X\sim\mathcal{N}(\mu_{0}+\mu_{x}X+\mu_{u}U,1)$, we get

$\displaystyle\gamma_{0}+\gamma_{a}A+\gamma_{x}X+\gamma_{u}U+\gamma_{au}AU=b_{0}+b_{a}A+b_{x}X+b_{ax}AX+(b_{w}+b_{aw}A)(\mu_{0}+\mu_{x}X+\mu_{u}U)$

Assigning values $A=0,1$, we obtain the following system

$\displaystyle 0$
$\displaystyle=\gamma_{0}-b_{0}-b_{w}\mu_{0}+(\gamma_{x}-b_{x}-\mu_{x}b_{w})X+(\gamma_{u}-b_{w}\mu_{u})U$
(36)
$\displaystyle\begin{split}0&=(\gamma_{0}+\gamma_{a})-(b_{0}+b_{a})-(b_{w}+b_{aw})\mu_{0}+\\\
&+(\gamma_{x}-b_{x}-b_{ax}-\mu_{x}(b_{w}+b_{aw}))X+(\gamma_{u}+\gamma_{au}-(b_{w}+b_{aw})\mu_{u})U\end{split}$
(37)

Multiplying (36) by $U$ and $X$ and taking the expectation in each resulting
equations yields

$\displaystyle 0$
$\displaystyle=\rho(\gamma_{x}-b_{x}-\mu_{x}b_{w})+(\gamma_{u}-b_{w}\mu_{u})$
$\displaystyle 0$
$\displaystyle=(\gamma_{x}-b_{x}-\mu_{x}b_{w})+\rho(\gamma_{u}-b_{w}\mu_{u})$

Since $\rho\in(-1,1)$, we obtain
$\gamma_{x}-b_{x}-\mu_{x}b_{w}=\gamma_{u}-b_{w}\mu_{u}=0$. From (36), this
additionally implies $\gamma_{0}-b_{0}-b_{w}\mu_{0}=0$.

Similarly, from (37) we obtain
$\gamma_{a}-b_{a}-\mu_{0}b_{aw}=-b_{ax}-\mu_{x}b_{aw}=\gamma_{au}-b_{aw}\mu_{u}=0$.
Solving for the coefficients of $h$, we obtain the unique solution

$(b_{0},b_{a},b_{x},b_{w},b_{ax},b_{aw})=\left(\gamma_{0}-\frac{\mu_{0}\gamma_{u}}{\mu_{u}},\gamma_{a}-\frac{\mu_{0}\gamma_{au}}{\mu_{u}},\gamma_{x}-\frac{\mu_{x}\gamma_{u}}{\mu_{u}},\frac{\gamma_{u}}{\mu_{u}},-\frac{\mu_{x}\gamma_{au}}{\mu_{u}},\frac{\gamma_{au}}{\mu_{u}}\right)$

##### Coefficients of $q$:

We have that
$\mathbb{P}[A|X,U]=\frac{1}{1+\exp\left\\{(-1)^{A}(\alpha_{0}+\alpha_{x}X+\alpha_{u}U)\right\\}}$,
such that (35) implies

$\displaystyle 1+\exp\\{(-1)^{A}(\alpha_{0}+\alpha_{x}X+\alpha_{u}U)\\}=$
$\displaystyle=1+\exp\\{(-1)^{1-A}(t_{0}+t_{a}A+t_{x}X)\\}\int\exp\\{(-1)^{1-A}t_{z}Z\\}dF(z|U,A,X)$

Since
$Z|U,A,X\sim\mathcal{N}(\theta_{0}+\theta_{a}A+\theta_{u}U+\theta_{x}X,1)$, we
get

$\displaystyle\quad 1$
$\displaystyle+\exp\\{(-1)^{A}(\alpha_{0}+\alpha_{x}X+\alpha_{u}U)\\}=$
$\displaystyle=1+\exp\\{(-1)^{1-A}(t_{0}+t_{a}A+t_{x}X)\\}\int\exp\\{(-1)^{1-A}t_{z}Z\\}dF(z|U,A,X)$
$\displaystyle=1+\exp\\{(-1)^{1-A}(t_{0}+t_{a}A+t_{x}X)\\}\int\frac{1}{\sqrt{2\pi}}\exp\\{(-1)^{1-A}t_{z}Z+0.5(Z-\theta_{0}-\theta_{a}A-\theta_{u}U-\theta_{x}X)^{2}\\}$
$\displaystyle=1+\exp\left\\{(-1)^{1-A}(t_{0}+t_{a}A+t_{x}X)+(-1)^{1-A}t_{z}(\theta_{0}+\theta_{a}A+\theta_{u}U+\theta_{x}X)+\frac{t_{z}^{2}}{2}\right\\}$

for each $A=0,1$. This is equivalent to

$\displaystyle(-1)^{A}(\alpha_{0}+\alpha_{x}X+\alpha_{u}U)=(-1)^{1-A}(t_{0}+t_{a}A+t_{x}X)+(-1)^{1-A}t_{z}(\theta_{0}+\theta_{a}A+\theta_{u}U+\theta_{x}X)+0.5t_{z}^{2}$

Assigning values $A=0,1$, we obtain the system

$\displaystyle 0$
$\displaystyle=\alpha_{0}+t_{0}+\theta_{0}t_{z}-0.5t_{z}^{2}+(\alpha_{x}+t_{x}+\theta_{x}t_{z})X+(\alpha_{u}+\theta_{u}t_{z})U$
(38) $\displaystyle 0$
$\displaystyle=\alpha_{0}+(t_{0}+t_{a})+(\theta_{0}+\theta_{a})t_{z}+0.5t_{z}^{2}+(\alpha_{x}+t_{x}+\theta_{x}t_{z})X+(\alpha_{u}+\theta_{u}t_{z})U$
(39)

As in the outcome bridge function case, it follows that the coefficients of 1
(the constant term), $X$, and $U$ must be identically 0. We then obtain
$\alpha_{0}+t_{0}+\theta_{0}t_{z}-0.5t_{z}^{2}=t_{a}+\theta_{a}t_{z}+t_{z}^{2}=\alpha_{x}+t_{x}+\theta_{x}t_{z}=\alpha_{u}+\theta_{u}t_{z}=0$,
which yields the unique solution

$\displaystyle(t_{0},t_{a},t_{x},t_{z})=\left(-\alpha_{0}+\frac{\theta_{0}}{\theta_{u}}\alpha_{u}+\frac{0.5}{\theta_{u}^{2}}\alpha_{u}^{2},-\frac{1}{\theta_{u}^{2}}\alpha_{u}^{2}+\frac{\theta_{a}}{\theta_{u}}\alpha_{u},\frac{\theta_{x}}{\theta_{u}}\alpha_{u}-\alpha_{x},-\frac{\alpha_{u}}{\theta_{u}}\right)$

##  Appendix B Proving violations of Completeness Assumption 7(a)

###  B.1 Violation of Assumption 7(a) under setup (18)

We will prove that completeness assumption 7(a) is violated under the DGP (18)
with
$\boldsymbol{\theta}_{u}=\begin{pmatrix}\theta_{u1}&\theta_{u2}\end{pmatrix}^{T}$,
$\theta_{u1}\neq 0$. We note that case $\theta_{u2}\neq 0$ can be treated
symmetrically, by appropriately exchanging $u1$ and $u2$ in the following
computations.

For any values $u,z,a,x$, we have that

$\displaystyle\mathbb{P}[U=u|Z=z,A=a,X=x]$
$\displaystyle=\frac{\mathbb{P}[U=u,Z=z|A=a,X=x]}{\mathbb{P}[Z=z|A=a,X=x]}$
$\displaystyle=\frac{\mathbb{P}[Z=z|U=u,A=a,X=x]\mathbb{P}[U=u|A=a,X=x]}{\mathbb{P}[Z=z|A=a,X=x]}$
$\displaystyle=\frac{\mathbb{P}[\epsilon_{1}=z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u]\frac{\mathbb{P}[A=a|U=u,X=x]\mathbb{P}[U=u|X=x]}{\mathbb{P}[A=a|X=x]}}{\mathbb{P}[Z=z|A=a,X=x]}$

Using

$\displaystyle\mathbb{P}[U=u|X=x]=\frac{\exp\left(\frac{\left(\rho_{2}u_{1}-\rho_{1}u_{2}\right)^{2}-\left(u_{2}-\rho_{2}x\right)^{2}-\left(u_{1}-\rho_{1}x\right)^{2}}{2\left(1-\rho_{1}^{2}-\rho_{2}^{2}\right)}\right)}{2\pi\sqrt{1-\rho_{1}^{2}-\rho_{2}^{2}}}$

we get

$\displaystyle\mathbb{P}[U=u|Z=z,A=a,X=x]=\frac{\frac{1}{\sqrt{2\pi}}\exp\left\\{-0.5\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u\right)^{2}\right\\}}{\mathbb{P}[Z=z,A=a|X=x]\left(1+\exp\left\\{\left(-1\right)^{a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right\\}\right)}\cdot$
$\displaystyle\cdot\frac{1}{2\pi\sqrt{1-\rho_{1}^{2}-\rho_{2}^{2}}}\exp\left\\{\frac{\left(\rho_{2}u_{1}-\rho_{1}u_{2}\right)^{2}-\left(u_{2}-\rho_{2}x\right)^{2}-\left(u_{1}-\rho_{1}x\right)^{2}}{2\left(1-\rho_{1}^{2}-\rho_{2}^{2}\right)}\right\\}=$
$\displaystyle=\frac{\exp\left\\{\frac{1}{2}\frac{\left(\rho_{2}u_{1}-\rho_{1}u_{2}\right)^{2}-\left(u_{2}-\rho_{2}x\right)^{2}-\left(u_{1}-\rho_{1}x\right)^{2}}{\left(1-\rho_{1}^{2}-\rho_{2}^{2}\right)}-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u\right)^{2}\right\\}}{\left(2\pi\right)^{3/2}\sqrt{1-\rho_{1}^{2}-\rho_{2}^{2}}\mathbb{P}[Z=z,A=a|X=x]\left(1+\exp\left\\{\left(-1\right)^{a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right\\}\right)}$

