# The Falsification Adaptive Set in Linear Models with Instrumental Variables
that Violate the Exogeneity or Exclusion Restriction

Nicolas Apfela and Frank Windmeijerb
aSchool of Economics, University of Surrey, UK
bDept of Statistics and Nuffield College, University of Oxford, UK

(December 9, 2022)

###### Abstract

For the classical linear model with an endogenous variable estimated by the
method of instrumental variables (IVs) with multiple instruments, Masten and
Poirier (2021) introduced the falsification adaptive set (FAS). When a model
is falsified, the FAS reflects the model uncertainty that arises from
falsification of the baseline model. It is the set of just-identified IV
estimands, where each relevant instrument is considered as the just-
identifying instrument in turn, whilst all other instruments are included as
controls. It therefore applies to the case where the exogeneity assumption
holds and invalid instruments violate the exclusion assumption only. We
propose a generalized FAS that reflects the model uncertainty when some
instruments violate the exogeneity assumption and/or some instruments violate
the exclusion assumption. This FAS is the set of all possible just-identified
IV estimands where the just-identifying instrument is relevant. There are a
maximum of $k_{z}2^{k_{z}-1}$ such estimands, where $k_{z}$ is the number of
instruments. If there is at least one relevant instrument that is valid in the
sense that it satisfies the exogeneity and exclusion assumptions, then this
generalized FAS is guaranteed to contain $\beta$ and therefore to be the
identified set for $\beta$.

Keywords: Instrumental variables, invalid instruments, falsification adaptive
set

JEL Codes: C26, C52

##  1 Introduction

Masten and Poirier (2021) (henceforth MP) introduced the falsification
adaptive set (FAS) for the classical linear model with an endogenous variable
estimated by the method of instrumental variables with multiple instruments.
For the model specification
$Y=X\beta+\boldsymbol{Z}^{T}\boldsymbol{\gamma}+U$, where $\boldsymbol{Z}$ is
a $k_{z}$-vector of putative instruments, the exclusion assumption implies
that $\boldsymbol{\gamma}=\boldsymbol{0}$, and the exogeneity assumption that
$\text{cov}\left(\boldsymbol{Z},U\right)=\boldsymbol{0}$. The FAS as proposed
by MP applies to the case where the baseline model is falsified due to
instruments violating the exclusion assumption whilst maintaining the
exogeneity assumption. We therefore denote this the FASexcl. It is the set of
just-identified, and hence non-falsifiable, IV estimands, where each relevant
instrument is considered as the just-identifying instrument in turn, whilst
all other instruments are included as controls. The FASexcl is thus an
expanded set compared to the baseline point estimand to account for the
uncertainty due to a violation of the exclusion assumption. MP specify the
falsification frontier as the set of smallest relaxations of the exclusion
restriction that are not falsified. They recommend to report estimates of
FASexcl under the assumption that the true model lies on this frontier, as
then the FASexcl contains $\beta$ and is therefore the identified set for
$\beta$.

An instrument is valid if it satisfies both the exogeneity and exclusion
assumption. As we show in Section 3, the FASexcl is guaranteed to contain
$\beta$ if there is at least one valid and relevant instrument if the
exogeneity assumption holds and invalid instruments violate the exclusion
assumption only. MP argue that, mathematically, the same technical analysis
can be used to relax both the exclusion assumption and exogeneity assumption.
However, we show in Section 4 that the FASexcl is no longer guaranteed to
contain $\beta$ if there is at least one valid and relevant instrument when
instead the exclusion assumption holds and invalid instruments violate the
exogeneity assumption only. We propose a different FAS, termed FASexo, for
this setting. FASexo is the set of just-identified IV estimands where each
relevant instrument is considered as the just-identifying instrument in turn,
with all other instruments removed from the instrument set. We show that
FASexo is guaranteed to contain $\beta$ when at least one of the instruments
is relevant and valid and invalid instruments can violate the exogeneity
assumption only.

From these results, and as the main contribution of this paper, we argue in
Section 5 that a FAS that properly takes into account possible violations of
both the exogeneity and exclusion assumptions when the baseline model is
falsified is the set of all possible just-identified IV estimands where the
just-identifying instrument is relevant. This is due to the fact that when an
instrument violates the exogeneity assumption it should be removed from the
instrument set, and when an instrument violates the exclusion assumption it
should be included as a control. Whereas FASexcl and FASexo consider a maximum
of $k_{z}$ just-identified estimands, our generalized FAS considers a maximum
of $k_{z}2^{k_{z}-1}$ just-identified estimands. Under the assumption that an
invalid instrument can either violate the exogeneity assumption or the
exclusion assumption, this generalized FAS is guaranteed to contain $\beta$
when there is at least one valid and relevant instrument.

In Section 5.1 we first introduce the generalized FAS for the case where there
are two instruments available, as for this case it is simply the union of the
FASexcl and FASexo. Section 5.2 provides the details for the generalized FAS
for a general number of instruments. In Section 6.1 we compare estimates of
the three falsification adaptive sets for an empirical example with two
instruments on the origin of gender roles taken from Alesina et al. (2013), as
considered in Masten and Poirier (2020) (henceforth MP20). Section 6.2
presents estimation results for an empirical analysis of roads and trade with
three instruments taken from Duranton et al. (2014), which was the main
application in MP.

##  2 Model and Assumptions

MP consider the classical linear model. In the main text they derive the
falsification adaptive set for a model with one endogenous explanatory
variable and we focus on that model here. $Y\left(x,\boldsymbol{z}\right)$
denotes the potential outcomes defined for values
$\left(x,\boldsymbol{z}\right)\in\mathbb{R}^{k_{z}+1}$. It is assumed that

$\displaystyle Y\left(x,\boldsymbol{z}\right)$
$\displaystyle=x\beta+\boldsymbol{z}^{T}\boldsymbol{\gamma}+U,$ (1)

where $\beta$ is an unknown constant, $\boldsymbol{\gamma}$ is an unknown
$k_{z}$-vector and $U$ is an unobserved random variable. Let $X$ be an
observed endogenous variable and let $\boldsymbol{Z}$ be an observed
$k_{z}$-vector of potentially invalid instruments. Individual elements of
$\boldsymbol{Z}$ are denoted $Z_{\ell}$, $\ell=1,\ldots,k_{z}$. The observed
outcome is $Y=Y\left(X,\boldsymbol{Z}\right)$,

$Y=X\beta+\boldsymbol{Z}^{T}\boldsymbol{\gamma}+U,$ (2)

and the joint distribution of $\left(Y,X,\boldsymbol{Z}\right)$ is assumed
known.

The exclusion assumption as detailed below is an assumption on the values of
$\gamma_{\ell}$. In order to state the exogeneity assumption in a similar way,
we specify

$\text{cov}\left(\boldsymbol{Z},U\right)=\boldsymbol{\alpha},$ (3)

where $\boldsymbol{\alpha}$ is an unknown $k_{z}$-vector.

The following relevance and sufficient variation assumptions are maintained:

######  Assumption 1.

Relevance: The $k_{z}$-vector
$\text{cov}\left(\boldsymbol{Z},X\right)\neq\boldsymbol{0}$.

######  Assumption 2.

Sufficient variation: The $k_{z}\times k_{z}$ matrix
$\boldsymbol{\Sigma}_{z}:=\text{var}\left(\boldsymbol{Z}\right)$ is
invertible.

For all instruments to be valid, the exogeneity and exclusion assumptions need
to be satisfied:

######  Assumption 3.

Exogeneity: $\alpha_{\ell}=0$ for all
$\ell\in$$\left\\{1,\ldots,k_{z}\right\\}.$

######  Assumption 4.

Exclusion: $\gamma_{\ell}=0$ for all
$\ell\in$$\left\\{1,\ldots,k_{z}\right\\}.$

Under model (1) and Assumptions 1-4 it follows that
$\text{cov}\left(\boldsymbol{Z},Y\right)=\text{cov}\left(\boldsymbol{Z},X\right)\beta$
and

$\left(\text{cov}\left(\boldsymbol{Z},X\right)\right)^{T}\boldsymbol{\Sigma}_{z}^{-1}\left(\text{cov}\left(\boldsymbol{Z},X\right)\right)^{-1}\left(\text{cov}\left(\boldsymbol{Z},X\right)\right)^{T}\boldsymbol{\Sigma}_{z}^{-1}\left(\text{cov}\left(\boldsymbol{Z},Y\right)\right)=\beta,$

and so the two-stage least squares (2sls) estimand is equal to $\beta$. For
each individual instrument, we have that
$\text{cov}\left(Z_{\ell},Y\right)=\text{cov}\left(Z_{\ell},X\right)\beta$.
Maintaining Assumptions 1 and 2, Proposition 1 in MP states that model (2)
with Assumptions 3 and 4 is not falsified if and only if

$\text{cov}\left(Z_{m},Y\right)\text{cov}\left(Z_{\ell},X\right)\neq\text{cov}\left(Z_{\ell},Y\right)\text{cov}\left(Z_{m},X\right)$

for all $m$ and $\ell$ in $\left\\{1,\ldots,k_{z}\right\\}$.

As discussed in MP, if the distribution of $\left(Y,X,\boldsymbol{Z}\right)$
is such that the model is falsified then this could be due to misspecification
of model (2), which assumes homogeneous linear treatment effects, and/or
instrument invalidity. As in MP, we maintain here model (2) and Assumptions 1
and 2 and focus on failure of instrument exogeneity or instrument exclusion as
reasons for falsifying the baseline model.

We define an instrument to be valid as follows.

######  Definition 1.

Valid instrument: An instrument $Z_{\ell}$ is a valid instrument if both the
exogeneity and exclusion assumptions hold, $\alpha_{\ell}=\gamma_{\ell}=0$.

We further make the assumption that an invalid instrument can either violate
the exclusion assumption or the exogeneity assumption, but not both:

######  Assumption 5.

Invalid instrument: An invalid instrument violates either the exogeneity
assumption, $\alpha_{\ell}\neq 0$, or the exclusion assumption,
$\gamma_{\ell}\neq 0$, but not both, $\gamma_{\ell}\alpha_{\ell}=0$.

Assumption 5 rules out that a variable considered to be an instrument for the
endogenous variable $X$ is itself an endogenous variable.

Note that a valid instrument by itself may not identify $\beta$ if there are
invalid instruments present and when instruments are correlated. A valid
instrument then identifies $\beta$ if the invalid instruments that violate the
exclusion restriction are added as controls as specified in model (2) and
instruments that violate the exogeneity assumption are removed from the
instrument set.

In the next section, we will present the identified set for $\beta$, the
falsification frontier and the falsification adaptive set as developed by MP
for the case where invalid instruments can violate the exclusion assumption
only.

