# Choosing the Right Approach at the Right Time: A Comparative Analysis of
Causal Effect Estimation using Confounder Adjustment and Instrumental
Variables

Roy S. Zawadzki
Department of Statistics
University of California, Irvine
Irvine, CA
Corresponding Author<EMAIL_ADDRESS>Daniel L. Gillen
Department of Statistics
University of California, Irvine
Irvine, CA

###### Abstract

In observational studies, unobserved confounding is a major barrier in
isolating the average causal effect (ACE). In these scenarios, two main
approaches are often used: confounder adjustment for causality (CAC) and
instrumental variable analysis for causation (IVAC). Nevertheless, both are
subject to untestable assumptions and, therefore, it may be unclear which
assumption violation scenarios one method is superior in terms of mitigating
inconsistency for the ACE. Although general guidelines exist, direct
theoretical comparisons of the trade-offs between CAC and the IVAC assumptions
are limited. Using ordinary least squares (OLS) for CAC and two-stage least
squares (2SLS) for IVAC, we analytically compare the relative inconsistency
for the ACE of each approach under a variety of assumption violation scenarios
and discuss rules of thumb for practice. Additionally, a sensitivity framework
is proposed to guide analysts in determining which approach may result in less
inconsistency for estimating the ACE with a given dataset. We demonstrate our
findings both through simulation and an application examining whether maternal
stress during pregnancy affects a neonate’s birthweight. The implications of
our findings for causal inference practice are discussed, providing guidance
for analysts for judging whether CAC or IVAC may be more appropriate for a
given situation.

##  1 Introduction

A common goal in observational studies is to estimate the average causal
effect (ACE) of a treatment or exposure on a specific outcome such as the
effect of maternal stress during pregnancy on a child’s birthweight. Due to
the exposure not being randomized, the presence of confounders may bias
estimates of the ACE. Confounders are factors that influence both the level of
exposure (or treatment assignment) and the outcome. If uncontrolled for,
confounders can create extraneous differences between the exposure groups that
make it difficult to isolate the causal effect. The effectiveness of
confounder adjustment for causality (CAC), however, is contingent on the
untestable assumption that all possible confounders, or suitable proxies, are
present in the dataset and have been accounted for appropriately. In other
words, we cannot use observed data to prove that the CAC is consistent for the
ACE. A simple example is if a mother’s obstetric risk, a known confounder for
stress and birthweight, and any proxies were missing from the dataset we would
like to analyze. Without a priori knowledge that obstetric risk was a
confounder, we would not be able to ascertain that the CAC assumptions were
violated.

An alternative approach that may avoid concerns about unobserved confounders
is the instrumental variable (IV) approach, or instrumental variable analysis
for causation (IVAC). An IV is defined by three main conditions: (i) it
influences the treatment assignment (relevance), (ii) it is not a cause of the
outcome after conditioning on treatment assignment (exclusion restriction),
and (iii) it is not associated with unobserved confounders (independence). In
the event that we have a variable that satisfies these conditions, we could
then use variation in the IV as a proxy for variation in the treatment and
measure the effect on the outcome. Importantly, by (iii), the variation in the
IV is independent from unobserved confounding and, therefore, unlike CAC, IVAC
does not require appropriately accounting for all possible confounding to
consistently estimate the ACE.[1]

When choosing to use CAC over the IVAC, and vice-versa, we trade one set of
untestable assumptions for another. For CAC, we cannot prove that we have
accounted for all confounders while, for IVAC, we cannot prove we have a valid
IV. For example, proving the exclusion restriction requires us to establish
the lack of a direct relationship between the IV and outcome, or a null
result, which is not possible with data. Furthermore, to be consistent for the
ACE, the IVAC must meet certain untestable conditions surrounding treatment
effect heterogeneity. We therefore have no guarantee that under either
approach our produced estimate is consistent for the ACE. Despite this, we
remain interested in estimating the ACE and subsequently direct our efforts
towards attempting to estimate a parameter with the least possible distance
from the ACE. In this vein, the key question and central premise of the
current manuscript focuses on addressing the following question in practice:
under the potential violation of the untestable assumptions, when is the
parameter estimated by CAC closer to the ACE than that of IVAC, and vice
versa?

In the literature, there exist general guidelines surrounding whether CAC is
more appropriate than IVAC.[1, 2] Generally, we contend that analysts should
weigh whether the potential degree of unobserved confounding outweighs the
potential for violations in the IV assumptions. There is, however, little by
way of theoretical research that directly compares the two approaches to
assess these trade-offs. Focusing on ordinary least squares (OLS) for CAC and
two-stage least squares (2SLS) for IVAC, there are two overarching goals we
seek to accomplish with this paper. Firstly, we seek to analytically compare
the inconsistency of 2SLS to OLS in a variety of scenarios involving
assumption violations. Next, we provide a sensitivity framework to guide
analysts in determining whether the inconsistency of 2SLS is more than that of
OLS, and vice versa.

In order to more succinctly express the relative performance of OLS and 2SLS
under assumption violations, we focus on the scenario where, for a given
variable, we must decide on whether the variable should be adjusted for as a
confounder in OLS, used as an IV in 2SLS, or not incorporated into the
analysis at all. Note that we allow confounders to be adjusted for in 2SLS –
an IV is only used in the first stage whereas confounders would need to be
present in both the first and second stage.

To our knowledge, there is no existing theoretical literature directly
comparing the trade-offs of pursuing CAC versus IVAC, though authors have
considered the impact of assumption violations in both settings. The first
area of literature relevant to the themes of our paper is bias amplification,
which refers to the fact that using certain variables as confounders may
increase pre-existing bias due to unobserved confounding. In the linear
setting, an IV is a bias amplifier.[3, 4, 5, 6] In these papers, the authors
compare the consistency for the ACE with and without adjusting for an IV.
Pearl (2012) provides an extension where there is an imperfect IV (in that the
exclusion restriction is violated) and shows that under certain conditions,
adjusting for the imperfect IV may actually reduce bias. Nevertheless, he does
not address whether this variable may be more appropriately used in 2SLS.

The bias amplification literature and, by extension, our findings have
important implications on applied practice. In particular, the rise of data-
driven variable selection approaches for the propensity score, or probability
of receiving the intervention, in confounder methods and first-stage for IV
analysis. Though IVs, by definition, influence the treatment assignment, they
may amplify bias if included in the model for the propensity score.[3] For IV
methods, though a variable may not be a perfect IV it may still be worth using
it as such. In addition, data-driven modeling of the first-stage may, for
example, shrink to zero an important confounder used to achieve the IV
assumptions. The gain in the strength of the first-stage may, however, offset
the penalty incurred by omitted variable bias in 2SLS. As it stands, these
intuitions are difficult to incorporate into variable selection procedures.
With this work, we seek to elucidate these complicated scenarios.

In another area of the literature, there has been some work related to
comparing OLS to 2SLS under the violation of IV assumptions. It is a well-
known result that if an IV is poorly predictive of the intervention (i.e.
weak), then small violations in the exclusion restriction and independence
assumption can lead to large inconsistencies for the IV estimand.[7, 8]. In
addition, to assess the independence assumption, one may compare the impact of
intentionally omitting an observed confounder on OLS and 2SLS in order to
compare the sensitivity of OLS to that of 2SLS in estimating the ACE.[9] The
main assumption behind this procedure is that the impact of omitting an
observed confounder on the consistency of OLS and 2SLS is similar to that of
omitting a correlated unobserved confounder. A similar assumption and
"benchmarking" procedure will be used in our sensitivity analysis.

There are several procedures in the literature regarding sensitivity analyses
for violations in IV assumptions. For example, Cinelli and Hazlett (2022)
provide a compelling framework and visualization scheme for omitted variable
bias in 2SLS based on several partial $R^{2}$ measures.[10] Their framework
addresses the question of how large the impact of an unobserved confounder
would have to be in order to qualitatively change the inferential conclusions
of a study, which covers both violations of the exclusion restriction and
independence assumptions. A similar procedure to this is the E-value.[11]
While we use many of the same tools – notably benchmarking unobserved $R^{2}$
measures with observed data from Cinelli and Hazlett (2022) – we do not focus
on this sensitivity analysis paradigm of hypothesis testing but instead
consider the relative inconsistencies of CAC and IVAC for the ACE.

Another complication to IVAC lies in treatment effect heterogeneity. In this
setting, Imbens and Angrist (1994) state that, under monotonicity conditions,
IVAC identifies the local average causal effect (LACE) or the causal effect of
the "compliers" subpopulation (those whose treatment assignment varies with
the IV).[12] If the factors that determine compliance also cause treatment
effect heterogeneity, then the LACE may not be equal to the ACE. Hartwig et
al. (2020) and Wang and Tchetgen Tchetgen (2018) give clear explanations of
the assumptions needed for the LACE to equal the ACE.[13, 14] Essentially, the
heterogeneity between the treatment and outcome should be independent of both
the IV and the effect modification between the treatment and outcome. For the
ease of parameterization in our paper, we will use Wang and Tchetgen
Tchetgen’s notion that one of two conditions must be met: (i) no unmeasured
confounders are additive effect modifiers of the relationship of both the
instrument and treatment or (ii) no unmeasured confounders are additive effect
modifiers the treatment and the outcome.

The rest of the paper is organized as follows. First, we provide the general
model setting of interest and introduce relative notation. We then present
results regarding the consistency of no adjustment, OLS, and 2SLS for the
scenarios of an exclusion restriction violation, independence violation, and
treatment effect heterogeneity relevant to IV estimation with and without
covariates. In all scenarios, we consider unobserved confounding and isolate
the impact of individual assumption violations (e.g. both an exclusion
restriction and independence violation). Following this, we present a
sensitivity analysis procedure based on partial $R^{2}$ and benchmarking
unobserved quantities with observed quantities. The goal of this procedure is
to give the analyst relevant information to assess the plausibility of whether
it may be more appropriate to adjust for a variable in OLS or 2SLS. Next, in
simulations, we verify our closed-form results and demonstrate the use of the
sensitivity analysis procedure in a variety of scenarios. Then, we apply the
procedure to a real world dataset studying a neonate’s birthweight as a
function of maternal stress during pregnancy. We conclude with a discussion
regarding the implications of our closed form results on the practice of
causal inference and, additionally, provide further guidance on how to use our
sensitivity analysis procedure.

##  2 Notation and Set-Up

Our goal is to estimate the ACE of some continuous treatment $X$ on an outcome
$Y$, denoted as $\beta_{1}=\frac{\partial}{\partial x}E[Y|do(x)]$ in Pearl’s
notation.[15] We depict the causal relationships in Figure 1, a directed
acyclic graph (DAG) with edge weights $c_{0},\dots,c_{3}$. We further consider
the following structural equations where $E[\epsilon_{1}|U,Z]=0$ and
$E[\epsilon_{2}|X,U]=0$:

$X=\alpha_{1}U+\alpha_{2}Z+\epsilon_{1},$ (1)
$Y=\beta_{1}X+\beta_{2}U+\epsilon_{2}.$ (2)

ZXYU$c_{3}$$c_{0}$$c_{1}$$c_{2}$ Figure 1: A directed acyclic graph with one
confounder and one instrumental variable.

Given that all variables in Figure (1) are standardized to have mean $0$ and
variance $1$, the edge weights are equivalent to the coefficients in Eqs. (1)
and (2). For example, the ACE, $\beta_{1}$, is the same as the edge weight
$c_{0}$. In addition, $\alpha_{1}=c_{1}$, $\alpha_{2}=c_{3}$,
$\beta_{1}=c_{0}$, and $\beta_{2}=c_{2}$. This equivalence will be helpful in
visually expressing assumptions surrounding different scenarios. Furthermore,
the edge weights are correlations and are bounded between $-1$ and $1$.

Throughout, we assume the stable unit treatment value assumption (SUTVA) of
consistency and no interference. Because $U$ is unobserved, we must estimate
the the following reduced form regression where the subscript $R$ indicates
these are the values related to the reduced regression

$Y=\beta_{1}^{R}X+\epsilon_{2}^{R}.$ (3)

$U$ is a confounder because it both influences $X$ and $Y$, and, therefore,
because we have failed to block the path through $U$, $\hat{\beta}_{1}^{R}$ is
inconsistent for the ACE. Alternatively, to estimate the ACE, we may utilize
$Z$, which is a valid IV if

  1. 1.

$\alpha_{2}>0$ (relevance)

  2. 2.

$Z$ is not a cause of $Y$ conditional on $X$ (exclusion restriction)

  3. 3.

$Z$ is not influenced by any unaccounted for confounders such that
$Z\mathchoice{\mathrel{\hbox to0.0pt{\kern
7.63889pt\kern-5.27776pt$\displaystyle\not$\hss}{\perp\mkern-9.5mu\perp}}}{\mathrel{\hbox
to0.0pt{\kern
7.63889pt\kern-5.27776pt$\textstyle\not$\hss}{\perp\mkern-9.5mu\perp}}}{\mathrel{\hbox
to0.0pt{\kern
2.61118pt\kern-4.11108pt$\scriptstyle\not$\hss}{\perp\mkern-9.5mu\perp}}}{\mathrel{\hbox
to0.0pt{\kern
1.11118pt\kern-3.3333pt$\scriptscriptstyle\not$\hss}{\perp\mkern-9.5mu\perp}}}Y|X$
(independence).

Supposing that there is no treatment effect heterogeneity or that the
conditions of Hartwig et al. (2020) or Wang and Tchetgen Tchetgen (2018) [13,
14] are met, we have
$\beta_{IV}=\frac{Cov(Z,Y)}{Cov(X,Z)}=\frac{\beta_{1}\alpha_{1}}{\alpha_{1}}=\beta_{1}$,
and hence 2SLS will provide a consistent estimate of the ACE.

We are interested in estimating and comparing the following estimands: the
causal effect i.e. Eq. (4), one that omits $Z$ i.e. Eq. (5), one that uses $Z$
as a confounder i.e. Eq. (6), and one that utilizes $Z$ as an IV i.e. Eq.(7):

$A_{1}=\frac{\partial}{\partial x}E[Y|do(x)]=c_{0},$ (4)
$A_{2}=\frac{\partial}{\partial x}E[Y|x],$ (5) $A_{3}=\frac{\partial}{\partial
x}E[Y|x,z],$ (6)
$A_{4}=\frac{Cov(Y,Z)}{Cov(X,Z)}=\frac{\frac{\partial}{\partial
z}E[Y|z]}{\frac{\partial}{\partial z}E[X|z]}.$ (7)

We define the degree of inconsistency for $A_{1}$ of estimates of $A_{2}$,
$A_{3}$, and $A_{4}$ as a set of absolute differences:
$\lambda_{2}=|A_{4}-A_{2}|$, $\lambda_{3}=|A_{4}-A_{3}|$, and
$\lambda_{4}=|A_{4}-A_{3}|$. The crux of our findings is to compare these
$\lambda$s. One approach "performs better" than the other if the respective
$\lambda$ is smaller. For example, 2SLS performs better than OLS if
$\lambda_{4}>\lambda_{3}$.

As a simple example of the types of calculations and comparisons that we will
do in the next section, we can use the conditions of Figure 1. From Pearl
(2012), we have that $A_{2}=c_{0}+c_{1}c_{2}$ by Wright’s rules of path
analysis and $A_{3}=c_{0}+c_{2}\frac{\partial}{\partial
x}E[U|x,z]=c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}$.[5] As a result of adjusting
for $Z$ as a confounder, we have bias amplification that increases with the
strength of the IV (i.e. the magnitude of $c_{3}$). In addition, we have that
$\hat{A}_{4}\overset{p}{\to}c_{0}$. The proof of this is in the first section
of the Appendix. Further, by Chebyshev’s inequality and assuming finite
variances hold, we also have that
$\hat{A}_{2}\overset{p}{\to}c_{0}+c_{1}c_{2}$ and
$\hat{A}_{3}\overset{p}{\to}c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}$. Thus,
$\lambda_{3}=\frac{c_{1}c_{2}}{1-c_{3}^{2}}>\lambda_{2}=c_{1}c_{2}>\lambda_{4}=0$
or, in words, 2SLS performs better than no adjustment, which performs better
than OLS.

##  3 Trade-offs Under Violations in Instrumental Variable Assumptions

In this section, we present results for the trade-offs between confounder and
IV methods for three scenarios: (i) violation of the exclusion restriction
assumption, (ii) violation of the independence assumption, and (iii) treatment
effect heterogeneity. In all scenarios, $U$ is unobserved, which provides the
realistic setting where we may be motivated to use 2SLS due to the concern of
unobserved confounding. For ease of exposition, we first derive the quantities
of interest without adjustment for observed confounding. We then present the
quantities with these observed covariates. For ease of comparison of the
quantities of interest, we further assume all regression slope parameters are
positive throughout this section and we will handle the general case in the
sensitivity analysis portion. Unless otherwise stated, all proofs for the
propositions in this section can be found in the Appendix.