Let us consider

$\begin{split}g(U)&=u_{2}\left(u_{2}^{2}-3-\alpha_{u2}^{2}-\frac{\alpha_{u1}^{2}\theta_{u2}^{2}}{\theta_{u1}^{2}}+\frac{2\alpha_{u1}\alpha_{u2}\theta_{u2}}{\theta_{u1}}\right)\exp\left(-\frac{u_{2}^{2}}{2}\right)\cdot\\\
&\cdot\exp\left(-\frac{\left(\rho_{2}u_{1}-\rho_{1}u_{2}\right)^{2}-\left(u_{2}-\rho_{2}x\right)^{2}-\left(u_{1}-\rho_{1}x\right)^{2}}{2\left(1-\rho_{1}^{2}-\rho_{2}^{2}\right)}\right)\cdot\\\
&\cdot\left(2+\exp\left(-\alpha_{0}-\alpha_{x}x-\alpha_{u}^{T}u\right)+\exp\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right)\end{split}$
(40)

We will prove that $\operatorname{\mathbb{E}}[g(U)|Z=z,A=a,X=x]=0$ for any
values $z,a,x$. We have

$\displaystyle\operatorname{\mathbb{E}}[g(U)|Z=z,A=a,X=x]=\int_{(-\infty,\infty)^{2}}g(u)\mathbb{P}[U=u|Z=z,A=a,X=x]du_{1}du_{2}$
$\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}\sqrt{1-\rho_{1}^{2}-\rho_{2}^{2}}\mathbb{P}[Z=z,A=a|X=x]}\int_{(-\infty,\infty)^{2}}u_{2}\left(u_{2}^{2}-3-\alpha_{u2}^{2}-\frac{\alpha_{u1}^{2}\theta_{u2}^{2}}{\theta_{u1}^{2}}+\frac{2\alpha_{u1}\alpha_{u2}\theta_{u2}}{\theta_{u1}}\right)\cdot$
$\displaystyle\cdot\exp\left\\{-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u\right)^{2}-\frac{u_{2}^{2}}{2}\right\\}\left(1+\exp\left\\{\left(-1\right)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right\\}\right)du_{1}du_{2}$

Let

$\displaystyle T_{1}$
$\displaystyle=\int_{(-\infty,\infty)^{2}}u_{2}\exp\left\\{-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u\right)^{2}-\frac{1}{2}u_{2}^{2}\right\\}\cdot$
$\displaystyle\cdot\left(1+\exp\left\\{\left(-1\right)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right\\}\right)du_{1}du_{2}$
$\displaystyle T_{2}$
$\displaystyle=\int_{(-\infty,\infty)^{2}}u_{2}^{3}\exp\left\\{-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u\right)^{2}-\frac{1}{2}u_{2}^{2}\right\\}\cdot$
$\displaystyle\cdot\left(1+\exp\left\\{\left(-1\right)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right\\}\right)du_{1}du_{2}$

We have that

$\displaystyle\int_{-\infty}^{\infty}\exp\left\\{-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u}^{T}u\right)^{2}\right\\}\left(1+\exp\left\\{\left(-1\right)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u}^{T}u\right)\right\\}\right)du_{1}=$
$\displaystyle=\exp\left\\{-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}\right)^{2}\right\\}\left(\int_{-\infty}^{\infty}\exp\left\\{\theta_{u1}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}\right)u_{1}-\frac{1}{2}\theta_{u1}^{2}u_{1}^{2}\right\\}du_{1}\right.+$
$\displaystyle\left.+\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u2}u_{2}\right)\right\\}\cdot\right.$
$\displaystyle\left.\cdot\int_{-\infty}^{\infty}\exp\left\\{\theta_{u1}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}-\frac{(-1)^{a}\alpha_{u1}}{\theta_{u1}}\right)u_{1}-\frac{1}{2}\theta_{u1}^{2}u_{1}^{2}\right\\}du_{1}\right)=$
$\displaystyle=\exp\left\\{-\frac{1}{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}\right)^{2}\right\\}\left(\frac{\sqrt{2\pi}}{|\theta_{u1}|}\exp\left\\{\frac{\theta_{u1}^{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}\right)^{2}}{2\theta_{u1}^{2}}\right\\}\right.+$
$\displaystyle+\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u2}u_{2}\right)\right\\}\cdot\left.\frac{\sqrt{2\pi}}{|\theta_{u1}|}\exp\left\\{\frac{\theta_{u1}^{2}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}-\frac{(-1)^{a}\alpha_{u1}}{\theta_{u1}}\right)^{2}}{2\theta_{u1}^{2}}\right\\}\right)=$
$\displaystyle=\frac{\sqrt{2\pi}}{|\theta_{u1}|}\left(1+\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\alpha_{u2}u_{2}+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x-\theta_{u2}u_{2}\right)\right)+\frac{\alpha_{u1}^{2}}{2\theta_{u1}^{2}}\right\\}\right)=$
$\displaystyle=\frac{\sqrt{2\pi}}{|\theta_{u1}|}\left(1+\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)+\frac{\alpha_{u1}^{2}}{2\theta_{u1}^{2}}+(-1)^{a}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)u_{2}\right\\}\right)$

which implies

$\displaystyle
T_{1}=\frac{\sqrt{2\pi}}{|\theta_{u1}|}\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)+\frac{\alpha_{u1}^{2}}{2\theta_{u1}^{2}}\right\\}\cdot$
$\displaystyle\cdot\int_{-\infty}^{\infty}u_{2}\exp\left\\{(-1)^{a}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)u_{2}-\frac{1}{2}u_{2}^{2}\right\\}du_{2}=$
$\displaystyle=\frac{\sqrt{2\pi}}{|\theta_{u1}|}\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)+\frac{\alpha_{u1}^{2}}{2\theta_{u1}^{2}}\right\\}\cdot$
$\displaystyle\cdot\sqrt{2\pi}(-1)^{a}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)\exp\left\\{\frac{1}{2}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)^{2}\right\\}=$
$\displaystyle=\frac{2\pi(-1)^{a}}{|\theta_{u1}|}\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)\right.+$
$\displaystyle+\left.\frac{\alpha_{u1}^{2}(1+\theta_{u2}^{2})}{2\theta_{u1}^{2}}-\frac{\theta_{u1}\alpha_{u1}\alpha_{u2}}{\theta_{u1}}-\frac{1}{2}\alpha_{u2}^{2}\right\\}\cdot\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)$

and

$\displaystyle
T_{2}=\frac{\sqrt{2\pi}}{|\theta_{u1}|}\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)+\frac{\alpha_{u1}^{2}}{2\theta_{u1}^{2}}\right\\}\cdot$
$\displaystyle\cdot\int_{-\infty}^{\infty}u_{2}^{3}\exp\left\\{(-1)^{a}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)u_{2}-\frac{1}{2}u_{2}^{2}\right\\}du_{2}=$
$\displaystyle=\frac{\sqrt{2\pi}}{|\theta_{u1}|}\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)+\frac{\alpha_{u1}^{2}}{2\theta_{u1}^{2}}\right\\}\cdot$
$\displaystyle\cdot\sqrt{2\pi}(-1)^{a}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)\left[3+\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)^{2}\right]\exp\left\\{\frac{1}{2}\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)^{2}\right\\}=$
$\displaystyle=\frac{2\pi(-1)^{a}}{|\theta_{u1}|}\exp\left\\{(-1)^{1-a}\left(\alpha_{0}+\alpha_{x}x+\frac{\alpha_{u1}}{\theta_{u1}}\left(Z-\theta_{0}-\theta_{a}a-\theta_{x}x\right)\right)\right.+$
$\displaystyle+\left.\frac{\alpha_{u1}^{2}(1+\theta_{u2}^{2})}{2\theta_{u1}^{2}}-\frac{\theta_{u1}\alpha_{u1}\alpha_{u2}}{\theta_{u1}}-\frac{1}{2}\alpha_{u2}^{2}\right\\}\cdot\left(\frac{\alpha_{u1}\theta_{u2}}{\theta_{u1}}-\alpha_{u2}\right)\left(3+\alpha_{u2}^{2}+\frac{\alpha_{u1}^{2}\theta_{u2}^{2}}{\theta_{u1}^{2}}-\frac{2\alpha_{u1}\alpha_{u2}\theta_{u2}}{\theta_{u1}}\right)$

using the fact that
$\int_{-\infty}^{\infty}u_{2}\exp\left\\{-\frac{1}{2}u_{2}^{2}\right\\}du_{2}=0$
and
$\int_{-\infty}^{\infty}u_{2}^{3}\exp\left\\{-\frac{1}{2}u_{2}^{2}\right\\}du_{2}=0$
(as integrals of odd functions). We then obtain

$\displaystyle\operatorname{\mathbb{E}}[g(U)|Z=z,A=a,X=x]=\frac{1}{\left(2\pi\right)^{3/2}\sqrt{1-\rho_{1}^{2}-\rho_{2}^{2}}\mathbb{P}[Z=z,A=a|X=x]}\cdot$
$\displaystyle\cdot\left(T_{2}-\left(3+\alpha_{u2}^{2}+\frac{\alpha_{u1}^{2}\theta_{u2}^{2}}{\theta_{u1}^{2}}-\frac{2\alpha_{u1}\alpha_{u2}\theta_{u2}}{\theta_{u1}}\right)T_{1}\right)=0$

for any $z,a,x$. However, we clearly do not have $g(U)\equiv 0$ a.s., so
completeness assumption 7(a) does not hold.