The FAS is the identified set for $\beta$ if it contains $\beta$. We specify
here the condition that the FAS is the identified set for $\beta$ if there is
at least one relevant valid instrument.

######  Condition 1.

Sufficient Condition: For the model specification given in (2) and (3),
suppose Assumptions 1, 2 and 5 hold and the joint distribution of
$\left(Y,X,\boldsymbol{Z}\right)$ is known. Then a sufficient condition for
the FAS to contain $\beta$ and thus be the identified set for $\beta$ is that
there is at least one instrument that is relevant and valid.

Relevance of an instrument will be further clarified below. We consider
Condition 1 to be an essential requirement for the FAS. We show in Section 3
that this sufficient condition for the FASexcl as proposed by MP requires the
validity of the exogeneity Assumption 3 instead of the weaker Assumption 5. In
other words, if invalid instruments violate the exogeneity assumption, FASexcl
is not guaranteed to contain $\beta$. We show this in Section 4, where we
develop the FASexo for the case where instruments may violate the exogeneity
Assumption 3 only. The sufficient condition for the FASexo requires the
validity of the exclusion Assumption 4 instead of the weaker Assumption 5.
Section 5 then combines these results and proposes the FAS for the case that
covers the possibility that there are invalid instruments that violate the
exclusion restriction and invalid instruments that violate the exogeneity
assumption and we show that this FAS satisfies Condition 1.

##  3 Failure of the Exclusion Assumption Only

MP derive the falsification adaptive set for the case of a failure of the
exclusion Assumption 4, whilst maintaining the exogeneity Assumption 3. They
argue that the same technical analysis can be used to relax both assumptions,
something we will discuss in greater detail below.

Let

$\boldsymbol{\pi}:=\boldsymbol{\Sigma}_{z}^{-1}\text{cov}\left(\boldsymbol{Z},X\right);\,\,\boldsymbol{\psi}:=\boldsymbol{\Sigma}_{z}^{-1}\text{cov}\left(\boldsymbol{Z},Y\right).$
(4)

As it is maintained that
$\text{cov}\left(\boldsymbol{Z},U\right)=\boldsymbol{\alpha}=\boldsymbol{0}$,
it follows from model (2) that
$\boldsymbol{\psi}=\boldsymbol{\pi}\beta+\boldsymbol{\gamma}$.

MP make the following partial exclusion assumption.

######  Assumption 6.

Partial exclusion: There are known constants $\delta_{\ell}\geq 0$ such that
$\left|\gamma_{l}\right|\leq\delta_{\ell}$ for $\ell=1,\ldots,k_{z}$.

Under Assumptions 1-3 and 6, the identified set for $\beta$ is then given in
MP, Theorem 1, by

$\mathcal{B}\left(\boldsymbol{\delta}\right)=\left\\{b\in\mathbb{R}:-\boldsymbol{\delta}\leq\left(\boldsymbol{\psi}-\boldsymbol{\pi}b\right)\leq\boldsymbol{\delta}\right\\},$
(5)

where the inequalities are componentwise. This follows straightforwardly as
$\boldsymbol{\psi}-\boldsymbol{\pi}\beta=\boldsymbol{\gamma}$ and
$-\boldsymbol{\delta}\leq\boldsymbol{\gamma}\leq\boldsymbol{\delta}$. The
model is falsified if and only if
$\mathcal{B}\left(\boldsymbol{\delta}\right)$ is empty. If
$\mathcal{B}\left(\boldsymbol{0}\right)$ is empty then the baseline IV model,
assuming $\boldsymbol{\gamma}=\boldsymbol{0}$, is falsified.

For the individual components of the identified set we have
$-\delta_{\ell}\leq\left(\psi_{\ell}-\pi_{\ell}b\right)\leq\delta_{\ell}$. As
stated in Corollary 1 in MP, it follows that

$\mathcal{B}\left(\boldsymbol{\delta}\right)=\cap_{\ell=1}^{k_{z}}B_{\ell}\left(\delta_{\ell}\right),$

where

$B_{\ell}\left(\delta_{\ell}\right)=\begin{cases}\left[\frac{\psi_{\ell}}{\pi_{\ell}}-\frac{\delta_{\ell}}{\left|\pi_{\ell}\right|},\frac{\psi_{\ell}}{\pi_{\ell}}+\frac{\delta_{\ell}}{\left|\pi_{\ell}\right|}\right]&\text{if
}\pi_{\ell}\neq 0,\\\ \mathbb{R}&\text{if }\pi_{\ell}=0\text{ and
$0\in\left[\psi_{\ell}-\delta_{\ell},\psi_{\ell}+\delta_{\ell}\right],$}\\\
\emptyset&\text{if }\pi_{\ell}=0\text{ and
$0\notin\left[\psi_{\ell}-\delta_{\ell},\psi_{\ell}+\delta_{\ell}\right].$}\end{cases}$

The falsification frontier, denoted FF, is the minimal set of
$\boldsymbol{\delta}$s which lead to a nonempty identified set. For a
$\boldsymbol{\delta}\in FF$ this means that for any other
$\boldsymbol{\delta}^{\prime}<\boldsymbol{\delta}$,
$B\left(\boldsymbol{\delta}^{\prime}\right)$ is empty and thus falsifies the
model, where $\boldsymbol{\delta}^{\prime}<\boldsymbol{\delta}$ means that
$\delta^{\prime}_{\ell}\leq\delta_{\ell}$ for all
$\ell\in\left\\{1,\ldots,k_{z}\right\\}$ and $\delta^{\prime}_{m}<\delta_{m}$
for some $m\in\left\\{1,\ldots,k_{z}\right\\}$, see Definition 1 in MP. As in
MP, let $\mathcal{L}_{rel}$ denote the set of relevant instruments,

$\mathcal{L}_{rel}=\left\\{\ell\in\left\\{1,\ldots,k_{z}\right\\}:\pi_{\ell}\neq
0\right\\}.$ (6)

For model (2), under Assumptions 1-3 and and 6, Proposition 2 in MP specifies
the falsification frontier as the set

$FF_{excl}=\left\\{\boldsymbol{\delta}\left(b\right)\in\mathbb{R}_{\geq
0}^{k_{z}}:\delta_{\ell}\left(b\right)=\left|\psi_{\ell}-b\pi_{\ell}\right|,\,\ell=1,\ldots,k_{z},\,b\in\left[\min_{\ell\in\mathcal{L}_{rel}}\frac{\psi_{\ell}}{\pi_{\ell}},\max_{\ell\in\mathcal{L}_{rel}}\frac{\psi_{\ell}}{\pi_{\ell}}\right]\right\\},$

where we have added the subscript “excl” to denote a relaxation of the
exclusion assumption only, for reasons that will become clear below, and also
defined $\boldsymbol{\delta}$ as a function of $b$.

It is instructive at this point to consider a small numerical example.

######  Example 1.

Let $\beta=1$, $k_{z}=3$, $\boldsymbol{\pi}=\left(1,1,1\right)^{T}$ and
$\boldsymbol{\gamma}=\left(-1,0,2\right)^{T}$, resulting in
$\boldsymbol{\psi}=\left(0,1,3\right)^{T}$. The values of
$\boldsymbol{\delta}\left(b\right)$ for values of
$b=\left\\{0,0.5,1,3\right\\}$, where
$\left\\{0,3\right\\}=\left\\{\min_{\ell}\frac{\psi_{\ell}}{\pi_{\ell}},\max_{\ell}\frac{\psi_{\ell}}{\pi_{\ell}}\right\\}$
are given by

$\boldsymbol{\delta}\left(0\right)=\left(\begin{array}[]{c}0\\\ 1\\\
3\end{array}\right);\,\,\boldsymbol{\delta}\left(0.5\right)=\left(\begin{array}[]{c}0.5\\\
0.5\\\
2.5\end{array}\right);\,\,\boldsymbol{\delta}\left(1\right)=\left(\begin{array}[]{c}1\\\
0\\\
2\end{array}\right);\,\,\boldsymbol{\delta}\left(3\right)=\left(\begin{array}[]{c}3\\\
2\\\ 0\end{array}\right)$

and for each value of $b\in\left[0,3\right]$, it follows that
$\mathcal{B}\left(\boldsymbol{\delta}\left(b\right)\right)=b$. For example,
for $\boldsymbol{\delta}$$\left(0.5\right)$,

$\left(\begin{array}[]{c}-0.5\\\ -0.5\\\
-2.5\end{array}\right)\leq\left(\left(\begin{array}[]{c}0\\\ 1\\\
3\end{array}\right)-\left(\begin{array}[]{c}1\\\ 1\\\
1\end{array}\right)b\right)\leq\left(\begin{array}[]{c}0.5\\\ 0.5\\\
2.5\end{array}\right)$

for $b=0.5$ only. For any
$\boldsymbol{\delta}^{\prime}<\boldsymbol{\delta}\left(0.5\right)$ it follows
that $\mathcal{B}\left(\boldsymbol{\delta}^{\prime}\right)$ is empty. It is
clear that for all values of $b\in\left[0,3\right]$ it holds that
$\mathcal{B}\left(\boldsymbol{\delta}^{\prime}\right)$ is empty for any
$\boldsymbol{\delta}^{\prime}<\boldsymbol{\delta}(b)$. For any
$b^{*}\notin\left[0,3\right]$ there is a
$\boldsymbol{\delta}^{\prime}<\boldsymbol{\delta}\left(b^{*}\right)$ for which
$\mathcal{B}\left(\boldsymbol{\delta}^{\prime}\right)$ is not empty, and hence
$\boldsymbol{\delta}\left(b^{*}\right)\notin FF$. For example, for $\eta>0$,
$\boldsymbol{\delta}\left(3+\eta\right)=\left(3+\eta,2+\eta,\eta\right)^{T}>\boldsymbol{\delta}\left(3\right)$.

