a potential confounder that motivates the use of the IV. The variables denoted
$U$ are unobserved _alternative path instrument variables_ (in short, API)
that pose threats to identification, in the sense that if they are related to
the outcome (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 instrument_ ($NC$) that is related to
the respective API variable ($U$) can be used to evaluate the presence of the
problematic link by testing whether $NC\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z$.

$Z$$X$$W$$Y$$U$$V$

Figure 3: Multiple Alternative Paths

Notes: This figure illustrates two potential violations of IV exogeneity,
through two alternative paths from the IV to the outcome, one through $U$ and
the other through $V$. Based on Definition 2, both $U$ and $V$ are APO
variables. As in all other examples, $X$ 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.

$Z$$W$$X$$Y$$U_{1}$$N$$Q$ A

$Z$$W$$X$$Y$$U_{2}$$N$ B

Figure 4: Violations of the Negative Control Outcome Assumption

Notes: This figure illustrates violations of the negative control outcome
assumption. As in all other examples, $X$ is the endogenous variable, $Y$ is
the outcome, $Z$ is the IV, and $W$ is a potential confounder that motivates
the use of an IV. In these examples, $U_{1},U_{2}$ are APO variables. However,
$N$ is not an NCO because it is related to $Z$ in other ways (marked by red
arrows), making $N$ uninformative about the APO variable. In Panel A, the IV
and $N$ are affected by an additional unobserved factor $Q$, which is
unrelated to the outcome. In Panel B, the IV directly affects the variable
$N$, violating the negative control assumption.

$Z$$W$$X$$Y$$NC$$U$

Figure 5: Unconditional NCI Example

Notes: This figure demonstrates a negative control instrument scenario where
conditioning on the IV is not necessary. As in all other examples, $X$ is the
endogenous variable, $Y$ is the outcome, $Z$ is the IV, and $W$ is a potential
confounder that motivates the use of an IV. In this example, $U$ is an
unobserved API variable that poses a threat to identification. If it is
related to the outcome (through the dashed arrow), the exclusion restriction
is violated. An observed NCI ($NC$) that affects the API variable ($U$) can be
used to evaluate the presence of the problematic link using an unconditional
independence test for the null hypothesis $H_{0}:NC\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y$.

Figure 6: Comparison of NCO Tests in Simulations of Outcome Independence
Violation

This figure presents the results of simulation studies using 1,000 samples of
10,000 observations each. In Panel A, the association between the APO and the
IV is linear. In Panel B, the association is highly non-linear. The data-
generating process is detailed in Appendix C. In both panels, NCO tests
applied include multiple pseudo-outcome regressions with Bonferroni
correction, a single linear regression (Equation (10)) with an F-test, and a
GAM with Wald test, both with and without smooth terms for continuous
controls. Each panel compares the rate of $H_{0}$ rejection of these tests as
a function of the strength of the IV-APO relationship ($\gamma_{U}$ in
Equation (15)). For details of the testing methods, see Section 4.

Figure 7: Comparison of NCO Tests in Simulations of Rich Covariates Violation

This figure presents the results of simulation studies using 1,000 samples
drawn with 10,000 observations each. The data generating process is detailed
in Appendix C. We use the following NCO tests: multiple pseudo-outcome
regressions with Bonferroni correction for multiple hypotheses, a single
linear regression (Equation (10)) with either an F-test, or a Wald test, and
GAM with Wald test with and without smooth terms for the continuous controls.
Each panel compares the rate of $H_{0}$ rejection as a function of the extent
of non-linearity of the IV in the control variable ($\eta$ in Equation (16)).
For details of the testing methods, see Section 4.

Figure 8: Steps for Implementing Negative Control Tests
1. Select Negative Control Variables • Consider all candidate NCOs or NCIs in the data that can serve as proxies for identification threats (APO or API variables). • Evaluate whether each candidate satisfies a negative control assumption (based on Definition 3 for NCO or Definition 5 for NCI). • Note that including ‘weak’ negative controls that are unlikely to be informative may reduce test power. • See Section 5.1 for examples of negative control variables. 2. Design the Negative Control Test • Select the test specification: When using 2SLS and when multiple NCOs are available, consider using an F-test, which tests both IV exogeneity and rich covariates. • Otherwise, depending on power, consider non-linear methods less sensitive to functional form assumptions. • In an NCI test, remember to condition on the IV when appropriate. • See Section 5.2 for discussion. 3. Interpret Results • Null rejected: – Test functional form directly (e.g., use Ramsey test). – Check which negative control is most related to both the outcome and the IV. • Null not rejected: – Consider power. – Acknowledge that untested threats may still exist.

$Z$$W$$X$$Y$$U$$NC$

Figure 9: A Cause of the IV is an NCI

Notes: This figure illustrates that a variable that causes the IV is a valid
negative control instrument. As in all other examples, $X$ is the endogenous
variable, $Y$ is the outcome, $Z$ is the IV, and $W$ is a potential confounder
that motivates the use of an IV. The variable $U$ is an unobserved API
variable that poses a threat to identification, in the sense that if they are
related to the outcome (through the links specified by dashed arrows) outcome
independence is violated. An observed NCI ($NC$) that is affecting the IV
($Z$) can be used to evaluate the presence of the problematic link by testing
$NC\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z$. Specifically, if
$NC\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z$ then the path $NC\rightarrow Z\leftarrow
U\rightarrow Y$ exists and hence $U\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z$. The square around $Z$ denotes
conditioning on it.

Figure 10: Correlations of NCOs with the IV and the Outcome in Deming (2014)

This figure shows a scatter plot of the absolute value of the correlation of
different NCOs with the IV (on the y-axis) and the outcome (on the x-axis).
Correlations are calculated using data from Deming (2014). The NCOs, the IV,
and the outcome were first residualized by regressing them on all control
variables and lottery fixed-effects. Each observation is one NCO. For
presentation purposes, NCOs are grouped into categories denoted by marker
shape. The different year markers refer to groups of students’ test scores
from that year. The VA marker denotes the value-added of the schools listed by
students as their 1st, 2nd, and 3rd submitted preferences, as well as their
neighborhood school’s VA (labeled Neighborhood School).

Table 1: Current Use of Falsification Tests for IV in Economics

Falsification Test Characteristics (Share of Papers that Included
Falsification Tests) Type of Test Test Specification Papers Reviewed Share
with Falsification Tests Negative Control Outcome Negative Control Instrument
Other Pseudo Outcome Pseudo IV # Negative Controls Used (median) (1) (2) (3)
(4) (5) (6) (7) (8) All Top-5 140 0.51 0.75 0.25 0.24 0.41 0.24 4 by Journal:
REStud 48 0.42 0.75 0.40 0.10 0.65 0.40 4 AER 42 0.50 0.86 0.24 0.14 0.52 0.24
4 JPE 21 0.62 0.54 0.15 0.31 0.15 0.08 3.5 QJE 19 0.63 0.83 0.25 0.33 0.17
0.25 5 ECMA 10 0.50 0.60 0.00 0.40 0.20 0.00 4.5

Notes: The table shows the result of our survey of highly cited articles
employing instrumental variable (IV) designs published in top economic
journals from 2013 to 2023. The sample includes all articles from this period
in the Review of Economic Studies (REStud), American Economic Review (AER),
Journal of Political Economy (JPE), Quarterly Journal of Economics (QJE), and
Econometrica (ECMA) that used IV designs and had significant citation counts
on Google Scholar (over 300 citations for papers until 2020, and over 100 for
those published after 2020). We examined these papers for their use of
falsification tests. Column 2 shows the proportion of papers employing any
falsification test. The subsequent columns detail various characteristics of
these tests. Columns 3–5 categorize the tests into negative control outcome,
negative control instrument, and other types of falsification tests,
respectively. Note that the fractions do not sum to one, as some papers
employed multiple test types. Columns 6–7 specify the proportion of papers
using pseudo-outcome and pseudo-IV designs. Column 8 reports the median number
of negative control variables used.