##  Appendix C Bias computations

###  C.1 Computing the (asymptotic) bias obtained through Method of Moments
estimator under setup (29)

We will compute the asymptotic bias obtained from the method of moments solver
using bridge function $h(W,A,0;b)=b_{0}+b_{a}A+b_{w}W+b_{aw}AW$ and vector
function $Q(A,Z,0)=(1,A,Z,AZ)^{T}$.

We define the moment restrictions
$H(D_{i};\theta)=\begin{Bmatrix}\\{Y_{i}-h(W_{i},A_{i},0;b)\\}\times
Q(A_{i},Z_{i},0)\\\ \Delta-(h(W_{i},1,0;b)-h(W_{i},0,0;b))\end{Bmatrix}$ and
let
$m(\theta)=\operatorname{\mathbb{E}}[H(D;\theta)]=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}h(D_{i};\theta)$.
The estimate of $\theta=(b,\Delta)$ is given by

$\hat{\theta}=\operatorname*{arg\,min}_{\theta}m^{T}(\theta)m(\theta)$

Using $\operatorname{\mathbb{E}}[U_{1}]=\operatorname{\mathbb{E}}[U_{2}]=0$,
$\operatorname{\mathbb{E}}[U_{1}^{2}]=\operatorname{\mathbb{E}}[U_{2}^{2}]=1$,
and $\operatorname{\mathbb{E}}[U_{1}U_{2}]=0$, we express the coordinates of
$\operatorname{\mathbb{E}}[h(D;\theta)]=(m_{1},m_{2},m_{3},m_{4},m_{5})$ as
follows:

$\begin{split}m_{1}&=-b_{0}-\operatorname{\mathbb{E}}[A]b_{a}-\mu_{0}b_{w}-(\operatorname{\mathbb{E}}[A]\mu_{0}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1})b_{aw}+\\\
&+\gamma_{0}+\operatorname{\mathbb{E}}[A]\gamma_{a}+\operatorname{\mathbb{E}}[AU_{1}]\gamma_{au1}\end{split}$
(41)
$\begin{split}m_{2}&=-\operatorname{\mathbb{E}}[A]b_{0}-\operatorname{\mathbb{E}}[A]b_{a}-(\operatorname{\mathbb{E}}[A]\mu_{0}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1})b_{w}-(\operatorname{\mathbb{E}}[A]\mu_{0}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1})b_{aw}+\\\
&+\left(\operatorname{\mathbb{E}}[A](\gamma_{0}+\gamma_{a})+\operatorname{\mathbb{E}}[AU_{1}](\gamma_{u1}+\gamma_{au1})\right)\end{split}$
(42)

$\begin{split}m_{3}&=-(\theta_{0}+\operatorname{\mathbb{E}}[A]\theta_{a})b_{0}-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU_{1}]\theta_{u1}\right)b_{a}-\\\
&-(\mu_{0}\theta_{0}+\mu_{u1}\theta_{u1}+\operatorname{\mathbb{E}}[A]\mu_{0}\theta_{a}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1}\theta_{a})b_{w}-\\\
&-\left(\operatorname{\mathbb{E}}[A]\mu_{0}(\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU_{1}]\left(\mu_{0}\theta_{u1}+\mu_{u1}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU_{1}^{2}]\mu_{u1}\theta_{u1}\right)b_{aw}+\\\
&+\gamma_{0}\theta_{0}+\gamma_{u1}\theta_{u1}+\operatorname{\mathbb{E}}[A]\left(\gamma_{0}\theta_{a}+\gamma_{a}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU_{1}]\left(\gamma_{a}\theta_{u1}+\gamma_{u1}\theta_{a}+\gamma_{au1}(\theta_{0}+\theta_{a})\right)+\\\
&+\operatorname{\mathbb{E}}[AU_{1}^{2}]\gamma_{au1}\theta_{u1}+\operatorname{\mathbb{E}}[AU_{2}]\gamma_{u2}\theta_{a}\end{split}$
(43)

$\begin{split}m_{4}&=-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU]\theta_{u1}\right)b_{0}-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU]\theta_{u1}\right)b_{a}-\\\
&-\left(\operatorname{\mathbb{E}}[A]\mu_{0}(\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU]\left(\mu_{0}\theta_{u1}+\mu_{u1}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU^{2}]\mu_{u1}\theta_{u1}\right)b_{w}-\\\
&-\left(\operatorname{\mathbb{E}}[A]\mu_{0}(\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU]\left(\mu_{0}\theta_{u1}+\mu_{u1}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU^{2}]\mu_{u1}\theta_{u1}\right)b_{aw}-\\\
&+\operatorname{\mathbb{E}}[A](\gamma_{0}+\gamma_{a})(\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU]\left(\gamma_{u1}\theta_{a}+\gamma_{au1}(\theta_{0}+\theta_{a})\right)+\\\
&+\operatorname{\mathbb{E}}[AU^{2}]\gamma_{au1}\theta_{u1}+\operatorname{\mathbb{E}}[AU_{2}]\gamma_{u2}\theta_{a}\end{split}$
(44)

Let

$\displaystyle R_{1}$
$\displaystyle=\frac{\operatorname{\mathbb{E}}[AU_{1}]\operatorname{\mathbb{E}}[AU_{2}]}{(1-\operatorname{\mathbb{E}}[A])(1-\operatorname{\mathbb{E}}[AU_{1}^{2}])-\operatorname{\mathbb{E}}[AU_{1}]^{2}}$
(45) $\displaystyle R_{2}$
$\displaystyle=\frac{1-\operatorname{\mathbb{E}}[A]-\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]-\operatorname{\mathbb{E}}[AU_{1}]^{2}}$
(46)

We obtain the estimated bridge function parameters

$\begin{split}\hat{b}_{0}&=\gamma_{0}+\frac{\mu_{0}}{\mu_{u1}}\gamma_{u1}+\left(\frac{\mu_{0}}{\mu_{u1}}-\frac{1-\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}\right)R_{1}\cdot\gamma_{u2}\\\
\hat{b}_{a}&=\gamma_{a}-\frac{\mu_{0}}{\mu_{u1}}\gamma_{au1}-\left(\frac{\mu_{0}}{\mu_{u1}}+\frac{\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}+\frac{\frac{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}-\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]-\operatorname{\mathbb{E}}[AU_{1}^{2}]}\right)R_{1}R_{2}\cdot\gamma_{u2}\\\
\hat{b}_{w}&=\frac{1}{\mu_{u1}}+\frac{R_{1}}{\mu_{u1}}\gamma_{u2}\\\
\hat{b}_{aw}&=\frac{1}{\mu_{u1}}\gamma_{au1}+\frac{R_{1}R_{2}}{\mu_{u1}}\gamma_{u2}\end{split}$
(47)

The estimated effect resulting from $\hat{h}(W,A,0;b)$ is then

$\displaystyle\hat{\Delta}$
$\displaystyle=\hat{b}_{a}+\hat{b}_{aw}\operatorname{\mathbb{E}}[W]=\hat{b}_{a}+\hat{b}_{aw}\mu_{0}=$
$\displaystyle=\gamma_{a}-\frac{\mu_{0}}{\mu_{u1}}\gamma_{au1}-\left(\frac{\mu_{0}}{\mu_{u1}}+\frac{\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}+\frac{\frac{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}-\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]-\operatorname{\mathbb{E}}[AU_{1}^{2}]}\right)R_{1}R_{2}\cdot\gamma_{u2}+\frac{\mu_{0}}{\mu_{u1}}\gamma_{au1}+\frac{\mu_{0}}{\mu_{u1}}R_{1}R_{2}\gamma_{u2}$
$\displaystyle=\gamma_{a}-\left(\frac{\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}+\frac{\frac{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}-\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]-\operatorname{\mathbb{E}}[AU_{1}^{2}]}\right)R_{1}R_{2}\cdot\gamma_{u2}$

which yields a bias equal to

$\delta=-\frac{\operatorname{\mathbb{E}}[AU_{1}^{2}]}{\operatorname{\mathbb{E}}[AU_{1}]}\left(1+\frac{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]-\operatorname{\mathbb{E}}[AU_{1}]^{2}}{\operatorname{\mathbb{E}}[AU_{1}^{2}]\left(1-\operatorname{\mathbb{E}}[A]-\operatorname{\mathbb{E}}[AU_{1}^{2}]\right)}\right)R_{1}R_{2}\cdot\gamma_{u2}$
(48)

