# Negative Controls for Instrumental Variable Designs††thanks: We thank Maya
Orenstein for excellent research assistance. We thank seminar participants at
Princeton University, the Technion - Israel Institute of Technology, the AEA
Meeting in New Orleans, Tel Aviv University, and Stanford University for their
helpful comments and suggestions. We acknowledge support from Tel Aviv
University Center for AI and Data Science (TAD) in collaboration with Google
as part of the initiative of AI and DS for social good.

Oren Danieli, Daniel Nevo, Itai Walk, Bar Weinstein, Dan Zeltzer111All authors
are from Tel Aviv University. Danieli is the corresponding author. Email:
<EMAIL_ADDRESS>
###### Abstract

Studies using instrumental variables (IV) often assess the validity of their
identification assumptions using falsification tests. However, these tests are
often carried out in an ad-hoc manner, without theoretical foundations. In
this paper, we establish a theoretical framework for _negative control_ tests,
the predominant category of falsification tests for IV designs. These tests
are conditional independence tests between negative control variables and
either the IV or the outcome (e.g., examining the “effect” on the lagged
outcome). We introduce a formal definition for threats to IV exogeneity
(_alternative path variables_) and characterize the necessary conditions that
proxy variables for such unobserved threats must meet to serve as negative
controls. The theory highlights prevalent errors in the implementation of
negative control tests and how they could be corrected. Our theory can also be
used to design new falsification tests by identifying appropriate negative
control variables, including currently underutilized types, and suggesting
alternative statistical tests. The theory shows that all negative control
tests assess IV exogeneity. However, some commonly used tests simultaneously
evaluate the 2SLS functional form assumptions. Lastly, we show that while
negative controls are useful for detecting biases in IV designs, their
capacity to correct or quantify such biases requires additional non-trivial
assumptions.

##  1 Introduction

The identification assumptions in instrumental variable (IV) designs cannot be
tested directly. Instead, researchers often use falsification (“placebo”)
tests, to indirectly test these assumptions. Surveying the most highly cited
papers published in top-five economics journals over the last decade, we find
that about half of the papers that used IV have implemented falsification
tests. Moreover, the large majority of falsification tests fall into two
categories: tests that the IV is not associated with certain variables (e.g.,
replace the outcome with a lagged outcome or present a “balance table”) or,
similarly, tests that the outcome is not associated with other variables
(e.g., future measurements of the original IV). Borrowing terminology from
other disciplines, we refer to these falsification tests and the involved
variables as _negative controls_.222In epidemiology and medical settings,
researchers use similar terminology for falsification tests for potential
confounding of an exposure-outcome relationship (Lipsitch et al., 2010; Arnold
and Ercumen, 2016; Shi et al., 2020).

Despite their widespread use in IV designs, negative control tests lack a
comprehensive theoretical foundation. Without such theory, it remains unclear
which of the IV-design assumptions negative control tests assess. It is also
unclear what the necessary conditions for negative control tests are. This
theoretical gap often results in practitioners making ad-hoc decisions when
selecting negative control variables and statistical tests, potentially
leading to inefficient utilization of available data and even incorrect
conclusions regarding the validity of the IV design.

In this paper, we develop a theory of negative control testing for IV designs.
The theory uncovers which of the IV design assumptions is tested in each
negative control test. We find that some negative control tests are not only
testing the IV exogeneity but are also testing the functional form assumptions
that were used in the IV analysis. Moreover, our theory explicates the
assumption that variables need to satisfy in order to serve as negative
controls. We apply our theory to show that some prevalent ways of using
negative controls are invalid. We also apply the theory to propose additional
negative control tests, using negative control variables or statistical
methods that are rarely used in practice. We conclude by discussing the
possibility of using negative controls for bias correction, as commonly done
in non-IV settings. We demonstrate that in IV settings, simple bias correction
approaches require stronger assumptions.

We characterize two types of negative control tests. We call the first, more
common type, a _negative control outcome_ (NCO) test. An NCO is a variable
that should be independent of the IV if the design is valid. Figure 1
illustrates how NCOs help evaluate the validity of IV designs.333Throughout
the paper, we use directed acyclic graphs (DAGs) as a way to visualize and
communicate certain aspects of potential underlying complex structures, as
advocated by Imbens (2020). In our proofs, we do not formally use DAG theory
(e.g. structural causal models, do-calculus and d-separation, Pearl (2009)).
However, we largely interpret DAGs accordingly. For concreteness, consider the
context of evaluating the impact of teacher quality (by some measure $X$) on
student performance (e.g., test scores $Y$). An IV is used, such as a quasi-
random assignment of teachers ($Z$). Panel A illustrates a case of a potential
violation of the independence assumption (Angrist and Imbens, 1995). For
example, unobserved student ability $U_{1}$, which affects test scores, might
also affect teacher assignments (e.g., if better teachers are assigned to more
able students). In such a case, there is an alternative path between the
teacher assignment and student performance, rendering the design invalid.
Although ability is unobserved, the researcher can indirectly test if it is
associated with teacher assignment by using observed NCOs as proxies, such as
students’ past test scores ($NC_{1}$). An association between students’ past
test scores and teacher assignments would indicate that an alternative path
exists between the IV and the outcome, violating the independence assumption.
Panel B shows a different example, illustrating how an NCO can also test the
exclusion restriction.

To develop a theory of NCOs, we introduce the concept of an _alternative path
outcome_ (APO) variable, which represent the potential threat to the
identification. Intuitively, an APO is a variable that if associated with the
IV, is also associated with the potential outcome. These associations create
an alternative path from the IV to the outcome. Thus, if an APO is associated
with the IV, it indicates a violation of IV exogeneity. In Figure 1, $U_{1}$
and $U_{2}$ are APO variables. If they are associated with the IV (through the
link represented by the dashed arrows), there is an alternative path between
the IV and the outcome.

Building on this definition, we define an NCO as a variable $NC$ that
satisfies the following NCO assumption: there exists an APO variable $U$ such
that if the IV design is valid (i.e., $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U$), then

$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|U.$

By contrast, variables that do not meet this assumption (e.g., because they
are directly affected by the IV), are not informative about the validity of
the IV design. We prove that if the IV is not independent of the NCO, the IV
design is invalid.

Contrary to NCO tests, which evaluate associations with the IV, the second
type of negative control tests, Negative Control Instrument (NCI) tests,
evaluate associations with the outcome. This approach is mostly applied using
pseudo-IV designs, where the original IV is replaced by a variable similar to
the IV, but presumed not to influence the treatment. For example, in Nunn and
Qian (2014), the US production of wheat, the original IV for the availability
of international aid, is substituted with the US production of other crop
types, which are not typically used in aid and thus serve as NCIs.

Figure 2 illustrates how NCIs evaluate the validity of IV designs. Consider
the same context of using a quasi-random allocation of teachers as an IV for
studying the impact of teacher quality on test scores. Panel A shows a
different kind of violation of the independence assumption. For example,
administrative considerations ($U_{3}$) affect the teacher assignment. We can
proxy for the unobserved administrative considerations using teacher
assignment in the next year, $NC_{3}$ (the IV measurement in the next period).
Here, it is known that the threat $U_{3}$ is associated with the IV and the
question is whether this threat is also associated with the potential outcome.
If the NCI ($NC_{3}$) is associated with the outcome ($Y$) after conditioning
on the IV ($Z$), then there exists an alternative path between the IV and the
outcome through $U_{3}$, in violation of the independence assumption. Panel B
shows a different example where NCIs help evaluate the exclusion restriction.

Variables serving as NCIs must satisfy the NCI assumption. We define this
assumption by introducing an Alternative Path Instrument (API) variable. An
API, when associated with the potential outcome, generates an alternative path
between the IV and the outcome. In Figure 2, $U_{3}$ and $U_{4}$ are API
variables. If they are directly associated with the outcome (through the
dashed arrow), there is an alternative path between the IV and the outcome. A
variable $NC$ satisfies the NCI assumption if there exists an API variable $U$
such that when the design is valid ($U\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z$),

$Y\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|U,Z.$

In contrast to the NCO assumption, the NCI assumption requires independence
conditional on the IV. Conditioning on the IV is required because the NCI and
the outcome are often associated through the IV, even in valid designs (as in
both panels of Figure 2).

NCI tests are sometimes improperly implemented because conditioning on the IV
is omitted. For a valid IV design, the outcome should be independent of the
NCI, conditional on the IV. However, we observe cases when such conditioning
was overlooked. This omission could falsely reject valid IVs.

Nevertheless, we also characterize scenarios when the conditioning on the IV
is not needed for an NCI test. Specifically, if the IV is independent of the
NCI, and the design is valid, then the NCI and the outcome are unconditionally
independent. Hence, conditioning on the IV is not required. Statistical
independence between the IV and the NCI is possible mostly when the NCI tests
a violation of the exclusion restriction.

Our theory reveals that different negative control tests examine different
assumptions. When the IV specification does not include control variables, an
association between an NCO and the IV or an NCI and the outcome (conditional
on the IV) indicates a violation of one of two assumptions. The first is
outcome independence (independence between the IV and the potential outcome),
and the second is the exclusion restriction. Other IV assumptions such as
treatment independence (independence between the IV and the potential
treatment), monotonicity, and relevance are not tested by the negative control
tests we analyze. When control variables are required for the IV design, a
negative control test can also assess functional form assumptions. NCO tests,
in particular, can evaluate the assumption of linearity of the IV in the
controls (_rich covariates_). This assumption is necessary when two-stage
least square (2SLS) is used for estimation (Blandhol et al., 2022). Similarly,
NCI tests can examine that the reduced-form equation is correctly specified.
However, unlike rich covariates, correctly specified reduced form is not a
necessary condition for 2SLS. Therefore, NCI tests can falsely reject valid
designs due to misspecification.

Next, we compare specific statistical tests that use negative controls,
examining the assumptions they test and their statistical properties. The most
common negative control tests use only one negative control in a single
regression. By contrast, under standard regularity conditions, researchers can
jointly test independence using multiple negative controls. For example,
researchers can regress the IV on multiple NCO variables and use an F-test to
examine the hypothesis that all NCO coefficients are zero. These regressions
examine both the IV exogeneity, as well as the functional form. To avoid
testing the functional form, researchers can opt for non-linear tests. Non-
linear tests can also capture more flexible associations but may have lower
statistical power if the linear model is approximately correct. We carry out
simulations to illustrate the differences between these tests under different
scenarios.

We build on our theoretical results to provide several concrete suggestions
for practitioners. First, we discuss examples of negative controls and the
conditions for their validity. This includes both commonly used and novel
types of negative controls, as suggested by our theory. For example, we show
that variables that cause the IV can be valid NCIs. Second, we summarize the
tradeoffs involved in the choice of testing method. Third, we propose
additional diagnostic tools that may assist in detecting the problem in the IV
design when the negative control test null hypothesis is rejected. We
demonstrate these principles on data from prior work.