The falsification adaptive set, denoted here FASexcl, is then given in Theorem
2 of MP as

$FAS_{excl}=\cup_{\boldsymbol{\delta}\in
FF}\mathcal{B}\left(\boldsymbol{\delta}\right)=\left[\min_{\ell\in\mathcal{L}_{rel}}\frac{\psi_{\ell}}{\pi_{\ell}},\max_{\ell\in\mathcal{L}_{rel}}\frac{\psi_{\ell}}{\pi_{\ell}}\right].$
(7)

As MP point out in their Lemma 1, for $\ell\in\mathcal{L}_{rel}$,
$\frac{\psi_{\ell}}{\pi_{\ell}}\left(=\beta+\frac{\gamma_{\ell}}{\pi_{\ell}}\right)$
is the IV/2sls estimand in the just identified model specification

$Y=X\beta_{\ell}+\boldsymbol{Z}_{\\{-\ell\\}}^{T}\boldsymbol{\gamma}_{\left\\{-\ell\right\\}}+U_{\ell}$
(8)

where
$\boldsymbol{Z}_{\\{-\ell\\}}=\boldsymbol{Z}\setminus\left\\{Z_{\ell}\right\\}$,
and using $Z_{\ell}$ as the excluded just-identifying instrument. This follows
directly from the model specification (2) and the linear projection
specification $X=\boldsymbol{Z}^{T}\boldsymbol{\pi}+V$. Hence,
$Z_{\ell}\pi_{\ell}=X-\boldsymbol{Z}_{\\{-\ell\\}}^{T}\boldsymbol{\pi}_{\left\\{-\ell\right\\}}-V$,
or
$Z_{\ell}\gamma_{\ell}=\left(X-\boldsymbol{Z}_{\\{-\ell\\}}^{T}\boldsymbol{\pi}_{\left\\{-\ell\right\\}}-V\right)\frac{\gamma_{\ell}}{\pi_{\ell}}$.
It therefore follows that

$\beta_{\ell}=\beta+\frac{\gamma_{\ell}}{\pi_{\ell}}=\frac{\psi_{\ell}}{\pi_{\ell}};\,\,\gamma_{\left\\{-\ell\right\\},j}=\gamma_{j}-\pi_{j}\frac{\gamma_{\ell}}{\pi_{\ell}};\,\,U_{\ell}=U-V\frac{\gamma_{\ell}}{\pi_{\ell}}$

and $\text{cov}\left(Z_{\ell},U_{\ell}\right)=0$. This is for
$\ell\in\mathcal{L}_{rel}$, $j=1,\ldots,k_{z},j\neq\ell$, so a slightly
unusual notation for $\boldsymbol{\gamma}_{\left\\{-\ell\right\\}}$. For
example, for $k_{z}=3$ and $\ell=2$, we have here that
$\boldsymbol{\gamma}_{\left\\{-2\right\\}}=\left(\gamma_{\left\\{-2\right\\},1},\gamma_{\left\\{-2\right\\},3}\right)^{T}$,
see also Windmeijer et al. (2021, Appendix A.5).

As just-identified models are not falsifiable, it follows that for
$\ell\in\mathcal{L}_{rel}$,
$\delta_{\ell}\left(\frac{\psi_{\ell}}{\pi_{\ell}}\right)=0$, as highlighted
in Example 1. From this, the results of the falsification frontier follow
straightforwardly, as when moving $b$ from
$\min_{j\in\mathcal{L}_{rel}}\frac{\psi_{j}}{\pi_{j}}$ to
$\max_{j\in\mathcal{L}_{rel}}\frac{\psi_{j}}{\pi_{j}}$ there is at least one
element in $\boldsymbol{\delta}\left(b\right)$ that decreases in value and at
least one that increases in value.

We can write the FASexcl alternatively as

$FAS_{excl}=\left[\beta+\min_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}},\beta+\max_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}}\right].$

It follows that the FASexcl contains $\beta$ if
$0\in\left[\min_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}},\max_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}}\right]$
and hence it is then the identified set for $\beta$. This is the case in
Example 1 where $\beta=1$ and FASexcl is given by $\left[0,3\right]$. MP, p
1456, state that the FASexcl “…is the identified set for $\beta$ under the
assumption that one of the points on the falsification frontier is true.”
Maintaining the exogeneity Assumption 3 that $\alpha_{\ell}=0$ for all
$\ell\in\left\\{1,\ldots,k_{z}\right\\}$, if there is at least one relevant
valid instrument, then $\beta\in FAS_{excl}$, as $\gamma_{\ell}=0$ for a valid
instrument $Z_{\ell}$. If none of the instruments are valid, the FASexcl
contains $\beta$ if
$\min_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}}<0$ and
$\max_{\ell\in\mathcal{L}_{rel}}\frac{\gamma_{\ell}}{\pi_{\ell}}>0$.
Therefore, under the exogeneity assumption, a sufficient, but not necessary
condition for FASexcl to be the identified set for $\beta$ is that at least
one of the instruments is valid and relevant. We state this in the following
condition.

######  Condition 2.

Exclusion Sufficient Condition: If the instruments satisfy the exogeneity
Assumption 3 that $\alpha_{\ell}=0$ for all
$\ell\in\left\\{1,\ldots,k_{z}\right\\}$, then a sufficient condition for the
FASexcl to be the identified set for $\beta$ is that there is at least one
instrument that is relevant and valid.

As specified for $\mathcal{L}_{rel}$ in (6), an instrument is relevant in this
setting if $\pi_{\ell}\neq 0$. Condition 2 requires the exogeneity Assumption
3 which is a stronger assumption than Assumption 5 that an invalid instrument
can violate the exogeneity or exclusion assumption for the general sufficient
Condition 1.

###  3.1 Related Instrument Selection Methods

Kang et al. (2016), Guo et al. (2018), Windmeijer et al. (2019) and Windmeijer
et al. (2021) considered the same setting where invalid instruments can only
violate the exclusion assumption. These papers then salvage falsified models
by selecting the set of valid instruments under either a majority rule or
plurality rule assumption. The majority rule states that more than 50% of the
relevant instruments are valid, whilst the plurality rule states that within
$\mathcal{L}_{rel}$ the valid instruments form the largest group, where a
group of instruments is defined by having the same value
$\frac{\psi_{\ell}}{\pi_{\ell}}$.

The latter can be seen in relation to the falsification frontier, by finding,
within $\mathcal{L}_{rel}$, the value of $b=\frac{\psi_{\ell}}{\pi_{\ell}}$
for which $\boldsymbol{\delta}\left(b\right)$ has the largest number of zeros.
To illustrate, if we add a fourth instrument to our numerical Example 1 above,
with $\pi_{4}=1$ and $\gamma_{4}=0$, then we have both $Z_{2}$ and $Z_{4}$ as
valid instruments and they form the largest group, as the values of
$\frac{\psi_{\ell}}{\pi_{\ell}}$ are given by $\left\\{0,1,3,1\right\\}$. For
$b=\beta=1$ we then clearly have that
$\boldsymbol{\delta}\left(1\right)=\left(1,0,2,0\right)^{T}$ and the selection
methods of Guo et al. (2018) and Windmeijer et al. (2021) consistently select
$Z_{2}$ and $Z_{4}$ as the valid instruments. These two selection methods use
dimension reduction techniques to deal with situations of a large number of
instruments. For a small number of instruments, as in this example, the
selection procedures of Andrews (1999), adjusted to deal with the violation of
the exclusion assumption only, can be used to select the set of valid
instruments.

###  3.2 Estimation of FASexcl and First-Stage Hard Thresholding

We have an i.i.d. sample of size $n$,
$\left\\{Y_{i},X_{i},\boldsymbol{Z}_{i}^{T}\right\\}_{i=1}^{n}$. The
$n$-vectors $\left(Y_{i}\right)$ and $\left(X_{i}\right)$ are denoted
$\boldsymbol{y}$ and $\boldsymbol{x}$ respectively, and here $\boldsymbol{Z}$
denotes the $k_{z}\times n$ matrix of observations on the instruments. MP
suggest to estimate the set of relevant instruments by

$\mathcal{\widehat{L}}_{rel}=\left\\{\ell\in\left\\{1,\ldots,k_{z}\right\\}:F_{\ell}\geq
C_{n}\right\\},$

where $F_{\ell}$ is the first-stage $F$-statistic for model (8), where
$Z_{\ell}$ is considered as an instrument and $Z_{\left\\{-\ell\right\\}}$ as
controls. For all values of $\ell$, the first-stage model is therefore given
by $\boldsymbol{x}=\boldsymbol{Z}\boldsymbol{\pi}+\boldsymbol{v}$ and so
$F_{\ell}$ is the same as the Wald statistic for testing $H_{0}:\pi_{\ell}=0$
based on the OLS estimator of $\boldsymbol{\pi}$, denoted
$\widehat{\boldsymbol{\pi}}$. The same first-stage hard thresholding was
proposed in Guo et al. (2018). Although $C_{n}\rightarrow\infty$ as
$n\rightarrow\infty$ and $C_{n}=o\left(n\right)$ for consistent selection, MP
choose $C_{n}=10$ as their default cutoff, or for the t-ratio,
$\left|\frac{\widehat{\pi}_{\ell}}{se\left(\widehat{\pi}_{\ell}\right)}\right|\geq\sqrt{10}=3.16$,
where $\widehat{\pi}_{\ell}$ is the OLS estimator of $\pi_{\ell}$ in the
first-stage model.

Let $\widehat{\beta}_{\ell}$ be the IV estimator of $\beta_{\ell}$ in just-
identified model specification (8). Then FASexcl is estimated by

$\widehat{FAS}_{excl}=\left[\min_{\ell\in\mathcal{\widehat{L}}_{rel}}\widehat{\beta}_{\ell},\max_{\ell\in\mathcal{\widehat{L}}_{rel}}\widehat{\beta}_{\ell}\right]$

and MP show that $\widehat{FAS}_{excl}$ is a consistent estimator of the
$FAS_{excl}$ under the conditions of their Proposition 3.

Note that instead of having to calculate
$k_{z,rel}=\left|\mathcal{\widehat{L}}_{rel}\right|$ IV estimates, the same
estimates are obtained from two OLS regressions. Let
$\widehat{\boldsymbol{\psi}}$ denote the OLS estimator of $\boldsymbol{\psi}$
in the reduced form model
$\boldsymbol{y}=\boldsymbol{Z}\boldsymbol{\psi}+\boldsymbol{v}_{y}$. Let
$\boldsymbol{z}_{\ell}$ denote the $n$-vector of observations on the $\ell$-th
instrument $\left(Z_{i,\ell}\right)$, the $\ell$-th column of
$\boldsymbol{Z}$, and

$\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}:=\boldsymbol{M}_{Z_{\left\\{-\ell\right\\}}}\boldsymbol{z}_{\ell},$

the residual vector after regressing $\boldsymbol{z}_{\ell}$ on
$\boldsymbol{Z}_{\left\\{-\ell\right\\}}$, where
$\boldsymbol{Z}_{\left\\{-\ell\right\\}}$ is the $n\times\left(k_{z}-1\right)$
matrix $\boldsymbol{Z}\setminus\boldsymbol{z}_{\ell}$, and where for a general
full column rank matrix $\boldsymbol{A}$,
$\boldsymbol{M}_{A}=\boldsymbol{I}_{n}-\boldsymbol{P}_{A}$, with
$\boldsymbol{P}_{A}=\boldsymbol{A}\left(\boldsymbol{A}^{T}\boldsymbol{A}\right)^{-1}\boldsymbol{A}^{T}$
and $\boldsymbol{I}_{n}$ is the $n$-dimensional identity matrix. Then it
follows that

$\widehat{\beta}_{\ell}=\frac{\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}^{T}\boldsymbol{y}}{\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}^{T}\boldsymbol{x}}=\frac{\left(\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}^{T}\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}\right)^{-1}\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}^{T}\boldsymbol{y}}{\left(\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}^{T}\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}\right)^{-1}\boldsymbol{z}_{\ell|\left\\{-\ell\right\\}}^{T}\boldsymbol{x}}=\frac{\widehat{\psi}_{\ell}}{\widehat{\pi}_{\ell}},$
(9)

see also Windmeijer et al. (2021, Appendix A.5). Note further that irrelevant
instruments $\boldsymbol{z}_{j}$ for $j\notin\mathcal{\widehat{L}}_{rel}$ are
included as control variables in this setting, as it is in Guo et al. (2018)
and Windmeijer et al. (2021).