We note that the expectations

$\begin{split}\operatorname{\mathbb{E}}[A]&=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[A|U_{1},U_{2}]]=\operatorname{\mathbb{E}}\left[\mathbb{P}[A=1|U_{1},U_{2}]\right]=\\\
&=\operatorname{\mathbb{E}}\left[\frac{1}{1+\exp\\{-\alpha_{0}-\alpha_{u1}U_{1}-\alpha_{u2}U_{2}\\}}\right]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\frac{1}{2\pi}\exp\left\\{-\frac{u^{2}+v^{2}}{2}\right\\}dudv}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u-\alpha_{u2}v\\}}\end{split}$
(49)
$\begin{split}\operatorname{\mathbb{E}}[AU_{1}]&=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[AU_{1}|U_{1},U_{2}]]=\operatorname{\mathbb{E}}[U_{1}\operatorname{\mathbb{E}}[A|U_{1},U_{2}]]=\\\
&=\operatorname{\mathbb{E}}\left[\frac{U_{1}}{1+\exp\\{-\alpha_{0}-\alpha_{u1}U_{1}-\alpha_{u2}U_{2}\\}}\right]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\frac{1}{2\pi}u\exp\left\\{-\frac{u^{2}+v^{2}}{2}\right\\}dudv}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u-\alpha_{u2}v\\}}\end{split}$
(50)
$\begin{split}\operatorname{\mathbb{E}}[AU_{2}]&=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[AU_{2}|U_{1},U_{2}]]=\operatorname{\mathbb{E}}[U_{2}\operatorname{\mathbb{E}}[A|U_{1},U_{2}]]=\\\
&=\operatorname{\mathbb{E}}\left[\frac{U_{2}}{1+\exp\\{-\alpha_{0}-\alpha_{u1}U_{1}-\alpha_{u2}U_{2}\\}}\right]=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\frac{1}{2\pi}v\exp\left\\{-\frac{u^{2}+v^{2}}{2}\right\\}dudv}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u-\alpha_{u2}v\\}}\end{split}$
(51)
$\begin{split}\operatorname{\mathbb{E}}\left[AU_{1}^{2}\right]&=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[AU_{1}^{2}|U_{1},U_{2}\right]\right]=\operatorname{\mathbb{E}}\left[U_{1}^{2}\operatorname{\mathbb{E}}\left[A|U_{1},U_{2}\right]\right]=\\\
&=\operatorname{\mathbb{E}}\left[\frac{U_{1}^{2}}{1+\exp\\{-\alpha_{0}-\alpha_{u1}U_{1}-\alpha_{u2}U_{2}\\}}\right]=\int_{-\infty}^{\infty}\frac{\frac{1}{2\pi}u^{2}\exp\left\\{-\frac{u^{2}+v^{2}}{2}\right\\}dudv}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u-\alpha_{u2}v\\}}\end{split}$
(52)

cannot be computed in closed form but can be obtained numerically using a
software like Mathematica or Maple once we provide the values of $\alpha_{0}$
and $\alpha_{u}$.

###  C.2 Computing the (asymptotic) bias obtained through Method of Moments
estimator under setup (24)

We will compute the asymptotic bias obtained from the method of moments solver
using bridge function $h(W,A,0;b)=b_{0}+b_{a}A+b_{w}W+b_{aw}AW$ and vector
function $Q(A,Z,0)=(1,A,Z,AZ)^{T}$.

We define the moment restrictions
$H(D_{i};\theta)=\begin{Bmatrix}\\{Y_{i}-h(W_{i},A_{i},0;b)\\}\times
Q(A_{i},Z_{i},0)\\\ \Delta-(h(W_{i},1,0;b)-h(W_{i},0,0;b))\end{Bmatrix}$ and
let
$m(\theta)=\operatorname{\mathbb{E}}[H(D;\theta)]=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}h(D_{i};\theta)$.
The estimate of $\theta=(b,\Delta)$ is given by

$\hat{\theta}=\operatorname*{arg\,min}_{\theta}m^{T}(\theta)m(\theta)$

Using $\operatorname{\mathbb{E}}[U_{1}]=\operatorname{\mathbb{E}}[U_{2}]=0$,
$\operatorname{\mathbb{E}}[U_{1}^{2}]=\operatorname{\mathbb{E}}[U_{2}^{2}]=1$,
and $\operatorname{\mathbb{E}}[U_{1}U_{2}]=0$, we express the coordinates of
$\operatorname{\mathbb{E}}[h(D;\theta)]=(m_{1},m_{2},m_{3},m_{4},m_{5})$ as
follows:

$\displaystyle m_{1}$
$\displaystyle=-b_{0}-\operatorname{\mathbb{E}}[A]b_{a}-\mu_{0}b_{w}-(\operatorname{\mathbb{E}}[A]\mu_{0}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1})b_{aw}+\gamma_{0}+\operatorname{\mathbb{E}}[A]\gamma_{a}+\operatorname{\mathbb{E}}[AU_{1}]\gamma_{au1}$
$\displaystyle m_{2}$
$\displaystyle=-\operatorname{\mathbb{E}}[A]b_{0}-\operatorname{\mathbb{E}}[A]b_{a}-(\operatorname{\mathbb{E}}[A]\mu_{0}+\operatorname{\mathbb{E}}[AU]\mu_{u1})b_{w}-(\operatorname{\mathbb{E}}[A]\mu_{0}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1})b_{aw}+$
$\displaystyle+\left(\operatorname{\mathbb{E}}[A](\gamma_{0}+\gamma_{a})+\operatorname{\mathbb{E}}[AU_{1}](\gamma_{u1}+\gamma_{au1})\right)$
$\displaystyle m_{3}$
$\displaystyle=-(\theta_{0}+\operatorname{\mathbb{E}}[A]\theta_{a})b_{0}-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU_{1}]\theta_{u1}\right)b_{a}-$
$\displaystyle-(\mu_{0}\theta_{0}+\mu_{u1}\theta_{u1}+\mu_{u2}\theta_{u2}+\operatorname{\mathbb{E}}[A]\mu_{0}\theta_{a}+\operatorname{\mathbb{E}}[AU_{1}]\mu_{u1}\theta_{a})b_{w}-$
$\displaystyle-\left(\operatorname{\mathbb{E}}[A]\left(\mu_{0}(\theta_{0}+\theta_{a})+\mu_{u2}\theta_{u2}\right)+\operatorname{\mathbb{E}}[AU_{1}]\left(\mu_{0}\theta_{u1}+\mu_{u1}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU_{1}^{2}]\mu_{u1}\theta_{u1}\right)b_{aw}+$
$\displaystyle+\gamma_{0}\theta_{0}+\gamma_{u1}\theta_{u1}+\operatorname{\mathbb{E}}[A]\left(\gamma_{0}\theta_{a}+\gamma_{a}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU_{1}]\left(\gamma_{a}\theta_{u1}+\gamma_{u1}\theta_{a}+\gamma_{au1}(\theta_{0}+\theta_{a})\right)+$
$\displaystyle+\operatorname{\mathbb{E}}[AU_{1}^{2}]\gamma_{au1}\theta_{u1}$

$\begin{split}m_{4}&=-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU]\theta_{u1}\right)b_{0}-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU_{1}]\theta_{u1}\right)b_{a}-\\\
&-\left(\operatorname{\mathbb{E}}[A]\left(\mu_{0}(\theta_{0}+\theta_{a})+\mu_{u2}\theta_{u2}\right)+\operatorname{\mathbb{E}}[AU_{1}]\left(\mu_{0}\theta_{u1}+\mu_{u1}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU_{1}^{2}]\mu_{u1}\theta_{u1}\right)b_{w}-\\\
&-\left(\operatorname{\mathbb{E}}[A]\left(\mu_{0}(\theta_{0}+\theta_{a})+\mu_{u2}\theta_{u2}\right)+\operatorname{\mathbb{E}}[AU_{1}]\left(\mu_{0}\theta_{u1}+\mu_{u1}(\theta_{0}+\theta_{a})\right)+\operatorname{\mathbb{E}}[AU_{1}^{2}]\mu_{u1}\theta_{u1}\right)b_{aw}-\\\
&+\operatorname{\mathbb{E}}[A](\gamma_{0}+\gamma_{a})(\theta_{0}+\theta_{a})+\operatorname{\mathbb{E}}[AU_{1}]\left(\gamma_{u1}\theta_{a}+\gamma_{au1}(\theta_{0}+\theta_{a})\right)+\\\
&+\operatorname{\mathbb{E}}[AU_{1}^{2}]\gamma_{au1}\theta_{u1}\end{split}$

Let

$\displaystyle S_{1}$
$\displaystyle=\frac{(1-\operatorname{\mathbb{E}}[A])^{2}}{(1-\operatorname{\mathbb{E}}[A])(1-\operatorname{\mathbb{E}}[AU_{1}^{2}])-\operatorname{\mathbb{E}}[AU_{1}]^{2}}$
$\displaystyle S_{2}$
$\displaystyle=\frac{\operatorname{\mathbb{E}}[A]^{2}}{\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU_{1}^{2}]-\operatorname{\mathbb{E}}[AU_{1}]^{2}}$