Finally, we examine the potential use of NCOs for bias correction in IV
designs. In non-IV settings, a rapidly growing literature discusses how
negative controls can be used not only to detect biases but also to correct
them (Sofer et al., 2016; Shi et al., 2020; Tchetgen Tchetgen et al., 2020).
In IV designs, bias correction is less common. We study the case when the
outcome from a pre-period, before the IV and treatment were initiated, is used
as an NCO to detect a violation of outcome independence. In this case, the NCO
can be used for bias correction under treatment effect homogeneity. In less
common scenarios, negative controls can be used for bias correction in cases
of heterogeneous treatment effect, or potential violation of the exclusion
restriction. Our results suggest that simple methods for bias correction in
non-IV settings do not apply to IV settings without additional non-trivial
assumptions.

This paper adds to prior econometrics work on tests for IV-design validity
and, more generally, the validity of causal designs. Recent work suggested
novel tests to examine the validity of IV designs (Kitagawa, 2015; Huber and
Mellace, 2015; Mourifié and Wan, 2017). Previous work has also discussed
robustness tests, not specific to IV, based on varying the set of controls
(Altonji et al., 2005; Oster, 2019; Diegert et al., 2022). We contribute to
this literature by providing a theoretical foundation for the most common type
of falsification test for IV designs.

This paper also contributes to the growing literature on negative controls by
extending it to IV designs. The notion of negative controls originated in
experimental biomedical research (e.g., Tanida et al., 2008; Power et al.,
2012; Lee and Schapira, 2021), where it considers the causal effects of a
treatment on an outcome, without the use of IVs. Eggers et al. (2021) discuss
the usage of placebo tests in the social sciences more broadly. Davies et al.
(2017) propose that a negative control approach may be used to compare IVs and
ordinary least square designs. However, they do not develop a theoretical
framework for such an approach. Our main contribution is extending the theory
of negative controls to IV designs.

We find several important differences in the analysis of negative controls in
IV analysis compared to non-IV settings. In particular, unconditional negative
control tests for independence with the outcome are unique to the IV setting.
Moreover, the reliance on treatment effect homogeneity for bias correction is
also a feature of the IV structure. We also contribute to this literature by
formally defining alternative path variables, which are essential for any
negative control theory.

The rest of this paper proceeds as follows. Section 2 survey the current
practice of falsification tests for IV designs. Section 3 develops the theory
underlying negative control tests. Section 4 discusses testing procedures.
Section 5 provides guidance for practitioners on implementing and interpreting
negative control tests. Section 6 discusses bias correction. Section 7
concludes.444Simulation and data analysis reproducibility materials are
available from https://github.com/barwein/NC_for_IV

##  2 Survey of Current Practice

In this section, we provide an overview of current practices in falsification
testing for IV designs. We find that falsification tests are widely employed
in IV analysis in economics. The predominant types of falsification tests are
the negative control tests which we discuss in this paper. Despite the
prevalence of negative control tests, we also see potential for further
broadening of their usage, as current practices tend to repeat the same type
of statistical tests and use only a small number of negative controls.

We surveyed the most highly cited articles with an IV analysis published
between 2013 and 2023 in top economics journals. Our survey included 140
papers.555Using Google Scholar in November 2023, we searched the terms
“instrumental variable,” “instrument,” “2SLS,” and “IV.” We restricted to all
articles published at the five top-ranked economic journals with over 300
citations or, if published after 2020, over 100 citations.  We then classified
the characteristics of the falsification tests used. The results are
summarized in Negative Controls for Instrumental Variable Designs††thanks: We
thank Maya Orenstein for excellent research assistance. We thank seminar
participants at Princeton University, the Technion - Israel Institute of
Technology, the AEA Meeting in New Orleans, Tel Aviv University, and Stanford
University for their helpful comments and suggestions. We acknowledge support
from Tel Aviv University Center for AI and Data Science (TAD) in collaboration
with Google as part of the initiative of AI and DS for social good.. We
highlight several findings from this survey.

First, falsification tests are widely used in IV analyses. Approximately half
(51%) of all surveyed articles employ some form of falsification test.

Second, most of these falsification tests fall within the negative control
framework outlined in this paper. We categorize negative control tests into
two types: _negative control outcome_ (NCO) tests, which examine associations
of the IV with variables it should not be associated with. This includes, for
example, the commonly used balance test. NCO tests were used in 75% of papers
with some falsification tests. The second type is _negative control
instrument_ (NCI) tests, which examine the association of the outcome
variables with variables it should not be associated with. Such tests were
used in 25% of papers that included some falsification test. Only 24% of
papers implemented other types of falsification tests (some used multiple
types).666Other falsification tests include negative control tests in non-IV
settings, examining only the first-stage or second-stage in a 2SLS estimation.
Other papers use ‘placebo population’ analyses (Eggers et al., 2021), which
involve repeating the analysis using a different population where the IV is
not expected to affect the outcome. Some papers validate the results remain
similar when including additional controls. Finally, when more than one IV is
available, researchers sometimes validate the results using an over-
identification test.

Third, current applied work predominantly restricts itself to two simple
specifications for negative control tests: pseudo-outcome tests and pseudo-IV
tests. Pseudo-outcome tests (Athey and Imbens, 2017) are a form of NCO tests;
they account for 55% of such tests. They involve estimating a revised reduced-
form equation using an alternative outcome (e.g., lagged outcome) and testing
if it is unrelated to the IV. The remaining NCO tests often follow a similar
logic. For example, balance tables often regress various NCOs on the
IV.777Most balance tables are not classified as pseudo-outcome in our analysis
since they do not use the exact same specification as the reduced form.
Pseudo-IV tests, the predominant specification of NCI tests, replace the
original IV with a similar but ostensibly unrelated variable (the pseudo-IV)
to assess its independence from the outcome (95% of papers using NCI tests
used a pseudo-IV specification). In Section 4, we discuss alternative tests
and specifications and explain that they may be more appropriate in certain
contexts.

Finally, papers using falsification tests usually utilize only a few negative
control variables. The median number of variables used is four, and a quarter
of the papers use only one variable. These findings suggest that researchers
only use a subset of the valid and relevant negative controls available in
their data. As we demonstrate in Section 5, our theory can guide a more
systematic search for negative control variables in existing data and suggest
novel types of negative control variables researchers can use to evaluate
their IV designs.

##  3 Theory of Negative Controls in IV Settings

In this section, we develop the theory for negative control tests for IV
designs. We characterize two distinct types of negative control variables:
NCOs and NCIs, each corresponding to a different type of falsification test.
We discuss the criteria required for variables to qualify as negative controls
in each category and clarify which IV assumptions are tested in each type of
test.

###  3.1 Setup

Consider i.i.d units indexed by $i=1,\ldots,n$. Denote the observed
(endogenous) treatment status by $X_{i}$, and the candidate IV by $Z_{i}$. Let
$Y_{i}(z,x)$ be the potential outcome for unit $i$ had $X_{i}$ and $Z_{i}$
been set to the values $x$ and $z$, respectively.888This formulation
implicitly assumes the stable unit treatment value assumption (SUTVA). We make
the standard assumption that the observed outcome $Y_{i}$ is
$Y_{i}=Y_{i}(Z_{i},X_{i})$. Because the units are assumed to be i.i.d, we omit
the subscript $i$ when it improves clarity. Unless stated otherwise, all
variables may be discrete or continuous.

A valid IV design relies on several assumptions. The negative control tests we
discuss in this paper focus on a subset of these assumptions. The first
assumption, _outcome independence_ , maintains that IV assignment is
independent of the potential outcomes.

######  Assumption 1 (Outcome independence).

For all $z,x$,

$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(z,x).$

This assumption is usually written as part of a more general independence
assumption (Angrist and Imbens, 1995). Here, we distinguish between outcome
independence and _treatment independence_ , which requires
$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}X(z)$ for every value of $z$. Only outcome
independence is tested in the negative control tests we discuss in this paper.
The examples in Panel A of Figure 1 and Panel A of Figure 2 illustrate a
potential violation of the outcome independence assumption.

The second assumption, _exclusion restriction_ , maintains that the IV has no
direct effect on the outcome.

######  Assumption 2 (Exclusion restriction).

For all $z,x$,

$Y(z,x)=Y(x).$

The examples in Panel B of Figure 1 and Panel B of Figure 2 illustrate a
potential violation of this assumption.

Together, outcome independence and exclusion restriction imply _IV exogeneity_
, that is,

$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)\text{ for all }x.$ (1)

In loose terms, IV exogeneity requires that there are no alternative paths
between the IV and the outcome except through the treatment. However, because
potential outcomes are never observed, neither these two assumptions nor the
ensuing IV exogeneity can be tested directly.

To identify a causal effect using an IV design, additional assumptions are
also necessary. Depending on the specific design, such assumptions may include
treatment independence, relevance, and monotonicity. However, the negative
control tests presented in this paper do not test these other assumptions.

In many cases, IVs are believed to be valid only conditional on certain
control variables (e.g., assignment of judges is quasi-random within time and
location; Kling, 2006). Let $C_{i}$ be the observed vector of controls.
Similar to the case without controls, outcome independence and exclusion
restriction together imply $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|\ C$. See Appendix B for the assumptions’
conditional forms. For simplicity of presentation, we sometimes omit control
variables. However, we include control variables in all our proofs.

While accounting for controls in an IV analysis can be done in a variety of
ways (e.g., Abadie, 2003), the large majority of applications use a two-stage
least squares (2SLS) specification. With some abuse of notation to avoid
clutter, let $C$ also denote the set of controls in a 2SLS
specification.999The vector $C$ may include, for example, a quadratic function
of one of the original controls or interactions. For ease of notation, $C$
would always include the intercept. The 2SLS specification requires the
following additional linearity assumption to yield a _causal estimate_
(Blandhol et al., 2022).101010Throughout the paper when referring to a causal
estimate we adopt the definition of weak causality from Blandhol et al.
(2022): positively weighted average of subgroup-specific treatment effects.

######  Assumption 3 (Rich covariates).

The IV $Z$ is linear in the control specification. Namely,

$\mathbb{E}[Z|C]=\gamma^{\prime}_{C}C,$ (2)

for some real vector $\gamma_{C}$.

We show that this assumption is also tested by some (but not all) negative
control tests.