Table 2: Illustrative Applications of Negative Control Outcome Tests

Original Analysis Alternative Analyses Type of Test: Pseudo- Outcome (Single)
Pseudo- Outcome (Single) Pseudo- Outcome (Multiple) F-test GAM, linear
controls GAM, non-linear controls Number of Observations (1) (2) (3) (4) (5)
(6) (7) Autor et al. (2013) $<$0.001 0.593† $<$0.001 $<$0.001 1.000 1.000 722
Deming (2014) - 0.686†† $<$0.001 $<$0.001 $<$0.001 $<$0.001 2343

Notes: This table reports the $p$-values of different negative control outcome
tests using replication data from Autor et al. (2013) and Deming (2014).
Column 1 replicates the original falsification analyses: Autor et al.
performed a variant of the pseudo-outcome test using the same 2SLS
specification as the original analysis with each of three pseudo-outcome
variables (as reported in part II of their Table 2). Deming did not report any
falsification tests. Column 2 reports the p-value from a single pseudo-outcome
test (Equation (8)) using only the main NCO. Columns 3 through 6 present
results using a broader set of NCOs that we identified in the original data
based on our theory (52 variables for Autor et al. and 37 for Deming), using
the same controls as in the most exhaustive specification in the original
analysis. Column 3 presents a Bonferroni-corrected p-value for multiple
pseudo-outcome tests using all the NCO variables. Column 4 presents an F-test
(Equation (10)) using all NCOs jointly in a single regression. Columns 5 and 6
detail two GAM test specifications (Equations (11) and (12)) using all NCOs
jointly. In Column 5, the control variables are assumed to be linear, while in
Column 6, the control variables are smoothed as well.

†The reported value corresponds to the lagged outcome (in 1970) NCO. This is
the same NCO as in the original paper (Column 1) once the control variables of
the full specification are included.

††The reported value corresponds to the lagged test scores (in 2002) NCO.

Table 3: Illustrative Applications of Negative Control Instrument Tests

Without Conditioning on the IV With Conditioning on the IV Pseudo IV (Single)
Pseudo IV (Single) Pseudo IV (Multiple) F-test GAM, Linear Controls Number of
Observations (1) (2) (3) (4) (5) (6) Nunn and Qian (2014) 0.007 0.123 0.636
0.138 <0.01 4572 Ashraf and Galor (2013) 0.234 0.778 1.000 0.994 1.000† 145

Notes: This table reports the $p$-values of different negative control
instrument tests using replication data from Nunn and Qian (2014) and Ashraf
and Galor (2013). We use the same sets of NCIs used in the original analyses
(which include three and ten NCI variables, respectively). Column 1 shows a
single pseudo IV test (Equation (9)) that, inappropriately, does not condition
on the IV. We use the NCI that generates the minimal p-value (grape production
for Nunn and Qian and distance from Mexico City for Ashraf and Galor). Columns
2 through 6 show different tests that do condition on the IV. Column 2 reports
the p-value from a single pseudo-IV test that controls for the IV, using the
same NCI as in Column 1. Column 3 presents a Bonferroni-corrected p-value for
multiple pseudo-IV tests using all the NCI variables. Column 4 presents an
F-test using all NCIs jointly in a single regression. Column 5 reports results
from a GAM test that smoothes the IV and the NCIs, but maintains linearity in
the control variables. Smoothing the control variables is impossible due to
the large number of controls relative to the sample size.

†In all three marginal GAM models.

On-Line Supplementary Appendices Negative Controls for Instrumental Variable
Designs

##  Appendix A Examples and Counterexamples

###  A.1 Non-Causal APO Variable

$U_{1}$$U_{2}$$NC$$Z$$X$$Y$$W$ Figure A1: An illustration of causal and non-
causal APOs.

In the example presented in Figure A1, $U_{2}$ is a valid APO variable, which
satisfies path indication, even though there is no direct causal effect on $Y$
(other than the potential effect through the IV). Path indication is satisfied
because both $U_{2}$ and $Y$ are commonly caused by $U_{1}$. For example,
consider $Z$ to be teacher assignment that is claimed to be quasi-random, $X$
the teacher value added of the teacher and $Y$ some test scores. The variable
$W$ represents the concern that some students can switch classrooms and end up
with different teachers. In this example, $U_{1}$ is unobserved ability, which
directly affects test scores. $U_{2}$ represents detailed test scores in some
previous exams that are also unobserved. The dashed arrow represents a concern
that principals allocate students to teachers based on the detailed test score
(e.g., students with low math scores are assigned to a specific teacher).

In this example, $U_{2}$ satisfies path indication even though it is not
causally affecting the outcome as detailed past test scores do not directly
affect future test scores. However, there is a path between the previous
detailed test scores and the current test scores, because both are affected by
ability ($U_{1}$).

The variable $NC$ is aggregated previous test scores, which averages past
scores in math with other subjects. In this setting, $NC$ is a negative
control, with $U_{2}$ as an APO variable. An association between the IV and
aggregated lagged test scores would imply an alternative path from the IV to
the outcome. Specifically, this will violate outcome independence as students
with different abilities would sort into different teachers based on their
previous math scores.

Of note is that in this scenario, $U_{1}$ is also an APV. However, $NC$ is a
valid negative control with respect to $U_{2}$ but not with respect to $U_{1}$
alone, as conditionally on the unobserved ability, there is still a
correlation between the negative control and the instrument
($Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|U_{1}$), i.e, the teacher assignment is
not independent of the aggregated test scores.

###  A.2 Violation of Path Indication: Multidimensional Variable

$U_{1}$$U_{2}$$N$$Z$$X$$Y$$W$$U$ Figure A2: Violation of path indication
(Definition 1 when $U$ has multiple components

The example presented in Figure A2 discusses a case where $U$ is a vector of
dummy variables. The ellipse around $U_{1},U_{2}$ is a visual marking of
taking $U=(U_{1},U_{2})$. For example, assume $Z$ is teacher assignment, which
is claimed to be quasi-random, $X$ is the teacher value added of the teacher
and $Y$ is test scores. The variable $W$ represents the concern that some
students can switch classrooms and end up with different teachers. Assume that
$U$ is the student’s hobbies, which they report to the school, but unobserved
to the researcher. Each student can have more than one hobby.

Assume that $U_{1}$ is having basketball as one of the student’s hobbies.
Assume also that basketball is correlated with the IV. For example, one
teacher also teaches basketball so basketball players (who list basketball as
their favorite hobby) are automatically assigned to her. However, a basketball
hobby does not affect test scores. Let $U_{2}$ indicate having math as one of
your hobbies. Assume that students that report math as their favorite hobby
tend to perform better in exams and that math lovers are randomly allocated
across teachers. Finally, assume that $U_{1}\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U_{2}$.

In this case, the vector $U$ is not an APO variable as it does not satisfy
path indication. The IV is exogenous ($Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$) as even though the IV is correlated
with having basketball as a hobby, this is irrelevant for test scores.

Defining $N$ as participating in an after-school basketball program, we find
that $N$ is not a proper NCO. Even though $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}N|U$, it is still not a NCO since $U$ is not
an APO variable.

###  A.3 Heterogeneity-Based Violation of Path Indication

$W$$Z$$X$$Y$$U$$N$ A Females

$W$$Z$$X$$Y$$U$$N$ B Males

Figure A3: Violation of path indication (Definition 1) due to heterogeneity

Figure A3 describes a random variable $U$ that is linked to both the IV $Z$
and the potential outcome $Y(x)$. Yet is not an APO variable since it does not
satisfy path indication. In Figure A3, the IV affects $U$ for males but not
for females. On the other hand, $U$ affects $Y$ for females but not for males.
Path indication is not satisfied since even though the IV is exogenous
($Z\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$), it is still associated with $U$
($Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U$).