We obtain the estimated bridge function parameters

$\begin{split}\hat{b}_{0}&=\gamma_{0}-\left(\frac{\mu_{0}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{S_{1}}{1-\operatorname{\mathbb{E}}[A]}\cdot\frac{\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}\\\
\hat{b}_{a}&=\gamma_{a}-\frac{\mu_{0}-\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{au1}+\left(\frac{\mu_{0}+\frac{\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]}S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}-\frac{\mu_{0}-\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}\\\
\hat{b}_{w}&=\frac{1}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{u1}\\\
\hat{b}_{aw}&=\frac{1}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{au1}-\left(\frac{1}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}-\frac{1}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}\end{split}$
(53)

The estimated effect resulting from $\hat{h}(W,A,0;b)$ is then

$\displaystyle\hat{\Delta}$
$\displaystyle=\hat{b}_{a}+\hat{b}_{aw}\operatorname{\mathbb{E}}[W]=\hat{b}_{a}+\hat{b}_{aw}\mu_{0}=$
$\displaystyle=\gamma_{a}+\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{au1}+\left(\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]}S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}=$
$\displaystyle=\gamma_{a}+\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A](1-\operatorname{\mathbb{E}}[A])}\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}\left[\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{au1}+\left(\frac{\operatorname{\mathbb{E}}[A]S_{1}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}\right]$

which yields a bias equal to

$\delta=\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A](1-\operatorname{\mathbb{E}}[A])}\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}\left[\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\gamma_{au1}+\left(\frac{\operatorname{\mathbb{E}}[A]S_{1}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}\right]$
(54)

In the particular case $\gamma_{au1}=0$, we obtain a bias equal to

$\delta=\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A](1-\operatorname{\mathbb{E}}[A])}\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}\left(\frac{\operatorname{\mathbb{E}}[A]S_{1}}{\mu_{u1}+S_{1}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}+\frac{(1-\operatorname{\mathbb{E}}[A])S_{2}}{\mu_{u1}+S_{2}\cdot\frac{\theta_{u2}}{\theta_{u1}}\mu_{u2}}\right)\gamma_{u1}$
(55)

Similarly to Proof C.1, we note that the expectations

$\displaystyle\operatorname{\mathbb{E}}[A]$
$\displaystyle=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[A|U_{1}]]=\operatorname{\mathbb{E}}\left[\mathbb{P}[A=1|U_{1}]\right]=$
$\displaystyle=\operatorname{\mathbb{E}}\left[\frac{1}{1+\exp\\{-\alpha_{0}-\alpha_{u1}U_{1}\\}}\right]=\int_{-\infty}^{\infty}\frac{\frac{1}{\sqrt{2\pi}}\exp\left\\{-\frac{u^{2}}{2}\right\\}}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u\\}}du$
$\displaystyle\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle=\operatorname{\mathbb{E}}[\operatorname{\mathbb{E}}[AU_{1}|U_{1}]]=\operatorname{\mathbb{E}}[U_{1}\operatorname{\mathbb{E}}[A|U_{1}]]=$
$\displaystyle=\operatorname{\mathbb{E}}\left[\frac{U_{1}}{1+\exp\\{-\alpha_{0}-\alpha_{u}U_{1}\\}}\right]=\int_{-\infty}^{\infty}\frac{\frac{1}{\sqrt{2\pi}}u\exp\left\\{-\frac{u^{2}}{2}\right\\}}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u\\}}du$
$\displaystyle\operatorname{\mathbb{E}}\left[AU_{1}^{2}\right]$
$\displaystyle=\operatorname{\mathbb{E}}\left[\operatorname{\mathbb{E}}\left[AU_{1}^{2}|U_{1}\right]\right]=\operatorname{\mathbb{E}}\left[U_{1}^{2}\operatorname{\mathbb{E}}\left[A|U_{1}\right]\right]=$
$\displaystyle=\operatorname{\mathbb{E}}\left[\frac{U_{1}^{2}}{1+\exp\\{-\alpha_{0}-\alpha_{u}U_{1}\\}}\right]=\int_{-\infty}^{\infty}\frac{\frac{1}{\sqrt{2\pi}}u^{2}\exp\left\\{-\frac{u^{2}}{2}\right\\}}{1+\exp\\{-\alpha_{0}-\alpha_{u1}u\\}}du$

cannot be computed in closed form but can be obtained numerically using a
software like Mathematica or Maple once we provide the values of $\alpha_{0}$
and $\alpha_{u}$.

###  C.3 Computing the estimator bias under setup (29)

Let $M=\begin{pmatrix}1&Z&W&A&AZ&AW\end{pmatrix}$. By the typical formula
$\hat{\boldsymbol{b}}=\left(M^{T}M\right)^{-1}M^{T}Y$ for the OLS estimator
and the following

$\displaystyle\operatorname{\mathbb{E}}[Z]$
$\displaystyle=\theta_{0}+\theta_{a}\operatorname{\mathbb{E}}[A]$
$\displaystyle\operatorname{\mathbb{E}}[W]$ $\displaystyle=\mu_{0}$
$\displaystyle\operatorname{\mathbb{E}}[AZ]$
$\displaystyle=(\theta_{0}+\theta_{a})\operatorname{\mathbb{E}}[A]+\theta_{u1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[AW]$
$\displaystyle=\mu_{0}\operatorname{\mathbb{E}}[A]+\mu_{u1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[Z^{2}]$
$\displaystyle=\theta_{0}^{2}+\theta_{u1}^{2}+\theta_{u2}^{2}+1+(\theta_{a}^{2}+2\theta_{0}\theta_{a})\operatorname{\mathbb{E}}[A]+2\theta_{0}\theta_{u1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[W^{2}]$
$\displaystyle=\mu_{0}^{2}+\mu_{u1}^{2}+\mu_{u2}^{2}+1$
$\displaystyle\operatorname{\mathbb{E}}[AZ^{2}]$
$\displaystyle=\left(1+\left(\theta_{0}+\theta_{a}\right)^{2}+\theta_{u2}^{2}\right)\operatorname{\mathbb{E}}[A]+2\left(\theta_{0}+\theta_{a}\right)\theta_{u1}\operatorname{\mathbb{E}}[AU_{1}]+\theta_{u1}^{2}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AW^{2}]$
$\displaystyle=\left(1+\mu_{0}^{2}+\mu_{u2}^{2}\right)\operatorname{\mathbb{E}}[A]+2\mu_{0}\mu_{u1}\operatorname{\mathbb{E}}[AU_{1}]+\mu_{u1}^{2}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AZW]$
$\displaystyle=\left(\left(\theta_{0}+\theta_{a}\right)\mu_{0}+\theta_{u2}\mu_{u2}\right)\operatorname{\mathbb{E}}[A]+\left(\left(\theta_{0}+\theta_{a}\right)\mu_{u1}+\theta_{u1}\mu_{0}\right)\operatorname{\mathbb{E}}[AU_{1}]+\theta_{u1}\mu_{u1}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[Y]$
$\displaystyle=\gamma_{0}+\gamma_{a}\operatorname{\mathbb{E}}[A]+\gamma_{au1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[ZY]$
$\displaystyle=\theta_{0}\gamma_{0}+\theta_{u1}\gamma_{u1}+\left(\left(\theta_{0}+\theta_{a}\right)\gamma_{a}+\theta_{a}\gamma_{0}\right)\operatorname{\mathbb{E}}[A]+\left(\theta_{u1}\gamma_{a}+\theta_{a}\gamma_{u1}+\left(\theta_{0}+\theta_{a}\right)\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}]+$
$\displaystyle+\theta_{u1}\gamma_{au1}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[WY]$
$\displaystyle=\mu_{0}\gamma_{0}+\mu_{u1}\gamma_{u1}+\mu_{0}\gamma_{a}\operatorname{\mathbb{E}}[A]+\left(\mu_{0}\gamma_{au1}+\mu_{u1}\gamma_{a}\right)\operatorname{\mathbb{E}}[AU_{1}]+\mu_{u1}\gamma_{au1}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AY]$
$\displaystyle=\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\left(\gamma_{u1}+\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[AZY]$
$\displaystyle=\left(\theta_{0}+\theta_{a}\right)\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\left(\left(\theta_{0}+\theta_{a}\right)\left(\gamma_{u1}+\gamma_{au1}\right)+\theta_{u1}\left(\gamma_{0}+\gamma_{a}\right)\right)\operatorname{\mathbb{E}}[AU_{1}]+$
$\displaystyle+\theta_{u1}\left(\gamma_{u1}+\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AWY]$
$\displaystyle=\mu_{0}\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\left(\mu_{0}\left(\gamma_{u1}+\gamma_{au1}\right)+\mu_{u1}\left(\gamma_{0}+\gamma_{a}\right)\right)\operatorname{\mathbb{E}}[AU_{1}]+\mu_{u1}\left(\gamma_{u1}+\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}^{2}]$