###  3.2 Negative Control Outcomes

####  3.2.1 Alternative Path Outcome Variables

Our theory characterizes negative controls as proxies for threats to the IV
design’s validity. To formalize the notion of a threat, we introduce the
concept of an _alternative path variable_ —a variable that is part of a
suspected alternative path between the IV and the outcome which, if existed,
would violate IV exogeneity. In this section, we formalize the first type of
identification threat by defining Alternative Path Outcome (APO) variables.

Typically, an APO variable is a variable that is associated with the potential
outcome, i.e., with the outcome but not through the treatment. Therefore, any
association between the APO variable and the IV implies the existence of an
alternative path between the IV and the outcome not through the treatment,
thereby violating IV exogeneity (1). Figure 1 illustrates such violations, for
outcome independence (Panel A) and exclusion restriction (Panel B).

##### Single Violation

For simplicity, we begin by assuming that only one potential threat to IV
exogeneity exists. We later provide a more general definition of an APO
variable, allowing for multiple potential threats, for which the following
definition is a special case.

######  Definition 1 (Alternative path outcome variable with a single
violation ).

A random variable $U$ is an APO variable if the following two conditions hold.

  1. 1.

Latent IV exogeneity: $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|U$.

  2. 2.

Path indication: If $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$ then $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U$.

The first condition, latent IV exogeneity, posits that had we observed and
conditioned on the APO variable, the IV design would have been valid. This
assumption implies that $U$ itself is part of the threat to the identification
— if the IV is invalid, conditioning on $U$ blocks the alternative path
between the IV and the outcome.111111Note that a single causal chain from $Z$
to $Y$ may go through several factors: $Z\rightarrow
U_{1}\rightarrow\ldots\rightarrow U_{k}\rightarrow Y(x)$. In such case, each
of $U_{1},\ldots,U_{k}$ would constitute an APO variable by our definition.
See also Mackie (1965). By this assumption, proxies for identification threats
cannot be APO variables, as controlling for a proxy does not make the IV and
the potential outcome conditionally independent of each other. Latent IV
exogeneity is analogous to the latent exchangeability assumption appearing in
recent literature on negative controls in epidemiology (Shi et al., 2020) and
statistics (Tchetgen Tchetgen et al., 2020).

The second condition, path indication, states that an exogenous IV is not
associated with the APO variable. Its contrapositive ensures that a link
between the IV and the APO variable implies an alternative path between the IV
and the potential outcome. We later show that NCO tests evaluate the
association between the IV and an APO variable. Path indication assures that
such association serves as a marker for the validity of the IV design.

Path indication guarantees that if there is a path from the IV to the APO, the
path continues from the APO variable to the outcome. Therefore, it excludes
variables unrelated to the outcome, as they can be associated with the IV
without implying anything about the design validity. While APO variables often
causally affect the outcome, it is not mandatory (as demonstrated in Appendix
A.1).

Path indication also rules out variables that could be related to both the IV
and the outcome without generating a correlation between them. One example is
when $U$ is a multidimensional variable, where the dimensions of $U$
associated with the outcome are not associated with the IV.121212A
multivariate variable $U$ could be an APO variable if it satisfies the
conditions of the definition. We elaborate on this in Appendix A.2. Path
indication will also be violated if a variable is correlated with the outcome
for some subpopulation, but potentially correlated with the IV only for a
separate subpopulation. We provide such an example in Appendix A.3.

##### Multiple Violations

In some applications, multiple potential alternative paths can exist between
the IV and the outcome. Figure 3 presents an example of two distinct variables
that are affecting the outcome, and could potentially affect the IV as well,
and thus may violate outcome independence. To accommodate the possibility of
multiple violations of IV exogeneity, we extend Definition 1. We introduce an
additional (potentially multivariate) random variable $V$, which represents
other potential threats that may exist alongside the APO variable $U$.

######  Definition 2 (Alternative path outcome variable).

A random variable $U$ is an APO variable if there exists a random variable $V$
such that the following conditions hold.

  1. 1.

Latent IV exogeneity. $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|U,V.$

  2. 2.

Path indication. If $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|V$ then $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|V.$

  3. 3.

Direct IV link. If $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|V$ then $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U.$

  4. 4.

V-validity. If $Z{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)$ then $Z{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|V$.

In this definition, latent IV exogeneity states that IV exogeneity holds
conditional not only on the APO variable $U$, but also on the additional
threat(s) $V$. In contrast to Definition 1, this more general version of
latent IV exogeneity also holds if $U$ is only a proxy of the actual threat to
the identification ($V$), and not the threat itself.

Therefore, to maintain the same interpretation of an APO variable as a threat
to IV exogeneity, we replace the assumption of path indication from Definition
1, and include two additional conditions. Together, combining conditions 2–4
yields Condition 2 from Definition 1. However, the three separate conditions
ensure that the APO is the threat itself and not a proxy. Specifically, path
indication and direct IV link each rules out a different type of proxy; see
Appendix A.4 and A.5 for counterexamples. The final property, V-validity, is a
more technical requirement for the variable $V$ that represents other threats.
It states that an exogenous IV remains exogenous conditional on $V$. See
Appendix A.6 for a counterexample. If there are no additional threats,
Definition 2 is equivalent to Definition 1.

####  3.2.2 Negative Control Outcome Assumption

Building on the definition of an APO variable, we are now ready to formalize
the assumption required for a random variable to serve as a _negative control
outcome_.

######  Definition 3 (Negative control outcome).

A random variable $NC$ is an NCO if it satisfies the NCO assumption: There
exists an APO variable $U$ such that if $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U$ then

$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|U.$

The NCO assumption guarantees that dependency between the IV and the NCO would
be informative about the threat $U$. More formally, this assumption implies
that there exists an APO variable $U$ such that any statistical dependence
between the IV and the NCO ($Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC$) implies a relation between the IV and
the APO variable ($Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U$). Hence, if the IV $Z$ is associated with
the NCO variable $NC$, an alternative path exists between the IV and the
outcome through $U$. Figure 1 demonstrates this idea using two examples for
NCOs that satisfy this assumption.

The NCO assumption is violated for variables that can be directly related to
the IV even if the IV design is valid (and therefore are uninformative of its
validity). Figure 4 shows two examples of such violations. In these examples,
$U_{1},U_{2}$ are APO variables. However, the variable $N$ is not an NCO
because it is related to $Z$ in other ways, making $N$ uninformative about the
APO variable. In Panel A, the IV and $N$ are affected by an additional factor
$Q$, which is unrelated to the outcome. In Panel B, the IV directly affects
$N$, beyond its effect through the APO.

However, the NCO assumption is not violated for variables that might be
directly related to the IV, as long as this relationship only occurs when
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U$. Hence, when
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U$ and therefore an alternative path between
the IV and the outcome exists, a valid NCO could still be directly associated
with the IV. An association between such an NCO and the IV is still
informative about the validity of the IV design since this association only
exists when the design is invalid. For example, there could be a concern that
an IV is not randomly assigned as argued and is instead affected by some
observed lagged variables. The researcher can then test if the IV is valid by
examining whether the IV is associated with these lagged variables. While the
lagged variables might directly affect the IV, they are still valid NCOs as
they are not directly associated with the IV ($Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|U$) when the design is potentially valid
($Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U$).

The NCO assumption formalizes discussions about the validity of NCOs that are
currently made ad-hoc. For example, Guidetti et al. (2021) demonstrates that
non-respiratory hospital admissions are not valid NCOs for IV designs that
evaluate the impact of air pollution exposure. It might be expected that non-
respiratory hospital admissions would only correlate with IVs for air
pollution if these IVs are flawed. That is, if both the IV and non-respiratory
admissions are affected by some APO variable. However, Guidetti et al. (2021)
show that non-respiratory admissions are affected by air pollution through
hospital congestion caused by an increase in respiratory admissions. Hence,
these admissions do not provide insights into IV exogeneity, as they are
correlated even with valid IVs, not through any APO variable. Formally, non-
respiratory admissions violate the NCO assumption.

####  3.2.3 Negative Control Outcome Test

A _negative control outcome test_ (NCO test, in short) is any statistical test
for independence between the IV and an NCO variable. The null hypothesis is
$H_{0}:Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC$. The following theorem formalizes the
implication of rejecting this null for the IV exogeneity.

######  Theorem 1.

Assume that a random variable $NC$ satisfies the NCO assumption. If
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC$, then either outcome independence or the
exclusion restriction is violated. That is, the IV design is invalid.

The proof is given in Appendix B for the more general case, discussed below,
when control variables are included. For the case without controls, the sketch
of the proof is as follows. By the NCO assumption, the dependency of the IV
and the NCO implies an association between the IV and an APO variable
($Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U$). It can be shown that by the definition
of the APO variable, $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U$ indicates an alternative path between the
IV and the outcome, so $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)$, i.e., IV exogeneity does not hold.

Although a failed NCO test indicates that the IV design is invalid, the
converse is not always true. An invalid IV design might still pass an NCO
test. This would happen if there are other alternative paths from the IV to
the outcome that are not captured by the NCO, or due to lack of statistical
power.

##### Control Variables

When the IV is presumably valid only conditionally on a vector of control
variables $C$, an NCO test is a test for the null hypothesis

$H_{0}:Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|C.$ (3)

See Appendix B for the theory of APO and NCO variables when controls are
included. As discussed in Section 3.1, in most cases, IV designs that are
conditional on control variables are estimated using a 2SLS specification,
which also entails the rich covariates assumption (Assumption 3). Combining
the null hypothesis (3) and rich covariates we will expect that

$\mathbb{E}[Z|C,NC]=\gamma_{C}^{\prime}C.$ (4)

This equation provides a more specific null hypothesis for conditional
independence testing. The following corollary formalizes this argument.

######  Corollary 1.

Assume that the random variable $NC$ satisfies the NCO assumption for some APO
variable. If there is no real vector $\gamma_{C}$ such that
$\mathbb{E}[Z|C,NC]=\gamma_{C}^{\prime}C$, then either outcome independence,
exclusion restriction, or rich covariates is violated.

The proof is immediate as outcome independence, exclusion restriction, and
rich covariates together yield Equation (4). Hence if this equation does not
hold for any real vector $\gamma$, at least one of the three assumptions must
not hold.

Corollary 1 implies that if a statistical test rejects the null that
$H_{0}:\mathbb{E}[Z|C,NC]=\gamma^{\prime}C$, then a 2SLS estimand that
linearly controls for $C$ is not informative. This is either because IV
exogeneity is violated, or because the 2SLS estimand cannot be interpreted as
a causal estimate because of a violation of rich covariates (Blandhol et al.,
2022).