Let $N$ be a proxy for the variable $U$. In this example, for a large enough
sample size we will conclude that $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}N$ (due to the affect among males), but from
this test, one cannot deduce that the IV is no exogenous since $U$ is not an
APO variable.

A remark is that had we considered an analysis among males only, path
indication would still be violated because $U$ is not associated with $Y(x)$.
However, among females only, $U$ is an APO variable, $N$ is a valid negative
control, and the IV design would pass the test (among the females,
$Z\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}N$).

###  A.4 Direct IV link rules out proxies of $V$

$V$$Z$$X$$Y$$W$$U_{2}$$U_{1}$ Figure A4: Violation of direct IV link
(Definition 2

Figure A4 presents the threat $V$ (violation of outcome independence) as well
as two proxies of $V$, $U_{1}$ and $U_{2}$. Note that latent IV exogeneity, as
stated in Definition 2, holds for either variable ($U_{1}$ or $U_{2}$), as the
further conditioning on $U_{1}$ or $U_{2}$ does not invalidate the IV,
conditionally on $V$. Note also that for both variables, path indication holds
as 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U_{1}|V$ (or $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U_{2}|V$). However, we rule out $U_{1}$ and
$U_{2}$ as APO variables because they do not represent a threat to IV
exogeneity. Condition 3 of Definition 2, direct IV link, does not hold,
because $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U_{1}$ while $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U_{1}|V$ (and similarly for $U_{2}$).

###  A.5 Path indication rules out proxies of $V$

$Z$$V$$X$$Y$$W$$U$ Figure A5: Violation of path indication (Definition 2)

Figure A5 presents the threat $V$ (violation of exclusion restriction) as well
as a proxy of $V$, $U_{1}$. Note that latent IV exogeneity holds, as the
further conditioning on $U_{1}$ does not invalidate the IV once we have
conditioned on $V$. Note also that direct IV link holds as 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|V$. However, we rule out $U_{1}$ as an APO
variable because it does not represent a threat to the IV exogeneity.
Condition 2 of Definition 2, path indication, does not hold, because
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U|V$ while $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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 the language of DAGs, $V$ is a
collider (Pearl, 2009) and conditioning on it creates a dependence between $U$
and $Z$.

Here $U$ is a cause of $V$, had we considered an effect of $V$, it would have
also been ruled out as an APO variable, but due to violation of direct IV
link.

###  A.6 Violation of V-validity

$U$$Z$$X$$Y$$W$$V$$N$ Figure A6: Violation of $V$-validity

Figure A6 presents a situation with no valid APO variable. We examine $U$ as a
candidate APO variable, and consider $V$ in the DAG as the potential $V$ in
Definition 2. We see that latent instrument validity holds: while
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|V$, because $V$ is a common effect (a
collider) of $U$ and $Y$, controlling for $U$ in addition to $V$ blocks the
flow of association (Pearl, 2009) resulting in $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|U,V$. Path indication holds trivially
because $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|V$. Direct IV link also holds because of
the effect of $U$ on $Z$. However, it is clear $U$ should not be an APO
variable. An association between $Z$ and $U$ does not imply the IV design is
invalid. Here comes $V-$validity to the rescue. In Figure A6, the IV is
exogenous ($Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$), but, as previously noted
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|V$ due to $V$ being a common effect of
both variables.

In this case, no other choice for $V$ exists to satisfy Definition 2.
Therefore $U$ is not an APV, and the random variable $N$ is not a negative
control.

###  A.7 Counterexample: A Vector of NCOs Which is Not an NCO

Let $R_{1},R_{2}$ be two independent Bernoulli random variables with
probabilities $\Pr(R_{j}=1)=p_{j}$ with $p_{1}=p_{2}=0.5$. Let $U$ be another
Bernoulli random variable, independent of $\\{R_{1},R_{2}\\}$. Let $Z$ be the
IV, and assume that

$Z=(R_{1}\oplus R_{2})+\theta U+\epsilon_{Z},$

where $\oplus$ is the XOR operator. Assume that $Y(x)=x+U+\epsilon_{Y}$, such
that $U$ is an APO variable. The IV design is valid if $\theta=0$.

Now, assume there are two observed negative controls $NC_{i}=U\oplus R_{i}$
for $i=1,2$. Both $NC_{1}$ and $NC_{2}$ are valid negative controls as they
satisfy the assumption $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC_{i}|U$. This is because for $i=1,2$,
$R_{i}\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(R_{1}\oplus R_{2})$, and therefore
$Z\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}R_{i}|U$. However
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}(NC_{1},NC_{2})|U$. Therefore
$(NC_{1},NC_{2})$ does not satisfy the NCO assumption. Indeed, even if the IV
is valid, we could still have $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}(NC_{1},NC_{2})$.

A small change in the data-generating process will break some of the
independencies discussed above. For example, changing the value of $p_{1}$ to
something different from 0.5 would imply that
$R_{2}\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}(R_{1}\oplus R_{2})$. In that case,
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC_{2}$ and $NC_{2}$ would no longer satisfy
the NCO assumption.

###  A.8 Alternative Path Variables Linked to Treatment

$Z$$W$$X$$Y$$U$$NC$ A $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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 violation of IV exogeneity. $U$
is an APO variable. $NC$ is an NCI.

$Z$$W$$X$$Y$$U$$N$ B $Y\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}N|Z$ does not necessarily implies violation
of IV exogeneity. $U$ is not an API variable. $N$ is not an NCI.

Figure A7: Direct association between potential alternative path variables and
the treatment

Path indication in Definition 4 implies that conditionally on the IV ($Z$), an
API variable ($U$) cannot be associated with the treatment ($X$). By contrast,
there is no such requirement from an APO variable. Figure A7 illustrates these
points. In Panel A, $U$ is a valid APO variable and the arrow $U\rightarrow X$
is allowed; Latent IV exogeneity holds, and so is path indication. Therefore,
$NC$ is a valid NCO, and an association between $NC$ and $Z$ implies the
dashed arrow between $U$ and $Z$ exists so IV exogeneity does not hold.
Conversely, in Panel B, $U$ is not a valid API variable. Path indication is
violated because $U\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z$ does not 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)$. Therefore, $N$ is not an NCI and
$Y\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}N|Z$ does not necessarily implies violation
of IV exogeneity. Intuitively, $N\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z$ even if the IV design is valid, because
of the association between $U$ and $Y$ through $X$. Conditioning on $X$ would
not solve this problem, because $X$ is a collider.
$N\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z,X$ because of the path $N\leftarrow
U\rightarrow X\leftarrow W\rightarrow Y$ (Pearl, 2009) .

##  Appendix B Additional Theory and Proofs

Throughout, we let $P(\cdot|\cdot)$ be the conditional probability or density
function. As a shorthand, we leave the random variables to be understood from
the arguments of $P$. For example, if $Y(x)$ is discrete, $P[y(x)|u]$ is a
shorthand for $\Pr[Y(x)=y(x)|U=u]$.

### Auxiliary Lemmas

######  Lemma 1.

Let $A,B,D,Q$ be four vectors of random variables. If
$A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B|D,Q$ and $B\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Q|D$ then $A\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B|D$.

###### Proof.

Because $A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B|D,Q$ we have for all $a,b,d,q$

$\displaystyle\begin{split}P(a,b|d,q)&=P(a|d,q)P(b|d,q)\\\
&=P(a|d,q)P(b|d),\end{split}$ (A1)

where the last line follows from $B\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Q|D$. Now,

$P(a,b|d)=\int P(a,b|d,q)P(q|d)dq=\left[\int
P(a|d,q)P(q|d)dq\right]P(b|d)=P(a|d)P(b|d),$

where the second move is by (A1). ∎

######  Lemma 2.

Let $A,B,D,Q$ be four vectors of random variables. If
$A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B|D,Q$ and
$A\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}B|D$ then
$A\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Q|D$ and $B\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Q|D$.