we obtain a linear regression estimator bias equal to

$\begin{split}\delta_{OR}&=\frac{\left(\frac{\operatorname{\mathbb{E}}[AU]}{\operatorname{\mathbb{E}}[A]}-\frac{(1-\operatorname{\mathbb{E}}[A])\theta_{a}}{S_{2}}\theta_{u1}\right)\left(1+\mu_{u2}^{2}\right)+\left(\frac{\operatorname{\mathbb{E}}[AU]}{\operatorname{\mathbb{E}}[A]}-\frac{(1-\operatorname{\mathbb{E}}[A])\theta_{a}}{S_{2}}\frac{\mu_{u1}\mu_{u2}}{\theta_{u2}}\right)\theta_{u2}^{2}}{\left(1+\frac{\theta_{u1}^{2}}{S_{2}}\right)\left(1+\mu_{u2}^{2}\right)+\left(1+\frac{\mu_{u1}^{2}}{S_{2}}\right)\left(1+\theta_{u2}^{2}\right)-\left(1+2\frac{\theta_{u1}\mu_{u1}\theta_{u2}\mu_{u2}}{S_{2}}\right)}\gamma_{u1}\
+\\\
&+\left[\frac{\left(1+\theta_{u2}^{2}+\mu_{u2}^{2}\right)\operatorname{\mathbb{E}}[AU]}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)}\left(\left(1+\theta_{u2}^{2}+\mu_{u2}^{2}\right)+\right.\right.\\\
&+\left.\left(1-\frac{\operatorname{\mathbb{E}}[AU^{2}]}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)}\right)\left(\theta_{u1}^{2}\left(1+\mu_{u2}^{2}\right)+\mu_{u1}^{2}\left(1+\theta_{u2}^{2}\right)-2\theta_{u1}\mu_{u1}\theta_{u2}\mu_{u2}\right)\right)+\\\
&+\theta_{a}\theta_{u1}\left(1+\mu_{u2}-\frac{\mu_{u1}\theta_{u2}\mu_{u2}}{\theta_{u1}}\right)\left(-\left(\frac{\operatorname{\mathbb{E}}[AU^{2}]\left(1-\operatorname{\mathbb{E}}[AU^{2}]\right)}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)}+\operatorname{\mathbb{E}}[AU]^{2}\left(\frac{1}{\operatorname{\mathbb{E}}[A]^{2}S_{1}}+\frac{1}{\left(1-\operatorname{\mathbb{E}}[A]\right)^{2}S_{2}}\right)\right)\cdot\right.\\\
&\cdot\left.\left(\theta_{u1}^{2}\left(1+\mu_{u2}^{2}\right)+\mu_{u1}^{2}\left(1+\theta_{u2}^{2}\right)-2\theta_{u1}\mu_{u1}\theta_{u2}\mu_{u2}\right)+\frac{\left(1+\theta_{u2}^{2}+\mu_{u2}^{2}\right)}{\operatorname{\mathbb{E}}[A]^{2}\left(1-\operatorname{\mathbb{E}}[A]\right)^{2}}\right.\cdot\\\
&\cdot\left.\left.\left(\operatorname{\mathbb{E}}[A]^{4}-\operatorname{\mathbb{E}}[A]^{3}(1+2\operatorname{\mathbb{E}}[AU^{2}])+3\operatorname{\mathbb{E}}[A]^{2}(\operatorname{\mathbb{E}}[AU^{2}]+\operatorname{\mathbb{E}}[AU]^{2})-\operatorname{\mathbb{E}}[A](3\operatorname{\mathbb{E}}[AU]^{2}+\operatorname{\mathbb{E}}[AU^{2}])+\operatorname{\mathbb{E}}[AU^{2}]\right)\right)\right]\\\
&\cdot\prod_{i=1,2}\frac{1}{\left(1+\frac{\theta_{u1}^{2}}{S_{i}}\right)\left(1+\mu_{u2}^{2}\right)+\left(1+\frac{\mu_{u1}^{2}}{S_{i}}\right)\left(1+\theta_{u2}^{2}\right)-\left(1+2\frac{\theta_{u1}\mu_{u1}\theta_{u2}\mu_{u2}}{S_{i}}\right)}\gamma_{au1}\end{split}$
(56)

In particular, for $\gamma_{au1}=0$, we obtain a bias equal to

$\delta_{OR}=\frac{\left(\frac{\operatorname{\mathbb{E}}[AU]}{\operatorname{\mathbb{E}}[A]}-\frac{(1-\operatorname{\mathbb{E}}[A])\theta_{a}}{S_{2}}\theta_{u1}\right)\left(1+\mu_{u2}^{2}\right)+\left(\frac{\operatorname{\mathbb{E}}[AU]}{\operatorname{\mathbb{E}}[A]}-\frac{(1-\operatorname{\mathbb{E}}[A])\theta_{a}}{S_{2}}\frac{\mu_{u1}\mu_{u2}}{\theta_{u2}}\right)\theta_{u2}^{2}}{\left(1+\frac{\theta_{u1}^{2}}{S_{2}}\right)\left(1+\mu_{u2}^{2}\right)+\left(1+\frac{\mu_{u1}^{2}}{S_{2}}\right)\left(1+\theta_{u2}^{2}\right)-\left(1+2\frac{\theta_{u1}\mu_{u1}\theta_{u2}\mu_{u2}}{S_{2}}\right)}\gamma_{u1}$
(57)

###  C.4 Computing the estimator bias under setup (24)

Let $M=\begin{pmatrix}1&Z&W&A&AZ&AW\end{pmatrix}$. Similarly to Appendix C.3,
we use the typical OLS estimator
$\hat{\boldsymbol{b}}=\left(M^{T}M\right)^{-1}M^{T}Y$ and the following values

$\displaystyle\operatorname{\mathbb{E}}[Z]$
$\displaystyle=\theta_{0}+\theta_{a}\operatorname{\mathbb{E}}[A]$
$\displaystyle\operatorname{\mathbb{E}}[W]$ $\displaystyle=\mu_{0}$
$\displaystyle\operatorname{\mathbb{E}}[AZ]$
$\displaystyle=(\theta_{0}+\theta_{a})\operatorname{\mathbb{E}}[A]+\theta_{u1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[AW]$
$\displaystyle=\mu_{0}\operatorname{\mathbb{E}}[A]+\mu_{u1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[Z^{2}]$
$\displaystyle=\theta_{0}^{2}+\theta_{u1}^{2}+1+(\theta_{a}^{2}+2\theta_{0}\theta_{a})\operatorname{\mathbb{E}}[A]+2\theta_{0}\theta_{u1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[W^{2}]$
$\displaystyle=\mu_{0}^{2}+\mu_{u1}^{2}+1$
$\displaystyle\operatorname{\mathbb{E}}[AZ^{2}]$
$\displaystyle=\left(1+\left(\theta_{0}+\theta_{a}\right)^{2}\right)\operatorname{\mathbb{E}}[A]+2\left(\theta_{0}+\theta_{a}\right)\theta_{u1}\operatorname{\mathbb{E}}[AU_{1}]+\theta_{u1}^{2}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AW^{2}]$
$\displaystyle=\left(1+\mu_{0}^{2}\right)\operatorname{\mathbb{E}}[A]+2\mu_{0}\mu_{u1}\operatorname{\mathbb{E}}[AU_{1}]+\mu_{u1}^{2}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AZW]$
$\displaystyle=\left(\theta_{0}+\theta_{a}\right)\mu_{0}\operatorname{\mathbb{E}}[A]+\left(\left(\theta_{0}+\theta_{a}\right)\mu_{u1}+\theta_{u1}\mu_{0}\right)\operatorname{\mathbb{E}}[AU_{1}]+\theta_{u1}\mu_{u1}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[Y]$
$\displaystyle=\gamma_{0}+\gamma_{a}\operatorname{\mathbb{E}}[A]+\gamma_{au1}\operatorname{\mathbb{E}}[AU_{1}]$
$\displaystyle\operatorname{\mathbb{E}}[ZY]$
$\displaystyle=\theta_{0}\gamma_{0}+\theta_{u1}\gamma_{u1}+\left(\left(\theta_{0}+\theta_{a}\right)\gamma_{a}+\theta_{a}\gamma_{0}\right)\operatorname{\mathbb{E}}[A]+\left(\theta_{u1}\gamma_{a}+\theta_{a}\gamma_{u1}+\left(\theta_{0}+\theta_{a}\right)\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}]+$
$\displaystyle+\theta_{u1}\gamma_{au1}\operatorname{\mathbb{E}}[AU_{1}^{2}]+\theta_{a}\gamma_{u2}\operatorname{\mathbb{E}}[AU_{2}]$
$\displaystyle\operatorname{\mathbb{E}}[WY]$
$\displaystyle=\mu_{0}\gamma_{0}+\mu_{u1}\gamma_{u1}+\mu_{0}\gamma_{a}\operatorname{\mathbb{E}}[A]+\left(\mu_{0}\gamma_{au1}+\mu_{u1}\gamma_{a}\right)\operatorname{\mathbb{E}}[AU_{1}]+\mu_{u1}\gamma_{au1}\operatorname{\mathbb{E}}[AU_{1}^{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AY]$
$\displaystyle=\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\left(\gamma_{u1}+\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}]+\gamma_{u2}\operatorname{\mathbb{E}}[AU_{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AZY]$
$\displaystyle=\left(\theta_{0}+\theta_{a}\right)\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\left(\left(\theta_{0}+\theta_{a}\right)\left(\gamma_{u1}+\gamma_{au1}\right)+\theta_{u1}\left(\gamma_{0}+\gamma_{a}\right)\right)\operatorname{\mathbb{E}}[AU_{1}]+$
$\displaystyle+\theta_{u1}\left(\gamma_{u1}+\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}^{2}]+\left(\theta_{0}+\theta_{a}\right)\gamma_{u2}\operatorname{\mathbb{E}}[AU_{2}]$
$\displaystyle\operatorname{\mathbb{E}}[AWY]$
$\displaystyle=\mu_{0}\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\left(\mu_{0}\left(\gamma_{u1}+\gamma_{au1}\right)+\mu_{u1}\left(\gamma_{0}+\gamma_{a}\right)\right)\operatorname{\mathbb{E}}[AU_{1}]+$
$\displaystyle+\mu_{u1}\left(\gamma_{u1}+\gamma_{au1}\right)\operatorname{\mathbb{E}}[AU_{1}^{2}]+\mu_{0}\gamma_{u2}\operatorname{\mathbb{E}}[AU_{2}]$