##### Multiple Negative Controls

Often, multiple variables individually satisfying the NCO assumption exist.
Let $\overline{NC}=\\{NC_{1},...,NC_{M}\\}$ be a vector of $M$ NCOs, all of
which individually satisfy the NCO assumption (with either the same or
distinct APO variables). Can the researcher test jointly whether
$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\overline{NC}$ in such cases, instead of
testing for independence one by one? We argue that yes, for all practical
purposes. While one can construct examples in which a valid IV that is
uncorrelated with any of the individual NCOs might be correlated with the
vector $\overline{NC}$ (see Appendix A.7 for an example), we argue that such
cases are knife-edge. Rejecting the independence $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\overline{NC}$ implies that the IV design is
invalid in nearly all realistic scenarios. Even in the above counterexample
(A.7) small perturbations to the data-generating process would generate
dependence with one of the NCOs.131313A valid IV design in which all
individual independence hypotheses are satisfied yet there is no independence
with the vector $\overline{NC}$ requires strong functional form assumptions
that are implausible in most IV settings. In the DAGs literature, the
exclusion of a small number of probability functions that generate these
unlikely independence structures is formalized via the notion of stability or
faithfulness; see Pearl (2009, Page 48). Therefore, when multiple NCOs exist,
it is possible to test the validity of the IV design using a single test,
avoiding the need for multiple hypotheses correction. In Section 4, we present
several examples of such tests.

###  3.3 Negative Control Instruments

In this section, we turn our attention to the second type of negative
controls, _negative control instruments_. In the previous section, we showed
that NCO tests examine whether the IV is associated with variables it should
not. By contrast, NCI tests begin with the assumption that the IV is
associated with certain variables and examine whether this association
matters. NCI tests seek to examine whether variables that are associated with
the IV are also associated with the potential outcome. To formalize this idea,
we first define a different type of alternative path variables. Building on
this definition, we characterize NCI variables and tests and explain how they
differ from our previous theory on NCOs.

####  3.3.1 Alternative Path Instrument Variables

As with NCOs, NCIs are typically proxies of threats to the identification. We
define such threats as _Alternative Path Instrument_ (API) variables. Much
like an APO variable, an API variable is also part of a suspected alternative
path between the IV and the outcome which could violate IV exogeneity.
However, API variables are known to be associated with the IV and researchers
are concerned about an association they might have with the outcome. This is
in contrast to APO variables that are known to be associated with the outcome,
and researchers are concerned about their potential association with the IV.
Figure 2 demonstrates two such cases. In Panel A, the IV is known to be non-
randomized, as it is affected by an unobserved variable $U_{3}$. In Panel B
the IV is known to affect another variable, $U_{4}$, in addition to its effect
on the treatment. In such cases, the validity of the IV design depends on
whether the additional variable (e.g. $U_{3}$ or $U_{4}$ in the examples
depicted in Figure 2) directly affects the outcome $Y$.

Formally, in the case of a single violation, an API variable satisfies the
following definition.

######  Definition 4 (Alternative path instrument variable with a single
violation).

A random variable $U$ is an API variable if the following two conditions hold.

  1. 1.

Latent IV exogeneity. $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|U$.

  2. 2.

Path indication. If $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$ then $U\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z$.

This definition resembles the definition of APO variables (Definition 1). The
first property (latent IV exogeneity) is exactly as before. The difference
between API and APO variables is encapsulated in the path indication
condition. For API variables, this condition requires that if IV exogeneity is
satisfied ($Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$), the API variable must be independent
of the observed outcome conditional on the IV ($U\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z$). This condition implies that an
association between the API variable and the outcome, not through the IV,
indicates that the IV is not exogenous. Typically, path indication is
satisfied when the API variable is associated with the IV, and so if also
directly associated with the outcome, an alternative path between the IV and
the outcome exists.

Path indication also rules out variables that are associated with the outcome
(conditionally on the IV) through the treatment. Such variables are not
informative about the validity of the IV design as they are associated with
the outcome through the treatment even when IV exogeneity is satisfied. This
is different from APO variables that could be associated with the treatment
(even conditionally on the IV). See Appendix A.8 for an example and further
discussion of this issue.

Similar to the definition of APO variables, the definition of API variable
with a single violation (Definition 4) can be extended to settings where
additional violations are potentially present. As before, such an extension
requires more nuanced assumptions. The API definition also easily generalizes
to a case where control variables are included in the IV design. The details
of both extensions are given in Appendix B.1.2.

####  3.3.2 Negative Control Instrument Assumption

A _negative control instrument_ is defined as a variable satisfying the
following NCI assumption.

######  Definition 5 (Negative control instrument).

A random variable $NC$ is an NCI if it satisfies the NCI assumption: There
exists an API variable $U$ such that if $U\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z$ then

$Y\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|Z,U.$ (5)

While similar, there are three key differences between the NCI assumption and
the NCO assumption (Definition 3). First, the alternative path variable $U$ is
an API variable instead of an APO variable. Second, the conditional
independence of the NCI is with the outcome, instead of the IV. These two
differences reflect that NCI tests, defined below, are testing an association
with the outcome, and not with the IV. The third difference is that the
independence condition is conditional on the IV as well. This is because the
NCI can be associated with the outcome through the IV, as we discuss in the
next section.

In many cases, the NCI assumption rules out a large class of variables that
appear in the data, because of their association with the outcome. For
example, demographic variables often exhibit an association with the outcome,
conditional on the IV, and therefore cannot serve as NCIs. Variables that are
associated with the treatment are also not NCIs as they are associated with
the outcome, conditional on the IV. Moreover, in cases where the IV effect on
the outcome is heterogeneous, any variable that is associated with the source
of heterogeneity cannot serve as an NCI. Generally, the NCI assumption imposes
a greater restriction than the NCO assumption. The NCO assumption requires
conditional independence with the IV, which is more prevalent than conditional
independence with the outcome, as IVs are typically quasi-random.

####  3.3.3 Negative Control Instrument Test

A _negative control instrument test_ examines whether the outcome and the NCI
are independent conditional on the IV. Formally, the statistical test is for
the following null hypothesis $H_{0}:Y\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|Z$. If the NCI is associated with the
outcome conditionally on the IV, this necessarily implies that the IV design
is not valid, as stated in the following theorem.

######  Theorem 2.

Assume that a random variable $NC$ satisfies the NCI assumption. If
$Y\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|Z$, then either outcome independence or
exclusion restriction is violated. That is, the IV design is invalid.

The proof is given in Appendix B.1.2 for the more general case that control
variables are included and additional threats are possible.

The conditioning on the IV is required since unconditionally, the NCI is
associated with the outcome even in many valid IV designs. This is because the
NCI is often associated with the IV, which in turn affects the outcome through
the treatment.141414A similar conditioning is not required for NCO variables,
even though they are often similarly associated with the outcome. This is
because of the direction of causality from the IV to the outcome, which
implies that an association with the IV would entail an association with the
outcome, but not the other way around. For example, in the two panels of
Figure 2, the NCI and the outcome are associated through the IV even if the IV
design is valid.

However, in some cases, an NCI test can be carried out without conditioning on
the IV. Figure 5 shows an example of such a case. In this example, the IV is
independent of the NCI ($Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC$), which was not the case in Figure 2. When
$Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC$, the researcher can use an unconditional
independence test, as formalized in the next theorem.

######  Theorem 3.

Assume that a random variable $NC$ satisfies the NCI assumption. If in
addition $Z\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC$, then if
$Y\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC$, either outcome independence or exclusion
restriction is violated. That is, the IV design is invalid.

The proof is given in Appendix B.1.2.

Unconditional NCI tests may arise when considering violations of the exclusion
restriction and are therefore unique to the IV setting. When the IV is known
to affect an API variable, potentially violating the exclusion restriction,
any other independent variable that affects this API variable as well can
serve as an NCI (as in Figure 5). Such NCIs are not associated with the IV and
therefore an unconditional test can be utilized.When a violation of outcome
independence is suspected, the IV and the NCI are typically associated as well
(as in Panel A of Figure 2). Therefore, unconditional independence tests
between a negative control and the outcome are unique for IV settings. In non-
IV settings, there is no exclusion restriction, and therefore negative control
tests for dependency with the outcome are always done conditionally.151515The
analog of NCI in a non-IV setting is a negative control exposure (NCE). An NCE
is a variable known not to causally affect the outcome (Shi et al., 2020). NCE
tests are always conducted conditional on the exposure.

##### Specification Testing

As we now discuss, NCI tests often examine not only the IV exogeneity but also
a functional form assumption which is unnecessary for the validity of the IV
design. In Appendix B.1.2 we show that when control variables are included, an
NCI test is a test for the null hypothesis

$H_{0}:Y\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|C,Z.$ (6)

As with NCOs, this null hypothesis can be combined with an assumption on the
functional form of the IV design, to generate a stronger null hypothesis that
is simpler to test. Specifically, NCI tests can examine the following
assumption that the reduced form in a 2SLS specification is correctly
specified.

######  Assumption 4 (Correctly Specified Reduced Form).

The conditional expectation of the outcome is linear in the IV and controls

$\mathbb{E}[Y|Z,C]=\theta_{Z}Z+\theta^{\prime}_{C}C.$

Combining the null hypothesis (6) with the correctly specified reduced form
(CSRF) assumption, we will expect that

$\mathbb{E}[Y|Z,C,NC]=\theta_{Z}Z+\theta^{\prime}_{C}C.$ (7)

This equation provides a more specific null hypothesis, that is often used in
practice. The following corollary shows that testing this hypothesis implies
testing IV exogeneity and CSRF jointly.

######  Corollary 2.

Assume that the random variable $NC$ satisfies the NCI assumption. If there is
no real vector $\theta=(\theta_{Z},\theta^{\prime}_{C}$) such that
$\mathbb{E}[Y|Z,C,NC]=\theta_{Z}Z+\theta^{\prime}_{C}C$, then either outcome
independence, the exclusion restriction, or CSRF is violated.

The proof is immediate as outcome independence, exclusion restriction, and
CSRF together yield Equation (7).

However, CSRF is not a necessary assumption for 2SLS analysis, implying that
testing this assumption could reject valid IV designs. In contrast to the rich
covariates assumption, the CSRF assumption is not necessary for a 2SLS
analysis to yield a causal estimate. For example, consider a case where the IV
is randomly assigned, so IV exogeneity and rich covariates hold, and the
design is valid. CSRF could still be violated if the IV has a non-linear
effect on the outcome or a heterogeneous effect across control vector values.
In such case, Equation (7) will not hold even when the design yields a causal
estimate.