###### Proof.

Assume by contradiction that $B\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Q|D$. Therefore, by Lemma 1, because
$A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B|D,Q$ we have that
$A\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}B|D$ which contradicts the assumption.
Similarly for $A\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Q|D$. ∎

###  B.1 Negative Controls for Instrumental Variable when Control Variables
are Included

This section includes the proofs of the theoretical results from Section 3. We
prove versions of the results when controls are included. Therefore, we first
review the IV assumptions when controls are included, and then discuss
separately APO variables and NCOs (Section B.1.1), and API variables and NCIs
(Section B.1.2).

We start by presenting the independence and exclusion restriction assumptions
when controls are included.

######  Assumption A1.

Outcome 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(z,x)|C$ for all possible $z,x$ values.

######  Assumption A2.

Exclusion restriction. $\Pr(Y(z,x)=Y(z,x^{\prime})=Y(x)|C=c)=1$ for all
possible $x,z,z^{\prime},c$ values.

Similar to the case without controls, outcome independence and exclusion
restriction together yield $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|\ C$.

####  B.1.1 Negative Control Outcomes when Control Variables are Included

We adapt the definitions of APO variables and negative controls as follows.

######  Definition A1 (Alternative path outcome variable with controls).

A random variable $U$ is an APO variable conditionally on a set of controls
$C$ if there exists a vector of variables $V$ such that

  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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|C,U,V$

  2. 2.

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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|C,V$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|C$.

  3. 3.

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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|C,V$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|C,V$.

  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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C,V$.

######  Definition A2 (Negative control outcome with controls).

A random variable $NC$ is an NCO if it satisfies the NCO assumption with
respect to controls $C$: 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|C$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|\ U,C$

We are now ready to state a version of Theorem 1 with controls and present its
proof. This theorem also covers the case no control are used (as in Theorem 1)
by letting $C$ to be degenerate.

######  Theorem A5.

Assume that a random variable $NC$ satisfies the NCO assumption with respect
to controls $C$ (Definition A2). 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|C$ then
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C$

###### Proof.

$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|C$ implies that
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U|C$. Else, 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}U|C$ then by the NCO assumption (see
Definition A2) $Z\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|\ U,C$. Based on Lemma 2,
$Z\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|\ U,C$ and
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|C$ imply that
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U|C$, a contradiction.

From direct IV link we get that $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U|C$ implies
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}U|V,C$. Then by path indication, this implies
that $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|V,C$. Finally, from V-validity we get
that

$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C$

∎

##### Proof of Corollary 3

###### Proof.

Let $\tilde{Z}=Z-C^{\prime}E[(CC^{\prime})]^{-1}E[CZ]$ and
$\widetilde{NC}=NC-C^{\prime}E[(CC^{\prime})]^{-1}E[CNC]$ be the residuals
from the linear regressions of $Z$ and $NC$ on $C$, respectively. From
Frisch–Waugh–Lovell Theorem, we can write $\beta_{Z}$ as

$\beta_{Z}=\frac{COV(\tilde{Z},\widetilde{NC})}{Var(\tilde{Z})}.$

If $\beta_{Z}\neq 0$ then it must be that $COV(\tilde{Z},\widetilde{NC})\neq
0$. Define $(\delta,\gamma^{\prime})$ to be the population-level solution of
the reverse OLS, with $Z$ as the independent variable,

$(\delta,\gamma^{\prime})=\operatorname*{arg\,min}_{d,q}\mathbb{E}(Z-dNC-q^{\prime}C)^{2}.$

Again, by Frisch–Waugh–Lovell Theorem we can write $\delta$ as

$\delta=\frac{COV(\tilde{Z},\widetilde{NC})}{Var(\widetilde{NC})}.$

Since $COV(\tilde{Z},\widetilde{NC})\neq 0$ then $\delta\neq 0$ as well.
Therefore, $\mathbb{E}[Z|C,NC]\neq\gamma^{\prime}C$. From Corollary 1, this
implies that either outcome independence, the exclusion restriction, or rich
covariates does not hold.

∎

####  B.1.2 Negative Control Instruments when Control Variables are Included

We first extend Definition 4 to allow for additional threats, as captured by
the possibly multivariate variable $V$ (as in Definition 2 for APO variables),
and to include controls $C$ (as in Section B.1.1).

######  Definition A3 (Alternative path instrument variable with controls).

A random variable $U$ is an API variable if the following three 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|C,U,V$.

  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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|C,V$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z,C,V$.

  3. 3.

Direct outcome link. 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z,C,V$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z,C$.

  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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C,V$.

Condition 1 is a conditional versions of Condition 1 in Definition 1,
conditionally on $C,V$. Conditions 2–4 together imply a conditional version of
Condition 2 from Definition 1. Similar to the theory of APO variables, the
conditional condition was broken to three independent conditions to exclude
proxies to threats being API variables and to maintain that $V$ is indeed a
threat (similar to Definition 2).

Next, we generalize the definition of an NCI to includes controls as follows.

######  Definition A4 (Negative control instrument with controls).

A random variable $NC$ is an NCI if it satisfies the NCI assumption with
respect to controls $C$: 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y|Z,C$ 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|Z,C,U.$ (A2)

We are now ready to state a version of Theorem 2 with controls and present its
proof. This theorem also covers the case no control are used (as in Theorem 2)
by letting $C$ to be degenerate.

######  Theorem A6.

Assume that a random variable $NC$ satisfies the NCI assumption with respect
to controls $C$ (Definition A4). 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|Z,C$, then either outcome independence or
exclusion restriction is violated. That is, the IV design is invalid.

###### Proof.

We divide the proof into two cases with respect to the API variable $U$ for
which the NCI assumption holds for NC:

  1. (a)

$U\cancel{{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}}Y|Z,C$,

  2. (b)

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

In case (a), by direct outcome link we obtain
$U\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z,C,V$, which gives by path indication
$Z\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C,V$. Therefore, by $V$-validity we get
that $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C$, i.e., IV exogeneity is violated.

Turning to case (b). From the NCI assumption (Definition A4), we have that
$Y{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|Z,C,U$ and
$Y\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|Z,C$ from the condition if the theorem.
Therefore, by Lemma 2 (with $D=\\{Z,C\\},Q=U$), we obtain that
$U\cancel{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z,C$, which contradicts the assumption
$U{\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y|Z,C$, which by the proof above of case (a),
implies $Z\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}Y(x)|C$.

∎

We turn to state and prove a conditional version (conditional on controls $C$)
of Theorem 3.

######  Theorem A7.

Assume that a random variable $NC$ satisfies the NCI assumption with respect
to controls $C$. 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|C$, 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|C$, then either outcome independence or
exclusion restriction is violated. That is, the IV design is invalid.

###### Proof.

Assume by way of contradiction that $Y\cancel{\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}}NC|C$ and 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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Y(x)|C$ holds. Because $NC$ is an NCI,
according to Theorem A6, we have $Y\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|Z,C$. Additionally, based on the assumption
that $Z\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|C$, Lemma 1 implies that
$Y\mathchoice{\mathrel{\hbox to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}NC|C$, a contradiction. ∎

###  B.2 Bias correction Appendix

####  B.2.1 Additional Scenarios

Our results in Section 6 discussed the scenario of an NCO $Y_{0}$, which is a
lagged outcome from a pre-period before the IV and the treatment were
initiated (i.e., $Z_{0}=X_{0}=0$ with probability one). In this section, we
study two additional scenarios for the lagged period. Proofs are given in
Section B.2.2.