###  3.3 2sls

For the model specification

$\boldsymbol{y}=\boldsymbol{x}\beta+\boldsymbol{u},$

and the $n\times k_{z}$ instrument matrix $\boldsymbol{Z}$, the 2sls estimator
is given by

$\displaystyle\widehat{\beta}_{2sls}$
$\displaystyle=\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{y}$
$\displaystyle=\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}\boldsymbol{x}^{T}\boldsymbol{Z}\widehat{\boldsymbol{\psi}}$
$\displaystyle=\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}\boldsymbol{x}^{T}\boldsymbol{Z}\boldsymbol{D}_{\widehat{\pi}}\widehat{\boldsymbol{\beta}}$

where $\widehat{\boldsymbol{\beta}}$ is the $k_{z}$-vector
$\left(\widehat{\beta}_{\ell}\right)$, and $\boldsymbol{D}_{\widehat{\pi}}$ is
a $k_{z}\times k_{z}$ diagonal matrix with $\ell$-th diagonal element equal to
$\widehat{\pi}_{\ell}$. Therefore $\widehat{\beta}_{2sls}$ is a linear
combination of the just-identified IV estimators,

$\widehat{\beta}_{2sls}=\sum_{\ell=1}^{k_{z}}w_{\ell}\widehat{\beta}_{\ell},$
(10)

with
$w_{\ell}=\widehat{\pi}_{\ell}\boldsymbol{z}_{\ell}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}$.
Let $\boldsymbol{\iota}_{k_{z}}$be a $k_{z}$-vector of ones. As
$\boldsymbol{x}^{T}\boldsymbol{Z}\boldsymbol{D}_{\widehat{\pi}}\boldsymbol{\iota}_{k_{z}}=\boldsymbol{x}^{T}\boldsymbol{Z}\widehat{\boldsymbol{\pi}}=\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}$
it follows that $\sum_{\ell=1}^{k_{z}}w_{\ell}=1$, but
$\widehat{\beta}_{2sls}$ is not necessarily a weighted average of the
$\widehat{\beta}_{\ell}$ as $w_{\ell}$ can be negative. This is the case when
$\widehat{\pi}_{\ell}$ and $\boldsymbol{z}_{\ell}^{T}\boldsymbol{x}$ have
opposite signs, or equivalently if $\widehat{\pi}_{\ell}$ and
$\widehat{\pi}_{\ell}^{*}$ have opposite signs, where
$\widehat{\pi}_{\ell}^{*}$ is the OLS estimator of $\pi_{\ell}^{*}$, the
coefficient in the linear specification
$\boldsymbol{x}=\boldsymbol{z}_{\ell}\pi_{\ell}^{*}+\boldsymbol{v}_{\ell}$.

##  4 Failure of the Exogeneity Assumption Only

We now maintain the exclusion Assumption 4, so
$\boldsymbol{\gamma}=\boldsymbol{0}$, but consider violations of the
exogeneity Assumption 3. The observed outcome model is therefore given by

$Y=X\beta+U,$ (11)

with the possible exogeneity violation specified as in (3),

$\text{cov}\left(\boldsymbol{Z},U\right)=\boldsymbol{\alpha}.$

MP, page 1453, argue that, mathematically, the same technical analysis can be
used to relax the exogeneity assumption as is used above for the relaxation of
the exclusion assumption. In MP20, Appendix G, it is argued that a linear
projection of $U$ on $\boldsymbol{Z}$ results in

$U=\boldsymbol{Z}^{T}\boldsymbol{\eta}+\dot{U},$ (12)

with $\text{cov}\left(\boldsymbol{Z},\dot{U}\right)=\boldsymbol{0}$ by
construction. Then

$\displaystyle Y$ $\displaystyle=X\beta+U$
$\displaystyle=X\beta+\boldsymbol{Z}^{T}\boldsymbol{\eta}+\dot{U},$

which is the same type of specification as that of model (2) and MP20 argue
that the results above for the relaxation of the exclusion assumption apply
here as well. MP20 stress that the key difference is the interpretation of the
coefficients on $\boldsymbol{Z}$. We argue here that it actually has an
implication for the FAS.

For the linear projection specification (12), we have that

$\boldsymbol{\eta}=\boldsymbol{\Sigma}_{z}^{-1}\text{cov}\left(\boldsymbol{Z},U\right)=\boldsymbol{\Sigma}_{z}^{-1}\boldsymbol{\alpha}.$
(13)

If the instruments are orthogonal or independent and $\boldsymbol{\Sigma}_{z}$
is a diagonal matrix, then the distinction between a relaxation of the
exclusion or exogeneity assumption is immaterial. However, for a general
$\boldsymbol{\Sigma}_{z}$, the two different relaxations need different
treatment. With correlated instruments, instruments that violate the exclusion
restriction will need to be included as controls in the model for a valid
instrument to identify $\beta$. Instruments that violate the exogeneity
assumption need to be excluded from the instrument set for a valid instrument
to identify $\beta$.

As stated in Condition 2, when considering a relaxation of the exclusion
assumption only, a sufficient condition for the FASexcl as defined in (7) to
contain $\beta$ and hence to be the identified set for $\beta$ is that at
least one instrument is relevant and valid, satisfying the exclusion
assumption $\gamma_{\ell}=0$. For a relaxation of the exogeneity assumption
only, there could be a relevant and valid instrument with $\alpha_{\ell}=0$,
but from (13) it follows that it could be the case that $\eta_{\ell}\neq 0$
for $\ell=1,\ldots,k_{z}$, due to the correlation of the instruments. So
although there is a valid instrument, the FASexcl of (7) is then not
guaranteed to contain $\beta$. A FAS that does is easily obtained.

Like the partial exclusion Assumption 6 as made in MP, we make the following
partial exogeneity assumption.

######  Assumption 7.

Partial exogeneity: There are known constants $\delta_{\ell}^{*}\geq 0$ such
that $\left|\alpha_{\ell}^{*}\right|\leq\delta_{\ell}^{*}$ for
$\ell=1,\ldots,k_{z}$, where
$\alpha_{\ell}^{*}=$$\left(\text{var}\left(Z_{\ell}\right)\right)^{-1}\alpha_{\ell}$.

Then define the $k_{z}$-vectors $\boldsymbol{\pi}^{*}$ and
$\boldsymbol{\psi}^{*}$ with $\ell$-th elements given by

$\displaystyle\pi_{\ell}^{*}$
$\displaystyle:=\left(\text{var}\left(Z_{\ell}\right)\right)^{-1}\text{cov}\left(Z_{\ell},X\right);\,\,\psi_{\ell}^{*}:=\left(\text{var}\left(Z_{\ell}\right)\right)^{-1}\text{cov}\left(Z_{\ell},Y\right),$
(14)

for $\ell=1,\ldots,k_{z}$.

Under Assumptions 1, 2, 4 and 7, the identified set for $\beta$ is given by

$\mathcal{B}\left(\boldsymbol{\delta}^{*}\right)=\left\\{b\in\mathbb{R}:-\boldsymbol{\delta}^{*}\leq\left(\boldsymbol{\psi^{*}}-\boldsymbol{\pi}^{*}b\right)\leq\boldsymbol{\delta}^{*}\right\\},$

which follows straightforwardly as
$\boldsymbol{\psi}^{*}-\boldsymbol{\pi}^{*}\beta=\boldsymbol{\alpha}^{*}$ and
$-\boldsymbol{\delta}^{*}\leq\boldsymbol{\alpha}^{*}\leq\boldsymbol{\delta}^{*}$.
The model is falsified if and only if
$\mathcal{B}\left(\boldsymbol{\delta}^{*}\right)$ is empty. If
$\mathcal{B}\left(\boldsymbol{0}\right)$ is empty then the baseline IV model
(11), assuming $\boldsymbol{\alpha}=\boldsymbol{0}$, is falsified.

Let $\mathcal{L}_{rel}^{*}$ denote the set of relevant instruments

$\mathcal{L}_{rel}^{*}=\left\\{\ell\in\left\\{1,\ldots,k_{z}\right\\}:\pi_{\ell}^{*}\neq
0\right\\}.$ (15)

For the IV model (11), under Assumptions 1, 2 and 4, known joint distribution
of $\left(Y,X,\boldsymbol{Z}\right)$ and following the same arguments as above
for the derivation of the falsification frontier for the relaxation of
exclusion restrictions, the falsification frontier here is given as the set

$FF_{exo}=\left\\{\boldsymbol{\delta}^{*}\left(b\right)\in\mathbb{R}_{\geq
0}^{k_{z}}:\delta_{\ell}^{*}\left(b\right)=\left|\psi_{\ell}^{*}-b\pi_{\ell}^{*}\right|,\,\ell=1,\ldots,k_{z},\,b\in\left[\min_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}},\max_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}}\right]\right\\}.$

The falsification adaptive set is therefore given

$\displaystyle FAS_{exo}$
$\displaystyle=\left[\min_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}},\max_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}}\right]$
(16)
$\displaystyle=\left[\beta+\min_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\alpha_{\ell}^{*}}{\pi_{\ell}^{*}},\beta+\max_{\ell\in\mathcal{L}_{rel}^{*}}\frac{\alpha_{\ell}^{*}}{\pi_{\ell}^{*}}\right].$

For $\ell\in\mathcal{L}_{rel}^{*}$, the ratio

$\beta_{\ell}^{*}:=\frac{\psi_{\ell}^{*}}{\pi_{\ell}^{*}}=\frac{\text{cov}\left(Z_{\ell},Y\right)}{\text{cov}\left(Z_{\ell},X\right)}$

is the IV estimand for the specification $Y=X\beta+U$, using $Z_{\ell}$ as the
just-identifying instrument for $X$, and treating the instruments
$\boldsymbol{Z}_{\left\\{-\ell\right\\}}$ as invalid and excluding them from
the analysis.

Under the exclusion assumption, a sufficient condition for FASexo to contain
$\beta$ is that there is at least one relevant and valid instrument as we
state in the following condition.