##  4 Negative Control Test Procedures

This section discusses statistical tests for negative controls in IV designs.
Our theory shows that for NCO tests without controls, and for some NCI tests,
IV exogeneity requires unconditional independence between the negative control
variable and the IV (Theorem 1) or the outcome (Theorem 3). Such independence
is testable via various established statistical tests (e.g., Székely et al.,
2007; Heller et al., 2013). However, for most NCI tests, and for all negative
control tests when controls are included, IV exogeneity requires conditional
independence, which, especially with continuous controls, has no established
one-size-fits-all solution (Shah and Peters, 2020).161616For a more general
survey of conditional independence tests see Heinze-Deml et al. (2018).
Therefore, we discuss the pros and cons of different conditional independence
tests in our context.

In the rest of this section, we delineate and compare different conditional
independence tests. We then illustrate their performance differences in
simulations tailored to capture different IV design scenarios.

###  4.1 Pseudo Outcomes, Pseudo IVs

As discussed in Section 2, the most common falsification tests used in
practice are _pseudo outcome_ tests. These tests estimate the original
reduced-form equation while replacing the outcome with a pseudo outcome (e.g.,
past outcomes). That is, a pseudo-outcome analysis estimates the model

$NC=\beta_{Z}Z+\beta^{\prime}_{C}C+\epsilon_{NC},$ (8)

and examines the estimated coefficient of the IV in this model (e.g., by
testing $H_{0}:\beta_{Z}=0$).171717The chosen testing procedure for $H_{0}$
may vary according to assumptions on $\epsilon_{NC}$.

The following result establishes that when the pseudo outcome is a valid NCO,
this practice can be used to jointly examine IV exogeneity and rich
covariates. The result follows from Corollary 1.

######  Corollary 3.

Assume that $NC$ satisfies the NCO assumption. Let

$\beta=(\beta_{Z},\beta^{\prime}_{C})=\operatorname*{arg\,min}_{b_{Z},b_{C}}\mathbb{E}[NC-
b_{Z}Z-b^{\prime}_{C}C]^{2}$

be the population-level OLS estimator. If $\beta_{Z}\neq 0$ then either
outcome independence, exclusion restriction, or rich covariates is violated.

The proof is given in Appendix B.1.1.

Unlike the pseudo-outcome approach, the common approach for NCI tests,
_pseudo-IV_ analysis, is not always a valid test for IV exogeneity. In its
simplest form, the pseudo-IV analysis substitutes the original IV with a
similar but ostensibly unrelated variable (the pseudo IV) in the regression
model and assesses its independence of the outcome. Such a test may not be
informative about IV exogeneity unless the pseudo IV is also independent of
the original IV (Theorem 3); otherwise, the original IV must be controlled for
(Theorem 2). Notably, this is not consistently done in empirical
studies.181818Surveying recent papers discussed in Section 2, reveals that
many studies are ambiguous when specifying their approach to pseudo-IV
testing. Moreover, those failing to condition on the original IV rarely offer
justifications for the assumed independence between the pseudo IV and the
original IV. Moreover, pseudo-IV tests that omit the IV might be substantially
more prevalent than what we observe in publications, if results that (wrongly)
find problems in the IV design are less likely to be reported.

A pseudo-IV analysis that adjusts for the IV and for controls may use the
linear regression

$Y=\theta_{Z}Z+\theta^{\prime}_{C}C+\theta_{NC}NC+\epsilon_{Y},$ (9)

where $NC$ is an NCI. The pseudo-IV test then examines the null hypothesis
that $H_{0}:\theta_{NC}=0$. 191919As with the pseudo-outcome test, the testing
procedure is determined according to additional assumptions on $\epsilon_{Y}$.

Despite the adjustment, this pseudo-IV analysis could reject the null
hypothesis for valid IV designs as well. Corollary 2 implies that even when
the IV is exogenous, the null could still be rejected due to a violation of
CSRF. As discussed in Section 3.3.3, the CSRF assumption is not necessary for
causal identification. Therefore, it is possible that an adjusted pseudo-IV
test would reject the null hypothesis for a valid IV design that can identify
a causal estimate with 2SLS. In Sections 4.3 and 4.4, we discuss tests that
could test the IV exogeneity without testing additional functional form
assumptions.

When multiple negative controls are available, multiple pseudo-outcome or
pseudo-IV analyses can be conducted, separately for each negative control.
This procedure requires multiple hypothesis testing corrections, such as
Bonferroni. Alternatively, in the rest of this section, we discuss procedures
that perform a joint test for multiple negative controls. A comparison between
the two approaches is part of our simulation study (Section 4.5).

###  4.2 Parametric Conditional Independence Tests

When multiple NCOs are available, one can test the null that the conditional
expectation of the IV is a linear function only of the controls

$H_{0}:\mathbb{E}[Z|C,NC]=\gamma^{\prime}C.$

Under the following linear model for $Z$

$Z=\gamma^{\prime}C+\gamma^{\prime}_{NC}NC+\epsilon_{Z},$ (10)

the commonly used F-test for the null hypothesis
$H_{0}:\gamma^{\prime}_{NC}=0^{\prime}$ can be used under the standard
assumptions. Alternatively, a Wald test might be preferred when robust or
clustered standard errors should be used. This procedure jointly tests IV
exogeneity, and the rich covariates assumption (Corollary 1).

Similarly, with multiple NCIs, an F-test (or other tests for a vector of
parameters) can be used in an adjusted pseudo-IV analysis. Using a vector of
NCIs in Equation (9), the researchers can carry out a valid statistical test
for the hypothesis that the coefficient vector of the NCIs is zero
($H_{0}:\theta^{\prime}_{NC}=0^{\prime}$). This procedure jointly tests IV
exogeneity and the CSRF assumption (Corollary 2). Therefore, it could reject
valid IV designs as well.202020More generally, any test that relies on a
functional form assumption could be rejected due to model misspecification.

###  4.3 Semi-parametric Conditional Independence Tests

In certain instances, researchers may opt for semi- or non-parametric tests
that rely on weaker or no assumptions on the functional forms. Avoiding
parametric assumption is particularly useful for NCO tests when methods other
than 2SLS are employed for IV estimation. In other IV methods (e.g., Abadie
(2003)) the rich covariates assumption is not necessary, and therefore
researchers may want to avoid testing it. Similarly, when employing an NCI
tests researchers may prefer a test that does not depend on the CSRF
assumption which could be violated in valid designs. Moreover, models (9) and
(10) do not capture non-linear associations with the negative controls.
Therefore they could fail to detect dependencies between the NCO(s) and the IV
or the NCI(s) and the outcome.

As an alternative to the parametric tests we discussed so far, consider, for
example, the use of General Additive Models (GAMs) (Wood, 2006). For NCO
tests, one such specification assumes that for $NC$ a vector of NCOs,

$Z=\phi(C,NC)+\epsilon_{Z}=\sum_{j}f_{j}(C_{j})+\sum_{k}g_{k}(NC_{k})+\epsilon_{Z},$
(11)

where $j$ and $k$ index the controls and negative control variables,
respectively; $f_{j}$, $g_{k}$ are smooth functions allowed to be different
for different variables; and $\epsilon_{Z}$ is an independent normally-
distributed error term. Under this model, $Z$ is assumed to be composed of
additive smooth functions that are typically estimated using splines (Wood,
2006). A Wald test for the spline coefficients can be used to test the null
hypothesis that $g_{k}=0$ for all $k$.212121Technical details of how to
construct such a test and conditions for its validity can be found in Section
6.12.1 in Wood (2006). Interactions can be accommodated by including
additional smooth functions of product terms.

In some cases, researchers may want to test the rich covariates assumptions,
while still allowing for a non-linear association between the NCOs and the IV.
For example, when using 2SLS (which requires rich covariates) while suspecting
a strong non-linear association between the NCO and the IV. In such cases, the
following model can be used

$Z=\phi(C,NC)+\epsilon_{Z}=\gamma^{\prime}C+\sum_{k}g_{k}(NC_{k})+\epsilon_{Z}.$
(12)

As with model (11), the NCO test is for the hypothesis that $g_{k}=0$ for all
$k$. When the hypothesis is rejected, $g_{k}$ may capture a residual non-
linear association between $C$ and $Z$, not captured by $\gamma$, which
violates rich covariates (due to an association between $C$ and $NC$).

For NCI tests, one can consider the following model (with some abuse of
notation) for a vector of NCI variables (denoted by NC),

$Y=\psi(Z,C,NC)+\epsilon_{Y}=h(Z)+\sum_{j}f_{j}(C_{j})+\sum_{k}g_{k}(NC_{k})+\epsilon_{Y},$
(13)

where $j$ and $k$ index the controls and the NCIs, respectively;
$h,f_{j},g_{k}$ are smooth functions; and $\epsilon_{Y}$ is an independent
normally-distributed error term. As before, the null hypothesis is that for
all $k$, $g_{k}=0$. Interactions between the IV and the control variables
could also be included. This test does not depend on the CSRF assumption, and
so a rejection of the null implies a violation of IV exogeneity.

###  4.4 Non-parametric Conditional Independence Tests

More generally, researchers can implement conditional independence test using
invariant target prediction (Heinze-Deml et al., 2018). In the context of an
NCO test, this method involves using a prediction algorithm to predict the IV
twice: once using only the controls and once using both the NCOs and the
controls. Under the null ($Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|C$), the out-of-sample performance of both
predictions should be similar, as the NCOs should not have any additional
predictive power if they are conditionally independent of the IV. A similar
approach can be constructed for NCI tests (while including the IV in the input
of both prediction algorithms). Various prediction algorithms can be employed
in this procedure.222222For example, Abramitzky et al. (2023) use a parametric
model (LASSO) to predict the IV with the controls and NCOs. With sufficient
data, flexible, non-parametric models or algorithms can be used to capture
non-linear associations.232323This is similar to the multiple-hypotheses test
suggested by Ludwig et al. (2017) when control variables are included. For
more details, see Appendix B.4 in Heinze-Deml et al. (2018).

Similar to the other methods we discussed so far, invariant target prediction
typically focuses on mean independence. However, more general approaches go
beyond mean independence using kernel measures of conditional independence
(Fukumizu et al., 2007; Zhang et al., 2011; Strobl et al., 2019).242424These
kernel approaches rely on the fact that, under certain conditions, conditional
independence is equivalent to a conditional cross-covariance operator defined
for reproducing kernel Hilbert spaces equaling zero. As a result, a test for
zero Hilbert-Schmidt norm of the conditional cross-covariance operator is a
test for conditional independence. See Zhang et al. (2011) for details.

###  4.5 Simulations

We use simulations to compare the different negative control tests we have
discussed. We consider two main scenarios. The first scenario considers
violations of outcome independence that involve either a linear or a non-
linear effect of the APO variable on the IV, with multiple NCOs. It
demonstrates that a joint hypothesis test could outperform multiple hypothesis
testing and that non-linear tests could outperform linear tests when the IV-
NCO association is highly non-linear. The second scenario considers violations
of the rich covariates assumption. It illustrates that tests based on linear
models are sensitive to a failure of this assumption, while some non-linear
tests are not.