  * •

No-IV: $Z_{0}=0,X_{0}=X_{0}(0)$

  * •

No IV-treatment link: $Z_{0}=Z_{1},X_{0}=X_{0}(0)$

In the no-IV scenario, the treatment has already started, however, the IV was
not been initiated yet. This corresponds to cases where the IV represents a
new policy to encourage participation in an ongoing treatment. For example,
the IV is a new subsidy for participation in an already-running program.

The no IV-treatment link scenario considers the case where only the link
between the IV and the treatment has not been initiated yet. In this scenario,
the IV was already initiated. Moreover, we assume that the IV is the same in
both periods. The treatment was also already initiated. However, the IV still
does not have a causal effect on the treatment. For example, the IV could be a
subsidy to encourage participation in the treatment. In contrast to the no-IV
scenario, here the subsidy is already given at the lagged period. Yet, it
still does not affect the treatment (e.g., there is a delay in joining the
treatment, which does not affect those already treated). While this scenario
would be most useful for bias correction, it is also quite rare to observe.

Denote $Y_{t}(z_{0},x_{0},z_{1},x_{1})$ for the potential outcomes at time
$t=0,1$. Whenever we write $Y_{t}(z,x)$ for some $z,x$, we refer to the
potential outcomes under the (hypothetical) intervention that sets $Z_{t}=z$
and $X_{t}=x$. The potential outcomes at time $t=1$ depend on the (random) way
$Z_{0}$ and $X_{0}$ are assigned, which varies across the three scenarios.

As before, we assume that the NCO satisfies similar assumptions to the
assumptions of DiD (Roth et al., 2023).

######  Assumption A3 (Differences-in-Wald Assumptions).

  1. 1.

_No Anticipation_ $Y_{0}(z_{1},x_{1},z_{0},x_{0})=Y_{0}(z_{0},x_{0})$.

  2. 2.

_Parallel Trends_

$\displaystyle E$
$\displaystyle\big{[}Y_{1}(z_{1}=Z_{0},x_{1}=X_{0}(Z_{0}))-Y_{0}(z_{0}=Z_{0},x_{0}=X_{0}(Z_{0}))\
|\ Z_{1}=1\big{]}$ $\displaystyle=E$
$\displaystyle\big{[}Y_{1}(z_{1}=Z_{0},x_{1}=X_{0}(Z_{0}))-Y_{0}(z_{0}=Z_{0},x_{0}=X_{0}(Z_{0}))\
|\ Z_{1}=0\big{]}.$

The first condition implies that the lagged outcome $Y_{0}$ is not affected by
the future treatment or instrument. The second condition generalizes the DiD
parallel-trends assumption for the IV setting. It is a counterfactual
statement that in the absence of changes in the IV and treatment between the
periods, the change in the outcome would be the same in expectation for
observations that are IV-treated and IV-untreated. In the case of no IV and no
treatment ($Z_{0}=X_{0}=0$), this assumption is equivalent to Condition 2 in
Assumption 5.

Since in these scenarios treatment is defined in the pre-period as well, we
need to make an additional assumption.

######  Assumption A4 (Same Type).

For every value of $z\in{0,1}$, $X_{0}(z)=X_{1}(z)$

This assumption states that for all observations, the type (always-taker,
compiler, never-taker) is the same in both periods. Hence, always-takers are
treated in both periods, never-takers are untreated in both periods, and
compliers are only treated when the IV equals one.

The following two theorems analytically calculate the value of $DiW$ in each
of the two scenarios.

######  Theorem A8 (No-IV).

Assume the no-IV scenario. Under treatment independence, monotonicity, same-
type (assumption A4) and assumption 5

  1. (i)

The Differences in Wald Estimator equals

$\mathbb{E}[Y_{1}(1,1)-Y_{1}(0,0)|X_{1}=1,X_{1}(1)>X_{1}(0)]+B_{2},$

where $B_{2}$ is given in Appendix B.2.2.

  2. (ii)

If, in addition, exclusion restriction holds, the Differences in Wald
Estimator equals

$\mathbb{E}[Y_{1}(x_{1}=1)-Y_{1}(x_{1}=0)|X_{1}=1,X_{1}(1)>X_{1}(0)]$

######  Theorem A9 (No IV-treatment link).

Assume the no IV-treatment link scenario. Under treatment independence,
monotonicity, same-type (Assumption A4) and Assumption 5, the Differences in
Wald Estimator estimator identifies the causal effects on the treated
compliers. That is,

$DiW=[Y_{1}(1,1)-Y_{1}(1,0)|X_{1}=1,X_{1}(1)>X_{1}(0)]$

Theorem A8 shows that in the no-IV scenario, $DiW$ can be used for bias
correction for violation of the independence assumption. However, violation of
the exclusion restriction would still bias this estimator. By contrast,
Theorem A9 shows that in the no IV-treatment link scenario, $DiW$ could
address violations of both independence and exclusion. Once again, the DiW
estimator estimates the causal effect only for the treated compliers.

####  B.2.2 Proofs

In all three proofs presented below, the denominator of the DiW estimator (the
first stage) is the proportion of compliers in the population, i.e.,

$\mathbb{E}[X_{1}|Z_{1}=1]-\mathbb{E}[X_{1}|Z_{1}=0]=\Pr[X_{1}(1)>X_{1}(0)],$

since we assume treatment independence and monotonicity.

##### Proof of Theorem 4

As previously noted, the parallel-trends assumption under this scenario is

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

The numerator equals to

$\displaystyle\mathbb{E}[Y_{1}-Y_{0}$
$\displaystyle|Z_{1}=1]-\mathbb{E}[Y_{1}-Y_{0}|Z_{1}=0]$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{0}(0,0)|Z_{1}=1\big{]}-E\big{[}Y_{1}(0,X_{1}(0))-Y_{0}(0,0)|Z_{1}=0\big{]}.$
(A3)

By parallel trends, we can replace $Y_{0}(0,0)$ with $Y_{1}(0,0)$ inside both
of the expectations of (A3) to obtain

$\mathbb{E}[Y_{1}(1,X(1))-Y_{1}(0,0)|Z_{1}=1]-\mathbb{E}[Y_{1}(0,X(0))-Y_{1}(0,0)|Z_{1}=0].$
(A4)

By the law of total expectation with respect to the type (compliers, for which
$X_{1}(1)>X_{1}(0)$; always-takers, for which $X_{1}(0)=1$; and never-takers,
for which $X_{1}(1)=0$), and by Treatment Independence
$\\{X(0),X(1)\\}\mathchoice{\mathrel{\hbox
to0.0pt{$\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$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}Z$ we can further write Equation (A4) as

$\displaystyle\Pr\big{[}X_{1}(1)>X_{1}(0)\big{]}\Big{(}E\big{[}Y_{1}(1,1)-Y_{1}(0,0)|Z_{1}=1,X_{1}(1)>X_{1}(0)\big{]}-E\big{[}Y_{1}(0,0)-Y_{1}(0,0)|Z_{1}=0,X_{1}(1)>X_{1}(0)\big{]}\Big{)}$
$\displaystyle+\Pr\big{(}X_{1}(1)=0\big{)}\Big{(}E\big{[}Y_{1}(1,0)-Y_{1}(0,0)\
|\ Z_{1}=1,X_{1}(1)=0\big{]}-E\big{[}Y_{1}(0,0)-Y_{1}(0,0)\ |\
Z_{1}=0,X_{1}(1)=0\big{]}\Big{)}$
$\displaystyle+\Pr\big{(}X_{1}(0)=1\big{)}\Big{(}E\big{[}Y_{1}(1,1)-Y_{1}(0,0)\
|\ Z_{1}=1,X_{1}(0)=1\big{]}-\mathbb{E}[Y_{1}(0,1)]-Y_{1}(0,0)\ |\
Z_{1}=0,X_{1}(0)=1\big{]}\Big{)}$
$\displaystyle=\Pr\big{[}X_{1}(1)>X_{1}(0)\big{]}E\big{[}Y_{1}(1,1)-Y_{1}(0,0)\
|\ Z_{1}=1,X_{1}(1)>X_{1}(0)\big{]}$
$\displaystyle+\Pr\big{(}X_{1}(1)=0\big{)}E\big{[}Y_{1}(1,0)-Y_{1}(0,0)\ |\
Z_{1}=1,X_{1}(1)=0\big{]}$
$\displaystyle+\Pr\big{[}X_{1}(0)=1\big{]}\Big{(}E\big{[}Y_{1}(1,1)-Y_{1}(0,0)\
|\ Z_{1}=1,X_{1}(0)=1\big{]}-E\big{[}Y_{1}(0,1)-Y_{1}(0,0)\
|Z_{1}=0,X_{1}(0)=1\big{]}\Big{)}$