######  Condition 3.

Exogeneity Sufficient Condition: If the instruments satisfy the exclusion
Assumption 4 that $\gamma_{\ell}=0$ for all
$\ell\in\left\\{1,\ldots,k_{z}\right\\}$, then a sufficient condition for the
FASexo to be the identified set for $\beta$ is that at least one instrument is
relevant and valid.

As specified for $\mathcal{L}_{rel}^{*}$ in (15), an instrument $Z_{\ell}$ is
relevant in this setting if $\pi_{\ell}^{*}\neq 0$. Relative to Assumption 5
for the general Condition 1, Condition 3 requires the stronger exclusion
assumption.

To contrast the condition for FASexo with the finding for FASexcl, consider
the following simple example where there is one valid instrument and one
invalid instrument that violates the exogeneity assumption. As Condition 2 is
not satisfied, the FASexcl is in this case not guaranteed to be the identified
set for $\beta$, even though one of the instruments is valid and relevant.

######  Example 2.

Let $k_{z}=2$, $\boldsymbol{\pi}=\left(\pi_{1},\pi_{2}\right)^{T}$,
$\pi_{\ell}\neq 0$ for $\ell=1,2$, $\boldsymbol{\gamma}=\boldsymbol{0}$,
$\boldsymbol{\alpha}=\left(0,\alpha_{2}\right)^{T}$, $\alpha_{2}\neq 0$, and

$\boldsymbol{\Sigma}_{z}=\left[\begin{array}[]{cc}1&\rho_{12}\\\
\rho_{12}&1\end{array}\right],$

with $\left|\rho_{12}\right|<1$. Then
$\boldsymbol{\pi}^{*}=\left(\pi_{1}+\rho_{12}\pi_{2},\pi_{2}+\rho_{12}\pi_{1}\right)^{T}$,
$\boldsymbol{\alpha}^{*}=\boldsymbol{\alpha}$, and so, provided
$\pi_{\ell}^{*}\neq 0$ for $\ell=1,2$,

$\displaystyle FAS_{exo}$
$\displaystyle=\beta+\left[\min\left\\{0,\frac{\alpha_{2}}{\pi_{2}+\rho_{12}\pi_{1}}\right\\},\max\left\\{0,\frac{\alpha_{2}}{\pi_{2}+\rho_{12}\pi_{1}}\right\\}\right]$

and $\beta\in FAS_{exo}$.

For $FAS_{excl}$, we have that
$\boldsymbol{\eta}=\boldsymbol{\Sigma}_{z}^{-1}\boldsymbol{\alpha}=\frac{\alpha_{2}}{1-\rho_{12}^{2}}\left(-\rho_{12},1\right)^{T}$,
and so

$FAS_{excl}=\beta+\left[\min\left\\{\frac{-\rho_{12}\alpha_{2}}{\pi_{1}\left(1-\rho_{12}^{2}\right)},\frac{\alpha_{2}}{\pi_{2}\left(1-\rho_{12}^{2}\right)}\right\\},\max\left\\{\frac{-\rho_{12}\alpha_{2}}{\pi_{1}\left(1-\rho_{12}^{2}\right)},\frac{\alpha_{2}}{\pi_{2}\left(1-\rho_{12}^{2}\right)}\right\\}\right],$

therefore, if $\frac{-\rho_{12}\alpha_{2}}{\pi_{1}}$ and
$\frac{\alpha_{2}}{\pi_{2}}$ have the same sign, $\beta\notin FAS_{excl}$.

We can of course adapt Example 2 to the case where the invalid instrument
violates the exclusion assumption instead, in which case the FASexo is not
guaranteed to be the identified set for $\beta$.

The selection methods of Guo et al. (2018) and Windmeijer et al. (2021) can be
adjusted to select the valid instruments consistently in the case of a
violation of the exogeneity assumption only if the plurality rule applies that
the valid instruments form the largest group within the set of relevant
variables. The Andrews (1999) selection method was originally designed for
this case.

###  4.1 Estimation

The set of relevant instruments can here be estimated by

$\mathcal{\widehat{L}}_{rel}^{*}=\left\\{\ell\in\left\\{1,\ldots,k_{z}\right\\}:F_{\ell}^{*}\geq
C_{n}\right\\},$

where $F_{\ell}^{*}$ is the F-statistic for testing $H_{0}:\pi_{\ell}^{*}=0$
in the first-stage linear specification

$\boldsymbol{x}=\boldsymbol{z}_{\ell}\pi_{\ell}^{*}+\boldsymbol{v}_{\ell}.$

Let $\widehat{\beta}_{\ell}^{*}$ be the IV estimators

$\widehat{\beta}_{\ell}^{*}=\frac{\boldsymbol{z}_{\ell}^{T}\boldsymbol{y}}{\boldsymbol{z}_{\ell}^{T}\boldsymbol{x}}=\frac{\widehat{\psi}_{\ell}^{*}}{\widehat{\pi}_{\ell}^{*}},$

where $\widehat{\psi}_{\ell}^{*}$ is the OLS estimator of $\psi_{\ell}^{*}$ in
the reduced-form specification
$\boldsymbol{y}=\boldsymbol{z}_{\ell}^{T}\psi_{\ell}^{*}+\boldsymbol{\varepsilon}_{\ell}$,
and $\widehat{\pi}_{\ell}^{*}$ the OLS estimator of $\pi_{\ell}^{*}$. Then the
consistent estimator of FASexo is given by

$\widehat{FAS}_{exo}=\left[\min_{\ell\in\mathcal{\widehat{L}}_{rel}^{*}}\widehat{\beta}_{\ell}^{*},\max_{\ell\in\mathcal{\widehat{L}}_{rel}^{*}}\widehat{\beta}_{\ell}^{*}\right].$

###  4.2 2sls

The 2sls estimator for $\beta$ for the specification
$\boldsymbol{y}=\boldsymbol{x}\beta+\boldsymbol{u}$ using the $n\times k_{z}$
instrument matrix $\boldsymbol{Z}$ is again a linear combination of the
individual estimators $\widehat{\beta}_{\ell}^{*}$,

$\displaystyle\widehat{\beta}_{2sls}$
$\displaystyle=\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{y}$
$\displaystyle=\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}\widehat{\boldsymbol{\pi}}^{T}\boldsymbol{D}_{Z^{T}x}\widehat{\boldsymbol{\beta}}^{*}$
$\displaystyle=\sum_{\ell=1}^{k_{z}}w_{\ell}^{*}\widehat{\beta}_{\ell}^{*}.$
(17)

where $\widehat{\boldsymbol{\beta}}^{*}$ is the $k_{z}$-vector
$\left(\widehat{\beta}_{\ell}^{*}\right)$ and $\boldsymbol{D}_{Z^{T}x}$ is the
$k_{z}\times k_{z}$ diagonal matrix with $\ell$-th diagonal element equal to
$\boldsymbol{z}_{\ell}^{T}\boldsymbol{x}$. It follows that
$\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}\widehat{\boldsymbol{\pi}}^{T}\boldsymbol{D}_{Z^{T}x}\boldsymbol{\iota_{k_{z}}}=1$
and so $\sum_{\ell=1}^{k_{z}}w_{\ell}^{*}=1$. Comparing the weights with those
in (10), where we had that
$\widehat{\beta}_{2sls}=\sum_{\ell=1}^{k_{z}}w_{\ell}\widehat{\beta}_{\ell}$,
with $\sum_{\ell=1}^{k_{z}}w_{\ell}=1$, we find that the weights are
identical, as

$w_{\ell}^{*}=\widehat{\pi}_{\ell}\boldsymbol{z}_{\ell}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}=w_{\ell},$

for $\ell=1,\ldots,k_{z}$, i.e. the 2sls weights for the just-identified
estimators $\widehat{\beta}_{\ell}$, where
$\boldsymbol{Z}_{\left\\{-\ell\right\\}}$ are included as controls, are the
same as those for the just-identified estimators $\widehat{\beta}_{\ell}^{*}$,
where $\boldsymbol{Z}_{\left\\{-\ell\right\\}}$ are excluded from the
instrument set.

##  5 Failure of the Exogeneity or Exclusion Assumption

If a model is falsified, a researcher is unlikely to know whether this is due
to a violation of the exogeneity or the exclusion assumption and the
falsification adaptive set should reflect this. In this section we generalize
the FAS, such that it reflects the fact that instruments can violate the
exogeneity or exclusion restriction. We therefore consider here the general
model specifications as in (2) and (3),

$\displaystyle Y$
$\displaystyle=X\beta+\boldsymbol{Z}^{T}\boldsymbol{\gamma}+U$
$\text{cov}\left(\boldsymbol{Z},U\right)=\boldsymbol{\alpha}.$

We obtain the falsification frontier and associated FAS that satisfies
Condition 1 by simply considering all possible just-identified model
specifications, as detailed below. We first show this for the $k_{z}=2$ case,
where the results are a simple combination of the results for the violation of
the exclusion restriction only and the violation of the exogeneity assumption
only.

###  5.1 $k_{z}=2$

When there are two instruments available, we can generalize the FAS by simply
combining the results for FASexcl and FASexo as presented in Sections 3 and 4.
There are in this case four just-identified model specifications, one with
$Z_{1}$ as control and $Z_{2}$ as excluded instrument for $X$, one with
$Z_{2}$ as control and $Z_{1}$ as excluded instrument for $X$, and the model
without controls and $Z_{1}$ or $Z_{2}$ as instrument for $X$. As above, let
$\boldsymbol{\pi}$ and $\boldsymbol{\psi}$ be as defined in (4) for the model
with instruments included as controls, and $\boldsymbol{\pi}^{*}$ and
$\boldsymbol{\psi}^{*}$ for the model without instruments included as
controls, as defined in (14).

From the combination of the results in Sections 3 and 4, it follows that we
get here

$\boldsymbol{\psi}-\boldsymbol{\pi}\beta=\boldsymbol{\gamma}+\boldsymbol{\eta},$

with, as in (13),
$\boldsymbol{\eta}=\boldsymbol{\Sigma}_{z}^{-1}\boldsymbol{\alpha}$. Further

$\boldsymbol{\psi}^{*}-\boldsymbol{\pi}^{*}\beta=\boldsymbol{\alpha}^{*}+\boldsymbol{\xi}^{*},$

with
$\boldsymbol{\xi}^{*}=\text{diag}\left(\left(\text{var}\left(Z_{\ell}\right)\right)^{-1}\right)\boldsymbol{\Sigma}_{z}\boldsymbol{\gamma}$,
where for general $d_{\ell}$, $\text{diag}\left(d_{\ell}\right)$ is a diagonal
matrix with $\ell$-th diagonal element $d_{\ell}$, and, as before,
$\alpha_{\ell}^{*}=\left(\text{var}\left(Z_{\ell}\right)\right)^{-1}\alpha_{\ell}$.
We can now make the partial exogeneity and exclusion assumption.