For all scenarios described in this section, and for each unique parameter
value, we simulated $1,000$ samples, each with $10,000$ observations.
Technical details about the data-generating processes (DGP) and specific
parameter values are given in Section C.

##### Violations of Outcome Independence

For the first study, the DGPs were parameterized as follows.

$\displaystyle\begin{split}Z&=\gamma_{0}+\gamma_{U}g(U)+\sum_{j=1}^{5}C_{j}+\epsilon_{Z},\\\
NC_{k}&=\beta U+\epsilon_{NC,k},\end{split}$ (14)

where $Z$ is the IV, $U$ is an APO variable, $C$ is a vector of five
independent control variables, $NC$ is a vector of ten NCOs indexed by $k$,
and $\epsilon_{Z},\epsilon_{NC,k}$ are independent normally-distributed error
terms.

We consider two alternative scenarios, one with a linear $g$ (the identity)
and the other with a non-linear $g$ (a truncated parabolic function). The
parameters $\beta$ and $\gamma_{U}$ control the magnitude of the association
between the APO variables and the IV and the NCOs, respectively. In our study,
we vary the values of $\gamma_{U}$.252525We do not specify the DGP of the
outcome and how exactly the APO variable is associated with it, as the NCO
test performance does not depend on this information.

In each simulated sample, we test the null hypothesis

$H_{0}:\mathbb{E}[Z|C,NC]=\mathbb{E}[Z|C]$ (15)

using the following tests: multiple pseudo-outcome regressions with Bonferroni
correction for multiple hypotheses, a single (multivariate) linear regression
(Equation (10)) with an F-test, and Wald tests from GAMs with and without
smooth terms for the continuous controls.262626We also considered a Wald test
from the single linear regression; the results were nearly identical to those
obtained by F-test and hence are not reported. We set the desired type-I error
to 5%.

Figure 6 summarizes the results, showing how the proportion of null rejections
varies with the type ($g$, the two panels) and strength ($\gamma_{U}$ on the
horizontal axis) of the IV-APO relationship (i.e., with the strength of
outcome independence violation). When the null is correct and there is no
violation of outcome independence (namely, $\gamma_{U}=0$) all tests appear
correctly sized (the null hypothesis is rejected in approximately 5% of the
samples). When the APO-IV relationship is linear and sufficiently strong
(large $\beta$), the power of all tests increases and approaches one. However,
the sensitivity of these tests to weak violations of outcome independence
varies, with the F-test from a single regression outperforming the GAMs and
the multiple pseudo-outcome linear regressions. By contrast, when the
relationship between IV and the APO is (highly) non-linear, only the two GAM
tests, which accommodate non-linearity, have any power at all.

##### Violations of Rich Covariates

Our second simulation study compares the performance of different methods for
conditional independence testing when IV exogeneity holds, but the rich
covariates assumption (Assumption 3) is violated. Namely, the NCOs are
conditionally independent of the IV given the controls but are still
correlated with the projection of the IV on higher degree polynomials of the
controls. We study the performance of different NCO tests, with the same null
as before (Equation (15)).

We use the same DGP and parameter values specified by Blandhol et al. (2022)
in their simulation study. Following Blandhol et al. (2022), $Z$ is a binary
IV with probability that can be written as a cubic polynomial in a single
control variable, that is
$\Pr(Z=1|C)=\gamma_{0}+\gamma_{1}C-\gamma_{2}C^{2}+\gamma_{3}C^{3}$. We
augment this DGP with an APO variable $U$, which is taken to be independent of
the IV as the IV is exogenous. We then define ten negative controls associated
with this APO. To that end, each NCO is simulated by

$NC=\beta_{1}C+\eta(\beta_{0}+\beta_{2}C^{2}+\beta_{3}C^{3})+\beta_{U}U+\epsilon_{NC},$
(16)

where $\epsilon_{NC}$ is a normally-distributed error term. In this DGP, the
parameter $\eta$ is a weight on the non-linear part of the association between
$C$ and $NC$. As $\eta$ increases the association becomes more non-linear.

Figure 7 displays the proportion of null hypothesis rejections for each test
within each scenario. In all the tests that assume a linear association
between the IV and the controls, the null hypothesis is rejected with a
sufficiently strong non-linear association between the NCOs and the controls
(large enough $\eta$). However, in the GAM model with smooth non-linear
function of the controls (Equation (12)), the null hypothesis is rarely
rejected even if $\eta$ is large. This result is expected because in contrast
to the other tests we consider, this GAM test is not sensitive to violation of
the rich covariates assumption (see Section 4.3).

##  5 Practical Guidance

This section offers guidelines for implementing negative control tests. We
recommend that researchers follow three steps, summarized in Figure 8. First,
use domain knowledge to identify suitable negative controls. Examples are
discussed in Section 5.1. Second, choose an appropriate statistical test based
on sample size, control variables, and functional form assumptions, detailed
in Section 5.2. Third, interpret the result and conduct further diagnostics if
the test rejects the null, as discussed in Section 5.3.

To illustrate these recommendations, we apply them to IV designs used in prior
work. We chose four widely cited papers published in the American Economic
Review with publicly posted replication data. We use Autor et al. (2013) and
Deming (2014) to discuss NCO tests and Ashraf and Galor (2013) and Nunn and
Qian (2014) to discuss NCI tests.272727Ashraf and Galor (2013) and Nunn and
Qian (2014) are the two most cited AER papers published since 2013 that use an
NCI test. Similarly, Autor et al. (2013) is the most cited AER paper published
after 2013 that uses an NCO test. Deming (2014) was selected to demonstrate
how our proposed follow-up analysis can be used to diagnose and correct
problems with the IV design in Section 5.3. D.4 briefly introduces these
papers’ IV designs and the negative controls they used in falsification tests.
Appendix D provides additional details on our analyses.

###  5.1 Choosing Negative Controls

In our survey of recent literature in Section 2, we found that researchers
often use only one, or very few negative controls. This may raise the concern
that valid and informative negative controls are sometimes left unused.
Building on our theory, we were able to find additional negative controls in
the replication data of some of the papers we analyzed (Column 6 of D.4).

We next discuss different types of negative controls, both commonly used in
practice and novel ones suggested by our theoretical framework.

####  5.1.1 Common Types of Negative Controls Outcomes

##### Predetermined variables.

Predetermined variables are frequently used in NCO tests, especially in
balance tests. Common examples include lagged outcomes and demographic factors
such as gender, race, and age. Predetermined variables can proxy for APO
variables that would violate outcome independence if they affect the IV. In
many cases, researchers could choose to use predetermined variables as NCO
while only vaguely knowing which APO they proxy for. The NCO assumption
requires that if w]the NCO is associated with the IV, the association with the
IV would (also) be through some APO variable. This is often the case as
predetermined variables are determined before the IV and thus unaffected by
it. However, not every predetermined variable is a valid NCO. Certain
predetermined variables may influence the IV, thus violating the NCO
assumption. For example, the parents’ quarter of marriage is not a valid NCO
when the IV is the child’s quarter of birth (Angrist and Krueger, 1991), as it
is likely affecting it. Such concerns are less relevant when the IV is based
on an alleged randomization (e.g., school lotteries).

Autor et al. (2013) used predetermined outcomes to evaluate a shift-share IV
for commute-zone exposure to imports. The NCO assumption seems reasonable in
this context (it rules out that past local labor market outcomes in the US are
associated with future exposure to Chinese imports in other countries in ways
unrelated to the outcome). Following this logic allowed us to discover many
additional predetermined variables in the original paper’s replication data
that could serve as NCOs. Similarly, we find multiple predetermined variables
in the replication data from Deming (2014). These are all NCO variables as the
IV is based on randomization (school lottery).

##### IV leads.

Certain IVs are based on random or chance occurrences (‘strikes of luck’). A
recurring pattern of these events could question the validity of such IV
designs. The NCO assumption here stipulates that subsequent IV measurements
should not be autocorrelated. For example, Jäger and Heining (2019) use a
worker’s premature death as an IV for employee turnover and argues that such
deaths are random across firms. One potential concern is that firms with
premature deaths are systematically different—that premature death is caused
by some APO variable that represents riskier conditions in that firm, which
could directly affect wages (the outcome). To rule this out, they use
subsequent premature deaths in the same firm as an NCO that proxies for the
APO (riskier conditions), and find no correlation over time.

##### Alternative Outcomes.

Unrelated or alternative outcomes can also serve as NCOs for two different
types of APO variables. First, when the APO variable is potentially violating
outcome independence (potentially affecting the IV), alternative outcomes can
serve as NCOs, provided they are unaffected by the IV. For example, Chetty et
al. (2014) use middle-school teachers’ moves across schools to evaluate
teacher value-added measures. They use test scores from other subjects, not
taught by the teacher, as an NCO. The APO variable in concern is unobserved
school quality. For example, high-quality teachers might tend to move to
schools where students’ quality simultaneously improves as well. The NCO test
they perform does not provide evidence for this conjecture. By contrast,
Chetty et al. also show that for elementary school teachers, test scores from
other subjects do not satisfy the NCO assumption, as teachers tend to teach
more than one topic.

The second type of APO potentially violates the exclusion restriction
(potentially affected by the IV). If the APO variable is not observed, a proxy
for it could serve as an NCO. In this case, the NCO assumption would not hold
if the IV affects the candidate NCO through the outcome—e.g., if improving one
of the main outcomes (reading scores) subsequently enhances the alternative
outcome (math scores).

####  5.1.2 Common Types of Negative Control Instruments

##### Pseudo IVs.

Researchers often use variables that are similar to the IV but are presumed
not to influence the treatment variable. These pseudo IVs usually share many
similarities to the IV and are thus likely to be correlated with the API
variable. For example, Ashraf and Galor (2013) use the distance from Addis
Ababa as their IV, while employing distance from other cities (unrelated to
the human origins hypothesis) as NCIs. Similarly, Nunn and Qian (2014) use
U.S. wheat production as their IV, and consider the U.S. production of other
crops (unrelated to U.S. aid, the treatment) as NCIs.

##### IV leads.

When IV measurements are autocorrelated, IV leads cannot serve as NCOs.
However, future instances of the IV often serve as effective NCIs, as in
Moretti (2021). As with pseudo IVs, IV leads share similarities with the IV
and are therefore likely to be associated with the API variable. IV leads
cannot influence the outcome as they occur later. Therefore, when IV leads are
observed, researchers should decide whether to use an NCO or NCI test
depending on whether autocorrelation is expected. Nevertheless, to satisfy the
NCI assumption, the outcome must not influence future realizations of the IV.