Dividing both sides by the first stage, we obtain

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

where

$\displaystyle B_{1}$
$\displaystyle=\frac{\Pr\big{[}X_{1}(1)=0\big{]}}{\Pr\big{[}X_{1}(1)>X_{1}(0)]}E\big{[}Y_{1}(1,0)-Y_{1}(0,0)|Z_{1}=1,X_{1}(1)=0\big{]}$
$\displaystyle+\frac{\Pr\big{(}X_{1}(0)=1\big{)}}{\Pr\big{[}X_{1}(1)>X_{1}(0)\big{]}}\Big{(}E\big{[}Y_{1}(1,1)-Y_{1}(0,0)\
|\ Z_{1}=1,X_{1}(0)=1\big{]}-E\big{[}Y_{1}(0,1)-Y_{1}(0,0)\ |\
Z_{1}=0,X_{1}(0)=1]\Big{)}.$

∎

##### Proof of Theorem A8

Under the no-IV scenario, $Z_{0}=0,X_{0}=X_{0}(0)$, and the parallel-trends
assumption takes the form

$\displaystyle E$
$\displaystyle\big{[}Y_{1}(z_{1}=0,x_{1}=X_{0}(0))-Y_{0}(z_{0}=0,x_{0}=X_{0}(0))\
|\ Z_{1}=1\big{]}$ $\displaystyle=E$
$\displaystyle\big{[}Y_{1}(z_{1}=0,x_{1}=X_{0}(0))-Y_{0}(z_{0}=0,x_{0}=X_{0}(0))\
|\ Z_{1}=0\big{]}.$

As before, we start with the numerator,

$\displaystyle E\big{[}Y_{1}-Y_{0}$ $\displaystyle\ |\
Z_{1}=1\big{]}-E\big{[}Y_{1}-Y_{0}\ |\ Z_{1}=0\big{]}$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{0}(0,X_{0}(0))\ |\
Z_{1}=1\big{]}-E\big{[}Y_{1}(0,X_{1}(0))-Y_{0}(0,X_{0}(0))\ |\ Z_{1}=0\big{]}$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(0,X_{0}(0))\ |\
Z_{1}=1\big{]}-E\big{[}Y_{1}(0,X_{1}(0))-Y_{1}(0,X_{0}(0))\ |\ Z_{1}=0\big{]}$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(0,X_{1}(0))\ |\
Z_{1}=1\big{]},$

where the second line is by definition, the third by parallel-trends and the
fourth follows from the fact that under “Same type” Assumption,
$X_{0}(0)=X_{1}(0)$.

As with the proof of Theorem 4, we continue by the law of total expectation
with respect to the type and treatment independence to obtain

$\displaystyle E\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(0,X_{1}(0))\ |\ Z_{1}=1\big{]}$
$\displaystyle=\Pr[X_{1}(1)>X_{1}(0)]E\big{[}Y_{1}(1,1)-Y_{1}(0,0)\ |\
Z_{1}=1,X_{1}(1)>X_{1}(0)\big{]}$
$\displaystyle+\Pr\big{[}X_{1}(1)=0\big{]}E\big{[}Y_{1}(1,0)-Y_{1}(0,0)\ |\
Z_{1}=1,X_{1}(1)=0\big{]}$
$\displaystyle+\Pr\big{[}X_{1}(0)=1\big{]}E\big{[}Y_{1}(1,1)-Y_{1}(0,1)\ |\
Z_{1}=1,X_{1}(0)=1\big{]}.$

Dividing by the first stage, we obtain

$\displaystyle\begin{split}DiW&=\mathbb{E}\big{[}Y_{1}(1,1)-Y_{1}(0,0)\ |\
Z_{1}=1,X_{1}(1)>X_{1}(0)\big{]}+B_{2}\\\ &=E\big{[}Y_{1}(1,1)-Y_{1}(0,0)\ |\
X_{1}=1,X_{1}(1)>X_{1}(0)\big{]}+B_{2},\end{split}$ (A5)

where

$\displaystyle\begin{split}B_{2}&=\frac{\Pr(X_{1}(1)=0)}{\Pr[X_{1}(1)>X_{1}(0)]}\mathbb{E}[Y_{1}(1,0)-Y_{1}(0,0)\
|\ Z_{1}=1,X_{1}(1)=0]\\\
&+\frac{\Pr(X_{1}(0)=1)}{\Pr[X_{1}(1)>X_{1}(0)]}E\big{[}Y_{1}(1,1)-Y_{1}(0,1)\
|\ Z_{1}=1,X_{1}(0)=1\big{]}.\end{split}$ (A6)

This concludes the proof of part (i) of Theorem A8. The second part follows
from that under the exclusion restriction assumption, the first term in (A5)
becomes

$E\big{[}Y_{1}(x_{1}=1)-Y_{1}(x_{1}=0)\ |\ X_{1}=1,X_{1}(1)>X_{1}(0)\big{]}$

and furthermore, $B_{2}=0$ because $Y_{1}(1,0)=Y_{1}(0,0)$ and
$Y_{1}(1,1)-Y_{1}(0,1)$. ∎

##### Proof of Theorem A9

Under the no IV-treatment link scenario, $Z_{0}=Z_{1},X_{0}=X_{0}(0)$, and the
parallel-trends Assumption takes the form

$\displaystyle E$
$\displaystyle\big{[}Y_{1}(z_{1}=1,x_{1}=X_{0}(0))-Y_{0}(z_{0}=1,x_{0}=X_{0}(0))\
|\ Z_{1}=1\big{]}$ $\displaystyle=E$
$\displaystyle\big{[}Y_{1}(z_{1}=0,x_{1}=X_{0}(0))-Y_{0}(z_{0}=0,x_{0}=X_{0}(0))\
|\ Z_{1}=0\big{]}.$

As before, we start with the numerator,

$\displaystyle E\big{[}Y_{1}-Y_{0}$ $\displaystyle\ |\
Z_{1}=1\big{]}-\mathbb{E}[Y_{1}-Y_{0}\ |\ Z_{1}=0\big{]}$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{0}(1,X_{0}(0))\ |\
Z_{1}=1\big{]}-E\big{[}Y_{1}(0,X_{1}(0))-Y_{0}(0,X_{0}(0))\ |\ Z_{1}=0]$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(1,X_{0}(0))\ |\
Z_{1}=1\big{]}-E\big{[}Y_{1}(0,X_{1}(0))-Y_{1}(0,X_{0}(0))\ |\ Z_{1}=0\big{]}$
$\displaystyle=E\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(0,X_{1}(0))\ |\
Z_{1}=1\big{]},$

where the second line is by definition, the third by parallel trends and the
third follows from the fact that under “Same type” Assumption,
$X_{0}(0)=X_{1}(0)$.