######  Assumption 8.

Partial exogeneity and exclusion: There are know constants $\delta_{\ell}\geq
0$ and $\delta_{\ell}^{*}\geq 0$ such that
$\left|\gamma_{\ell}+\eta_{\ell}\right|\leq\delta_{\ell}$ and
$\left|\alpha_{\ell}^{*}+\xi_{\ell}^{*}\right|\leq\delta_{\ell}^{*}$ for all
$\ell\in\left\\{1,\ldots,k_{z}\right\\}$.

Then define

$\tilde{\boldsymbol{\pi}}:=\left(\begin{array}[]{c}\boldsymbol{\pi}\\\
\boldsymbol{\pi}^{*}\end{array}\right);\,\,\tilde{\boldsymbol{\psi}}:=\left(\begin{array}[]{c}\boldsymbol{\psi}\\\
\boldsymbol{\psi}^{*}\end{array}\right);\,\,\tilde{\boldsymbol{\delta}}:=\left(\begin{array}[]{c}\boldsymbol{\delta}\\\
\boldsymbol{\delta}^{*}\end{array}\right).$

Extending the results as outlined in Sections 3 and 4, under Assumptions 1, 2,
5 and 8, the identified set for $\beta$ is given by

$\mathcal{B}\left(\tilde{\boldsymbol{\delta}}\right)=\left\\{b\in\mathbb{R}:-\tilde{\boldsymbol{\delta}}\leq\left(\tilde{\boldsymbol{\psi}}-\tilde{\boldsymbol{\pi}}b\right)\leq\tilde{\boldsymbol{\delta}}\right\\}.$

Let $\mathcal{\tilde{L}}_{rel}$ denote the set of relevant instruments in this
setting, specified as

$\tilde{\mathcal{L}}_{rel}=\left\\{j\in\left\\{1,\ldots,4\right\\}:\tilde{\pi}_{j}\neq
0\right\\}.$ (18)

Then we get for the falsification frontier,

$FF=\left\\{\tilde{\boldsymbol{\delta}}\left(b\right)\in\mathbb{R}_{\geq
0}^{4}:\tilde{\delta}_{j}\left(b\right)=\left|\tilde{\psi}_{j}-b\tilde{\pi}_{j}\right|,\,j=1,\ldots,4,\,b\in\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}},\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right]\right\\}.$

As discussed in Section 3, this follows as, for
$j\in\tilde{\mathcal{L}}_{rel}$,
$\tilde{\delta}_{j}\left(\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right)=0$
and so when moving $b$ from
$\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}$
to
$\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}$
there is at least one element in $\tilde{\boldsymbol{\delta}}\left(b\right)$
that decreases in value and at least one that increases in value. For any
$\tilde{\boldsymbol{\delta}}^{\prime}<\text{$\tilde{\boldsymbol{\delta}}\left(b\right)$}$,
$b\in\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}},\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right]$,
$\mathcal{B}\left(\tilde{\boldsymbol{\delta}}^{\prime}\right)$ is empty.

We then get our main result that the falsification frontier is given by

$FAS=\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}},\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right].$
(19)

These results are clear. In order to consider possible violations of the
exogeneity or exclusion assumptions, we need to consider all just-identified
model specifications.

When there are two instruments available, the model is falsified and one
instrument is valid, then from Assumption 5 it follows that the invalid
instrument does either violate the exogeneity or the exclusion restriction. If
it violates the exogeneity assumption, as in Example 2, with
$\boldsymbol{\alpha}=\left(0,\alpha_{2}\right)^{T}$, $\alpha_{2}\neq 0$, then
$Z_{1}$ identifies $\beta$ provided $Z_{2}$ is omitted from the instrument set
as
$\frac{\tilde{\psi}_{3}}{\tilde{\pi}_{3}}=\frac{\psi_{1}^{*}}{\pi_{1}^{*}}=\beta$,
provided $Z_{1}$ is a relevant instrument when $Z_{2}$ is omitted, or
$\tilde{\pi}_{3}=\pi_{1}^{*}\neq 0$. If alternatively the invalid instrument
violates the exclusion assumption, with
$\boldsymbol{\gamma}=\left(0,\gamma_{2}\right)^{T}$, $\gamma_{2}\neq 0$, then
$Z_{1}$ identifies $\beta$ when $Z_{2}$ is included as a control variable as
$\frac{\tilde{\psi}_{1}}{\tilde{\pi}_{1}}=\frac{\psi_{1}}{\pi_{1}}=\beta$,
provided $Z_{1}$ is a relevant instrument when $Z_{2}$ is included as a
control and hence in the instrument set, or
$\tilde{\pi}_{1}=\text{$\pi_{1}$}\neq 0$. In the words of MP, p 1452, when the
baseline model is falsified, the falsification adaptive set expands to account
for the uncertainty about which assumption along the frontier is true. The FAS
as defined in (19) does this for violations of the exogeneity and exclusion
assumptions. As it contains $\beta$ if there is a valid relevant instrument,
it satisfies Condition 1.

####  5.1.1 Alternative Representation

Instead of specifying different just identified models as in the previous
sections, we can simply focus on the transformed just-identifying instruments,
which will enable us to easily generalize the approach to the setup for a
general number of instruments. Define the population linear projection errors
$Z_{1|2}$ and $Z_{2|1}$ as

$Z_{1|2}:=Z_{1}-Z_{2}\phi_{21};\,\,Z_{2|1}:=Z_{2}-Z_{1}\phi_{12},$ (20)

where
$\phi_{21}=\left(\text{var}\left(Z_{2}\right)\right)^{-1}\text{cov}\left(Z_{2},Z_{1}\right)$
and
$\phi_{12}=\left(\text{var}\left(Z_{1}\right)\right)^{-1}\text{cov}\left(Z_{1},Z_{2}\right)$.
It follows straightforwardly that for the elements of
$\boldsymbol{\psi}=\boldsymbol{\Sigma}_{z}^{-1}\text{cov}\left(\boldsymbol{Z},Y\right)$,
we have that

$\displaystyle\psi_{1}$
$\displaystyle=\left(\text{var}\left(Z_{1|2}\right)\right)^{-1}\text{cov}\left(Z_{1|2},Y\right)$
$\displaystyle\psi_{2}$
$\displaystyle=\left(\text{var}\left(Z_{2|1}\right)\right)^{-1}\text{cov}\left(Z_{2|1},Y\right),$

and equivalently for the elements of $\boldsymbol{\pi}$. These are the
population equivalents of the sample estimation results in (9).

We can therefore get the vector of just-identifying transformed instruments as

$\tilde{\boldsymbol{Z}}=\left(Z_{1|2},Z_{2|1},Z_{1},Z_{2}\right)^{T}.$ (21)

Then the vectors $\tilde{\boldsymbol{\pi}}$ and $\tilde{\boldsymbol{\psi}}$
have the elements

$\tilde{\pi}_{j}=\left(\text{var}\left(\tilde{Z}_{j}\right)\right)^{-1}\text{cov}\left(\tilde{Z}_{j},X\right);\,\,\tilde{\psi}_{j}=\left(\text{var}\left(\tilde{Z}_{j}\right)\right)^{-1}\text{cov}\left(\tilde{Z}_{j},Y\right),$
(22)

for $j=1,\ldots,4$.

For $j\in\tilde{\mathcal{L}}_{rel}$, the IV estimands are then

$\tilde{\beta}_{j}=\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}=\frac{\text{cov}\left(\tilde{Z}_{j},Y\right)}{\text{cov}\left(\tilde{Z}_{j},X\right)}=\beta+\frac{\text{cov}\left(\tilde{Z}_{j},\tilde{U}\right)}{\text{cov}\left(\tilde{Z}_{j},X\right)},$

where $\tilde{U}=Y-X\beta=\boldsymbol{Z}^{T}\boldsymbol{\gamma}+U$. The FAS is
given by

$FAS=\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\tilde{\beta}_{j},\max_{j\in\tilde{\mathcal{L}}_{rel}}\tilde{\beta}_{j}\right].$

This representation makes the definition of a relevant instrument generic,
with an instrument $\tilde{Z}_{j}$ here relevant if $\tilde{\pi}_{j}\neq 0$,
confirming the definition as in (18).

####  5.1.2 Estimation

For estimation of the FAS as defined in (19) the set of relevant instruments
is estimated by

$\mathcal{\widehat{\tilde{L}}}_{rel}=\left\\{j\in\left\\{1,\ldots,4\right\\}:\tilde{F}_{j}\geq
C_{n}\right\\},$

where $\tilde{F}_{j}$ is the F-statistic for testing $H_{0}:\tilde{\pi}_{j}=0$
in the first-stage linear specification
$\boldsymbol{x}=\tilde{\boldsymbol{z}}_{j}\tilde{\pi}_{j}+\boldsymbol{v}_{j}$,
where $\tilde{\boldsymbol{z}}_{j}$ is the $j$-th column of the $n\times 4$
matrix of just-identifying transformed instruments

$\tilde{\boldsymbol{Z}}=\left[\boldsymbol{M}_{z_{2}}\boldsymbol{z}_{1}\,\,\boldsymbol{M}_{z_{1}}\boldsymbol{z}_{2}\,\,\boldsymbol{z}_{1}\,\,\boldsymbol{z}_{2}\right].$

Let $\widehat{\tilde{\beta}}_{j}$ be the IV estimators

$\widehat{\tilde{\beta}}_{j}=\frac{\tilde{\boldsymbol{z}}_{j}^{T}\boldsymbol{y}}{\tilde{\boldsymbol{z}}_{j}^{T}\boldsymbol{x}}=\frac{\widehat{\tilde{\psi}}_{j}}{\widehat{\tilde{\pi}}_{j}},$

where $\widehat{\tilde{\psi}}_{j}$ is the OLS estimator of $\tilde{\psi}_{j}$
in the reduced-form specification
$\boldsymbol{y}=\tilde{\boldsymbol{z}}_{j}^{T}\psi_{j}+\boldsymbol{\varepsilon}_{j}$,
and $\widehat{\tilde{\pi}}_{j}$ the OLS estimator of $\tilde{\pi}_{j}$. Then a
consistent estimator of $FAS$ is given by

$\widehat{FAS}=\left[\min_{j\in\mathcal{\widehat{\tilde{L}}}_{rel}}\widehat{\tilde{\beta}}_{j},\max_{j\in\mathcal{\widehat{\tilde{L}}}_{rel}}\widehat{\tilde{\beta}}_{j}\right].$