###  3.1 Exclusion Restriction Violation

Figure 2 directly reproduces Figure 2 from Pearl (2012) and presents a
violation of the exclusion restriction assumption for $Z$ if $c_{ER}\neq 0$.
We use this quantity to denote the degree of violation. By traditional logic,
one would define $Z$ as a confounder because it is a cause both of $X$ and $Y$
and use it as such. It is not, however, unequivocally true that one should use
$Z$ as a confounder. To see this, suppose we have the following structural
equations:

$X=c_{1}U+c_{3}Z+\epsilon_{3}$ (8) $Y=c_{0}X+c_{2}U+c_{ER}Z+\epsilon_{4}.$ (9)

ZXYU$c_{3}$$c_{0}$$c_{1}$$c_{2}$$c_{ER}$ Figure 2: Exclusion Restriction
Violation using $Z$ as an IV.

######  Proposition 3.1.

Under the conditions of Eqs. (8) and (9),
$\hat{A}_{2}\overset{p}{\to}c_{0}+c_{1}c_{2}+c_{3}c_{ER}$,
$\hat{A}_{3}\overset{p}{\to}c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}$, and
$\hat{A}_{4}\overset{p}{\to}c_{0}+\frac{c_{ER}}{c_{3}}$.

The derivations for $A_{2}$ and $A_{3}$ can be found in Pearl (2012) so we
omit them. The proof for $A_{4}$ can be found in the Appendix. We see that
adjusting for $Z$ decreases inconsistency compared to not adjusting for $Z$ if
$\frac{c_{ER}}{c_{3}}\geq\frac{c_{1}c_{2}}{1-c_{3}^{2}}$. This inequality
could be difficult to attain if the instrument is strong.[5] Interestingly,
the left term of this inequality is the inconsistency of 2SLS and thus the IV
being strong is relatively advantageous for the use of $Z$ as an IV. We can
re-arrange the inequality between $A_{3}$ and $A_{4}$ as
$\frac{c_{ER}(1-c_{3}^{2})}{c_{3}}\geq c_{1}c_{2}$ where $c_{1}c_{2}$
indicates the impact of unobserved confounding in the relationship between $X$
and $Y$. Here, it becomes more clear that the strength of the IV can be large
enough such that the degree of exclusion restriction violation (i.e. $c_{ER}$)
is offset and is smaller than the impact of unobserved confounding.

The trade-offs between $A_{2}$, $A_{3}$, and $A_{4}$ can be visualized with a
3-D contour plot. In Figure 3, letting $c_{0}=0.3,c_{1}=0.7,$ and $c_{2}=0.7$,
we can vary the values of $c_{3}$ and $c_{ER}$. Note the plausible coefficient
values for $c_{3}$ and $c_{ER}$ are restricted due to the requirement that the
variances for the variables to sum to one (see "Notes about Simulations"
section in the Appendix). This image gives us the visual intuition that when
the IV is stronger, moderate violations in the exclusion restriction violation
do not preclude the use of $Z$ as an IV. Furthermore, adjusting for $Z$ will
be inferior compared to not adjusting for $Z$. When the IV is weak, 2SLS
predictably performs very poorly in all cases.

Figure 3: Analytical comparison of no adjustment, OLS adjusting for the IV,
and 2SLS using the IV under varying IV strength and degree of exclusion
restriction violation.

###  3.2 Independence Violation

ZXYU$c_{3}$$c_{0}$$c_{1}$$c_{2}$$c_{I}$ Figure 4: $Z$ is Correlated with an
Unobserved Confounder $U$

Figure 4 represents one violation of the independence assumption where the
residual confounder $U$ is a cause of $Z$. Here, $c_{I}$ represents the degree
of violation or, in this case, the effect of $U$ on $Z$. Alternative
specifications create a new confounder that is only associated with $Z$ but
not $X$. Nonetheless, our parameterization provides a useful case where both
the independence assumption is violated and there is confounding in the
relationship between $Z$ and $X$. For structural equations, we can re-use Eqs
1 and 2 as well as add an extra structural equation given by

$Z=c_{I}U+\epsilon_{5}.$ (10)

######  Proposition 3.2.

Under Eqs. (1), (2), and (10),
$\hat{A}_{2}\overset{p}{\to}c_{0}+c_{1}c_{2}+c_{2}c_{3}c_{I}$,
$\hat{A}_{3}\overset{p}{\to}c_{0}+\frac{c_{1}c_{2}(1-c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}$,
and $\hat{A}_{4}\overset{p}{\to}c_{0}+\frac{c_{2}c_{I}}{c_{3}+c_{1}c_{I}}$.

The derivation for $A_{2}$ is a straightforward application of Wright’s path
analysis so only the proofs for $A_{3}$ and $A_{4}$ are provided.[16] To
better interpret these quantities, we can think about both paths on the DAG
and remaining variance after orthogonalizing variables via the Frisch-Waugh-
Lovell Theorem (FWL).[17] Looking at the pathways, $c_{1}c_{2}$ is the
backdoor path from $X$ to $Y$ through $U$, $c_{2}c_{3}c_{I}$ is the backdoor
path through $Z$, and $c_{2}c_{I}$ is the path from $Z$ to $Y$ via $U$.
$c_{3}+c_{1}c_{I}$ is the unconditional correlation between $Z$ and $X$, which
includes both the direct and backdoor path via $U$. $1-c_{I}^{2}$ depicts the
remaining (stochastic) variation in $Z$ after orthogonalizing $U$ while
$1-(c_{3}+c_{1}c_{I})^{2}$ is the remaining variance in $X$ after
orthogonalizing $Z$.

Of particular interest is the trade-off between using $Z$ in 2SLS and using
$Z$ in OLS. We find that adjusting for $Z$ is superior to using $Z$ for 2SLS
if
$\frac{c_{I}}{c_{3}+c_{1}c_{I}}>\frac{c_{1}(1-c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}$.
We can begin to interpret this inequality with the reoccurring theme that if
the IV is strong then attaining this inequality is more difficult: a strong IV
will cause $c_{3}+c_{1}c_{I}$ to be large which inflates the inconsistency in
$A_{3}$ while it decreases the consistency for $A_{4}$. The remaining
variation in $Z$ not caused by $U$, or $c_{I}$, is an important quantity
because as $c_{I}$ decreases, the inconsistency in $A_{3}$ will increase while
the inconsistency in $A_{4}$ will decrease. In this sense, a simple
sensitivity analysis procedure could be to benchmark the variation in the IV
that is explained by the covariates. One could use this benchmark to
conjecture how much of the variation in the IV is explained by an unobserved
confounder. If this quantity is small then one could plausibly assume a fair
amount of variation in $Z$ free from unobserved confounding and, thus, $c_{I}$
is small. These trade-offs can be visualized in Figure 5 where
$c_{0}=0.3,c_{1}=0.7,$ and $c_{2}=0.7$.

Figure 5: Analytical comparison of no adjustment, OLS adjusting for the IV,
and 2SLS using the IV under varying IV strength and degree of the independence
violation.

###  3.3 Treatment Effect Heterogeneity

We return the setting of Figure 1 where we have a perfect IV but now we omit
the edge weights and introduce treatment effect heterogeneity. In this case,
we do not know of any literature that gives us a salient way to to represent
treatment heterogeneity using DAGs and edge weights. Therefore, the nuance is
in the structural equations:

$X=\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1},$ (11)
$Y=\beta_{1}X+\beta_{2}U+\beta_{3}XU+\epsilon_{2}.$ (12)

The estimand of interest remains the ACE or the average of the individual
treatment effects, which is denoted by $\beta_{1}$. This parameterization is
consistent with a violation of Wang and Tchetgen Tchetgen (2018) assumptions
5a and 5b if $\alpha_{3}\neq 0$ and $\beta_{3}\neq 0$. The key parameter for
measuring the degree of "assumption violation" is $\alpha_{3}$ because we aim
to quantify how large an unobserved confounder needs to be in order to modify
the effect on $Z$ in the first-stage to render $\lambda_{4}>\lambda_{3}$ and
$\lambda_{4}>\lambda_{2}$, or that IVAC is inferior to CAC.

######  Proposition 3.3.

Under the conditions of Eqs. (11) and (12) as well as assuming $E[U^{3}]=0$,
$\hat{A}_{2}\overset{p}{\to}\beta_{1}+\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3},\hat{A}_{3}\overset{p}{\to}\beta_{1}+\frac{\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{3}\beta_{3}}{1-a_{1}^{2}},$
and
$\hat{A}_{2}\overset{p}{\to}\beta_{1}+\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}$

When comparing the relative trade-off between not adjusting for $Z$ and
adjusting for $Z$, the latter is inferior if
$\frac{\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{3}\beta_{3}}{1-a_{1}^{2}}>\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}$.
Unlike the case of no effect modification, it is not always true that
adjusting for an IV will amplify bias. Specifically, by rearranging terms we
see the inequality holds if
$2\alpha_{1}^{2}+\frac{\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}>1$. If
$|\alpha_{1}|\gtrapprox 0.7$, or the IV is strongly associated with $X$, then
this inequality will hold; however, this would not be the case if the IV is
sufficiently weak and the multiplication of the coefficients for the
interactions are larger than the multiplication of the coefficients
representing the main effect.

In the more realistic case, if an analyst is aware $Z$ is an IV and assuming
that the IV was sufficiently strong, the main point of concern surrounds
whether the LACE estimate is more inconsistent than the ACE estimate from the
unadjusted OLS. This notion is true if
$\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}>\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}$
or
$\frac{\alpha_{1}\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}+2\alpha_{1}^{2}<1$.
Firstly, a strong IV, $|\alpha_{1}|\gtrapprox 0.7$, precludes this inequality
from being attained. Alternatively, this inequality could fail to be attained
if the ratio of main effects to interaction effects, scaled by the IV
strength, is large. Setting $\alpha_{2}=0.15$, $\beta_{1}=0.1$,
$\beta_{2}=0.2$, and $\beta_{3}=0.1$, we can visualize the above inequalities
in Figure 6.

A final takeaway from these results is that the strength of an IV has
implications on the the OLS inconsistencies even though it is independent from
$U$ (e.g. via the term $2\alpha_{1}\alpha_{3}\beta_{3}$ in $A2$). Therefore,
accounting for any present IVs, even if never utilized, is important for
understanding the degree of inconsistency present in confounder methodology.
The results of the previous three Propositions are summarized in Table 1.

Figure 6: Analytical comparison of no adjustment, OLS adjusting for the IV, and 2SLS using the IV under varying IV strength and degree of the treatment effect heterogeneity assumption. Table 1: Summary of Results Scenario | $A_{2}$ | $A_{3}$ | $A_{4}$
---|---|---|---
No Violation |  |  |
ZXYU$c_{3}$$c_{0}$$c_{1}$$c_{2}$ |  |  |
$X=c_{1}U+c_{3}Z+\epsilon_{1}$ $Y=c_{0}X+c_{2}U+\epsilon_{2}$ | $c_{0}+c_{1}c_{2}$ | $c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}$ | $c_{0}$
Exclusion Restriction Violation |  |  |
ZXYU$c_{3}$$c_{0}$$c_{1}$$c_{2}$$c_{ER}$ |  |  |
$X=c_{1}U+c_{3}Z+\epsilon_{3}$ $Y=c_{0}X+c_{2}U+c_{ER}Z+\epsilon_{4}.$ | $c_{0}+c_{1}c_{2}+c_{3}c_{ER}$ | $c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}}$ | $c_{0}+\frac{c_{ER}}{c_{3}}$
Independence Assumption |  |  |
ZXYU$c_{3}$$c_{0}$$c_{1}$$c_{2}$$c_{I}$ |  |  |
$X=\alpha_{1}U+\alpha_{2}Z+\epsilon_{1}$ $Y=\beta X+\theta U+\epsilon_{2}$ $Z=c_{I}U+\epsilon_{5}.$ | $c_{0}+c_{1}c_{2}+c_{2}c_{3}c_{I}$ | $c_{0}+\frac{c_{1}c_{2}(1-c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}$ | $c_{0}+\frac{c_{2}c_{I}}{c_{3}+c_{1}c_{I}}$
Treatment Effect Heterogeneity |  |  |
$X=\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1},$ $Y=\beta_{1}X+\beta_{2}U+\beta_{3}XU+\epsilon_{2}.$ | $\beta_{1}+\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}$ | $\beta_{1}+\frac{\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{3}\beta_{3}}{1-a_{1}^{2}}$ | $\beta_{1}+\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}$

###  3.4 Adding Additional Confounders

In the vast majority of data analyses, an analyst will have access to observed
confounders that will be adjusted for to mitigate confounding both in OLS and
2SLS. Temporarily, we consider a single observed confounder, $W$, that is
continuous and has mean $0$ and variance $1$. If we have multiple confounders,
$\\{W_{1},W_{2},...W_{J}\\}$, that are in the same form as $W$, we can simply
redefine $W$ as some function of the confounders, $f(W_{1},W_{2},...W_{J})$.
For example, this might take a form linear combination derived from the first
principal component of the $W$ matrix. Therefore, the edge weights and
regression coefficients will be the joint effect of the confounders.
Therefore, operationally, our updated results can accommodate multiple
covariates.

Because we would like to use $W$ to benchmark relationships involving $U$, we
must update the DAGs and structural equations such that $W$ has a similar form
to $U$ in each assumption violation. Because of this, our previous results
will change. In all cases, $W$ is orthogonal to $U$ or, in other words, $U$
represents confounding in ACE unrelated to $U$. With the exception of $A_{1}$,
the quantities of interest are updated to

$\displaystyle A_{2}$ $\displaystyle=\frac{\partial}{\partial x}E[Y|x,w]$ (13)
$\displaystyle A_{3}$ $\displaystyle=\frac{\partial}{\partial x}E[Y|x,z,w]$
(14) $\displaystyle A_{4}$
$\displaystyle=\frac{Cov(Y,Z|W)}{Cov(X,Z|W)}=\frac{\frac{\partial}{\partial
z}E[Y|z,w]}{\frac{\partial}{\partial z}E[X|z,w]}.$ (15)

####  3.4.1 Exclusion Restriction

Figure 7 serves as an example of a covariate that has no direct impact on $Z$
or $U$. Nevertheless, if we were to condition upon $X$ and nothing else, $U$
would no longer be independent of $W$ or $Z$ due to the collider effect. A
collider effect induces an association between two variables that point to
(i.e. cause) a single variable that has been conditioned.[15] However,
conditioning upon $W$ breaks this association.

ZXYWU$c_{3}$$c_{0}$$c_{1}$$c_{2}$$c_{ER}$$c_{ER}$$c_{5}$$c_{6}$ Figure 7: A
DAG with observed confounder and violation of exclusion restriction ($W$).

The updated structural equations are

$X=c_{1}U+c_{3}Z+c_{5}W+\epsilon_{3}$ (16)
$Y=c_{0}X+c_{2}U+c_{ER}Z+c_{6}W+\epsilon_{4}.$ (17)

######  Proposition 3.4.

Under Eqs. (16) and
(17)$,\hat{A}_{2}\overset{p}{\to}c_{0}+\frac{c_{1}c_{2}+c_{3}c_{ER}}{1-c_{5}^{2}}$
and
$\hat{A}_{3}\overset{p}{\to}c_{0}+\frac{c_{1}c_{2}}{1-c_{3}^{2}-c_{5}^{2}}$

The proof is very similar to that of Proposition 3.1 so it is omitted. As
expected, adjusting for $W$ leads to decreased variance in $X$, which leads to
a higher proportional contribution of $U$, at the benefit of eliminating the
backdoor path via $c_{5}c_{6}$. For 2SLS, the results for $A_{4}$ are not
affected because the assumption violations vis-a-vis using $Z$ as an IV are
not influenced by $W$. Nevertheless, in practical settings, adjusting for $W$
will usually increase precision of $\hat{A}_{4}$.

####  3.4.2 Independence

ZXYUW$c_{3}$$c_{0}$$c_{1}$$c_{2}$$c_{I}$$c_{5}$$c_{6}$$c_{7}$ Figure 8: $W$
mimics $U$ in the DAG.