####  5.1.3 Novel Types of Negative Control Instruments

Our theory can highlight potential new types of NCIs beyond those commonly
used today, including the following two types.

##### Causes of the IV.

Variables that causally affect the IV can serve as effective NCIs for
assessing potential violations of outcome independence. For example, if the IV
is quarter of birth, then quarter of marriage qualifies as a valid NCI. As
previously discussed, such a variable cannot serve as an NCO since the direct
effect on the IV violates the NCO assumption. In such scenarios (illustrated
in Figure 9), both the API and the NCI influence the IV, leading to a
correlation between the API and NCI, conditional on the IV. The IV is what is
known in the literature of DAGs as a _collider_ (Pearl, 2009). If the NCI is
not independent of the outcome conditionally on the IV, this implies that
there is a path between the NCI and the outcome, through the IV and the API.
Therefore, the API affects the outcome, and the IV design is invalid.

##### IV Side Effect Proxies.

An IV that affects an API variable in addition to its effect on the treatment
may violate the exclusion restriction if this API also affects the outcome.
Therefore, proxies for such APIs can serve as NCIs. Panel B of Figure 2
illustrates a scenario where the IV influences the NCI via the API, so the NCI
is an alternative outcome. Figure 5 presents another case where the NCI and
the IV both cause the API. These “side effect” proxy variables must not be
linked to the outcome other than through the IV, as this would violate the NCI
assumption.

####  5.1.4 Power Considerations

Power considerations suggest excluding variables that meet a negative control
assumption but have weak associations with the alternative path variables, as
they can lower test power. This mirrors the effect of irrelevant controls in
OLS. The issue is more acute in NCI tests that require conditioning on the IV,
as some NCIs, especially pseudo-IVs, can be strongly correlated with the IV
but only weakly correlated with the API conditional on the IV.282828For
example, in Nunn and Qian (2014), the correlation between the NCIs and the IV
exceeds 0.9. Therefore, after controlling for the IV, the residual variation
in the NCI may be limited.

###  5.2 Choosing a Statistical Test

As discussed in Section 4, when implementing a negative control test,
researchers could choose from a variety of statistical methods that test for
(conditional) independence between the negative control and the IV or the
outcome. The selection of an appropriate statistical test depends on three
primary considerations: the design assumptions under scrutiny, the expected
functional relationship between the negative control and either the IV or the
outcome, and statistical power considerations.

For NCO tests, when the main analysis utilizes 2SLS with controls, a test that
is linear in the controls is a leading option. Such tests evaluate not only
the IV exogeneity but also the rich covariates assumption (Assumption 3).
Blandhol et al. (2022) show that this assumption is necessary for identifying
a causal estimate in a 2SLS setting. Specifically, estimating model (10) and
using F-test or Wald to test the hypothesis the NCOs are unrelated to the IV,
can aggregate information from multiple NCOs jointly. With sufficient data, a
GAM model as in (12) can allow for non-linear associations with the NCO while
still testing the rich covariates assumption. Alternatively, researchers might
opt for a pseudo-outcome test based on model (8). This test uses the same
inferential framework as the original study. Therefore, it can expose errors
in the inference method as well (Eggers et al., 2021). However, it demands
accounting for multiple hypothesis testing when multiple NCOs are available.

When the main analysis does not use 2SLS (e.g., in quantile regressions as in
Chernozhukov and Hansen (2008)), and data is sufficiently large, non-linear
tests would be preferred. This includes semi-parametric methods like GAM (see
(11)–(13)), or other methods discussed in sections 4.3 and 4.4. These tests
can capture more complex associations that would be missed in linear methods.
However, their flexibility implies larger data requirements. As demonstrated
in our simulations in Section 4.5, these tests have reduced power compared to
an F-test when the linear model (10) is (approximately) correct.

Negative Controls for Instrumental Variable Designs††thanks: We thank Maya
Orenstein for excellent research assistance. We thank seminar participants at
Princeton University, the Technion - Israel Institute of Technology, the AEA
Meeting in New Orleans, Tel Aviv University, and Stanford University for their
helpful comments and suggestions. We acknowledge support from Tel Aviv
University Center for AI and Data Science (TAD) in collaboration with Google
as part of the initiative of AI and DS for social good. presents the results
of all the above-discussed NCO tests using the data from Autor et al. (2013)
and Deming (2014). Column 2 shows that in both papers, using only one NCO, we
cannot reject the null hypothesis.292929In Column 1 we show the results from a
replication of the original NCO test conducted by Autor et al. (2013). They
find a significant correlation between the IV and lagged outcome. However,
this correlation becomes insignificant when control variables are included.
However, when using multiple NCOs in Column 3 (with Bonferroni adjustment for
multiple hypothesis testing) the null hypothesis is rejected in both designs.
The null is also rejected when using a joint F-test, in Column 4. Although
these results may not significantly impact the conclusions of the original
studies, they underscore that adding additional NCOs—using existing data and
guided by theory—can yield more powerful tests. In Columns 5 and 6 we
implement a GAM test with and without assuming linearity of the control
variables. As expected, the GAM tests seem less effective in smaller samples.

For NCI tests, researchers should first consider whether they need to control
for the IV in their test. Section 3.3.3 shows that this depends on whether the
NCI is independent of the IV. In most cases, controlling for the IV would be
required. One noticeable exception is when the API represents a violation of
the exclusion restriction, and the NCI is not affected by it.

Regarding the statistical test applied, the commonly used tests that involve
adding the NCIs to the reduced form equation are often problematic, as they
are contingent on the CSRF assumption (Assumption 4). The CSRF assumption is a
strong parametric assumption that can be attractive in smaller datasets.
However, this assumption is not necessary for the validity of the IV design
and is likely to be violated in many valid designs used in practice.
Consequently, a linear test might reject a valid IV design. With sufficiently
large data, researchers can instead use a semi- or non-parametric test, as
discussed above.

Negative Controls for Instrumental Variable Designs††thanks: We thank Maya
Orenstein for excellent research assistance. We thank seminar participants at
Princeton University, the Technion - Israel Institute of Technology, the AEA
Meeting in New Orleans, Tel Aviv University, and Stanford University for their
helpful comments and suggestions. We acknowledge support from Tel Aviv
University Center for AI and Data Science (TAD) in collaboration with Google
as part of the initiative of AI and DS for social good. presents results from
the above NCI tests using data from Nunn and Qian (2014) and Ashraf and Galor
(2013). Columns 1 and 2 show the results of a pseudo-IV test with and without
such conditioning on the IV, using one of the NCIs from the original paper. In
both contexts, conditioning on the IV is required as the NCI is likely
correlated with the outcome through the IV.303030While not explicitly stated
in Nunn and Qian (2014), their code reveals conditioning on the IV. Ashraf and
Galor (2013) did not condition for the IV. For Nunn and Qian, we find that the
null hypothesis is only rejected when conditioning on the IV, indicating that
neglecting to control for the IV could lead to a false rejection of their IV
design. Using data from Ashraf and Galor, the null is not rejected, even
without IV control, potentially due to insufficient statistical power.

In Columns 3 and 4 we show the null is not rejected when using the full set of
NCIs from the original papers in either multiple pseudo-IV tests with
Bonferroni correction or in a joint F-test. In Column 5, we use a GAM to allow
for non-linearities in both the IV and the NCI. However, due to the small
sample size compared to the large number of control variables, we could not
implement a GAM test with smooth controls. We find that for Nunn and Qian, the
null hypothesis is rejected. Since we maintain linearity in the control
variables, this rejection could still be the result of a violation of CSRF
(Assumption 4), as we indeed find using the tests discussed in the next
section.

###  5.3 Interpreting the Test Results

#### Rejection of the Null

A rejection of the null in a parametric linear negative control test, such as
an F-test, could suggest violations of either IV exogeneity or linearity (rich
covariates for NCO, CSRF for NCI). In such cases, researchers can further
explore the reason for the rejection by testing the linearity directly. For
example, linearity can be examined with a RESET test using control variables
only, without negative controls (Ramsey, 1969).313131For example, we used a
RESET test to explore the reason for the GAM test rejecting the NCI test null
when using data from Nunn and Qian (2014) (Column 5 of Negative Controls for
Instrumental Variable Designs††thanks: We thank Maya Orenstein for excellent
research assistance. We thank seminar participants at Princeton University,
the Technion - Israel Institute of Technology, the AEA Meeting in New Orleans,
Tel Aviv University, and Stanford University for their helpful comments and
suggestions. We acknowledge support from Tel Aviv University Center for AI and
Data Science (TAD) in collaboration with Google as part of the initiative of
AI and DS for social good.). Our RESET test rejected the null that the reduced
form is correctly specified. This is perhaps not surprising, given the many
control variables used. As discussed in Section 3.2.3, CSRF is not a necessary
requirement when using 2SLS.  With sufficient data, researchers can also use
semi- or non-parametric tests, which rely on less or no functional form
assumptions and focus on IV exogeneity.

When multiple negative controls are used, researchers may want to explore
which negative controls are driving the rejection of the null. To do so,
researchers can plot the strengths of the correlations for each negative
control with the IV and the outcome on a two-dimensional scatter plot. This
exercise can help identify negative controls with high predictive power with
both the IV and outcome, which may suggest the existence of a related
alternative path. Figure 10 displays such diagnostics for Deming (2014),
plotting each NCO’s correlation with the IV (vertical axis) against its
correlation with the outcome (horizontal axis). The NCO with the strongest
correlation with the IV is the neighborhood school’s value-added. This is
because the IV is constructed from interactions of the lottery results with
neighborhood school VA. In this case, the post-analysis exercise identifies a
fixable problem in the IV design, as using the original lottery results or
controlling for the neighborhood school VA can solve this problem. Using this
alternative IV does not substantially change the main results. This is
unsurprising, as the NCO is only weakly correlated with the outcome,
suggesting the alternative path from the IV to the outcome is likely weak.

One caveat for this exercise is that the correlation with negative control may
not always reflect how strong is the alternative path. Since negative controls
are just proxies for alternative path variables, their correlation with the IV
or outcome depends on the strength of their correlation to the alternative
path variable. Therefore, a negative control might show a weak correlation
with the IV or outcome, even if the actual association between the IV and
outcome through the alternative path is strong, or vice versa.

#### Non-Rejection of the Null

A successful negative control test (i.e., when the null hypothesis is not
rejected) provides supporting evidence but does not guarantee the validity of
the IV design. There are at least two reasons for this. First, IV exogeneity
may be invalidated by alternative path variables not captured by the NCO or
NCI used for the test. For example, a quasi-random allocation to teachers that
is found to be uncorrelated with students’ neighborhoods could still be
associated with students’ abilities within neighborhoods.