###  5.2 General $k_{z}$

The extension to the definition of the FAS for a general number of instruments
that can violate the exogeneity or exclusion assumption is obtained by
considering all possible just-identified model specifications. There are
$s_{k_{z}}=k_{z}2^{k_{z}-1}$ such specifications, resulting in e.g. $s_{2}=4$,
$s_{3}=12$ and $s_{4}=32$. This is established as follows. For an instrument
$Z_{\ell}$, $\ell\in\left\\{1,\ldots,k_{z}\right\\}$, let the set
$A_{-\ell}=\left\\{r:r\in\left\\{1,\ldots,k_{z}\right\\},r\neq\ell\right\\}$
and let $\mathcal{C_{-\ell}}=\left\\{C_{-\ell,m}\right\\}_{m=1}^{M}$ denote
the collection of all subsets of $A_{-\ell}$. The number of elements in
$C_{-\ell,m}$ range from $0,1,\ldots,k_{z}-1$, so $\mathcal{C}_{-\ell}$
includes $\emptyset$ and $A_{-\ell}$. There are $M=2^{k_{z}-1}$ such subsets.
Define as in (20) the linear projection error

$Z_{\ell|C_{-\ell,m}}:=Z_{\ell}-\boldsymbol{Z}_{C_{-\ell,m}}^{T}\boldsymbol{\phi}_{C_{-\ell,m}\ell},$

then $Z_{\ell|C_{-\ell,m}}$ is the transformed just-identifying instrument in
the model specification where the instruments $\boldsymbol{Z}_{C_{-\ell,m}}$
are included as controls and the instruments
$\boldsymbol{Z}_{A_{-\ell}\setminus C_{-\ell,m}}$ are omitted from the
instrument set. For each $\ell\in\left\\{1,\ldots,k_{z}\right\\}$ there are
$M=2^{k_{z}-1}$ such transformed just-identifying instruments and so the total
number of just-identifying specifications is given by
$s_{k_{z}}=k_{z}2^{k_{z}-1}$.

As an example, for $k_{z}=3$, $\ell=1$ and $C_{-1,m}=\left\\{3\right\\}$,
$Z_{1|3}$ is the just-identifying instrument $Z_{1}$ for the model with
$Z_{3}$ included as control and $Z_{2}$ excluded from the instrument set. The
full sequence for $Z_{1}$ is given by the $2^{2}=4$ just identifying
instruments $\left\\{Z_{1},Z_{1|2},Z_{1|3},Z_{1|2,3}\right\\}$. This applies
to all $3$ instruments, so there are a total of $12$ just-identifying model
specifications. For model specification (2) and (3), let
$\boldsymbol{\gamma}=\left(0,0,\gamma_{3}\right)^{T}$ and
$\boldsymbol{\alpha}=\left(0,\alpha_{2},0\right)^{T}$ with $\gamma_{3}\neq 0$
and $\alpha_{2}\neq 0$. Here $Z_{1}$ is a valid instrument, but with
correlated instruments, $Z_{3}$ needs to be included as a control for it to
identify $\beta$. Therefore $Z_{1|3}$ is here the valid instrument that
identifies $\beta$ if it is relevant. If the latter is the case, then the FAS
as constructed below will contain $\beta$.

Let $\tilde{\boldsymbol{Z}}$ be the $s_{k_{z}}$-vector of transformed just-
identifying instruments

$\displaystyle\tilde{\boldsymbol{Z}}$
$\displaystyle=\left(Z_{\ell|C_{-\ell,m}}\right)_{\ell=1,\ldots,,k_{z};m=1,\ldots,2^{k_{z}-1}}$
(23) $\displaystyle=\left(\tilde{Z}_{j}\right)_{j=1,\ldots,s_{k_{z}}}.$

Then define as in (22) the $s_{k_{z}}$-vectors $\tilde{\boldsymbol{\pi}}$ and
$\tilde{\boldsymbol{\psi}}$ with elements

$\tilde{\pi}_{j}=\left(\text{var}\left(\tilde{Z}_{j}\right)\right)^{-1}\text{cov}\left(\tilde{Z}_{j},X\right);\,\,\tilde{\psi}_{j}=\left(\text{var}\left(\tilde{Z}_{j}\right)\right)^{-1}\text{cov}\left(\tilde{Z}_{j},Y\right),$
(24)

for $j=1,\ldots,s_{k_{z}}$.

It follows from model specification (2) and (3) that

$\tilde{\psi}_{j}-\tilde{\pi}_{j}\beta=\left(\text{var}\left(\tilde{Z}_{j}\right)\right)^{-1}\text{cov}\left(\tilde{Z}_{j},\tilde{U}\right),$

where $\tilde{U}=Y-X\beta=\boldsymbol{Z}^{T}\boldsymbol{\gamma}+U$. We can now
make the general partial exogeneity and exclusion assumption,

######  Assumption 9.

General partial exogeneity and exclusion: There are known constants
$\tilde{\delta}_{j}\geq 0$ such that
$\left|\left(\text{var}\left(\tilde{Z}_{j}\right)\right)^{-1}\text{cov}\left(\tilde{Z}_{j},\tilde{U}\right)\right|\leq\tilde{\delta}_{j}$
for all $j\in\left\\{1,\ldots,s_{k_{z}}\right\\}$.

It then follows from the arguments given in Sections 3, 4 and 5.1 that under
Assumptions 1, 2, 5 and 9, the identified set for $\beta$ is given by

$\mathcal{B}\left(\tilde{\boldsymbol{\delta}}\right)=\left\\{b\in\mathbb{R}:-\tilde{\boldsymbol{\delta}}\leq\left(\tilde{\boldsymbol{\psi}}-\tilde{\boldsymbol{\pi}}b\right)\leq\tilde{\boldsymbol{\delta}}\right\\}.$

The set of relevant instruments is specified as

$\tilde{\mathcal{L}}_{rel}=\left\\{j\in\left\\{1,\ldots,s_{k_{z}}\right\\}:\tilde{\pi}_{j}\neq
0\right\\},$ (25)

and the falsification frontier is given by

$FF=\left\\{\tilde{\boldsymbol{\delta}}\left(b\right)\in\mathbb{R}_{\geq
0}^{s_{k_{z}}}:\tilde{\delta}_{j}\left(b\right)=\left|\tilde{\psi}_{j}-b\tilde{\pi}_{j}\right|,\,j=1,\ldots,s_{k_{z}},\,b\in\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}},\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right]\right\\}.$

Then the generalized falsification adaptive set is

$\displaystyle FAS$
$\displaystyle=\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}},\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right]$
$\displaystyle=\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\tilde{\beta}_{j},\max_{j\in\tilde{\mathcal{L}}_{rel}}\tilde{\beta}_{j}\right],$

where, for $j\in\tilde{\mathcal{L}}_{rel}$, the IV estimands are given by

$\tilde{\beta}_{j}=\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}=\frac{\text{cov}\left(\tilde{Z}_{j},Y\right)}{\text{cov}\left(\tilde{Z}_{j},X\right)}=\beta+\frac{\text{cov}\left(\tilde{Z}_{j},\tilde{U}\right)}{\text{cov}\left(\tilde{Z}_{j},X\right)}.$

We summarize our main result, the generalization of the falsification adaptive
set, in the following proposition. This generalizes Theorem 2 in MP, with the
further result that this FAS satisfies Condition 1. This latter follows
straightforwardly, as for each instrument all possible permutations of
invalidity of the other instruments under Assumption 5 are considered. Hence,
if there is at least one valid instrument, then at least one of these
permutations will identify $\beta$, as long as the associated transformed
just-identifying instrument is relevant.

######  Proposition 1.

For the linear model specification (2) and (3), suppose Assumptions 1, 2, 5
and 9 hold and the distribution of $\left(Y,X,\boldsymbol{Z}\right)$ is known.
i) The falsification adaptive set is given by

$FAS=\left[\min_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}},\max_{j\in\tilde{\mathcal{L}}_{rel}}\frac{\tilde{\psi}_{j}}{\tilde{\pi}_{j}}\right],$
(26)

where $\tilde{\psi}_{j}$ and $\tilde{\pi}_{j}$ are defined in (24) for
$j=1,\ldots,s_{k_{z}}$, with $s_{k_{z}}=k_{z}2^{k_{z}-1}$ the number of all
possible just-identified IV model specifications. The $\tilde{Z}_{j}$ in (24)
are the $s_{k_{z}}$ just-identifying transformed instruments as defined in
(23). An instrument $\tilde{Z}_{j}$ is relevant if $\tilde{\pi}_{j}\neq 0$,
and $\tilde{\mathcal{L}}_{rel}$ is the set of relevant instruments as defined
in (25). ii) A sufficient condition for the FAS to contain $\beta$ and thus be
the identified set for $\beta$ is that there is at least one instrument that
is relevant and valid.

For estimation, the same procedure is followed as in Section 5.1.2. The set of
relevant instruments is estimated by

$\mathcal{\widehat{\tilde{L}}}_{rel}=\left\\{j\in\left\\{1,\ldots,s_{k_{z}}\right\\}:\tilde{F}_{j}\geq
C_{n}\right\\},$

where $\tilde{F}_{j}$ is the F-statistic for testing $H_{0}:\tilde{\pi}_{j}=0$
in the first-stage linear specification
$\boldsymbol{x}=\tilde{\boldsymbol{z}}_{j}\tilde{\pi}_{j}+\boldsymbol{v}_{j}$,
where $\tilde{\boldsymbol{z}}_{j}$ is the $j$-th column of the $n\times
s_{k_{z}}$ matrix of just-identifying transformed instruments

$\displaystyle\tilde{\boldsymbol{Z}}$
$\displaystyle=\left[\boldsymbol{M}_{Z_{C_{-\ell},m}}\boldsymbol{z}_{\ell}\right]_{\ell=1,\ldots,k_{z},m=1,\ldots,2^{k_{z}-1}}$
$\displaystyle=\left[\tilde{\boldsymbol{z}}_{j}\right].$

Let $\widehat{\tilde{\beta}}_{j}$ be the IV estimators

$\widehat{\tilde{\beta}}_{j}=\frac{\tilde{\boldsymbol{z}}_{j}^{T}\boldsymbol{y}}{\tilde{\boldsymbol{z}}_{j}^{T}\boldsymbol{x}}=\frac{\widehat{\tilde{\psi}}_{j}}{\widehat{\tilde{\pi}}_{j}},$

then the consistent estimator of FAS is given by

$\widehat{FAS}=\left[\min_{j\in\mathcal{\widehat{\tilde{L}}}_{rel}}\widehat{\tilde{\beta}}_{j},\max_{j\in\mathcal{\widehat{\tilde{L}}}_{rel}}\widehat{\tilde{\beta}}_{j}\right].$

##  6 Some Empirical Examples

###  6.1 An Empirical Example with Two Instruments

MP20, Section 4.2, calculate $\widehat{FAS}{}_{excl}$ for the cross-country
study by Alesina et al. (2013) on the origin of gender roles. We will focus
here on the results for the model with the outcome variable “Female labor
force participation in 2000”. Their endogenous treatment variable is the
“estimated proportion of citizens with ancestors that traditionally used the
plough in pre-industrial agriculture” and they use geo-climatic area
conditions for suitability for growing “plough-positive” crops and “plough-
negative” crops as two instruments. The full estimation results are given in
Table VIII in Alesina et al. (2013), which also gives details of other control
variables that have been included in the model. In MP20, these results and the
$\widehat{FAS}{}_{excl}$ are reported in their Table 4.