Besides introducing $W$ as a confounder in the $X-Y$ relationship, Figure 8
extends $W$ to be a confounder in the $Z-Y$ and $Z-X$ relationships.
Therefore, the validity of $Z$ as an IV is contingent on conditioning upon
both $W$ and $Z$. Thus, the structural equations are updated to

$Z=c_{I}U+c_{7}W+\epsilon_{5}$ (18) $X=c_{1}U+c_{3}Z+c_{5}W+\epsilon_{1},$
(19) $Y=c_{0}X+c_{2}U+c_{6}W+\epsilon_{2}.$ (20)

The results for all quantities, $A_{2}$, $A_{3}$, and $A_{4}$, can now be
updated per the following proposition:

######  Proposition 3.5.

Under Eqs. (19), (20), and (18):

$\displaystyle\hat{A}_{2}\overset{p}{\to}c_{0}+\frac{c_{1}c_{2}+c_{2}c_{3}c_{I}}{1-(c_{5}+c_{3}c_{7})^{2}},$
(21)
$\displaystyle\hat{A}_{3}\overset{p}{\to}c_{0}+\frac{\frac{c_{1}c_{2}(1-c_{7}^{2}-c_{I}^{2})}{1-c_{7}^{2}}}{1-(c_{5}+c_{3}c_{7})^{2}-(1-c_{7}^{2})(c_{3}+\frac{c_{1}c_{I}}{1-c_{7}^{2}})^{2}},$
(22)
$\displaystyle\hat{A}_{4}\overset{p}{\to}c_{0}+\frac{c_{2}c_{I}}{(1-c_{7}^{2})(c_{3}+\frac{c_{1}c_{I}}{(1-c_{7}^{2})})}.$
(23)

These results bear some resemblance to Proposition 3.2 when we did not have
$W$ present. For $A_{2}$, in the numerator, because $W$ does not mitigate the
influence of $U$ in the DAG, we still have two backdoor paths from $X$ to $Y$
that go through $U$. Meanwhile, in the denominator the variance of $X$ is
reduced via controlling for $W$, which has a direct path to $X$ as well as an
indirect path via $Z$.

For $A_{3}$, the quantity is similar conceptually to our findings in
Proposition 3.2 except that we must additionally account for controlling for
$W$. In the numerator, $c_{1}c_{2}(1-c_{7}^{2}-c_{I}^{2})$ represents the
magnitude of unobserved confounding reduced (i.e. multiplied) by the exogenous
variance of $Z$ due to there no longer being a backdoor path to $Y$ via $Z$.
We must account for the cost of adjusting $W$ therefore this quantity is
amplified (i.e. divided) by $1-c_{7}^{2}$, the variance in $Z$ free of $W$. In
the denominator, we have the remaining variance of $X$ after adjusting for $W$
and $Z$. The first subtracting term represents the unconditional $R^{2}$
between $X$ and $Z$ while the second takes the variation of $Z$ free of $W$
and multiplies it by the partial $R^{2}$ between $Z$ and $X$, adjusting for
$W$. Because we haven’t adjusted for $U$, the backdoor path via $U$ remains
and, furthermore, because $W$ does not mitigate this, $W$ essentially acts
like a bias amplifier for $c_{1}c_{I}$. Lastly, $A_{4}$ represents the
association between $Z$ and $X$ via $U$ as well as the variation exogenous
from $W$ directly from $Z$.

####  3.4.3 Treatment Effect Heterogeneity

Because we require our observed confounder to be of the same form of $U$ for
benchmarking purposes, we will set $W$ as both an effect modifier of the
treatment on the outcome and of the instrument on the treatment assignment
with the following structural equations:

$X=\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\alpha_{4}W+\alpha_{5}ZW+\epsilon_{1},$
(24)
$Y=\beta_{1}X+\beta_{2}U+\beta_{3}XU+\beta_{4}W+\beta_{5}XW+\epsilon_{2}.$
(25)

The presence of $XW$, which is observed but still endogenous, means that in
OLS, we must adjust for it and in 2SLS, must provide an an additional IV. In
particular, we will choose $ZW$. Therefore we modify the quantities of
interest to

$\displaystyle A_{2}$ $\displaystyle=\frac{\partial}{\partial x}E[Y|x,w,xw]$
(26) $\displaystyle A_{3}$ $\displaystyle=\frac{\partial}{\partial
x}E[Y|x,z,w,xw]$ (27) $\displaystyle A_{4}$
$\displaystyle=\frac{\partial}{\partial\hat{x}}E[Y|\hat{x},\widehat{xw},w]$
(28)

where $\hat{x}$ and $\widehat{xw}$ represent the fitted values from using $Z$,
$ZW$, and $W$ as regressors for $X$ and $XW$, respectively. Note that now the
main effect is obtained after orthogonalizing $XW$ and $W$, or $\widehat{XW}$
and $W$ in 2SLS, which we can find via FWL by treating $XW$ as any other
covariate. Therefore, the corresponding interpretation of the main effect is
when $W=0$, or we are at the average value of the covariate due to centering
the variable.

######  Proposition 3.6.

Under Eqs. (24) and (25) as well as $E[U^{3}]=0$, $E[W^{3}]=0$, $E[W^{4}]=3$,
and $E[U^{4}]=3$ we have
$\hat{A}_{2}\overset{p}{\to}\beta_{1}+\frac{\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}}{1-\alpha_{4}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{1+\alpha_{4}^{2}+2\alpha_{5}^{2}}}$
and
$\hat{A}_{4}\overset{p}{\to}\beta_{1}+\frac{\alpha_{1}\alpha_{3}\beta_{3}-\frac{2\alpha_{1}\alpha_{3}\alpha_{5}^{2}\beta_{3}}{\alpha_{1}^{2}+\alpha_{5}^{2}}}{\alpha_{1}^{2}+\alpha_{5}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{(\alpha_{1}^{2}+\alpha_{5}^{2}})}$.

The interpretation of $A_{2}$ is consistent with Proposition 3.3 with the
denominator reflecting the fact that we are adjusting for $W$ and $XW$, which
reduces the remaining variance of $X$. For $A_{4}$, because we are no longer
computing the ratio of coefficients the interpretation is not directly
comparable to Proposition 3.3. Nevertheless, we can see the influence of the
$Z-U$ interaction on the inconsistency and observe that because $\hat{XU}$ is
correlated with $\widehat{XW}$, adjusting for $\widehat{XW}$ will reduce the
influence of $\widehat{XU}$, hence the subtraction terms.

One may notice that we have omitted the derivation for the result of $A_{3}$
and this is because we only wish to compare $A_{2}$ and $A_{4}$. From our
discussion of Proposition 3.3, assuming
$sign(\alpha_{2}\beta_{2})=sign(\alpha_{3}\beta_{3})$ for sake of simplicity,
bias amplification will hold in the effect modification case if
$2\alpha_{1}^{2}+\frac{\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}>1$. When we
have a strong IV, or $|\alpha_{1}|\gtrapprox 0.7$, then this inequality holds.
If we instead lower the strength of the IV to where we must consider
$\frac{\alpha_{2}\beta_{2}}{\alpha_{3}\beta_{3}}>1-2\alpha_{1}^{2}$, we would
require the multiplication interaction effects to be at least as large as the
main effects with this requirement increasing as the IV strength increases. We
argue that in this circumstance one should avoid using $Z$ altogether because
concern over this ratio would imply that the IV is weak and the LACE is likely
to be far from the ACE due to large heterogeneity. Thus, a comparison between
OLS and 2SLS is not warranted. As we will describe later on, one may
"benchmark" this ratio using the observed confounders and instrument as an
initial check if an IV is appropriate in their circumstance for targeting the
ACE.

##  4 Sensitivity Analysis

In this section, we use the closed form derivations of the previous section to
develop a set of sensitivity analysis procedures that provide analysts with
information surrounding whether OLS or 2SLS may be more appropriate given the
observed data and hypothesized assumption violations. We focus on comparing
$A_{3}$ and $A_{4}$, or using $Z$ as a confounder versus as an IV, by
graphically presenting the relative inconsistency
$\frac{\lambda_{4}}{\lambda_{3}}$ as measured by the degree of IV assumption
violations and unobserved confounding.

The graphical depiction of the relative inconsistencies is largely motivated
by Cincelli and Hazlett (2020) [18] and Cincelli and Hazlett (2022)[10] where
they use a set of partial $R^{2}$’s to characterize how large the assumption
violations in confounder and 2SLS analyses must be to render statistically
significant results null. Instead of focusing on hypothesis testing, however,
we examine how large unobserved confounding and the IV assumption violations
could be in order for $\frac{\lambda_{4}}{\lambda_{3}}>1$ or
$\frac{\lambda_{4}}{\lambda_{3}}<1$. Following this, like Cincelli and
Hazlett, we use benchmarking to estimate the unobserved sensitivity quantities
across a variety of scenarios.

To briefly summarize the benchmarking procedure, suppose we have the quantity
$R^{2}_{Y\sim U|X,W}$, or the additional gain in $R^{2}$ by adding $U$ to the
regression of $Y$ on treatment $X$ and observed covariate $W$. We must assume
that for the individual confounders that make up the composite confounder $W$,
the magnitude of the $W_{j}-X$ relationship and $W_{j}-Y$ relationship for
$j\in\\{1,2,...J\\}$ are similar to the magnitude of the $U-X$ and $U-Y$
relationship, respectively. As a result, we can therefore benchmark
$R^{2}_{Y\sim U|X,W}$ as conservatively as possible with a quantity such as
$\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j}}$ where $W_{-j}$ represents all
other observed confounders besides $W_{j}$. In the remainder of this section,
we will focus on the sensitivity analysis procedure in the scenario of the
exclusion restriction. Then, using the same principles, present the results
for the independence and heterogeneity assumption violations more succinctly.

###  4.1 Exclusion Restriction Violation

######  Proposition 4.1.

$\lambda_{4}=\frac{|c_{4}|}{|c_{3}|}=\frac{c_{1}c_{2}}{1-c_{3}^{2}-c_{5}^{2}}=\lambda_{3}$
can be re-written as $\frac{\sqrt{R^{2}_{Y\sim Z|X,U,W}}\sqrt{1-R^{2}_{Y\sim
U|X,W}}(1-R^{2}_{X\sim W+Z})}{\sqrt{1-R^{2}_{Y\sim Z|X,W}}\sqrt{R^{2}_{X\sim
Z}}sd(Z^{\perp X,W})}>\frac{\sqrt{R^{2}_{X\sim U}R^{2}_{Y\sim
U|X,W,Z}}}{\sqrt{1-R^{2}_{U\sim X|W,Z}}}$ where $sd(Z^{\perp X,W})$ is the
standard deviation resulting from the residuals of regressing $Z$ on $X$ and
$W$.

In order to simplify the description of the above quantity, we will decompose
the terms in the above inequality into three factors: symbolically, we let

$\displaystyle\phi$ $\displaystyle=\frac{1-R^{2}_{X\sim
W+Z}}{\sqrt{1-R^{2}_{Y\sim Z|X,W}}\sqrt{R^{2}_{X\sim Z}}sd(Z^{\perp X,W})}$
(29) $\displaystyle\gamma$ $\displaystyle=\frac{\sqrt{R^{2}_{X\sim
U}}\sqrt{R^{2}_{Y\sim U|X,W,Z}}}{\sqrt{1-R^{2}_{U\sim
X|W,Z}}\sqrt{1-R^{2}_{Y\sim U|X,W}}}$ (30) $\displaystyle\theta$
$\displaystyle=\sqrt{R^{2}_{Y\sim Z|X,W,U}}.$ (31)

Here, $\phi$ represents all observed quantities that can be directly
estimated, $\gamma$ is a function that is monotonically increasing as the
unobserved confounding increases, and $\theta$ is the degree of the exclusion
restriction violation. We will need to benchmark $\gamma$ and $\theta$, which
we will refer to as $\gamma^{B}$ and $\theta^{B}$. These observed benchmarks
are detailed in Table 2.

Table 2: Benchmarking for $\gamma$ and $\theta$ Quantities for Exclusion Restriction Quantity | Benchmark
---|---
$R^{2}_{Y\sim Z|X,W,U}$ | $R^{2}_{Y\sim Z|X,W}$
$R^{2}_{X\sim U}$ | $\max_{j}R^{2}_{X\sim W_{j}}$
$R^{2}_{Y\sim U|X,W,Z}$ | $\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j},Z}$
$R^{2}_{U\sim X|W,Z}$ | $\max_{j}R^{2}_{W_{j}\sim X|\mathcal{W}_{-j},Z}$
$R^{2}_{Y\sim U|X,W}$ | $\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j}}$

We begin our sensitivity analysis procedure with ambivalence between OLS and
2SLS, or that
$\frac{\lambda_{4}}{\lambda_{3}}=1\implies\frac{\theta\phi}{\gamma}=1$. Then,
given $\hat{\phi}$ and $\theta^{B}$, we can solve for how large $\gamma$, or
the degree of unobserved confounding, must be in order for the equality to be
maintained. We call this $\gamma^{I}$ where "I" indicates the $\gamma$
"implied" by assuming ambivalence. $\gamma^{I}$ serves as an anchor point for
the next step when we compute $\gamma^{B}$.

Now using the data, we will find $\gamma^{B}$ through benchmarking and combine
this with $\phi$ and $\theta_{B}$ to form $\frac{\theta^{B}\phi}{\gamma^{B}}$.
We will refer to this fraction as the "inconsistency ratio," or "IR", which
may be used to judge the use of 2SLS or OLS. For example, a ratio of $<1$
implies that $\lambda_{3}>\lambda_{4}$ and suggests that we should use 2SLS as
opposed to OLS. Of course, the degree of unobserved confounding could be
different than the observed benchmark. To mitigate this, we may additionally
add a multiplier, $M*\gamma^{B}$, if we believe the observed confounders
underestimate ($M<1$) or overestimate ($M>1$) the degree of unobserved
confounding. For example, if $M=0.5$, we are assuming that the unobserved
confounding is half as much as the benchmarked confounding. Lastly, we compute
$\theta^{I}$ or the degree of the exclusion restriction violation to satisfy
$\frac{\theta^{I}\phi^{B}}{\gamma^{B}}=1$, or returning to ambivalence.

The above quantities culminate in the graphical visualization in Figure 9.
This Figure was produced under a simulation where
$\frac{\lambda_{4}}{\lambda_{3}}<1$, implying that 2SLS is a better choice
than OLS in reducing inconsistency for the ACE. The x-axis shows the IR, while
the y-axis tells us the degree of unobserved confounding as captured by
$\gamma$. Focusing on the y-axis, the position of the black dotted line, our
anchor point, gives us the plausibility of $\gamma^{I}$ by comparing to
$0.5\times\gamma_{B}$ (red), $\gamma_{B}$ (purple), and $1.5\times\gamma_{B}$
(blue). We can see that the degree of unobserved confounding required for
ambivalence is roughly 60% of the benchmarked confounding. We obtain 60% by
comparing the y-intercept of the purple line ($\approx$ 0.25) with the
y-intercept of the dotted black line ($\approx$ 0.15). Looking at the x-axis,
this works in favor of 2SLS as the corresponding inconsistency calculated by
$\frac{\theta^{B}\phi^{B}}{\gamma^{B}}<1$ unless the true degree of unobserved
confounding is less than that of the benchmarked $\gamma$.

Lastly, we will address the legend labels for which we have one for each line.
"Benchmarked ER Violation" is the benchmarked exclusion restriction violation
as quantified by $R^{2}_{Y\sim Z|X,W,U}$ while "ER Violation Required" tells
us the value of $R^{2}_{Y\sim Z|X,W,U}$ required to return to ambivalence at
the given multiplier. The comparison of the required numbers and the
benchmarked violation gives analysts information to judge the plausibility of
ambivalence. For example, on Figure 9, the degree violation required to return
to ambivalence given $M=1$ (0.061) is roughly 2.3 times the observed benchmark
of 0.026.