Now, by the law of total expectation the last expression equals to

$\Pr\big{[}X_{1}(1)>X_{1}(0)\big{]}E\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(1,X_{0}(0))\
|\ Z_{1}=1,X_{1}(1)>X_{1}(0)\big{]},$

because for the always-takers and for the never-takers the “Same type”
Assumption implies $Y_{1}(1,X_{1}(1))=Y_{1}(1,X_{0}(0))$. Dividing the
numerator by the first stage once again, yields

$\mathbb{E}\big{[}Y_{1}(1,X_{1}(1))-Y_{1}(1,X_{0}(0))\ |\
Z_{1}=1,X_{1}(1)>X_{1}(0)\big{]}.$

∎

##  Appendix C Further Details of the Simulations

This section provides additional details about the DGPs used for the
simulations in Section 4.5.

##### Violations of Outcome Independence

For this study, the DGP was parameterized as in Equation (14), which is
repeated here for clarity

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

The APO variable was simulated as $U\sim U[-3,3]$. The vector of the five
independent control variables $C$ included $C_{1},C_{2},C_{3}\sim N(1,3)$,
$C_{4}\sim Ber(0.3)$, and $C_{5}\sim Ber(0.5)$. The intercept was taken to be
$\gamma_{0}=1$. For $\gamma_{U}$, the parameter controlling the magnitude of
the APO variable association with the IV, we took the values
$\gamma_{U}=0,0.1,0.2,...,1$. The error term for $Z$ was simulated from
$\epsilon_{Z}\sim\mathcal{N}(0,5)$, and for each $k$, $\epsilon_{NC,k}$ was
simulated independently as $\epsilon_{NC,k}\sim\mathcal{N}(0,\sigma^{2})$.

For $g(u)$, we considered two alternative scenarios. In the first, “linear”
scenario (Panel A of Figure 6), $g(u)=u$ (the identity) and $\sigma^{2}=25$.
In the second, “non-linear” scenario, (Panel B), $g(u)=8\min\\{u^{2},1,5\\}$
(a truncated parabolic function) and $\sigma^{2}=3$.

##### Violations of Rich Covariates

We use the same DGP and parameter values specified by Blandhol et al. (2022)
in their simulation study. For convenience, we repeat here the information
given in Section 4.5, while providing more technical details.

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, drawn from a
uniform distribution $C\sim U(0,1)$. That is,
$\Pr(Z=1|C)=\gamma_{0}+\gamma_{1}C-\gamma_{2}C^{2}+\gamma_{3}C_{i}^{3}$.
Following Blandhol et al. (2022), we took
$(\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3})=(0.119,1.785,-1.534,0.597)$.

We augment this DGP with an APO variable $U$. We then define ten negative
controls associated with this APO. To that end, each NCO is simulated by

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

where $\epsilon_{NC}$ is simulated from a standard normal distribution,
$\epsilon_{NC}\sim N(0,1)$. We took
$(\beta_{0},\beta_{1},\beta_{2},\beta_{3})=(0.119,1.785,-1.534,0.597)$ and
$\beta_{U}=0.3$

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. For this parameter, we took the values
$\eta=0.0.5,1,1.5,...,5$.

##  Appendix D Details of the Implementation of Negative Control Tests Using
Data from Prior Studies

This section provides additional details about our application of our proposed
method to the instrumental variable design used in Autor et al. (2013) and
Deming (2014) (discussed in Section 5). We are grateful to the authors of
these prior works for making these analyses possible by publicly posting their
data and code. In each case, we first used the publicly posted data to
replicate the related original study results (this step is not further
discussed here). We then applied our method, which attempts to predict the
instrument using negative control variables from the original study data.

###  D.1 Example I: Shift-Share Instruments

##### Sample construction.

For this analysis, we use the original study data from Autor et al. (2013,
henceforth ADH) which is taken from the US Census. The unit of analysis is a
commuting zone. The sample included 722 commuting zones.

##### Main variables.

For each commuting zone, we observe all variables from the original study
replication data, including the original instrument, controls, and additional
variables not used in the original study, some of which we use as negative
controls in our current analysis. Our analysis uses the same control variables
that were used in the original analysis. The controls we used are from the
year 2000. The outcome, exposure, and instrument are differences between the
years 2007 and 2000. Controls are the lagged year 2000 values. We use all
other variables in the ADH data set that were measured in or before 1990
(excluding the instrument and controls) as negative controls. Note that ADH
also used another version of the instrument, measured between 1990–2000. We do
not evaluate this version, because we do not observe variables that were
measured early enough before 1990 to serve as negative controls.

##### Original Falsification Tests

ADH conducted falsification exercises to evaluate the concern that a secular
decline in US manufacturing might be the underlying reason for the increased
imports from China. To that end, they regress past changes in the
manufacturing employment share on future changes in import exposure (See
columns 4–6 of Table 2 in ADH). This relationship was found to be significant
only for 1970–1980, but not for 1980–1990 or 1970–1990, leading the authors to
conclude that this is unlikely to be a major concern. We replicated this
analysis and obtained a similar result. This original exercise is similar in
spirit to our proposed approach, although it uses these different negative
controls separately and not jointly, as we propose. The rest of this section
discusses additional falsification tests that we performed using alternative
negative control variables.

##### Methods.

The main step in our analysis is testing for conditional independence of the
original instrument and different negative controls, with adjustment for
different sets of control variables that were used in the original study. We
report the results from the specification of all control variables (column 6
of table 3 of ADH).

We repeat this exercise for each of the five different sets of control
variables that were used in the original study and detailed in columns 2–6 of
Table 3 of ADH. We weighted all regressions using the same weights used by ADH
in the original study (timepwt48).

Using the set of $52$ NCO, $16$ control variables and original IV, we
performed the following.

  1. 1.

F-test. Run $IV\sim NCO+C$ using linear regression and report the p-value of
the F-test that tested whether the coefficient of all the NCO is different
than zero.

  2. 2.

Bonforeni. Run $NCO_{i}\sim Z+C$ for $i=1,\dots,52$ and report the minimal
p-value of $Z$ coefficients adjusted for multiple comparisons using Bonferroni
correction.

  3. 3.

GAM. Here we tested two specifications: (1)
$Z\sim\sum_{i=1}^{52}f_{i}(NCO_{i})+C$; and (2)
$Z\sim\sum_{i=1}^{52}f_{i}(NCO_{i})+\sum_{j}f_{j}(C_{j})$. That is, with or
without smooth terms for the control variables (smooth terms are used only for
numeric NCO or control variables and not for binary/categorical variables).
Each of $f_{i}$ is estimated using GAM with “thin-plate" regression splines,
as implemented in the mgcv package. We report the p-values from a Wald test
which tested whether all the NCO’s splines basis coefficients are different
than zero.

All regressions are weighted (timepwt48) and variance is estimated using a
cluster-robust estimator with states (statefip) as a cluster (same as ADH).

Moreover, we repeat the analysis with only one NCO variable (1970-1980
outcome) and a linear specification model (F-test and Bonforeni are equivalent
here).

###  D.2 School Choice Lotteries

##### Sample construction.

Our second replication analyzes data from Deming (2014). We use the study
data, originally sourced from a public school choice lottery in Charlotte-
Mecklenburg (CMS). In these data, we observe the original study outcome,
instrumental and control variable.

##### Main variables

VAM calculation was done from the 1996–1997 through 2001–2002 school years.
The outcome in the second stage is the school year 2002-2003. In the code,
each school year is called by the year in which it ends, for example, school
year 2002-2003 is called 2003.