to compute the regression estimator bias $\delta_{OR}$. We then obtain

$\begin{split}\delta_{OR}&=\left[\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}-\frac{1-\operatorname{\mathbb{E}}[A]}{S_{2}}\theta_{a}\theta_{u1}}{1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{2}}}+\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]}-\frac{\operatorname{\mathbb{E}}[A]}{S_{1}}\theta_{a}\theta_{u1}}{1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{1}}}\right]\gamma_{u1}+\\\
&+\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}-\frac{1-\operatorname{\mathbb{E}}[A]}{S_{2}}\theta_{a}\theta_{u1}}{1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{2}}}\gamma_{au1}+\\\
&+\operatorname{\mathbb{E}}[AU_{1}]\operatorname{\mathbb{E}}[AU_{2}]\theta_{a}\theta_{u1}\left[\frac{1-\operatorname{\mathbb{E}}[A]}{\operatorname{\mathbb{E}}[A]^{2}\left(1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{2}}\right)}+\frac{\operatorname{\mathbb{E}}[A]}{\left(1-\operatorname{\mathbb{E}}[A]\right)^{2}\left(1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{1}}\right)}\right]\gamma_{u2}\end{split}$
(58)

In particular, for $\gamma_{au1}=0$, we obtain a bias equal to

$\begin{split}\delta_{OR}&=\left[\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{\operatorname{\mathbb{E}}[A]}-\frac{1-\operatorname{\mathbb{E}}[A]}{S_{2}}\theta_{a}\theta_{u1}}{1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{2}}}+\frac{\frac{\operatorname{\mathbb{E}}[AU_{1}]}{1-\operatorname{\mathbb{E}}[A]}-\frac{\operatorname{\mathbb{E}}[A]}{S_{1}}\theta_{a}\theta_{u1}}{1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{1}}}\right]\gamma_{u1}+\\\
&+\operatorname{\mathbb{E}}[AU_{1}]\operatorname{\mathbb{E}}[AU_{2}]\theta_{a}\theta_{u1}\left[\frac{1-\operatorname{\mathbb{E}}[A]}{\operatorname{\mathbb{E}}[A]^{2}\left(1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{2}}\right)}+\frac{\operatorname{\mathbb{E}}[A]}{\left(1-\operatorname{\mathbb{E}}[A]\right)^{2}\left(1+\frac{\theta_{u1}^{2}+\mu_{u1}^{2}}{S_{1}}\right)}\right]\gamma_{u2}\end{split}$
(59)

###  C.5 Comparison of Proximal and Unadjusted Estimator Biases under Setup
(24)

We begin by proving that both $S_{1},S_{2}>0$:

##### Proof that $S_{1},S_{2}>0$:

We have that

$\displaystyle\left|Cov(A,AU)\right|=\left|\operatorname{\mathbb{E}}\left[A^{2}U\right]\right|=\left|\operatorname{\mathbb{E}}[AU]\right|$
$\displaystyle\leq\sqrt{Var(A)Var(AU)}=\sqrt{Var(A)}\sqrt{\operatorname{\mathbb{E}}[AU^{2}]-\operatorname{\mathbb{E}}[AU]^{2}}$

which implies $\operatorname{\mathbb{E}}[AU]^{2}\leq
Var(A)\left(\operatorname{\mathbb{E}}\left[AU^{2}\right]-\operatorname{\mathbb{E}}[AU]^{2}\right)$.
It follows that

$\displaystyle\operatorname{\mathbb{E}}[A]\operatorname{\mathbb{E}}[AU^{2}]$
$\displaystyle\geq\operatorname{\mathbb{E}}[A]\cdot\frac{\operatorname{\mathbb{E}}[AU]^{2}(1+Var(A))}{Var(A)}=\operatorname{\mathbb{E}}[A]\cdot\frac{\operatorname{\mathbb{E}}[AU]^{2}(1+Var(A))}{\operatorname{\mathbb{E}}[A]-\operatorname{\mathbb{E}}[A]^{2}}$
$\displaystyle=\frac{\operatorname{\mathbb{E}}[AU]^{2}(1+Var(A))}{1-\operatorname{\mathbb{E}}[A]}$
$\displaystyle\geq\operatorname{\mathbb{E}}[AU]^{2}$

since $1+Var(A)\geq 1$ and $1-\operatorname{\mathbb{E}}[A]\in(0,1)$. Thus,
$S_{2}>0$.

Similarly, if we consider $\bar{A}=1-A$ (such that $\bar{A}^{2}=\bar{A}$,
$\operatorname{\mathbb{E}}[\bar{A}]=1-\operatorname{\mathbb{E}}[A]$,
$Var(\bar{A})=Var(A)$,
$\operatorname{\mathbb{E}}[\bar{A}U]=-\operatorname{\mathbb{E}}[AU]$, and
$\operatorname{\mathbb{E}}[\bar{A}U^{2}]=1-\operatorname{\mathbb{E}}[AU^{2}]$),
we obtain

$\displaystyle(1-\operatorname{\mathbb{E}}[A])(1-\operatorname{\mathbb{E}}[AU^{2}])=\operatorname{\mathbb{E}}[\bar{A}]\operatorname{\mathbb{E}}[\bar{A}U^{2}]\geq\operatorname{\mathbb{E}}[\bar{A}U]^{2}=\operatorname{\mathbb{E}}[AU]^{2}$

Thus, $S_{1}>0$ as well. ∎

Taking the ratio of magnitudes for the two biases, we have

$\displaystyle\left|\frac{\delta_{POR}}{\delta_{unadj}}\right|=\left|\operatorname{\mathbb{E}}[A]\cdot\frac{S_{1}}{\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}+S_{1}}+\left(1-\operatorname{\mathbb{E}}[A]\right)\cdot\frac{S_{2}}{\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}+S_{2}}\right|$

Let
$f(r)=\operatorname{\mathbb{E}}[A]\cdot\frac{S_{1}}{r+S_{1}}+\left(1-\operatorname{\mathbb{E}}[A]\right)\cdot\frac{S_{2}}{r+S_{2}}$
for $r\in\left(-\infty,-\min\\{S_{1},S_{2}\\}\right)$. We note that $f(r)$ is
strictly increasing in $r$, that $\lim_{r\to-\infty}f(r)=0$, and that $f(r)=1$
has the unique solution
$r^{\ast}=-S_{1}(1-\operatorname{\mathbb{E}}[A])-S_{2}\operatorname{\mathbb{E}}[A]<0$.
We consider the following four cases:

  * I.

If $\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}\geq 0$, then
$S_{1},S_{2}>0$ imply that
$\frac{S_{i}}{\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}+S_{i}}\in(0,1)$
for $i=1,2$. Since $\operatorname{\mathbb{E}}[A]\in(0,1)$, it follows that
$0<\left|\frac{\hat{\delta}_{POR}}{\hat{\delta}_{unadj}}\right|<1$.

  * II.

If
$-\min\\{S_{1},S_{2}\\}\leq\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}<0$,
then $S_{1},S_{2}>0$ imply that
$\frac{S_{i}}{\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}+S_{i}}>1$ for
$i=1,2$. Similarly, it follows that
$\left|\frac{\hat{\delta}_{POR}}{\hat{\delta}_{unadj}}\right|>1$. In
particular, if
$\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}=-\min\\{S_{1},S_{2}\\}$, the
proximal estimator bias can be arbitrarily large.

  * III.

If
$r^{\ast}\leq\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}<-\min\\{S_{1},S_{2}\\}$,
then $\left|\frac{\hat{\delta}_{POR}}{\hat{\delta}_{unadj}}\right|\geq 1$.

  * IV.

If $\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}<r^{\ast}$, then
$0\leq\left|\frac{\hat{\delta}_{POR}}{\hat{\delta}_{unadj}}\right|<1$. In
particular, $\frac{\theta_{u1}\mu_{u1}}{\theta_{u2}\mu_{u2}}=-\infty$ implies
that the proximal estimator is unbiased (as either $\theta_{u2}=0$ or
$\mu_{u2}=0$).