Figure 9: A case where the inconsistency of OLS is more than that of 2SLS
under an exclusion restriction violation

###  4.2 Independence Violation

######  Proposition 4.2.

$\lambda_{4}=\Big{|}\frac{c_{2}c_{I}}{c_{3}(1-c_{7}^{2})+c_{1}c_{I}}\Big{|}=\Big{|}\frac{\frac{c_{1}c_{2}(1-c_{7}^{2}-c_{I}^{2})}{1-c_{7}^{2}}}{1-(c_{5}+c_{3}c_{7})^{2}-(1-c_{7}^{2})(c_{3}+\frac{c_{1}c_{I}}{1-c_{7}^{2}})^{2}}\Big{|}=\lambda_{3}$
can be re-written as $\frac{\sqrt{R^{2}_{Z\sim U}}}{\sqrt{1-R^{2}_{Z\sim
U|W}}}=\sqrt{R^{2}_{X\sim U|Z,W}}\Bigg{[}\frac{\sqrt{R^{2}_{X\sim
Z|W}}sd(X^{\perp Z,W})sd(X^{\perp W})sd(Z^{\perp W})}{1-R^{2}_{X\sim
W}-(1-R^{2}_{Z\sim W})R^{2}_{X\sim Z|W}}\Bigg{]}$

Once again, we partition the quantities in Proposition as follows:

$\displaystyle\phi$ $\displaystyle=\frac{\sqrt{R^{2}_{X\sim Z|W}}sd(X^{\perp
Z,W})sd(X^{\perp W})sd(Z^{\perp W})}{1-R^{2}_{X\sim W}-(1-R^{2}_{Z\sim
W})R^{2}_{X\sim Z|W}}$ (32) $\displaystyle\gamma$
$\displaystyle=\sqrt{R^{2}_{X\sim U|Z,W}}$ (33) $\displaystyle\theta$
$\displaystyle=\frac{\sqrt{R^{2}_{Z\sim U}}}{\sqrt{1-R^{2}_{Z\sim U|W}}}$ (34)
$\displaystyle R^{2}_{Z\sim U}$
$\displaystyle=\frac{\phi^{2}\gamma^{2}(1-R^{2}_{Z\sim
W})^{2}}{1+\phi^{2}\gamma^{2}}$ (35)

The IR in this case will be $\frac{\phi\gamma}{\theta}$ with $\theta$ being a
monotonic function with the degree of the independence violation increasing.
Though we may solve for $R^{2}_{Z\sim U}$, as in line 35, we leave $\theta$ in
its current form to maintain the clean interpretation $\gamma$. Then, given a
value for $\gamma$, $\phi$, and the observed $R^{2}_{Z\sim W}$, we calculate
the value of $R^{2}_{Z\sim U}$ in the legend entry for the corresponding line
in the sensitivity plot. An example of the sensitivity plot for the
independence violation is presented in Figure 10. The construction and
interpretation of this figure is the same as in the exclusion restriction
violation except for the fact that we now measure how large does an
independence violation need to be to return to ambivalence. This is denoted by
"U-Z B" and "U-Z Req." as in the $R^{2}$ magnitude of the edge from $U$ to $Z$
as in the DAG in Figure 8. In this case, we simulated the data such that
$\lambda_{4}=\lambda_{3}$ so the purple line (multiplier of 1) and the
required violation is close to the black anchor line and the observed
benchmark. The corresponding benchmarks are detailed in Table 3.

Figure 10: A case where the inconsistency of OLS is roughly equal to that of 2SLS under an independence violation Table 3: Benchmarking for $\gamma$ and $\theta$ Quantities for Independence Violation Quantity | Benchmark
---|---
$R^{2}_{X\sim U|Z,W}$ | $\max_{j}R^{2}_{X\sim W_{j}|\mathcal{W}_{-j},Z}$
$R^{2}_{Z\sim U}$ | $\max_{j}R^{2}_{Z\sim W_{j}}$
$R^{2}_{Z\sim W}$ | $\max_{j}R^{2}_{Z\sim W_{j}|\mathcal{W}_{-j}}$

###  4.3 Treatment Effect Heterogeneity

######  Proposition 4.3.

$\lambda_{2}=\beta_{1}+\frac{\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}}{1-\alpha_{4}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{1+\alpha_{4}^{2}+2\alpha_{5}^{2}}}$
and
$\lambda_{4}=\frac{\alpha_{1}\alpha_{3}\beta_{3}-\frac{2\alpha_{1}\alpha_{3}\alpha_{5}^{2}\beta_{3}}{\alpha_{1}^{2}+\alpha_{5}^{2}}}{\alpha_{1}^{2}+\alpha_{5}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{(\alpha_{1}^{2}+\alpha_{5}^{2}})}$
can be rewritten as

$\displaystyle\Bigg{|}\frac{1/2-\frac{R^{2}_{X\sim
ZW}}{Var(\hat{XW})}}{R^{2}_{X\sim W}+R^{2}_{X\sim ZW}-R^{2}_{XW\sim
X}}\Bigg{|}Var(X^{\perp W,XW})$ (36)
$\displaystyle=\Bigg{|}\frac{\sqrt{R^{2}_{X\sim U}}\sqrt{R^{2}_{Y\sim
U|X,W,XU,XW}}}{2\sqrt{R^{2}_{X\sim Z}}\sqrt{R^{2}_{X\sim
ZU}}\sqrt{R^{2}_{Y\sim XU|X,W,U,XW}}}$ (37)
$\displaystyle\frac{\sqrt{1-R^{2}_{Y\sim XU|X,W,XW}}\sqrt{1-R^{2}_{XU\sim
U|W,XW}}\sqrt{1-R^{2}_{XW\sim XU}}}{\sqrt{1-R^{2}_{Y\sim
XU|X,W,XW}}\sqrt{1-R^{2}_{U\sim
X}}}sign(\beta_{2}\alpha_{2})+sign(\alpha_{1}\alpha_{3}\beta_{3})\Bigg{|}$
(38)

Note that because we are using $A_{2}$ to measure the CAC, $\lambda_{2}$
represents the inconsistency. Our partitioned quantities are as follows

$\displaystyle\phi_{1}$ $\displaystyle=\Bigg{|}\frac{1/2-\frac{R^{2}_{X\sim
ZW}}{Var(\hat{XW})}}{R^{2}_{X\sim W}+R^{2}_{X\sim ZW}-R^{2}_{XW\sim
X}}\Bigg{|}Var(X^{\perp W,XW})$ (39) $\displaystyle\phi_{2}$
$\displaystyle=2\sqrt{R^{2}_{X\sim Z}}$ (40) $\displaystyle\gamma_{1}$
$\displaystyle=\frac{\sqrt{R^{2}_{X\sim U}}\sqrt{R^{2}_{Y\sim
U|X,W,XU,XW}}}{\sqrt{1-R^{2}_{Y\sim U|X,W,XW}}\sqrt{1-R^{2}_{X\sim U}}}$ (41)
$\displaystyle\gamma_{2}$ $\displaystyle=\frac{\sqrt{1-R^{2}_{Y\sim
XU|X,W,XW}}\sqrt{1-R^{2}_{XU\sim U|W,XW}}\sqrt{1-R^{2}_{XW\sim
XU}}}{\sqrt{R^{2}_{Y\sim XU|X,W,U,XW}}}$ (42) $\displaystyle\theta$
$\displaystyle=\sqrt{R^{2}_{X\sim ZU}}.$ (43)

We divide $\phi$ into $\phi_{1}$ and $\phi_{2}$ because the sign of
$sign(\phi_{2})$ may change with what the analyst will impute for
$sign(\alpha_{2}\beta_{2})$ and $sign(\alpha_{1}\alpha_{3}\beta_{3})$. We
initially will discuss the case where
$sign(\alpha_{2}\beta_{2})=sign(\alpha_{1}\alpha_{3}\beta_{3})$ and then
further expand upon the more complicated cases. $\gamma$ is essentially
divided into two parts: quantities related to the main effect of unobserved
confounding ($\gamma_{1}$) and quantities related to treatment effect
heterogeneity ($\gamma_{2}$). To maintain a consistent interpretation of
weighing potential unobserved confounding in OLS to assumption violations in
2SLS across our suite of sensitivity analysis tools, we will only put
multipliers on the main effects of $U$ or $\gamma_{1}$. Therefore, holding
treatment effect heterogeneity constant, $\gamma=\gamma_{1}\gamma_{2}$ is a
monotonically increasing function of the strength of unobserved confounding.
Lastly, $\theta$ quantifies the degree of violation of the IV assumption of
treatment effect heterogeneity by measuring the $R^{2}$ of the interaction
between the unobserved confounder in the first stage.

Under our sign assumptions, the IR is defined as
$\frac{(\phi_{1}-1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}$ where $M$ is the
multiplier on benchmarked residual confounding. Figure 11 shows an example
plot where $\lambda_{4}>\lambda_{2}$. In other words, this depicts the case
where the interaction in the first stage between the instrument and unobserved
effect modifier is large enough such that the inconsistency of LACE for the
ACE is larger than OLS under unobserved confounding. As such, the required
magnitude of this interaction is less than that of the benchmark. We define
the benchmarking quantities in Table 4.

Figure 11: A case where the inconsistency of OLS is less than that of 2SLS under a treatment effect heterogeneity violation Table 4: Benchmarking for $\gamma$ and $\theta$ Quantities for Treatment Effect Heterogeneity Quantity | Benchmark
---|---
$R^{2}_{X\sim U}$ | $\max_{j}R^{2}_{X\sim W_{j}}$
$R^{2}_{Y\sim U|X,W,XU,XW}$ | $\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j},XW}$
$R^{2}_{Y\sim U|X,W,XW}$ | $\max_{j}R^{2}_{Y\sim W_{j}|X,\mathcal{W}_{-j},XW}$
$R^{2}_{Y\sim XU|U,X,W,XW}$ | $\max_{j}R^{2}_{Y\sim XW_{j}|X,W,X\mathcal{W}_{-j}}$
$R^{2}_{Y\sim XU|X,W,XW}$ | $\max_{j}R^{2}_{Y\sim XW_{j}|X,W,X\mathcal{W}_{-j}}$
$R^{2}_{XU\sim U|W,XW}$ | $\max_{j}R^{2}_{XW_{j}\sim W_{j}|\mathcal{W}_{-j},X\mathcal{W}_{-j}}$
$R^{2}_{XW\sim XU}$ | $\max_{i\neq j}R^{2}_{XW_{i}\sim XW_{j}}$
$R^{2}_{X\sim ZU}$ | $\max_{j}R^{2}_{X\sim ZW_{j}}$

Now, we will address the case where it may not necessarily be true that
$sign(\alpha_{2}\beta_{2})=sign(\alpha_{1}\alpha_{3}\beta_{3})$. Although we
know $sign(\alpha_{1})$, we do not know the signs of the remaining
coefficients, which are associated with $U$. Though there may be many
combinations of what the signs of the individual coefficients may be, there
are only three possible IRs

$IR=\begin{cases*}\frac{(\phi_{1}-1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}&if
$sign(\alpha_{2}\beta_{2})=sign(\alpha_{1}\alpha_{3}\beta_{3})$,\\\
\frac{(\phi_{1}+1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}&if
$sign(\alpha_{2}\beta_{2})\neq sign(\alpha_{1}\alpha_{3}\beta_{3})$ and
$\frac{|\alpha_{2}\beta_{2}|}{|\alpha_{3}\beta_{3}|}>|2\alpha_{1}|$,\\\
-\frac{(\phi_{1}-1)\phi_{2}\theta}{M\gamma_{1}\gamma_{2}}&if
$sign(\alpha_{2}\beta_{2})\neq sign(\alpha_{1}\alpha_{3}\beta_{3})$ and
$\frac{|\alpha_{2}\beta_{2}|}{|\alpha_{3}\beta_{3}|}\leq|2\alpha_{1}|$.\end{cases*}$

Practically, there are only two IRs for us to consider because the first and
third cases are opposite signs and we are concerned with the magnitude of the
IR rather than the sign. Whether the condition of
$\frac{|\alpha_{2}\beta_{2}|}{|\alpha_{3}\beta_{3}|}>|2\alpha_{1}|$ is met can
be investigated through benchmarking. As an example of this condition, suppose
$\alpha=0.5$ – a fairly strong instrument in this context – then we require
the treatment effect heterogeneity, $\alpha_{3}\beta_{3}$, to be no stronger
than the effect of $U$ on $X$ when $Z=0$ ($\alpha_{2}$) as well as $U$ on $Y$
when $X=0$ ($\beta_{2}$). Note that because $X$ and $Z$ are centered,
$\alpha_{2}$ and $\beta_{3}$ are at the average value of these variables. We
suggest that two graphs are computed for both cases and, following this, if
the conclusions agree, then no further action should be taken. If they
disagree, then one may benchmark the condition to give a sense of which
condition is more likely or, alternatively, average the ICs assuming equal
probability of each case.

##  5 Simulation

In this section, we present results verifying the closed form derivations for
all three assumption violation scenarios with and without covariates.
Additionally, we further present the use of our sensitivity analysis
procedure. All variables were generated via the structural equations provided
in the earlier sections with the error terms following a normal distribution
with mean zero. When the variables were independent from all other variables
in the system, such as $U$, they were standard normal. When the variable was
determined by other variables in the system, such as $X$, it was still normal
but with the variance of the error term being equal to one minus the variance
of the other variables in the structural equation. This is so that the total
variance of terms like $Var(X)$ will still equal one (see the "Notes about
Simulations" section in the appendix). For example, in Eq 1,
$\sigma_{\epsilon_{1}}=1-c_{1}^{2}-c_{3}^{2}$ and, thus, $\epsilon_{1}\sim
N(0,1-c_{1}^{2}-c_{3}^{2})$.

To obtain the simulated numbers for OLS, we use the built-in lm from R
function while for 2SLS, we use the ivreg function from the ivreg R package.
In the exclusion restriction and independence setting, we present the Monte
Carlo averages over 500 simulations of 500 observations generated. For
treatment effect heterogeneity because the empirical results take more samples
to converge, we used 500 simulations of 3000 observations. Note that for all
simulations, for demonstration purposes, we set the IV to be sufficiently
strong such that the estimates would converge on the population value within a
reasonable sample size. Nevertheless, our results otherwise hold for weak IVs
if the number of observations in each simulation increased significantly.

To demonstrate the sensitivity analysis plots, we used one generation from the
relevant set of structural equations with a sample size of 500 for the
exclusion restriction and independence violations while we have a sample size
of 1000 for the heterogeneity violation. To simulate a set of covariates that
we may use to benchmark the unobserved quantities, we generated three
independent covariates ($W_{1},W_{2},W_{3}$) from a standard normal
distribution and constructed $W$ via the first principal component.

###  5.1 Exclusion Restriction

For the case with no covariates, we set the following structural parameters
$c_{0}=0.3$, $c_{1}=0.5$, $c_{2}=0.5$, $c_{3}=0.5$, and $c_{ER}=0.25$. For the
case with covariates, we have $c_{0}=0.25$, $c_{1}=0.4$, $c_{2}=0.4$,
$c_{3}=0.7$, $c_{ER}=0.25$, $c_{5}=0.4$, and $c_{6}=0.4$. The results are
presented in Table 5.

Table 5: Theoretical and Simulated Results for the Exclusion Restriction Violation |  | Method
---|---|---
|  | OLS without Z | OLS with Z | 2SLS with Z
Without covariates | Closed Form Result | 0.375 | 0.333 | 0.5
Simulated Result | 0.374 | 0.334 | 0.496
With covariates | Closed Form Result | 0.399 | 0.457 | 0.357
Simulated Result | 0.398 | 0.459 | 0.354

Using the values $c_{1}=c_{2}=c_{5}=c_{6}=0.4$ and $c_{3}=0.3$, an IV of
relative moderate strength, we present how the sensitivity plots change in
response to an increasing exclusion violation in Figure 12. We see that since
the IV is not very strong, it will take only a small degree of the exclusion
restriction violation to reach ambivalence and suggest that OLS is more
appropriate. Note that for cases that there is no violation, benchmarking will
pick up a small signal so the IC will not be 0 though small.

Figure 12: Increasing Exclusion Restriction Violation Sensitivity Analysis
Results with Benchmarking

(a) $c_{ER}=0$ (no violation); True IR = 0

(b) $c_{ER}=0.07$; True IR = 1.09

(c) $c_{ER}=0.09$; True IR = 1.4

###  5.2 Independence

For the case with no covariates, we set the following structural parameters
$c_{0}=0.3$, $c_{1}=0.5$, $c_{2}=0.5$, $c_{3}=0.5$, $c_{ER}=0.25$. For the
case with covariates, we have $c_{0}=0.3$, $c_{1}=0.4$, $c_{2}=0.4$,
$c_{3}=0.5$, $c_{ER}=0.25$, $c_{5}=0.4$, $c_{6}=0.4$, and $c_{7}=0.25$. The
results are presented in Table 6.

Table 6: Theoretical and Simulated Results for the Independence Violation |  | Method
---|---|---
|  | OLS without Z | OLS with Z | 2SLS with Z
Without covariates | Closed Form Result | 0.312 | 0.384 | 0.2
Simulated Result | 0.311 | 0.383 | 0.200
With covariates | Closed Form Result | 0.290 | 0.391 | 0.176
Simulated Result | 0.290 | 0.394 | 0.178

In Figure 13, we present the results of increasing the independence violation
with $c_{1}=c_{2}=c_{5}=c_{6}=0.4$ and $c_{3}=0.4$, a relatively strong IV. As
we vary $c_{I}$, we set $c_{7}$ equivalently under the assumption that our
observed benchmark is roughly equal to that of unobserved violation. We see
that a relatively large violation in the independence assumption is needed to
render a strong IV more inconsistent than OLS with unobserved confounding.