In the article, the endogenous treatment is the value added (VAM) of the
student’s school. Based on the replication code, the instrument can be written
as

$IV_{i}=L_{i}VAM_{i}^{1}+\left(1-L_{i}\right)VAM_{i}^{N}$ (A7)

where we denote the binary school lottery outcome by $L$, the value added of
the first-choice school by $VAM^{1}$ and the value added of the default
neighborhood school by $VAM^{N}$. These original variables are included in the
study replication data.373737In the original study, the lottery outcome is
denoted $W$, which we replace with $L$ here to avoid abuse of notation.

We use Deming’s variants of the school value-added estimates that are posted
in the replication data. Specifically, the original paper used different
variants of the VAM estimates that differ across three dimensions: 1) the VAM
model specification (equation 1 in the original study): mixed-effects, fixed-
effects, or AR; we use the posted version, which only includes mixed-effects
specifications. 2) the controls used for VAM calculations; we use the posted
version, which includes past grades.383838The richer specifications (models
3-4 in the original study) include personal information that was not made
publicly available. 3) the number of lagged years of the grades used; we use
the maximum number of four.

As negative controls, $NC_{ijt}$, we use lagged test scores from the school
years 1998-2001, and lagged outcome (2001-2002 test score difference). We also
used the VAM of the 3 schools that the student applied to in the lottery, and
the neighborhood school’s VAM.393939Note that only lagged test scores from
year $-1$ are used as controls (in both the original study and in our
replication), so there is no overlap between these controls and the earlier
test score we use as negative controls. In total, we used $37$ NCO variables.
Control variables are the year 2002 test scores. The sample size is $n=2343$.

We ran the same specification as explained in ADH analysis (A7). All the
regressions are unweighted and without robust s.e. estimators. In the F-test
and Bonforoeni, we perform a fixed-effect regression with lottery_FE variable.
In the GAM models, fixed effects are accounted for by taking lottery_FE as a
categorical variable without a smooth term.

Moreover, we repeat the analysis with only one NCO variable (the year 2002
test scores) and a linear specification model (F-test and Bonforeni are
equivalent here).

###  D.3 Nunn and Qian (2014)

##### Sample construction.

Our third replication analyzes data from Nunn and Qian (2014). We use the
study data, which consists of annual panel data of 125 non-OECD countries over
36 years. In the dataset, we observe the original study outcome, IV, control,
and negative control variables.

##### Main Variables.

The IV of the study is the lagged (one year) US wheat production. We limit our
analysis to the main outcome variable of the study which is the intrastate
conflict indicator. We utilize the extended set of $238$ control variables
(same as “baseline specification" columns in Table 2). The sample size is
$n=4572$.

##### Negative controls.

As in the original study, we used a set of ten NCIs. The NCI are the lagged US
production of various products that are not sent as foreign aid.

Nunn and Qian (2014) performed a falsification test (Table 5) with the NCI by
running a reduced-form specification $Y\sim NCI_{i}+IV+C$ for each of the
$NCI$ and the “baseline specification" of the control variables. We extended
the analysis by performing the following.

  1. 1.

F-test. Run $Y\sim NCI+IV+C$ using all the ten NCI combined. $Y$ is the
intrastate outcome. We report the p-value from the F-test that tested whether
all the NCI coefficients are different from zero.

  2. 2.

Bonforeni. Run $Y\sim NCI_{i}+IV+C$ for $i=1,\dots,10$ and report the minimal
p-value of NCI coefficient adjusted for multiple comparisons using Bonferroni
correction.

  3. 3.

GAM. Run $Y\sim\sum_{i=1}^{10}f_{i}(NCI_{i})+f(IV)+C$ with smooth terms
$f_{i}$ and $f$ estimated with GAM and “thin plates" spline regression. We
report the Wald test p-value that tested if all the NCI’s splines basis
coefficients are different from zero.

In all regression, cluster-robust (by country) s.e. estimator is used for
variance matrix estimation.

To test the functional form specification of the linear model $Y\sim
NCI+IV+C$, we’ve run three types of Ramsey’s RESET test for functional
specification error with quadratic and cubic fitted values: (1) $Y\sim
NCI+IV+C$; (2) $Y\sim IV+C$; (3) $IV\sim C$. All tests resulted in a p-value
lower than $1\%$ implying a specification error.

Furthermore, we analyze the marginal models (as in 2.) with and without
adjusting for the main IV and report the NCI coefficient with the minimal
p-value (without IV adjustment) and the same p-value in the model we adjust
for the IV.

###  D.4 Ashraf and Galor (2013)

##### Sample construction.

Our fourth replication analyzes data from Ashraf and Galor (2013). The study
data consists of a sample of 145 countries.

##### Main variables.

The outcome of the study is the historical population density, which is
defined as the log population density in 1500 CE. The main IV is the migratory
distance from Africa. We use the same set of four control variables included
in the study.

##### Negative controls.

As in the original study, we use a set of three NCI which are the migratory
distance from London, Tokyo, and Mexico.

Ashraf and Galor (2013) performed a falsification test (Table 4) by running a
reduced-form linear regression with form: $Y\sim
NCI_{i}+NCI_{i}^{2}+IV+IV^{2}+C$ for each of the NCI variables (IV is always
taken as linear + quadratic in the regressions). All NCI coefficients (both
linear and quadratic terms) were not statistically significant at the $5\%$
level. We extend the analysis by performing the following.

  1. 1.

F-test. Run $Y\sim\sum_{i=1}^{3}\Big{(}NCI_{i}+NCI_{i}^{2}\Big{)}+IV+IV^{2}+C$
using all the three NCI combined (linear + quadratic terms). We report the
p-value from the F-test that tested whether all the NCI coefficients are
different from zero.

  2. 2.

Bonforeni. Run $Y\sim NCI_{i}+NCI_{i}^{2}+IV+IV^{2}+C$ for each of the NCIs
and report the minimal p-value of NCI coefficient adjusted for multiple
comparisons using Bonferroni correction.

  3. 3.

GAM. Run $Y\sim f_{i}(NCI_{i})+f(IV)+C$ with smooth terms $f_{i}$ and $f$
estimated with GAM and “thin plates" spline regression. Here we had three
different GAM models, one for each NCI (due to sample size considerations). We
report the minimal (across NCI) Wald test p-value that tested if the NCI
splines basis coefficients are different from zero.

In all the regressions we compute the variance-matrix using a robust s.e.
estimator (“HC3").

In addition, we tested how adjusting for the IV impacts the results by running
$Y\sim NCI_{i}+NCI_{i}^{2}+C$ for each NCI and reporting the minimal p-value
of the NCI coefficients. For comparison, we report the p-value of the same
coefficient when adjusting for the IV as well.

Table A1: Papers Used to Illustrate Applications of Negative Control Tests

Original Variable Description N of Negative Controls Outcome Treatment IV
Negative Controls Original Paper Our Analyses (1) (2) (3) (4) (5) (6) A.
Negative Control Outcome Tests Autor et al. local labor market outcomes import
competition from China other countries’ import competition from China lagged
local labor market manufacturing employment 3 52 Deming student test scores
school value-added school assignment lottery, interacted with school VA - - 37
B. Negative Control Instrument Tests Ashraf and Galor level of economic
development population genetic diversity migratory distance from East Africa
migratory distance from alternative locations not associated with genetic
diversity (e.g., London) 3 3 Nunn and Qian level of civil conflict receipt of
U.S. food aid variation in U.S. wheat production variation in U.S. production
of crops not associated with aid (e.g., grapes) 10 10

Notes: This table provides contextual details on the papers we use for our
negative control application examples 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. and 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.. Columns 1,2, and 3 specify which outcome, treatment, and IV were
used in these papers, respectively. Column 4 depicts the negative controls
used in the original analysis, and column 5 presents the number of negative
controls used. Column 6 presents the number of negative controls used in the
analysis for 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. and
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., including the
original negative controls, as well as additional valid negative controls we
found in the original data.

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