Second, invalid IVs could pass the test due to lack of power. In negative
control tests, the roles of type I and type II errors are inverted. If the IV
design is invalid, the data’s ability to uncover this depends on the test’s
power, which, unlike the type I error, the researcher cannot control.323232See
Bilinski and Hatfield (2018) for a discussion of this issue and an alternative
approach.

##  6 Bias Correction

So far we have only discussed negative controls in the context of bias
detection. In this section, we turn to discuss the possibility of using them
for bias correction. Consider a scenario where a pseudo-outcome test reveals a
correlation between the IV and a pseudo-outcome (an NCO), with the
correlation’s sign being opposite to that in the main analysis. Could this
suggest that the overall effect is larger than estimated? In the articles we
surveyed in Section 2, negative control test failures often prompted authors
to discuss the sign and magnitude of the association. This suggests that
authors view the sign and size as meaningful for evaluating the bias direction
and size. However, in this section, we show that such logic is justified only
under additional non-trivial assumptions.

In non-IV contexts, negative controls are frequently used for bias correction.
For example, the Differences-in-Differences (DiD) estimator can be seen as a
form of bias correction using an NCO. In that context, the NCO is the lagged
period outcome (Sofer et al., 2016). For simple DiD, the average difference in
the NCO between treated and untreated units is an estimate for the confounding
bias. Hence, by subtracting it, the bias is corrected under the well-known
“parallel trends” assumption (Roth et al., 2023). More generally, when the NCO
does not directly quantify the bias, another type of negative control can be
used to scale the coefficient on the NCO such that it could be subtracted (Shi
et al., 2020)333333This design, known as double-negative control design, is
part of a more general theory and methods to use proxies to study causal
effects. This approach has been recently termed “causal proximal learning”
(Tchetgen Tchetgen et al., 2020).. In this section, we will consider a simple
case where the bias could be directly subtracted, and therefore scaling is not
necessary.

For this section, we assume the following simple IV setting. Let $Z_{1}$,
$X_{1}$, and $Y_{1}$ be the IV, treatment, and outcome, respectively. Assume
that an NCO $Y_{0}$ is also observed from a pre-period. We refer to $Y_{0}$ as
the lagged outcome; however, it could also represent an alternative pseudo-
outcome.343434For example, if $Y_{1}$ represents test scores in some subject,
$Y_{0}$ could measure test scores in an alternative subject. In this section,
we focus on the simple case where both the IV ($Z_{1}$) and the treatment
($X_{1}$) are binary. Assume treatment independence
($Z_{1}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}X_{1}(z)$) and monotonicity ($X_{1}\geq
X_{0}$) both hold. Hence, the first stage identifies the share of compilers in
the population. That is,
$\mathbb{E}[X_{1}|Z_{1}=1]-\mathbb{E}[X_{1}|Z_{1}=0]=\Pr[X_{1}(1)>X_{1}(0)]$,
even if the reduced form is biased.353535In many contexts, when the reduced
form is biased, the first stage would also be biased, and these assumptions
would not hold. Without these assumptions, the first-stage estimates would
require correction as well.

Similar to the canonical DiD model (without an IV), we further assume that
neither treatment nor the IV has been initiated in the pre-period
($Z_{0}=X_{0}=0$). This is a typical scenario for studies of new
interventions. Other pre-period scenarios are analyzed in Appendix B.2.

Our goal is to investigate whether the pseudo effect on the NCO can be
subtracted from the reduced form to correct the bias. The pseudo effect here
is the average difference in the NCO between IV-treated and IV-untreated
units, $\mathbb{E}[Y_{0}|Z_{1}=1]-\mathbb{E}[Y_{0}|Z_{1}=0]$. We consider the
following _differences-in-Wald_ (DiW) estimator363636This estimator can be
written as the difference between two Wald estimators with different outcome
variables ($Y_{0},Y_{1}$) but the same IV and treatment variables
($Z_{1},X_{1}$).

$DiW=\frac{\mathbb{E}\left[Y_{1}-Y_{0}|Z_{1}=1\right]-\mathbb{E}\left[Y_{1}-Y_{0}|Z_{1}=0\right]}{\mathbb{E}\left[X_{1}|Z_{1}=1\right]-\mathbb{E}\left[X_{1}|Z_{1}=0\right]}.$

We assume that the NCO satisfies assumptions analogous to DiD assumptions.
Denote $Y_{t}(x_{1},z_{1})=Y_{t}(0,0,x_{1},z_{1})$ for the potential outcomes
at time $t=0,1$.

######  Assumption 5 (Differences-in-Wald Assumptions).

  1. 1.

_No Anticipation_. For all $x_{1},z_{1},\tilde{x}_{1},\tilde{z}_{1}$,
$Y_{0}(x_{1},z_{1})=Y_{0}(\tilde{z}_{1},\tilde{x}_{1})$.

  2. 2.

_Parallel Trends_.
$\mathbb{E}[Y_{1}(0,0)-Y_{0}(0,0)|Z_{1}=1]=\mathbb{E}[Y_{1}(0,0)-Y_{0}(0,0)|Z_{1}=0].$

The first condition implies that the lagged outcome $Y_{0}$ is not affected by
the future treatment or IV (similiar to standard DiD, Roth et al., 2023). A
violation of this condition might imply that $Y_{0}$ does not satisfy the NCO
assumption. The second condition generalizes the DiD parallel-trends
assumption for the IV setting. It is the counterfactual statement that in the
absence of IV and treatment, the trend in the mean outcome would have been the
same for the IV-treated and IV-untreated observations.

Even under these strong assumptions, the DiW estimator does not yield a
consistent causal effect estimator.

######  Theorem 4.

Under treatment independence, monotonicity, and assumption 5, the Differences
in Wald Estimator equals

$\displaystyle
DiW=\mathbb{E}[Y_{1}(1,1)-Y_{1}(0,0)|X_{1}=1,X_{1}(1)>X_{1}(0)]+B_{1}$

where $B_{1}\neq 0$ and is given in Appendix B.2.

The theorem shows that DiW is an asymptotically biased estimator of the joint
causal effect of the IV and the treatment among the treated compliers. In
Appendix B.2 we show that the term $B_{1}$ is composed of two separate bias
terms. First, it captures violations of the exclusion restriction. This is
expected, as in this setting the NCO $Y_{0}$ cannot even detect such
violations. Second, $B_{1}$ includes the difference in treatment effect for
the always-takers with and without the IV. If the treatment effect for the
always-takers is substantially different across observations with and without
the IV, $B_{1}$ can receive large values and potentially even change the sign
of $DiW$. Therefore, observing the lagged outcome data before the IV and the
treatment were initiated does not suffice to sign or bound the bias; the bias
can be arbitrarily large.

Moreover, even if $B_{1}=0$, the estimator measures the average treatment
effect on treated compliers and does not capture the effect on untreated
compliers, which can generally be very different.

This DiW estimator is unbiased only with additional assumptions, as stated in
the following corollary.

######  Corollary 4.

Assume treatment independence, monotonicity, exclusion restriction, and
assumption 5 holds. Assume also that for every observation
$Y_{1}(1)-Y_{1}(0)=\tau$ for some constant $\tau$. The Differences in Wald
Estimator identifies $\tau$, $DiW=\tau$.

Under these (arguably strong) assumptions, the DiW is an unbiased estimate for
treatment effects, and the pseudo effect on the NCO can be used for bias
correction.

In Appendix B.2, we discuss two alternative and less common scenarios for
which the DiW estimator is unbiased under fewer assumptions. We show that
under different pre-period regimes (i.e., different assignments of
$Z_{0},X_{0}$) treatment homogeneity is not required, and violation of the
exclusion restriction can be corrected. However, such scenarios are less
frequent in practical applications.

##  7 Conclusion

This paper develops a theoretical framework for negative control tests for IV
designs. Our results uncover the underlying assumptions behind practices that
are frequently used in practice and give formal justification to common
intuitions. In this last section, we want to highlight four implications of
our findings that, in our impression, are not commonly known, and could have
practical implications on how negative control tests are used in practice.

First, our theory shows that some negative control tests are not only testing
the exogeneity of the IV (outcome independence and the exclusion restriction),
but are also testing functional form assumptions. When a negative control test
fails because of functional form assumption, the researcher could potentially
change the functional form (e.g., include non-linear controls) and correct the
validity of the design. Second, our theory clarifies which variables can serve
as negative controls. This includes variables that are rarely used in practice
such as as variables that causally affect the IV, which are valid NCIs.
Moreover, in some cases, variables that satisfy one of the negative control
assumptions are readily available in the data and can be used to construct
more powerful negative control tests. Third, for NCI tests, our theory spells
out in which cases the researcher needs to control for the IV. Failing to
control for the IV when necessary, which sometimes happens in practice, could
lead to the rejection of the null hypothesis and the discard of valid IV
designs. Finally, we find that negative control tests are limited in their
ability to correct the bias. Specifically, when the researcher finds a
correlation between the NCO and the IV, the sign and the magnitude of this
correlation is not informative about the bias without additional non-trivial
assumptions.

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$Z$$W$$X$$Y$$U_{1}$$NC_{1}$ A $NC_{1}\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Z$ implies that the dashed link exists, in
violation of outcome independence.

$Z$$W$$X$$Y$$NC_{2}$$U_{2}$ B $NC_{2}\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Z$ implies that the dashed link exists, in
violation of the exclusion restriction.

Figure 1: Negative Control Outcome: Graphical Illustrations

Notes: This figure illustrates the use of negative control outcome tests for
evaluating the validity of IV designs. In both panels, $X$ is the endogenous
variable, $Y$ is the outcome, $Z$ is the IV, and $W$ is a potential confounder
that motivates the use of the IV. The variables denoted $U$ are unobserved
_alternative path outcome variables_ (in short, APO) that pose threats to
identification, in the sense that if they are related to the IV (through the
links specified by dashed arrows), either outcome independence (Panel A) or
the exclusion restriction (Panel B) is violated. In each case, an observed
_negative control outcome_ ($NC$) that is related to the respective APO
variable ($U$) can be used to evaluate the presence of the problematic link,
by testing whether $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC$.

$Z$$W$$X$$Y$$U_{3}$$NC_{3}$ A $NC_{3}\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z$ implies violation of outcome
independence.

$Z$$W$$X$$Y$$NC_{4}$$U_{4}$ B $NC_{4}\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z$ implies violation of the exclusion
restriction.

Figure 2: Negative Control Instrument: Graphical Illustrations

Notes: This figure illustrates the use of negative control instrument tests
for evaluating the validity of instrumental-variable designs. In both panels,
$X$ is the endogenous variable, $Y$ is the outcome, $Z$ is the IV, and $W$ is