Figure 13: Increasing Independence Violation Sensitivity Analysis Results with
Benchmarking

(a) $c_{I}=c_{7}=0$ (no violation); True IR = 0

(b) $c_{I}=c_{7}=0.4$; True IR = 0.98

(c) $c_{I}=c_{7}=0.6$; True IR = 1.45

###  5.3 Treatment Effect Heterogeneity

For the case with no covariates, we set the following structural parameters
$\beta_{1}=0.1$, $\beta_{2}=0.2$, $\beta_{3}=0.1$, $\alpha_{1}=0.45$,
$\alpha_{2}=0.15$, and $\alpha_{3}=0.1$. For the case with covariates, we have
$\beta_{1}=0.1$, $\beta_{2}=0.2$, $\beta_{3}=0.1$, $\beta_{4}$, $\beta_{5}$,
$\alpha_{1}=0.45$, $\alpha_{2}=0.15$, $\alpha_{3}=0.1$, $\alpha_{4}=0.15$, and
$\alpha_{5}=0.1$. The results are presented in Table 7.

Table 7: Theoretical and Simulated Results for Treatment Effect Heterogeneity |  | Method
---|---|---
|  | OLS without Z | OLS with Z | 2SLS with Z
Without covariates | Closed Form Result | 0.039 | 0.044 | 0.022
Simulated Result | 0.039 | 0.044 | 0.021
With covariates | Closed Form Result | 0.040 | 0.037 | 0.021
Simulated Result | 0.039 | 0.046 | 0.021

Figure 14 reflects the how, with increasing treatment effect heterogeneity
related to the IV, 2SLS could become more inconsistent than OLS. The
structural parameters are the same as the above except that
$\alpha_{2}=\alpha_{3}=0.2$.

Figure 14: Heterogeneity Violation Sensitivity Analysis Results with
Benchmarking

(a) $\alpha_{3}=0$ (no violation); True IR = 0

(b) $\alpha_{3}=0.2$; True IR = 1

(c) $\alpha_{3}=0.3$; True IR = 1.25

##  6 Applied Example

We demonstrate the application of our sensitivity analysis in studying
neonatal outcomes as a function impacts to the mother during pregnancy.
Specifically, we seek to measure the impact of perceived maternal stress
during the pregnancy, as measured by the perceived stress scale (PSS),[19] on
the birthweight of the neonate (in pounds). The PSS asks the participant
several questions related to different symptoms of stress in her life where,
in each item, they give a response from zero (almost never) to four (very
often) on the Likert scale. Higher scores indicated a larger degree of stress
and we treat the overall PSS score as continuous for the purposes of our
analysis. From the literature, our hypothesis is that higher stress would lead
to lower birthweights;[20] however, due to the observational nature of such
data, we are concerned about unmeasured confounding. That is, there may be
uobserved factors that affect both perceived maternal stress during pregnancy
and birthweight. Therefore, we are motivated to use IVs to isolate the causal
effect.

We study 147 mother-child dyads from ecological momentary assessment data
collected at at University of California, Irvine. More details regarding the
cohort inclusion-exclusion criteria and characteristics can be found in
Lazarides et al.[21] For our purposes, at three points during their pregnancy
(early, mid, and late), the mothers were asked to fill out a questionnaire on
their mobile devices about regarding their stress at ten times throughout the
day. Furthermore, each mother repeated the questionnaire for four consecutive
days, two of these days being weekends. For each trimester, day, and
timepoint, we calculate the current PSS and then calculate the median PSS for
that day, which is our exposure of interest for perceived stress. For example,
a mother would fill out the questionnaire ten times during Thursday, Friday,
Saturday, and Sunday. At each timepoint, we would calculate the PSS and
subsequently have four median PSS measurements for each day the data was
collected. For the purposes of the application of our sensitivity analysis
tool, we only focus on data from the second collection point (i.e. mid-
pregnancy).

A potential IV that we utilized was a binary indicator of whether the stress
measurements were collected on a weekday or a weekend with the hypothesis
being that stress is generally increased on weekdays compared to weekends.
Because we had data for both weekdays and weekends, for each mother, we
randomly sampled a single day to use as her stress measurement and recorded if
this day was a weekday or weekend. This represents a realistic sampling scheme
where the mothers may be asked only once to fill the questionnaire and they
are randomized to a weekday or weekend. Regressing median PSS upon the weekday
indicator IV corroborated our hypothesized relevance and found that weekdays
were associated with 0.213 higher points, on average, compared to weekdays
(95% CI: [0.038,0.388],$p=0.017$). Nevertheless, the IV was fairly weak with
the first-stage $F=5.775$.

For the confounders we adjusted for in both OLS and 2SLS, we utilized the
neonate’s sex, the mother’s age, how many previous children the mother has had
(i.e. parity), and an indicator for obstetric risk, which summarizes many
factors that may result in adverse pregnancy outcomes such as low birth
weight. OLS found an insignificant positive relationship where as median
stress increased by one, on average, birthweight increased by 0.05 pounds (95%
CI: [-0.212,0.313],$p=0.702$). 2SLS found an insignificant relationship but of
the opposite sign where a one point increase in median PSS was associated with
an 0.19 pound decrease in average birthweight (95% CI: [-2.587, 2.193],
$p=0.87$).

The demonstration of our approach is presented in Figure 15. All variables
were standardized before inputting it into our tool and, to form the general
confounder $W$, the confounders were combined using the first principal
component. We may first look at the benchmarks of each assumption violation,
which appear to be quite small. Even so, the benchmarked strength of
unobserved confounding is relatively small, which overall, in conjunction with
the weakness of the IV, leads us to empirically prefer using OLS rather than
2SLS.

Examining each assumption violation individually, if one believes the
unobserved confounding is double the observed confounding (the blue line), we
are in ambivalence with respect to the exclusion restriction violation but not
the independence assumption. Theoretically, there may not be a basis for the
weekday indicator to violate the IV assumptions and this is a factor the
analyst should weigh against the observed evidence. If the analyst decides to
use the IV, there may be concern that, due to exposure effect heterogeneity,
the LACE will be far from the ACE. In this specific example, we would be
concerned that there is an unobserved covariate that modifies the relationship
between median PSS and birthweight in addition to modifying the effect of the
weekday indicator on median PSS. Fortunately, the corresponding plot in Figure
15 shows that benchmarked heterogeneity related to the IV is low and will not
cause large inconsistency in comparison to benchmarked unobserved confounding
(unless we assign a multiplier of $0.5$). Note that we only have one plot for
treatment effect heterogeneity because both cases related to the signs yield
similar plots and conclusions.

Figure 15: Examining the Three Assumptions for the Weekday Indicator IV

##  7 Discussion

In this paper, we have investigated two predominant ways that analysts may
isolate causal effects: confounders, or the CAC, and via IV, or the IVAC.
Furthermore, we have based our study on the notion that each approach may work
imperfectly due to assumption violation as is most plausible in the vast
majority of real world settings. Our closed-form results that capture
interpretable rules of thumb based on DAG edge weights and coefficients, as
well as our sensitivity analysis procedure, help guide analysts towards a
practical philosophy on how one may execute observational studies. If the goal
is to obtain an estimate of the ACE and there is a set of tools that one must
choose from, such as the CAC and IVAC, one must ask: when is one tool more
advantageous than the other? We have ultimately broken down this question by
juxtaposing the degree of unobserved confounding to the degree IV assumption
violation, shifting away from perfectionism and providing results for analysts
to offer evidence that the estimate computed was the least inconsistent it
could have been given the scenario.

Our results show that properly defining confounders and IVs in the study
paradigm is only a starting point. Whether a variable will be used in CAC or
as an IV in IVAC is not necessarily congruent with its formal definition. In
fact, we have presented analytically that, relative to OLS, there are
scenarios where a variable should be used as an IV even though it does not
meet the strict definition of an IV. For example, in an exclusion restriction
violation the variable used as an IV is, by definition, a confounder and,
although such a variable is not a "perfect IV," the relative performance of
2SLS would be better than that of OLS. Our sensitivity analysis procedure
assists in detecting this by using the observed data.

Another relevant scenario lies in the fact that even though we may have a
valid IV, the LACE estimand from 2SLS may be further away from the ACE than an
estimand from OLS that is impacted by unobserved confounding. Through our
closed form results and sensitivity analysis, we allow an analyst to judge how
large heterogeneity would need to be in order for this to occur. Whether the
resulting IV estimand remains scientifically useful is a subject of debate
that we will not discuss here.[22] Rather, we are interested in directly
targeting the ACE and will assume that treatment effect heterogeneity may
provide a barrier to this goal. In this sense, the results from our paper may
provide case-by-case evidence for and against those who may argue IV analyses
may still be useful for the ACE.

Moving to the use of our sensitivity analysis tools, we essentially provide
three separate plots for three distinct assumption violation scenarios. If all
three plots all agree that either 2SLS is superior or OLS is superior then the
suggested approach is clear to the analyst as was largely the case with the
birthweight example. However, one may encounter a scenario that the graphs
disagree: for example, the exclusion restriction and independence show 2SLS is
superior while the treatment effect heterogeneity graph does not. In this
case, one should rank the importance of each assumption and consider the
observed benchmarks for each assumption violation. Using the multipliers
provided or inputting a custom multiplier, one could consider the degree of
unobserved confounding across the board in order for all three graphs to agree
and evaluate whether these findings are reasonable in the particular study
scenario. Similarly, one may further use the implied assumption violations in
the legend relative to the benchmarked assumption violations for these
purposes. Ultimately, sensitivity analyses are matters of judgement and we aim
to provide quantitative tools for analysts to navigate these scenarios.

The results in this paper also have implications on more sophisticated, data-
driven variable selection techniques. Users of CAC often will opt to capture
confounding by adjusting for the propensity score (PS) using the fitted values
of the exposure regressed on the confounders. In order to reduce Type I error,
one would model the PS away from the outcome and, thus, it appears reasonable
to use penalization or machine learning (ML) techniques to optimize prediction
error. There are two main issues with this automated procedure: (1) strong
though imperfect IVs would be selected leading to possible bias amplification
and (2) omitted variable bias may occur due to shrinkage to zero of important
confounders. These points have been mentioned by others[5, 3, 23] and our
results further provide analysts information to act in the face of such
issues. We may use our results to quantify how strong the confounder or IV
should be in order to avoid adjustment altogether or, instead, use it in 2SLS.
Future directions may include extending the framework developed in this paper
to techniques such as augmented inverse probability of treatment weighting
(AIPTW), post-LASSO (PL), targeted minimum loss estimation (TMLE), and double
machine learning (DML) to weigh the approaches against one another.[24, 25,
26]

There are several avenues for future research. Firstly, in this paper, we
focus only on consistency but in estimation, we may also want to know whether
CAC may more efficient than IVAC or vice-versa. Furthermore, if the confidence
intervals overlap on the point estimate, which would cause ambivalence. We may
additionally wish to move beyond the setting with a continuous exposure and
outcome. Nevertheless, if one believes linear probability models (LPMs) are
appropriate for the study’s context, then one may extend our results to a
binary exposure and outcome. It has been shown that if the probabilities
produced by the LPM are not outside of the range of $[0,1]$, or that the
probabilities of exposure or outcome are not extreme in the population, then
OLS and 2SLS may still give consistent results.[27, 28] For the sensitivity
analysis procedure, the results are only as useful as variables available and
chosen for benchmarking, which is partially mitigated by using the
multipliers. Certainly, other benchmarking quantities for the unobserved
quantities than the ones we chose could be used. The aggregation of the
covariates into one general confounder $W$ could be done via other methods
besides PCA, which is limited when there are categorical variables. Using non-
continuous variables also is limited when capturing associations using $R^{2}$
due to the Frechet bounds on the correlation between non-continuous variables
being potentially far narrower than $[-1,1]$.[29]

Isolating causal effects in observational data presents many challenges, which
foremost include the effect of unobserved confounding. The CAC and IVAC offer
potential avenues to mitigate this confounding. Even so, under untestable
assumptions, choosing the optimal approach for the problem at hand involves
much conjecture. Our closed-form findings and sensitivity analysis approach
helps analysts quantitatively justify the approach that they ultimately
believe produces an estimate closest to the ACE. The upshot is that, with the
information we provide, results from observational studies will both be more
transparent and more useful in their interpretation.

##  8 Acknowledgements

RSZ and DLG are supported by NIH/NIA P30AG066519 and NIH/NIA 1RF1AG075107.

##  9 Bibliography

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##  10 Appendix

###  10.1 Proof for the Consistency $\hat{A}_{4}$ under Figure 1

###### Proof.

We begin by defining following system of structural equations under Figure 1
where $E[\epsilon_{1}|U,Z]=0$ and $E[\epsilon_{2}|X,U]=0$.

$X=\alpha_{1}U+\alpha_{2}Z+\epsilon_{1}$ (44)
$Y=\beta_{1}X+\beta_{2}U+\epsilon_{2}$ (45)

By the standard definition of the IV estimator,
$\hat{A}_{4}=\frac{\widehat{Cov}(Y,Z)}{\widehat{Cov}(X,Z)}$. Where $n$ is the
sample size, by finite variance and Slutsky’s theorem we can write:

$\displaystyle\frac{\widehat{Cov}(Y,Z)}{\widehat{Cov}(X,Z)}$
$\displaystyle=\frac{n^{-1}\sum_{i=1}^{n}y_{i}z_{i}}{n^{-1}\sum_{i=1}^{n}x_{i}z_{i}}\overset{p}{\to}\frac{E[YZ]}{E[XZ]}=\frac{Cov(Y,Z)}{Cov(X,Z)}$
(46)

Clearly, $Cov(X,Z)=\alpha_{2}=c_{3}$ because $Z\perp\\!\\!\\!\\!\perp U$.
Moving onto $Cov(Y,Z)$ we write:

$\displaystyle E[Y|z]$ $\displaystyle=\sum_{u}E[Y|z,u]P(u|z)$ (47)
$\displaystyle=\sum_{u}[c_{0}c_{3}z+c_{2}u]P(u)$ (48)
$\displaystyle=c_{0}c_{3}z\sum_{u}u+c_{2}\sum_{u}uP(u)$ (49)
$\displaystyle=c_{0}c_{3}z$ (50)

∎

###  10.2 Proof for Proposition 3.1

###### Proof.

Using an equivalent definition of the IV estimator,
$\hat{A}_{4}=(Z^{T}X)^{-1}(Z^{T}Y)$ we can substitute 9 for $Y$

$\displaystyle(Z^{T}X)^{-1}(Z^{T}Y)$
$\displaystyle=(Z^{T}X)^{-1}(Z^{T}(c_{0}X+c_{2}U+c_{ER}Z+\epsilon_{3}))$ (51)
$\displaystyle=c_{0}+c_{2}(Z^{T}X)^{-1}Z^{T}U+c_{ER}(Z^{T}X)^{-1}Z^{T}Z+(Z^{T}X)^{-1}Z^{T}\epsilon_{3}$
(52)
$\displaystyle=c_{0}+c_{2}\frac{n^{-1}\sum_{i=1}^{n}z_{i}u_{i}}{n^{-1}\sum_{i=1}^{n}z_{i}x_{i}}+c_{ER}\frac{n^{-1}\sum_{i=1}^{n}z_{i}^{2}}{n^{-1}\sum_{i=1}^{n}z_{i}x_{i}}+\frac{n^{-1}\sum_{i=1}^{n}\epsilon_{3i}x_{i}}{n^{-1}\sum_{i=1}^{n}z_{i}x_{i}}$
(53)
$\displaystyle\overset{p}{\to}c_{0}+c_{2}\frac{Cov(Z,U)}{Cov(Z,X)}+c_{ER}\frac{Var(Z)}{Cov(Z,X)}+\frac{Cov(Z,\epsilon_{3})}{Cov(Z,X}$
(54) $\displaystyle=c_{0}+\frac{c_{ER}}{c_{3}}$ ($Z\perp\\!\\!\\!\\!\perp U$
and $Cov(X,Z)=c_{3}$) (55)

The last line follows from the fact that $Z\perp\\!\\!\\!\\!\perp U$ and
$E[\epsilon_{3}Z]=0$ by iterated expectation. ∎

###  10.3 Proofs for Proposition 3.2

####  10.3.1 Proof for the Consistency of $\hat{A}_{3}$

###### Proof.

We need to find the value of $\frac{\partial}{\partial
x}E[Y|x,z]=c_{0}+c_{2}\frac{\partial}{\partial x}E[U|x,z]$. We will use the
FWL theorem to find $\frac{\partial}{\partial x}E[U|x,z]$ by orthogonalizing
$Z$ and using consistency. Assuming no intercept, we can write the quantity as
$\frac{X^{T}M_{Z}U}{X^{T}M_{Z}X}$ where $M_{Z}=I-Z^{T}(Z^{T}Z)^{-1}Z^{T}$
(residual making matrix). Noting that $Cov(X,Z)=c_{3}+c_{1}c_{I}$ and
$Cov(U,Z)=c_{I}$, we have:

$\displaystyle\frac{X^{T}M_{Z}U}{X^{T}M_{Z}X}$
$\displaystyle=\frac{\sum_{i=1}^{n}[x_{i}-(c_{3}+c_{1}c_{I})z_{i}][u_{i}-c_{I}z_{i}]}{\sum_{i=1}^{n}[x_{i}-(c_{3}+c_{1}c_{I})z_{i}]^{2}}$
$\displaystyle\overset{p}{\to}\frac{Cov[X-(c_{3}+c_{1}c_{I})Z,U-c_{I}Z]}{Var[X-(c_{3}+c_{1}c_{I})Z]}$
(56)
$\displaystyle=\frac{c_{1}+c_{3}c_{I}-c_{I}(c_{3}+c_{1}c_{I})}{1-(c_{3}+c_{1}c_{I})^{2}}=\frac{c_{1}(1-c_{I}^{2})}{1-(c_{3}+c_{1}c_{I})^{2}}$
(57)

∎

####  10.3.2 Proof for the Consistency of $\hat{A}_{4}$

###### Proof.

In a similar derivation Proposition 3.1 but plugging in
$c_{0}X+c_{2}U+\epsilon_{3}$ for the structural equation of $Y$, we obtain

$\displaystyle\hat{A}_{4}$
$\displaystyle\overset{p}{\to}c_{0}+c_{2}\frac{Cov(Z,U)}{Cov(Z,X)}+\frac{Cov(Z,\epsilon_{3})}{Cov(Z,X)}$
(58) $\displaystyle=c_{0}+c_{2}\frac{Cov(Z,U)}{Cov(Z,X)}$ (59)
$\displaystyle=c_{0}+\frac{c_{2}c_{I}}{c_{3}+c_{1}c_{I}}$ (60)

Note that the unconditional association between $X$ and $Z$ goes has two
paths: the direct path $X\to Z$ and the indirect path $X\leftarrow U\to Z$. ∎

###  10.4 Proofs for Proposition 3.3

####  10.4.1 Proof for the consistency of $\hat{A}_{2}$

###### Proof.

We simply will compute the regression of $Y$ on $X$ where
$\hat{A}_{2}\overset{p}{\to}Cov(Y,X)$.