###  C.6 Computing the Proximal Estimator Bias under $\gamma_{au}=0$ and
$h(W,A,X)=b_{0}+b_{a}A+b_{x}^{T}X+b_{w}^{T}W$

We will compute the asymptotic bias obtained from the method of moments solver
using bridge function $h(W,A,X;b)=b_{0}+b_{a}A+b_{w}^{T}W+b_{x}^{T}X$ and
vector function $q(A,Z,X)=\left(1,A,Z,X\right)$. We assume the general case of
multi-dimensional $U,Z,W,X$ with $Z\in\operatorname{\mathbb{R}}^{m}$,
$W\in\operatorname{\mathbb{R}}^{n}$, $U\in\operatorname{\mathbb{R}}^{p}$,
$X\in\operatorname{\mathbb{R}}^{q}$. Throughout this section, we use the
shorthand
$\operatorname{\mathbb{E}}[AU]=\left(\operatorname{\mathbb{E}}[AU_{1}],\ldots,\operatorname{\mathbb{E}}[AU_{p}]\right)$
and
$\operatorname{\mathbb{E}}[AX]=\left(\operatorname{\mathbb{E}}[AX_{1}],\ldots,\operatorname{\mathbb{E}}[AX_{q}]\right)$.

We define the moment restrictions
$H(D_{i};\theta)=\begin{Bmatrix}\\{Y_{i}-h(W_{i},A_{i},X_{i};b)\\}\times
Q(A_{i},Z_{i},X_{i})\\\
\Delta-(h(W_{i},1,X_{i};b)-h(W_{i},0,X_{i};b))\end{Bmatrix}$ and let
$m(\theta)=\operatorname{\mathbb{E}}[H(D;\theta)]=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}h(D_{i};\theta)$.
The estimate of $\theta=(b,\Delta)$ is given by

$\hat{\theta}=\operatorname*{arg\,min}_{\theta}m^{T}(\theta)m(\theta)$

Using $\operatorname{\mathbb{E}}[U_{i}]=0$,
$\operatorname{\mathbb{E}}[U_{i}^{2}]=1,\forall i=1,\ldots,p$, and
$\operatorname{\mathbb{E}}[U_{i}U_{j}]=0,\forall i,j=1,\ldots,p,\ i\neq j$, as
well as $\operatorname{\mathbb{E}}[X_{j}]=0$,
$\operatorname{\mathbb{E}}[XX^{T}]=\Sigma_{x}$, and
$\operatorname{\mathbb{E}}[UX^{T}]=\rho$, we express the coordinates of
$\operatorname{\mathbb{E}}[h(D;\theta)]=(m_{1},m_{2},\boldsymbol{m}_{3},\boldsymbol{m}_{4})$
with $m_{1},m_{2}\in\operatorname{\mathbb{R}}$,
$\boldsymbol{m}_{3}\in\operatorname{\mathbb{R}}^{m}$,
$\boldsymbol{m}_{4}\in\operatorname{\mathbb{R}}^{q}$ as follows:

$\displaystyle m_{1}$
$\displaystyle=-b_{0}-\operatorname{\mathbb{E}}[A]b_{a}-\mu_{0}^{T}b_{w}+\gamma_{0}+\operatorname{\mathbb{E}}[A]\gamma_{a}$
$\displaystyle m_{2}$
$\displaystyle=-\operatorname{\mathbb{E}}[A]b_{0}-\operatorname{\mathbb{E}}[A]b_{a}-\left(\operatorname{\mathbb{E}}[A]\mu_{0}^{T}+\operatorname{\mathbb{E}}[AU]^{T}\mu_{u}+\operatorname{\mathbb{E}}[AX]^{T}\mu_{x}\right)b_{w}-\operatorname{\mathbb{E}}[AX]^{T}b_{x}+$
$\displaystyle+\left(\gamma_{0}+\gamma_{a}\right)\operatorname{\mathbb{E}}[A]+\operatorname{\mathbb{E}}[AU]^{T}\gamma_{u}+\operatorname{\mathbb{E}}[AX]^{T}\gamma_{x}$
$\displaystyle m_{3}$
$\displaystyle=-(\theta_{0}+\theta_{a}\operatorname{\mathbb{E}}[A])b_{0}-\left(\operatorname{\mathbb{E}}[A](\theta_{0}+\theta_{a})+\theta_{u}^{T}\operatorname{\mathbb{E}}[AU]+\theta_{x}^{T}\operatorname{\mathbb{E}}[AX]\right)b_{a}-$
$\displaystyle-\left(\left(\theta_{0}+\operatorname{\mathbb{E}}[A]\theta_{a}\right)\mu_{0}^{T}+\left(\theta_{a}\operatorname{\mathbb{E}}[AU]^{T}+\theta_{u}^{T}+\theta_{x}^{T}\rho^{T}\right)\mu_{u}+\left(\theta_{a}\operatorname{\mathbb{E}}[AX]^{T}+\theta_{x}^{T}\Sigma_{x}+\theta_{u}^{T}\rho\right)\mu_{x}\right)b_{w}-$
$\displaystyle-\left(\theta_{a}\operatorname{\mathbb{E}}[AX]^{T}+\theta_{u}^{T}\rho+\theta_{x}^{T}\Sigma_{x}\right)b_{x}+\left(\theta_{0}+\theta_{a}\operatorname{\mathbb{E}}[A]\right)\gamma_{0}+$
$\displaystyle+\left(\left(\theta_{0}+\theta_{a}\right)\operatorname{\mathbb{E}}[A]+\theta_{u}^{T}\operatorname{\mathbb{E}}[AU]+\theta_{x}^{T}\operatorname{\mathbb{E}}[AX]\right)\gamma_{a}+$
$\displaystyle+\left(\theta_{a}\operatorname{\mathbb{E}}[AU]^{T}+\theta_{u}^{T}+\theta_{x}^{T}\rho^{T}\right)\gamma_{u}+\left(\theta_{a}\operatorname{\mathbb{E}}[AX]^{T}+\theta_{u}^{T}\rho+\theta_{x}^{T}\Sigma_{x}\right)\gamma_{x}$
$\displaystyle m_{4}$
$\displaystyle=-\operatorname{\mathbb{E}}[AX]b_{a}-\left(\Sigma_{x}\mu_{x}+\rho^{T}\mu_{u}\right)b_{w}-\Sigma_{x}b_{x}+\operatorname{\mathbb{E}}[AX]\gamma_{a}+\Sigma_{x}\gamma_{x}+\rho^{T}\gamma_{u}$

Under assumption $m=n$ and $p>m$: Let us define

$\displaystyle\beta$
$\displaystyle=\frac{\operatorname{\mathbb{E}}[AU]^{T}-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\rho^{T}}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]}$
$\displaystyle B$
$\displaystyle=\left(I_{p}-\rho\Sigma_{x}^{-1}\rho^{T}-\frac{\left(\operatorname{\mathbb{E}}[AU]-\rho\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]\right)\left(\operatorname{\mathbb{E}}[AU]^{T}-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\rho^{T}\right)}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]}\right)\theta_{u}.$

Setting $m_{1}=m_{2}=m_{3i}=m_{4i}=0$ for all $i=1,\ldots,m$, $j=1,\ldots,q$,
we get solution

$\displaystyle b_{0}$
$\displaystyle=\gamma_{0}-\operatorname{\mathbb{E}}[A]\beta\gamma_{u}-\left(\mu_{0}^{T}-\operatorname{\mathbb{E}}[A]\beta\mu_{u}\right)\left(B^{T}\mu_{u}\right)^{-1}B^{T}\gamma_{u}$
$\displaystyle b_{a}$
$\displaystyle=\gamma_{a}+\beta\left(I_{p}-\mu_{u}\left(B^{T}\mu_{u}\right)^{-1}B^{T}\right)\gamma_{u}$
$\displaystyle b_{w}$
$\displaystyle=\left(B^{T}\mu_{u}\right)^{-1}B^{T}\gamma_{u}$ $\displaystyle
b_{x}$
$\displaystyle=\gamma_{x}+\Sigma_{x}^{-1}\left(\rho^{T}-\operatorname{\mathbb{E}}[AX]\beta\right)\gamma_{u}-\left(\mu_{x}+\Sigma_{x}^{-1}\left(\rho^{T}-\operatorname{\mathbb{E}}[AX]\beta\right)\mu_{u}\right)\left(B^{T}\mu_{u}\right)^{-1}B^{T}\gamma_{u}$

(assuming $B^{T}\mu_{u}$ is invertible). The estimated effect resulting from
$\hat{h}(W,A,X;b)$ is then

$\displaystyle\hat{\Delta}=\hat{b}_{a}=\gamma_{a}+\frac{\operatorname{\mathbb{E}}[AU]^{T}-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\rho^{T}}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]}\left(I_{p}-\mu_{u}\left(B^{T}\mu_{u}\right)^{-1}B^{T}\right)\gamma_{u}$

which yields a bias equal to

$\delta=\frac{\operatorname{\mathbb{E}}[AU]^{T}-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\rho^{T}}{\operatorname{\mathbb{E}}[A]\left(1-\operatorname{\mathbb{E}}[A]\right)-\operatorname{\mathbb{E}}[AX]^{T}\Sigma_{x}^{-1}\operatorname{\mathbb{E}}[AX]}\left(I_{p}-\mu_{u}\left(B^{T}\mu_{u}\right)^{-1}B^{T}\right)\gamma_{u}$