$\displaystyle Cov(Y,X)$
$\displaystyle=Cov(\beta_{1}X+\beta_{2}U+\beta_{3}XU+\epsilon_{2},X)=\beta_{1}+\beta_{2}\alpha_{2}+\beta_{3}Cov(XU,X)$
(61)
$\displaystyle=\beta_{1}+\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}$
(62)

Where $Cov(XU,X)=2\alpha_{1}\alpha_{3}$ because

$\displaystyle Cov(XU,X)$
$\displaystyle=Cov(U(\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1}),X)$
(63)
$\displaystyle=\alpha_{1}Cov(UZ,X)+\alpha_{2}Cov(U^{2},X)+\alpha_{3}Cov(ZU^{2},X)$
(64)
$\displaystyle=\underbrace{\alpha_{1}E[UZX]}_{B_{1}}+\underbrace{\alpha_{2}E[U^{2}X]}_{B_{2}}+\underbrace{\alpha_{3}E[ZU^{2}X]}_{B_{3}}$
(65)

$B_{1}=\alpha_{1}\alpha_{3}$ because

$\displaystyle E[XUZ]$
$\displaystyle=E[ZE[XU|Z]]=E[ZE[(\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1})U|Z]]$
(66)
$\displaystyle=E[Z(\alpha_{1}E[UZ|Z]+\alpha_{2}E[U^{2}|Z]+\alpha_{3}E[ZU^{|}Z]+E[U\epsilon_{1}|Z)]$
(67)
$\displaystyle=E[\alpha_{1}Z^{2}E[U]+\alpha_{2}ZE[U^{2}]+\alpha_{3}ZE[U^{2}]+E[U\epsilon_{1}]]$
(68) $\displaystyle=E[\alpha_{2}Z+\alpha_{3}Z^{2}]=\alpha_{3}$ (69)

$B_{2}=0$ because $E[U^{2}X]=E[U^{2}E[X|U]]=\alpha_{2}E[U^{3}]=0$ because
$E[U^{3}]=0$.

$B_{3}=\alpha_{1}\alpha_{3}$ because

$\displaystyle E[ZU^{2}X]$
$\displaystyle=E[ZU^{2}(\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1})]$
(70)
$\displaystyle=\alpha_{1}E[Z^{2}U^{2}]+\alpha_{2}E[ZU^{3}]+\alpha_{3}E[Z^{2}U^{3}]$
(71) $\displaystyle=\alpha_{1}$ (72)

Where the last equality follows because $Z\perp\\!\\!\\!\\!\perp U$ and
$E[U^{3}]=0$. ∎

####  10.4.2 Proof for the consistency of $\hat{A}_{3}$

###### Proof.

Similar to proposition 3.2, we will proceed via FWL to find the value of
$A_{3}=\frac{X^{T}M_{Z}Y}{X^{T}M_{Z}X}$. First, we need to find $Cov(Z,X)$ and
$Cov(Z,Y)$:

$\displaystyle Cov(Z,X)$
$\displaystyle=Cov(Z,\alpha_{1}Z+\alpha_{2}U+\alpha_{3}ZU+\epsilon_{1})=\alpha_{1}$
(73) and (74) $\displaystyle Cov(Z,Y)$
$\displaystyle=Cov(Z,\beta_{1}X+\beta_{2}U+\beta_{X}U+\epsilon_{2})=E[Z(\beta_{1}X+\beta_{2}U+\beta_{X}U+\epsilon_{2})]$
(75)
$\displaystyle=E[\beta_{1}XZ+\beta_{2}UZ+\beta_{3}XUZ+Z\epsilon_{2}]=\alpha_{1}\beta_{1}+\alpha_{3}\beta_{3}$
(76)

Going back to FWL, letting $\lambda=\alpha_{1}\beta_{1}+\alpha_{3}\beta_{3}$,
we now have

$\displaystyle\frac{X^{T}M_{Z}Y}{X^{T}M_{Z}X}$
$\displaystyle=\frac{(X-\alpha_{1}Z)^{T}(Y-\lambda
Z)}{(X-\alpha_{1}Z)^{T}(X-\alpha_{1}Z)}=\frac{\sum_{i=1}^{n}X_{i}Y_{i}-\lambda
X_{i}Z_{i}-\alpha_{1}Z_{i}Y_{i}+\alpha_{1}\lambda
Z_{i}^{2}}{\sum_{i=1}^{n}(X_{i}-\alpha_{1}Z_{i})^{2}}$ (77)
$\displaystyle\overset{p}{\to}\frac{E[X_{i}Y_{i}-\lambda
X_{i}Z_{i}-\alpha_{1}Z_{i}Y_{i}+\alpha_{1}\lambda
Z^{2}]}{E[(X_{i}-\alpha_{1}Z_{i})^{2}]}$ (78)
$\displaystyle=\frac{Cov(X,Y)-\lambda
Cov(X,Z)-\alpha_{1}Cov(Z,Y)+\alpha_{1}\lambda Var(X)}{1-\alpha_{1}^{2}}$ (79)
$\displaystyle=\frac{(\beta_{1}+\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3})-\lambda\alpha_{1}-\alpha_{1}\lambda+\alpha_{1}\lambda}{1-\alpha_{1}^{2}}$
(80)
$\displaystyle=\frac{\beta_{1}+\alpha_{2}\beta_{2}+2\alpha_{1}\alpha_{3}\beta_{3}-\alpha_{1}^{2}\beta_{1}+\alpha_{3}\beta_{3}\alpha_{1}}{1-\alpha_{1}^{2}}=\beta_{1}+\frac{\alpha_{2}\beta_{2}+\alpha_{1}\alpha_{3}\beta_{3}}{1-\alpha_{1}^{2}}$
(81)

∎

####  10.4.3 Proof for the consistency of $\hat{A}_{4}$

###### Proof.

We can compute the Wald estimator from the quantities computed in the proof
for $\hat{A}_{3}$. In particular, $Cov(Z,Y)$ and $Cov(Z,X)$.

$\displaystyle\hat{A}_{4}$
$\displaystyle\overset{p}{\to}\frac{Cov(Z,Y)}{Cov(Z,X)}=\frac{\alpha_{1}\beta_{1}+\alpha_{3}\beta_{3}}{\alpha_{1}}=\beta_{1}+\frac{\alpha_{3}\beta_{3}}{\alpha_{1}}$
(82)

∎

###  10.5 Proofs for Proposition 3.5

####  10.5.1 Proof for the consistency of $\hat{A}_{2}$

###### Proof.

Our goal is to find $\frac{\partial}{\partial x}E[Y|x,w]$, which, by FWL, is
equivalent to the convergence in probability of
$\frac{X^{T}M_{W}Y}{X^{T}M_{W}X}$. Because we have already demonstrated the
use of FWL, we will skip several intermediate steps. Letting
$\eta=c_{6}+c_{5}c_{0}$

$\displaystyle\frac{X^{T}M_{W}Y}{X^{T}M_{W}X}$
$\displaystyle=\frac{(X-c_{5}W)^{T}(Y-\eta
W}{((X-c_{5}W))}\overset{p}{\to}\frac{Cov(X,Y)-\eta
Cov(W,Y)+c_{5}\eta}{1-c_{5}^{2}}$ (83)
$\displaystyle=\frac{c_{0}+c_{1}c_{2}+c_{2}c_{3}c_{I}-\eta c_{5}-\eta
c_{5}+c_{5}\eta)}{1-c_{5}^{2}}=c_{0}+\frac{c_{1}c_{2}+c_{2}c_{3}c_{I}}{1-c_{5}^{2}}$
(84)

∎

####  10.5.2 Proof for the consistency of $\hat{A}_{3}$

###### Proof.

Our goal is to find $\frac{\partial}{\partial x}E[Y|x,w,z]$, which, by FWL, is
equivalent to the limit of $\frac{X^{T}M_{\mathbb{B}}Y}{X^{T}M_{\mathbb{B}}X}$
where $\mathbb{B}=[W,Z]$ or the residuals of regressing both $W$ and $Z$. To
simplify matters, we can expand $Y$

$\displaystyle\frac{X^{T}M_{\mathbb{B}}Y}{X^{T}M_{\mathbb{B}}X}$
$\displaystyle=\frac{X^{T}M_{\mathbb{B}}(c_{0}+c_{2}U+c_{6}W+\epsilon_{2})}{X^{T}M_{\mathbb{B}}X}$
(85)
$\displaystyle=c_{0}+c_{2}\frac{X^{T}M_{\mathbb{B}}U}{X^{T}M_{\mathbb{B}}X}$
(86)

Where the last line follows from the orthogonality of $\epsilon_{2}$ and the
result of $W$ being regressed upon itself being that the residuals are $0$. So
we are focused on two regressions: $E[X|w,z]=\gamma_{1}W+\gamma_{2}Z$ and
$E[U|w,z]=\pi_{1}W+\pi_{2}Z$. Using FWL, we find the value of the coefficients
to be

$\displaystyle\gamma_{1}$
$\displaystyle=\frac{W^{T}M_{Z}X}{W^{T}M_{Z}W}\overset{p}{\to}c_{5}-\frac{c_{1}c_{I}c_{7}}{1-c_{7}^{2}},$
(87) $\displaystyle\gamma_{2}$
$\displaystyle=\frac{Z^{T}M_{W}X}{Z^{T}M_{W}Z}\overset{p}{\to}c_{3}+\frac{c_{1}c_{I}}{1-c_{7}^{2}},$
(88) $\displaystyle\pi_{1}$
$\displaystyle=\frac{W^{T}M_{Z}U}{W^{T}M_{Z}W}\overset{p}{\to}\frac{-c_{I}c_{7}}{1-c_{7}^{2}},$
(89) $\displaystyle\pi_{2}$
$\displaystyle=\frac{Z^{T}M_{W}U}{Z^{T}M_{W}Z}\overset{p}{\to}\frac{c_{I}}{1-c_{7}^{2}}$
(90)

Putting this altogether, we write
$A_{3}=c_{0}+c_{2}\frac{Cov(X-\gamma_{1}W-\gamma_{2}Z,U-\pi_{1}W-\pi_{2}Z)}{Var(X-\gamma_{1}W-\gamma_{2}Z)}$
(note that we are now in the limit). Expanding the numerator we get

$\displaystyle c_{2}$
$\displaystyle\bigg{[}Cov(X,U)-\pi_{1}Cov(X,W)-\pi_{2}Cov(X,Z)-\gamma_{1}Cov(W,U)+\gamma_{1}\pi_{1}$
(91)
$\displaystyle+\gamma_{1}\pi_{2}Cov(W,Z)-\gamma_{2}Cov(Z,U)+\gamma_{2}\pi_{1}Cov(W,Z)+\gamma_{2}\pi_{2}\bigg{]}$
(92)

Noting that $Cov(X,U)=c_{1}+c_{3}c_{I}$, $Cov(X,W)=c_{5}+c_{3}c_{7}$,
$Cov(X,Z)=c_{3}+c_{1}c_{I}+c_{5}c_{7}$, $Cov(W,U)=0$, $Cov(W,Z)=c_{7}$, and
$Cov(Z,U)=c_{I}$, we can simplify the above expression to

$\displaystyle
c_{1}c_{2}+\frac{c_{1}c_{2}c_{I}^{2}c_{7}}{1-c_{7}^{2}}-\frac{c_{1}c_{2}c_{I}^{2}c_{7}}{(1-c_{7}^{2})^{2}}+\frac{c_{1}c_{2}c_{I}c_{7}^{3}}{(1-c_{7}^{2})^{2}}-\frac{c_{1}c_{2}c_{I}^{2}}{1-c_{7}^{2}}$
(93)
$\displaystyle=c_{1}c_{2}+\frac{c_{1}c_{2}c_{I}^{2}c_{7}-c_{1}c_{2}c_{I}^{2}}{1-c_{7}^{2}}+\frac{c_{1}c_{2}c_{I}^{2}c_{7}(c_{7}^{2}-1}{(1-c_{7}^{2})^{2}}=c_{1}c_{2}+\frac{c_{1}c_{2}c_{I}^{2}c_{7}-c_{1}c_{2}c_{I}^{2}-c_{1}c_{2}c_{I}^{2}c_{7}}{1-c_{7}^{2}}$
(94) $\displaystyle=c_{1}c_{2}-\frac{c_{1}c_{2}c_{I}^{2}}{1-c_{7}^{2}}$ (95)

For the denominator

$\displaystyle Var(X-\gamma_{1}W-\gamma_{2}Z)$
$\displaystyle=1+\gamma_{1}^{2}+\gamma_{2}^{2}-2\gamma_{1}Cov(X,W)-2\gamma_{2}Cov(X,Z)+2\gamma_{1}\gamma_{2}Cov(W,Z)$
(96)

After combining like terms we obtain

$\displaystyle
1-c_{5}^{2}-c_{3}^{2}+\frac{c_{1}^{2}c_{I}^{2}}{(1-c_{7}^{2})^{2}}-2c_{3}c_{5}c_{7}-2c_{1}c_{3}c_{I}-2\frac{c_{1}^{2}c_{I}^{2}}{1-c_{7}^{2}}-\frac{c_{1}^{2}c_{I}^{2}c_{7}^{2}}{(1-c_{7}^{2})^{2}}$
(98)
$\displaystyle=1-c_{5}^{2}-c_{3}^{2}-2c_{3}c_{5}c_{7}-2c_{1}c_{3}c_{I}-\frac{c_{1}^{2}c_{I}^{2}}{1-c_{7}^{2}}+[c_{3}^{2}c_{7}^{2}-c_{3}^{2}c_{7}^{2}]$
(99)
$\displaystyle=1-(c_{5}+c_{3}c_{7})^{2}+c_{3}^{2}c_{7}^{2}-c_{3}^{2}-2c_{1}c_{3}c_{I}-\frac{c_{1}^{2}c_{I}^{2}}{1-c_{7}^{2}}$
(100)
$\displaystyle=1-(c_{5}+c_{3}c_{7})^{2}-(1-c_{7}^{2})(c_{3}^{2}+\frac{2c_{1}c_{3}c_{I}}{1-c_{7}^{2}}+\frac{c_{1}^{2}c_{I}^{2}}{1-c_{7}^{2}})$
(101)
$\displaystyle=1-(c_{5}+c_{3}c_{7})^{2}-(1-c_{7}^{2})(c_{3}+\frac{c_{1}c_{I}}{1-c_{7}^{2}})^{2}$
(102)

∎

####  10.5.3 Proof for the consistency of $\hat{A}_{4}$

###### Proof.

Via FWL, $\frac{\frac{\partial}{\partial z}E[Y|z,w]}{\frac{\partial}{\partial
z}E[X|z,w]}=\frac{}{}=\frac{Z^{T}M_{W}Y}{Z^{T}M_{W}X}$. Re-using quantities
from the proof for $A_{3}$ and, additionally
$Cov(Z,Y)=c_{0}c_{3}+c_{2}c_{I}+c_{6}c_{7}+c_{0}c_{5}c_{7}+c_{0}c_{1}c_{I}$
and $Cov(Y,W)=c_{0}c_{5}+c_{6}+c_{0}c_{3}c_{7}$ we have

$\displaystyle\frac{Z^{T}M_{W}Y}{Z^{T}M_{W}X}$
$\displaystyle\overset{p}{\to}\frac{Cov(Z,Y)-Cov(Z,W)Cov(Y,W)}{Cov(X,Z)-Cov(Z,W)Cov(X,W)}$
(103)
$\displaystyle=\frac{c_{0}c_{3}+c_{2}c_{I}+c_{6}c_{7}+c_{0}c_{5}c_{7}+c_{0}c_{1}c_{I}-c7(c_{0}c_{5}+c_{6}+c_{0}c_{3}c_{7})}{c_{3}+c_{1}c_{I}+c_{5}c_{7}-c_{7}(c_{5}+c_{3}c_{7})}$
(104)
$\displaystyle=\frac{c_{0}(c_{3}+c_{1}c_{I}-c_{7}^{2}c_{3})+c_{2}c_{I}}{c_{3}+c_{1}c_{I}-c_{7}^{2}c_{3}}=\frac{c_{2}c_{I}}{c_{3}(1-c_{7}^{2})+c_{1}c_{I}}$
(105)

∎

###  10.6 Proof for Proposition 3.6

####  10.6.1 Results for $A_{2}$

###### Proof.

Our goal is to find the value of $\frac{\partial}{\partial
x}\frac{X^{T}M_{\mathbb{C}}Y}{X^{T}M_{\mathbb{C}}X}$ where
$\mathbb{C}=[W,XW]$. Because $Cov(W,XW)=0$, we can write the following linear
conditional expectations

$\displaystyle
E[X|W,XW]=Cov(X,W)W+\frac{Cov(X,XW)}{Var(XW)}XW=\alpha_{4}W+\frac{2\alpha_{1}\alpha_{5}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}XW,$
(106) $\displaystyle
E[Y|W,XW]=Cov(Y,W)W+\frac{Cov(Y,XW)}{Var(XW)}XW=(\beta_{1}\alpha_{4}+\beta_{4})W+\frac{2\alpha_{1}\alpha_{5}\beta_{1}+\beta_{5}(\alpha_{4}^{2}+2\alpha_{5}^{2}+1)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}XW.$
(107)

Returning back to the FWL expression, we now plug in the above conditional
expectations. Focusing on the numerator, which is rewritten as
$Cov(X-E[X|W,XW],Y-E[X|W,XW])$, we have

$\displaystyle
Cov(X-\alpha_{4}W-\frac{2\alpha_{1}\alpha_{5}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}XW,Y-\beta_{1}\alpha_{4}+\beta_{4}W-\frac{2\alpha_{1}\alpha_{5}\beta_{1}+\beta_{5}(\alpha_{4}^{2}+2\alpha_{5}^{2}+1)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}XW$
(108)
$\displaystyle=Cov(X,Y)-\frac{\frac{2\alpha_{1}\alpha_{5}\beta_{1}+\beta_{5}(\alpha_{4}^{2}+2\alpha_{5}^{2}+1)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}Cov(X,XW)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}-\alpha_{4}Cov(W,Y)-$
(109)
$\displaystyle\frac{2\alpha_{4}\alpha_{5}Cov(XW,Y)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}+\frac{\frac{2\alpha_{1}\alpha_{5}\beta_{1}+\beta_{5}(\alpha_{4}^{2}+2\alpha_{5}^{2}+1)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}2\alpha_{1}\alpha_{5}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}$
(110)
$\displaystyle=\beta_{1}+\beta_{2}\alpha_{2}+2\alpha_{1}\alpha_{3}\beta_{3}+\beta_{4}\alpha_{4}+2\alpha_{1}\alpha_{5}\beta_{5}-\alpha_{4}(\beta_{1}\alpha_{4}+\beta_{4})-\frac{2\alpha_{1}\alpha_{5}\frac{2\alpha_{1}\alpha_{5}\beta_{1}+\beta_{5}(\alpha_{4}^{2}+2\alpha_{5}^{2}+1)}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}$
(111)
$\displaystyle=\beta_{1}+\beta_{2}\alpha_{2}+2\alpha_{1}\alpha_{3}\beta_{3}-\alpha_{4}^{2}\beta_{1}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}\beta_{1}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}.$
(112)

Focusing on the denominator:

$\displaystyle
Var(X-\alpha_{4}W-\frac{2\alpha_{1}\alpha_{5}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}XW)=1+\alpha^{4}+\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}-2\alpha_{4}^{2}-\frac{8\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}$
(113)
$\displaystyle=1-\alpha_{4}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}.$
(114)

Thus, we obtain the final result
$\beta_{1}+\frac{\beta_{2}\alpha_{2}+2\alpha_{1}\alpha_{3}\beta_{3}}{1-\alpha_{4}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}}$.
∎

####  10.6.2 Results for $A_{4}$

Due to there being two endogenous variables, $X$ and $XW$, we will need to
utilize two instruments, which are $Z$ and $ZW$, respectively. We will have
the following series of linear conditional expectations, noting that
$Cov(XW,W)=0$

$\displaystyle E[X|W,Z,ZW]=Cov(X,W)W+Cov(Z,X)Z+Cov(ZW,X)ZW,$ (115)
$\displaystyle E[XW|Z,ZW]=Cov(XW,Z)Z+Cov(XW,ZW)ZW,$ (116) $\displaystyle
E[Y|\hat{X},\hat{XW},W]=\frac{Cov(Y,\hat{X})}{Var(\hat{X})}\hat{X}+\frac{Cov(Y,\hat{XW})}{Var(\hat{XW})}\hat{XW}+Cov(Y,W)W$
(117)

where $\hat{X}$ and $\hat{XW}$ are the fitted values from the $E[X|W,Z,ZW]$
and $E[XW|Z,ZW]$, respectively. The coefficient of interest from 2SLS is thus
$\frac{Cov(Y,\hat{X})}{Var(\hat{X})}$ or, returning to FWL, denoted as
$\frac{\partial}{\partial\hat{x}}\frac{\hat{X}M_{\mathbb{D}}Y}{\hat{X}M_{\mathbb{D}}\hat{X}}$
where $\mathbb{D}=[W,\hat{XW}]$.

First, we will write out the relevant covariances and variances:

$\displaystyle Cov(W,\hat{X})$ $\displaystyle=\alpha_{4}$ (118) $\displaystyle
Cov(\hat{XW},\hat{X})$ $\displaystyle=2\alpha_{1}\alpha_{5}$ (119)
$\displaystyle Cov(W,Y)$ $\displaystyle=\beta_{1}\alpha_{4}+\beta_{4}$ (120)
$\displaystyle Cov(\hat{XW},Y)$
$\displaystyle=\alpha_{5}\beta_{1}Cov(Z,X)+\alpha_{5}\beta_{3}Cov(Z,XU)+\beta_{5}\alpha_{5}Cov(Z,XW)$
(121)
$\displaystyle+\alpha_{1}\beta_{1}Cov(ZW,X)+\alpha_{1}\beta_{3}Cov(ZW,XU)+\alpha_{1}\beta_{5}Cov(ZW,XW)$
(122)
$\displaystyle=2\alpha_{1}\alpha_{5}\beta_{1}+\alpha_{3}\alpha_{5}\beta_{3}+\beta_{5}\alpha_{5}^{2}+\alpha_{1}^{2}\beta_{5}$
(123) $\displaystyle Var(\hat{X)}$
$\displaystyle=\alpha_{4}^{2}+\alpha_{1}^{2}+\alpha_{5}^{2}$ (124)
$\displaystyle Var(\hat{XW})$ $\displaystyle=\alpha_{5}^{2}+\alpha_{1}^{2}$
(125) $\displaystyle Cov(\hat{X},Y)$
$\displaystyle=\alpha_{4}Cov(W,Y)+\alpha_{1}Cov(Z,Y)+\alpha_{5}Cov(ZW,Y)$
(126)
$\displaystyle=\alpha_{4}(\beta_{1}\alpha_{4}+\beta_{4})+\alpha_{1}(\alpha_{1}\beta_{1}+\alpha_{3}\beta_{3}+\alpha_{5}\beta_{5})+\alpha_{5}(\beta_{1}\alpha_{5}+\beta_{5}\alpha_{1})$
(127) $\displaystyle Cov(ZW,Y)=\beta_{1}\alpha_{5}+\beta_{5}\alpha_{1}$ (128)

Because $Cov(\hat{XW},W)=0$ we can simply substitute the covariances in the
FWL expression as such

$\displaystyle\frac{Cov(\hat{X}-\alpha_{4}W-\frac{2\alpha_{1}\alpha_{5}}{\alpha^{5}+\alpha_{1}^{2}}\hat{XW},Y-(\beta_{1}\alpha_{4}+\beta_{4})W-Cov(\hat{XW},Y)\hat{XW})}{\hat{X}-\alpha_{4}W-\frac{2\alpha_{1}\alpha_{5}}{\alpha^{5}+\alpha_{1}^{2}}\hat{XW}}$
(129)

Now focusing on the numerator, can further simplify to

$\displaystyle
Cov(\hat{X},Y)-(\beta_{1}\alpha_{4}+\beta_{4})Cov(\hat{X}W)-\frac{Cov(\hat{XW},Y)}{\alpha_{5}^{2}+\alpha_{1}^{2}}-\frac{2\alpha_{1}\alpha_{5}Cov(\hat{XW},Y)}{\alpha^{5}+\alpha_{1}^{2}}$
(130)
$\displaystyle+\frac{2\alpha_{1}\alpha_{5}Cov(\hat{XW},Y)Var(\hat{XW})}{(\alpha^{5}+\alpha_{1}^{2})^{2}}$
(131)
$\displaystyle=\alpha_{1}(\alpha_{1}\beta_{1}+\alpha_{3}\beta_{3}+\alpha_{5}\beta_{5})+\alpha_{5}(\beta_{1}\alpha_{5}+\beta_{5}\alpha_{1})-\frac{2\alpha_{1}\alpha_{5}Cov(\hat{XW},Y)}{\alpha^{5}+\alpha_{1}^{2}}$
(132)
$\displaystyle=\alpha_{1}\beta_{1}+\alpha_{1}\alpha_{3}\beta_{3}+2\alpha_{1}\alpha_{5}\beta_{5}+\beta_{1}\alpha_{5}^{2}$
(133)
$\displaystyle-\frac{4\alpha_{1}\alpha_{5}^{2}\beta_{1}+2\alpha_{1}\alpha_{3}\alpha_{5}^{2}\beta_{3}+2\alpha_{1}\alpha_{5}^{3}\beta_{5}+2\alpha_{1}^{3}\alpha_{5}\beta_{5}}{\alpha_{5}^{2}+\alpha_{1}^{2}}$
(134)
$\displaystyle=\beta_{1}\bigg{[}\alpha_{1}^{2}+\alpha_{5}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}\beta_{1}}{\alpha^{5}+\alpha_{1}^{2}}\bigg{]}+\alpha_{1}\alpha_{3}\beta_{3}-\frac{2\alpha_{1}\alpha_{3}\alpha_{5}^{2}\beta_{3}}{\alpha_{1}^{2}+\alpha_{5}^{2}}.$
(135)

For the denominator, we have

$\displaystyle
Var(\hat{X}-\alpha_{4}W-\frac{2\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{1}^{2}+\alpha_{5}^{2}}\hat{XW})$
$\displaystyle=\alpha_{4}^{2}+\alpha_{1}^{2}+\alpha_{5}^{2}+\alpha_{4}^{2}+\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{1}^{2}+\alpha_{5}^{2}}$
(136)
$\displaystyle-2\alpha_{4}^{2}-\frac{8\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{1}^{2}+\alpha_{5}^{2}}$
(137)
$\displaystyle=\alpha_{1}^{2}+\alpha_{5}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{1}^{2}+\alpha_{5}^{2}}.$
(138)

Therefore, all together we obtain the final result of

$\displaystyle\beta_{1}+\frac{\alpha_{1}\alpha_{3}\beta_{3}-\frac{2\alpha_{1}\alpha_{3}\alpha_{5}^{2}\beta_{3}}{\alpha_{1}^{2}+\alpha_{5}^{2}}}{\alpha_{1}^{2}+\alpha_{5}^{2}-\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{(\alpha_{1}^{2}+\alpha_{5}^{2}})}$
(139)

###  10.7 Proof for Exclusion Restriction $R^{2}$ Proposition 4.1

###### Proof.

We will first rewriting each edge weight as an $R^{2}$ quantities:

$\displaystyle|c_{1}|$ $\displaystyle=\sqrt{R^{2}_{X\sim U}}$ (140)
$\displaystyle|c_{2}|$ $\displaystyle=\frac{Cov(Y^{\perp X,Z,W},U^{\perp
X,W,Z})}{Var(U^{\perp X,W,Z})}=\frac{Cor(Y^{\perp X,Z,W},U^{\perp
X,W,Z}))sd(Y^{\perp X,W,Z})}{sd(U^{\perp X,W,Z}))}$ (141)
$\displaystyle=\sqrt{R^{2}_{Y\sim U|X,W,Z}}\frac{sd(Y^{\perp
X,W,Z})}{sd(U^{\perp X,W,Z})}$ (142) $\displaystyle|c_{3}|$
$\displaystyle=\sqrt{R^{2}_{X\sim Z}}$ (143) $\displaystyle|c_{ER}|$
$\displaystyle=\frac{Cov(Y^{\perp X,U,W},Z^{\perp X,U,W})}{Var(Z^{\perp
X,U,W})}=\frac{Cor(Y^{\perp X,U,W},Z^{\perp X,U,W})sd(Y^{\perp
X,U,W})}{sd(Z^{\perp X,U,W})}$ (144) $\displaystyle=\sqrt{R^{2}_{Y\sim
Z|X,U,W}}\frac{sd(Y^{\perp X,U,W})}{sd(Z^{\perp X,U,W})}$ (145)
$\displaystyle|c_{5}|$ $\displaystyle=\sqrt{R^{2}_{X\sim W}}$ (146)

where $\perp$ represents orthogonalizing the variables in the superscript.
Because $W\perp\\!\\!\\!\\!\perp Z$, $1-c_{3}^{2}-c_{5}^{2}=1-R^{2}_{X\sim
W+Z}$. Substituting this into the expression for the trade-off we obtain

$\displaystyle\frac{\sqrt{R^{2}_{Y\sim Z|X,U,W}}\frac{sd(Y^{\perp
X,U,W})}{sd(Z^{\perp X,U,W})}}{\sqrt{R^{2}_{X\sim
Z}}}>\frac{\sqrt{R^{2}_{X\sim U}R^{2}_{Y\sim U|X,W,Z}}\frac{sd(Y^{\perp
X,W,Z})}{sd(U^{\perp X,W,Z})}}{1-R^{2}_{X\sim W+Z}}$ (147)

We can further simplify the expression by canceling out like terms and noting
that

$\displaystyle sd(U^{\perp X,W,Z})=\frac{sd(U^{\perp X,W,Z})sd(U^{\perp
W,Z})}{sd(U^{\perp W,Z})}=\sqrt{1-R^{2}_{U\sim X|W,Z}}$ (148)

because $sd(U^{\perp W,Z})=1$ due to $U\perp\mkern-9.5mu\perp Z$ and
$U\perp\mkern-9.5mu\perp W$. Thus, our final expression is

$\displaystyle\frac{\sqrt{R^{2}_{Y\sim Z|X,U,W}}\sqrt{1-R^{2}_{Y\sim
U|X,W}}(1-R^{2}_{X\sim W+Z})}{\sqrt{1-R^{2}_{Y\sim Z|X,W}}\sqrt{R^{2}_{X\sim
Z}}sd(Z^{\perp X,W})}>\frac{\sqrt{R^{2}_{X\sim U}R^{2}_{Y\sim
U|X,W,Z}}}{\sqrt{1-R^{2}_{U\sim X|W,Z}}}$ (149)

∎

###  10.8 Proof for Independence $R^{2}$ Proposition 4.2

###### Proof.

We will first translate the edge weights or the linear combination of the edge
weights as $R^{2}$ quantities. Note that $c_{2}$ cancels out in the inequality
so we do not need to re-write this quantity.

$\displaystyle|c_{1}|$ $\displaystyle=\sqrt{R^{2}_{X\sim
U|Z,W}}\frac{sd(X^{\perp Z,W})}{sd(U^{\perp Z,W})}$ (150)
$\displaystyle|c_{3}|$ $\displaystyle=\sqrt{R^{2}_{X\sim
Z|U,W}}\frac{sd(X^{\perp U,W})}{sd(Z^{\perp U,W})}$ (151)
$\displaystyle|c_{I}|$ $\displaystyle=\sqrt{R^{2}_{Z\sim U}}$ (152)
$\displaystyle|c_{7}|$ $\displaystyle=\sqrt{R^{2}_{Z\sim W}}$ (153)
$\displaystyle(c_{5}+c_{3}c_{7})^{2}$ $\displaystyle=R^{2}_{X\sim Z}$ (154)
$\displaystyle(c_{3}+\frac{c_{1}c_{4}}{1-c_{7}^{2}})^{2}$
$\displaystyle=R^{2}_{X\sim Z|W}\frac{Var(X^{\perp W})}{Var(Z^{\perp W})}$
(155) $\displaystyle\frac{1-c_{4}^{2}-c_{7}^{2}}{1-c_{7}^{2}}$
$\displaystyle=\frac{1-R^{2}_{Z\sim W+U}}{1-R^{2}_{Z\sim W}}=1-R^{2}_{Z\sim
U|W}$ (156)

Substituting these quantities

$\displaystyle\frac{\sqrt{R^{2}_{Z\sim U}}}{\sqrt{R^{2}_{X\sim
Z|W}}sd(X^{\perp W}sd(Z^{\perp W})}$ $\displaystyle=\frac{\sqrt{R^{2}_{X\sim
U|Z,W}}\frac{sd(X^{\perp Z,W})}{sd(U^{\perp Z,W})}(1-R^{2}_{Z\sim
U|W})}{1-R^{2}_{X\sim W}-(1-R^{2}_{Z\sim W})R^{2}_{X\sim Z|W}}$ (157)
$\displaystyle\sqrt{R^{2}_{Z\sim U}}$ $\displaystyle=\sqrt{R^{2}_{X\sim
U|Z,W}}\frac{(1-R^{2}_{Z\sim U|W})}{sd(U^{\perp Z,W})}\phi$ (158)

where $\phi=\frac{\sqrt{R^{2}_{X\sim Z|W}}sd(X^{\perp Z,W})sd(X^{\perp
W})sd(Z^{\perp W})}{1-R^{2}_{X\sim W}-(1-R^{2}_{Z\sim W})R^{2}_{X\sim Z|W}}$
consists of observed quantities. Now further simplifying, we have

$\displaystyle\frac{(1-R^{2}_{Z\sim U|W})}{sd(U^{\perp Z,W})}$
$\displaystyle=\frac{(1-R^{2}_{U\sim Z|W})}{sd(U^{\perp
Z,W})}=\frac{var(U^{\perp Z,W})}{var(U^{\perp W})sd(U^{\perp
Z,W})}=sd(U^{\perp Z,W})=\sqrt{1-R^{2}_{Z\sim U|W}}$ (159)

because $var(U^{\perp W})=1$. Thus the final expression is

$\displaystyle\frac{\sqrt{R^{2}_{Z\sim U}}}{\sqrt{1-R^{2}_{Z\sim
U|W}}}=\sqrt{R^{2}_{X\sim U|Z,W}}\phi$ (160)

To solve for $R^{2}_{Z\sim U}$ is a matter of straightforward algebraic
manipulation as well as exploiting the fact that $U\perp Z$. Beginning with
squaring both sides, we have

$\displaystyle\frac{R^{2}_{Z\sim U}}{1-R^{2}_{Z\sim U|W}}$
$\displaystyle=R^{2}_{X\sim U|Z,W}\phi^{2}$ (161)
$\displaystyle\frac{R^{2}_{Z\sim U}}{\frac{1-R^{2}_{Z\sim W}-R^{2}_{Z\sim
U}}{1-R^{2}_{Z\sim W}}}$ $\displaystyle=R^{2}_{X\sim U|Z,W}\phi^{2}$ (162)
$\displaystyle\frac{R^{2}_{Z\sim U}}{1-R^{2}_{Z\sim W}-R^{2}_{Z\sim U}}$
$\displaystyle=\frac{R^{2}_{X\sim U|Z,W}\phi^{2}}{1-R^{2}_{Z\sim W}}$ (163)
$\displaystyle R^{2}_{Z\sim U}$ $\displaystyle=\frac{R^{2}_{X\sim
U|Z,W}\phi^{2}}{1-R^{2}_{Z\sim W}}(1-R^{2}_{Z\sim W}-R^{2}_{Z\sim U})$ (164)
$\displaystyle R^{2}_{Z\sim U}$ $\displaystyle=R^{2}_{X\sim
U|Z,W}\phi^{2}-\frac{R^{2}_{X\sim U|Z,W}\phi^{2}R^{2}_{Z\sim
U}}{1-R^{2}_{Z\sim W}}$ (165) $\displaystyle R^{2}_{Z\sim U}$
$\displaystyle=\frac{R^{2}_{X\sim U|Z,W}\phi^{2}}{1+\frac{R^{2}_{X\sim
U|Z,W}\phi^{2}}{1-R^{2}_{Z\sim W}}}.$ (166)

∎

###  10.9 Proof for Heterogeneity $R^{2}$ Proposition 4.3

###### Proof.

First, as we have done with the previous proofs, we will re-write the
coefficients and combinations as $R^{2}$:

$\displaystyle\alpha_{1}^{2}$ $\displaystyle=R^{2}_{X\sim Z}$ (167)
$\displaystyle\alpha_{5}^{2}$ $\displaystyle=R^{2}_{X\sim ZW}$ (168)
$\displaystyle\alpha_{4}^{2}$ $\displaystyle=R^{2}_{X\sim W}$ (169)
$\displaystyle|\alpha_{2}|$ $\displaystyle=\sqrt{R^{2}_{X\ simU}}$ (170)
$\displaystyle|\beta_{2}|$ $\displaystyle=\sqrt{R^{2}_{Y\sim
U|X,W,XU,XW}}\frac{sd(Y^{\perp X,W,XU,XW})}{U^{\perp X,W,XU,XW})}$ (171)
$\displaystyle\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{4}^{2}+2\alpha_{5}^{2}+1}$
$\displaystyle=\frac{Cov(XW,X)^{2}}{Var(XW)}=R^{2}_{XW\sim X}$ (172)
$\displaystyle\frac{4\alpha_{1}^{2}\alpha_{5}^{2}}{\alpha_{4}^{1}+\alpha_{5}^{2}}$
$\displaystyle=\frac{Cov(XW,X)^{2}}{Var(\hat{XW})}\frac{Var(XW)}{Var(XW)}=R^{2}_{XW\sim
X}\frac{Var(\hat{XW})}{Var(\hat{XW})}$ (173) $\displaystyle|\beta_{3}|$
$\displaystyle=\sqrt{R^{2}_{Y\sim
XU|X,W,U,XW}}\frac{sd(Y^{\perp}{X,W,U,XW})}{sd(XU^{\perp}{X,W,U,XW})}$ (174)
$\displaystyle|\alpha_{3}|$ $\displaystyle=\sqrt{R^{2}_{X\sim ZU}}.$ (175)

Now plugging in selected terms and dividing both sides by
$|2\alpha_{1}\alpha_{3}\beta_{3}|$, we have

$\displaystyle\Bigg{|}\frac{1/2-\frac{R^{2}_{X\sim
ZW}}{Var(\hat{XW})}}{R^{2}_{X\sim W}+R^{2}_{X\sim ZW}-R^{2}_{XW\sim
X}}\Bigg{|}$
$\displaystyle=\frac{\Big{|}\frac{|\beta_{2}\alpha_{2}|sign(\beta_{2}\alpha_{2})}{|2\alpha_{1}\alpha_{3}\beta_{3}|}+sign(\alpha_{1}\alpha_{3}\beta_{3})\Big{|}}{Var(X^{\perp
W,XW})}$ (176)

Note that only the right hand side contains unobserved quantities, so we will
focus on simplifying

$\displaystyle\frac{|\beta_{2}\alpha_{2}|}{|2\alpha_{1}\alpha_{3}\beta_{3}|}$
$\displaystyle=\frac{\sqrt{R^{2}_{X\sim U}}\sqrt{R^{2}_{Y\sim
U|X,W,XU,XW}}\frac{sd(Y^{\perp X,W,XU,XW})}{U^{\perp
X,W,XU,XW})}}{2R^{2}_{X\sim Z}\sqrt{R^{2}_{X\sim ZU}}\sqrt{R^{2}_{Y\sim
XU|X,W,U,XW}}\frac{sd(Y^{\perp}{X,W,U,XW})}{sd(XU^{\perp}{X,W,U,XW})}}.$ (177)

Simplifying $\frac{\frac{sd(Y^{\perp X,W,XU,XW})}{U^{\perp
X,W,XU,XW})}}{\frac{sd(Y^{\perp}{X,W,U,XW})}{sd(XU^{\perp}{X,W,U,XW})}}$ we
have

$\displaystyle\frac{\frac{sd(Y^{\perp X,W,XU,XW})}{sd(U^{\perp
X,W,XU,XW})}}{\frac{sd(Y^{\perp}{X,W,U,XW})}{sd(XU^{\perp}{X,W,U,XW})}}$
$\displaystyle=\frac{sd(Y^{\perp
X,W,XU,XW})sd(XU^{\perp}{X,W,U,XW})}{sd(U^{\perp
X,W,XU,XW})sd(Y^{\perp}{X,W,U,XW})}\frac{sd(Y^{\perp X,W,XW})}{Y^{\perp
X,W,XW})}$ (178) $\displaystyle=\frac{\sqrt{1-R^{2}_{Y\sim
XU|X,W,XW}}sd(XU^{\perp}{X,W,U,XW})}{\sqrt{1-R^{2}_{Y\sim
U|X,W,XW}}sd(U^{\perp X,W,XU,XW})}\frac{sd(XU^{\perp
X,W,XW})sd(U^{\perp}{X,W,XW})}{sd(XU^{\perp X,W,XW})sd(U^{\perp}{X,W,XW})}$
(179) $\displaystyle=\frac{\sqrt{1-R^{2}_{Y\sim
XU|X,W,XW}}\sqrt{1-R^{2}_{XU\sim U|W,X,XW}}sd(XU^{\perp
X,W,XW})}{\sqrt{1-R^{2}_{Y\sim U|X,W,XW}}\sqrt{1-R^{2}_{U\sim
XU|W,X,XW}}sd(U^{\perp X,W,XW})}$ (180)
$\displaystyle=\frac{\sqrt{1-R^{2}_{Y\sim XU|X,W,XW}}sd(XU^{\perp
X,W,XW})}{\sqrt{1-R^{2}_{Y\sim U|X,W,XW}}sd(U^{\perp
X,W,XW})}\frac{sd(XU^{\perp W,XW})sd(U^{\perp W,XW})}{sd(XU^{\perp
W,XW)sd(U^{\perp W,XW})}}$ (181) $\displaystyle=\frac{\sqrt{1-R^{2}_{Y\sim
XU|X,W,XW}}\sqrt{1-R^{2}_{XU\sim U|W,XW}}sd(XU^{\perp
W,XW})}{\sqrt{1-R^{2}_{Y\sim XU|X,W,XW}}\sqrt{1-R^{2}_{U\sim
X|W,XW}}sd(U^{\perp W,XW})}$ (182)

Because $Cov(U,W)Cov(U,XW)=0$, then we have that $\sqrt{U^{\perp W,XW})}=1$
and $\sqrt{1-R^{2}_{U\sim X|W,XW}}=\sqrt{1-R^{2}_{U\sim X}}$. Furthermore,
$Cov(W,XW)=Cov(W,XU)=0$ So our expression thus far, after re-arranging line
176 we obtain

$\displaystyle\Bigg{|}\frac{1/2-\frac{R^{2}_{X\sim
ZW}}{Var(\hat{XW})}}{R^{2}_{X\sim W}+R^{2}_{X\sim ZW}-R^{2}_{XW\sim
X}}\Bigg{|}Var(X^{\perp W,XW})$ (183)
$\displaystyle=\Bigg{|}\frac{\sqrt{R^{2}_{X\sim U}}\sqrt{R^{2}_{Y\sim
U|X,W,XU,XW}}}{2\sqrt{R^{2}_{X\sim Z}}\sqrt{R^{2}_{X\sim
ZU}}\sqrt{R^{2}_{Y\sim XU|X,W,U,XW}}}$ (184)
$\displaystyle\frac{\sqrt{1-R^{2}_{Y\sim XU|X,W,XW}}\sqrt{1-R^{2}_{XU\sim
U|W,XW}}\sqrt{1-R^{2}_{XW\sim XU}}}{\sqrt{1-R^{2}_{Y\sim
XU|X,W,XW}}\sqrt{1-R^{2}_{U\sim
X}}}sign(\beta_{2}\alpha_{2})+sign(\alpha_{1}\alpha_{3}\beta_{3})\Bigg{|}$
(185)

∎

###  10.10 Notes about Simulations

Specifically, we have restricted the variance for all variables to 1.
Therefore, choosing values of the structural coefficients for simulations to
verify our derivations is subject to constraints. Importantly, the variance of
the stochastic error (e.g. $\sigma_{\epsilon_{1}}$ in Eq 1) must be chosen
carefully such that we simulate the random variables properly. We detail these
constraints for each scenario below, which are calculated by taking variances
of $X$ and $Y$ and computing the relevant covariances.

####  10.10.1 Perfect IV

We are subject to the following constraints:

  1. 1.

$\sigma_{X}^{2}=1=c_{3}^{2}+c_{1}^{2}+\sigma_{\epsilon_{1}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=c_{0}^{2}+c_{2}^{2}+2c_{0}c_{1}c_{2}+\sigma_{\epsilon_{2}}^{2}$

####  10.10.2 Exclusion Restriction

Without Covariates

  1. 1.

$\sigma_{X}^{2}=1=c_{3}^{2}+c_{1}^{2}+\sigma_{\epsilon_{1}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=c_{0}^{2}+c_{2}^{2}+c_{ER}^{2}+2c_{0}c_{1}c_{2}+2c_{0}c_{3}c_{ER}+\sigma_{\epsilon_{2}}^{2}$

With Covariates

  1. 1.

$\sigma_{X}^{2}=1=c_{1}^{2}+c_{3}^{2}+c_{5}^{2}+\sigma_{\epsilon_{3}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=c_{0}^{2}+c_{2}^{2}+c_{ER}^{2}+c_{6}^{2}+2c_{0}c_{1}c_{2}+2c_{0}c_{3}c_{ER}+2c_{0}c_{5}c_{6}+\sigma_{\epsilon_{4}}^{2}$

####  10.10.3 $U$ is a cause of $Z$

Without Covariates

  1. 1.

$\sigma_{X}^{2}=1=c_{1}^{2}+c_{3}^{2}+2c_{1}c_{3}c_{I}+\sigma_{\epsilon_{1}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=c_{0}^{2}+c_{2}^{2}+2c_{0}c_{2}(c_{1}+c_{3}c_{I})+\sigma_{\epsilon_{2}}^{2}$

  3. 3.

$\sigma_{Z}^{2}=1=c_{I}^{2}+\sigma_{\epsilon_{3}}^{2}$

With Covariates

  1. 1.

$\sigma_{X}^{2}=1=c_{1}^{2}+c_{3}^{2}+2c_{1}c_{3}c_{I}+2c_{3}c_{5}c_{7}+\sigma_{\epsilon_{4}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=c_{0}^{2}+c_{2}^{2}+2c_{0}c_{2}(c_{1}+c_{3}c_{I})+2c_{0}c_{6}(c_{5}+c_{3}c_{7})+\sigma_{\epsilon_{5}}^{2}$

  3. 3.

$\sigma_{Z}^{2}=1=c_{I}^{2}+c_{7}^{2}+\sigma_{\epsilon_{6}}^{2}$

####  10.10.4 Treatment Effect Heterogeneity

Without Covariates

  1. 1.

$\sigma_{X}^{2}=1=\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\sigma_{\epsilon_{1}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+2\beta_{1}\beta_{2}\alpha_{2}+4\beta_{1}\beta_{2}\beta_{3}\alpha_{1}\alpha_{3}+\sigma_{\epsilon_{2}}^{2}$

With Covariates

  1. 1.

$\sigma_{X}^{2}=1=\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+\alpha_{4}^{2}+\alpha_{5}^{2}+\sigma_{\epsilon_{3}}^{2}$

  2. 2.

$\sigma_{Y}^{2}=1=\beta_{1}^{2}+\beta_{2}^{2}+\beta_{3}^{2}+2\beta_{1}\beta_{2}\alpha_{2}+4\beta_{1}\beta_{2}\beta_{3}\alpha_{1}\alpha_{3}+2\beta_{1}\alpha_{4}\beta_{4}+4(\beta_{1}\beta_{5}\alpha_{1}\alpha_{5})+2\beta_{3}\beta_{5}(\alpha_{2}\alpha_{4}+2\alpha_{3}\alpha_{5})+\sigma_{\epsilon_{4}}^{2}$