Table 1 reports the results for the model specification without continent
fixed effects. The 2sls estimation results using both instruments show a
significant negative effect of traditional plough use on female labor force
participation, but this model is falsified with the p-value of the Hansen
(1982) $J$-test being equal to $0.0009$. With $Z_{1}$ being the “plough-
positive environment” instrument, and $Z_{2}$ the “plough-negative
environment” one, it is found that the F-statistic for $H_{0}:\pi_{2}=0$ is
equal to $0.95$ and hence $Z_{2|1}$ is not in the relevant set. As reported in
MP20, the F-statistic for $H_{0}:\pi_{1}=0$ is equal to $78.20$ and so it
follows that the $\widehat{FAS}{}_{excl}=-14.31$, which is the IV estimate
using $Z_{1|2}$ as the just-identifying and relevant instrument. So, although
the two instruments produce significantly different results, falsifying the
model, the $\widehat{FAS}{}_{excl}$ is a single point. As mentioned before,
the falsification adaptive set should expand to account for the uncertainty
about which assumption along the frontier is true, which is not the case here.
For the FASexo, we get $\widehat{FAS}{}_{exo}=\left[-46.98,-22.65\right]$,
which is further expanded to the estimate of FAS,
$\widehat{FAS}=\left[-46.98,-14.31\right]$, which is the range of IV estimates
of the three just-identifying relevant instruments, $Z_{1|2}$, $Z_{1}$ and
$Z_{2}$. $\widehat{FAS}$ fully shows the uncertainty of the results due to the
falsification of the model.

Table 1:  Country level IV estimation results, Alesina et al. (2013, Table
VIII, column 1)

| Instruments
---|---
| $Z_{1},Z_{2}$ | $Z_{1|2}$ | $Z_{2|1}$ | $Z_{1}$ | $Z_{2}$
Traditional plough use | $\begin{array}[]{c}-21.63\\\ \left(5.25\right)\end{array}$ | $\begin{array}[]{c}-14.31\\\ \left(5.15\right)\end{array}$ | $\begin{array}[]{c}159.6\\\ \left(176.9\right)\end{array}$ | $\begin{array}[]{c}-22.65\\\ \left(5.32\right)\end{array}$ | $\begin{array}[]{c}-46.98\\\ \left(10.77\right)\end{array}$
F | $40.21$ | $78.20$ | $0.95$ | $74.01$ | $19.00$
$J$-test p-value | $0.0009$ |  |  |  |
$\widehat{FAS}{}_{excl}$ | $-14.31$ |  |  |
$\widehat{FAS}{}_{exo}$ | $\left[-46.98,-22.65\right]$ |  |  |
$\widehat{FAS}$ | $\left[-46.98,-14.31\right]$ |  |  |
$\tilde{\psi}$ |  | $\begin{array}[]{c}-10.64\\\ \left(3.82\right)\end{array}$ | $\begin{array}[]{c}18.93\\\ \left(6.51\right)\end{array}$ | $\begin{array}[]{c}-16.08\\\ \left(3.50\right)\end{array}$ | $\begin{array}[]{c}29.66\\\ \left(6.28\right)\end{array}$
$\tilde{\pi}$ |  | $\begin{array}[]{c}0.744\\\ \left(0.084\right)\end{array}$ | $\begin{array}[]{c}0.119\\\ \left(0.122\right)\end{array}$ | $\begin{array}[]{c}0.710\\\ \left(0.083\right)\end{array}$ | $\begin{array}[]{c}-0.631\\\ \left(0.144\right)\end{array}$

Notes: Outcome variable “Female labor force participation rate in 2000”,
$n=160$, 2sls estimation results for model without continent fixed effects.
Heteroskedasticity robust test statistics and (standard errors). $Z_{1}$ is
instrument “Plough-pos. environment”, $Z_{2}$ is “Plough-neg. environment”,
$Z_{s|r}$ is just identifying instrument $Z_{s}$ when $Z_{r}$ is added as a
control.

Table 1 also provides the estimates of $\tilde{\psi}_{j}$ and
$\tilde{\pi}_{j}$, for $j=1,\ldots,4$. It show that
$\widehat{\tilde{\pi}}_{2}\left(=\widehat{\pi}_{2}\right)$ and
$\widehat{\tilde{\pi}}_{4}\left(=\widehat{\pi}_{2}^{*}\right)$ have opposite
signs. Therefore, with
$\widehat{\beta}_{2sls}=\sum_{\ell=1}^{2}w_{\ell}\widehat{\beta}_{\ell}=\sum_{\ell=1}^{2}w_{\ell}\widehat{\beta}_{\ell}^{*}$,
as shown in Sections 3.3 and 4.2, it follows that the weight $w_{2}$ is
negative. The weights are here given by $w_{1}=1.042$ and
$w_{2}=1-w_{1}=-0.042$. Note that using any pair of the four just-identifying
instruments results in the same 2sls estimator, which is then of course also a
linear combination of the associated pair of the just-identified estimators,
with the weights adding up to $1$. For example, when using $Z_{1|2}$ and
$Z_{1}$ as the instruments, we have here that
$\widehat{\beta}_{2sls}=0.12\widehat{\beta}_{1}+0.88\widehat{\beta}_{1}^{*}$.
For all the six pairs, we find that111A proof for the equivalence of the
weights for $\widehat{\beta}_{1},\widehat{\beta}_{1}^{*}$ and
$\widehat{\beta}_{2},\widehat{\beta}_{2}^{*}$ is provided in the Appendix.

$\displaystyle\widehat{\beta}_{2sls}$
$\displaystyle=1.042\widehat{\beta}_{1}-0.042\widehat{\beta}_{2}=1.042\widehat{\beta}_{1}^{*}-0.042\widehat{\beta}_{2}^{*}$
$\displaystyle=0.12\widehat{\beta}_{1}+0.88\widehat{\beta}_{1}^{*}=0.12\widehat{\beta}_{2}+0.88\widehat{\beta}_{2}^{*}$
$\displaystyle=0.78\widehat{\beta}_{1}+0.22\widehat{\beta}_{2}^{*}$
$\displaystyle=0.006\widehat{\beta}_{2}+0.994\widehat{\beta}_{1}^{*}.$

Use of any of any of these pairs of instruments also results in the same value
for the $J$-statistic, which is equivalent to a minimum-distance criterion,
see Windmeijer (2019). The small p-value therefore indicates that the
estimands, for example $\beta_{1}^{*}$ and $\beta_{2}^{*}$, are different from
each other.

Although $Z_{2|1}$ is found to be an irrelevant instrument, this is not the
case for $Z_{2}$ when used as the just-identifying instrument by itself. This
is due to the strong negative correlation between the two instruments, with
the (partial) sample correlation between them given by
$\widehat{\rho}_{12}=-0.54$. Therefore if $Z_{2}$ is a valid instrument, with
$\gamma_{2}=\alpha_{2}=0$, it can identify $\beta$ if $Z_{1}$ is invalid due
to a violation of the exogeneity assumption, $\gamma_{1}=0$, $\alpha_{1}\neq
0$, even when $\pi_{2}=0$, as long as $\rho_{12}\neq 0$. But clearly, when
$\pi_{2}=0$, $Z_{2}$ cannot identify $\beta$ if $Z_{1}$ instead violates the
exclusion assumption.

###  6.2 An Empirical Example with Three Instruments

One of the main examples in MP is the empirical analysis of roads and trade by
Duranton et al. (2014). The outcome variable is a measure of how much a city
exports. The one considered in MP is called the “propensity to export weight”.
The treatment variable is the log number of kilometers of interstate highway
within a city in 2007. Duranton et al. (2014) estimate the causal effect of
within city highways on the propensity to export weight using instrumental
variables. There are three potential instruments: $Z_{1}=$ Plan is the log
number of kilometers of highway in the city according to a planned highway
construction map, approved by the federal government in 1947; $Z_{2}=$
Railroads is the log number of kilometers of railroads in the city in 1898;
and $Z_{3}=$ Exploration is a measure of the quantity of historical
exploration routes that passed through the city. For a fuller description see
Duranton et al. (2014) and MP.

We will focus here on the model specification and estimation results as
displayed in column 2 of Table 5 in Duranton et al. (2014) and in column 2 of
Table I in MP. This model specification includes the additional control
variables “log employment” and “Market access (export)”. Estimation results
for the two additional control variables have been omitted and notation-wise
these variables have been partialled out. The first column in Table 2
replicates the 2sls estimation results using all 3 instruments. This
specification is falsified by the $J$-statistic, which has a p-value of 0.043.
The next columns give the estimation results for all twelve just-identified
model specifications. The instruments $Z_{2|1,3}$ and $Z_{2|1}$ are found to
be not relevant, and the resulting estimates of the falsification adaptive
sets are given by $\widehat{FAS}{}_{excl}=\left[-0.32,0.28\right]$ as in MP,
$\widehat{FAS}{}_{exo}=\left[0.13,1.09\right]$ and
$\widehat{FAS}=\left[-0.61,1.18\right]$.

The $\widehat{FAS}$ has quite a wide range of values here, as it correctly
takes into account possible violations of exogeneity and exclusion
restrictions. If for example
$\boldsymbol{\gamma}=\left(0,0,\gamma_{3}\right)^{T}$ and
$\boldsymbol{\alpha}=\left(\alpha_{1},0,0\right)^{T}$ with $\gamma_{3}\neq 0$
and $\alpha_{1}\neq 0$, then $Z_{2|3}$ is a valid instrument. It is found to
be a relevant instrument, and the associated IV estimate is given by $1.18$,
which is the largest coefficient estimate of all just-identified
specifications. $Z_{2|1,3}$ and $Z_{2|1}$ are found to be not relevant, with
their F-statistics less than $10$. In contrast, $Z_{2}$ as well as $Z_{2|3}$
are found to be relevant. As in the empirical example of Section 6.1, this can
be explained by the strong correlation of the instruments. The sample partial
correlation coefficients are here given by $\widehat{\rho}_{12}=0.57$,
$\widehat{\rho}_{13}=0.34$ and $\widehat{\rho}_{23}=0.11$.

The 2sls estimator is again the same when using any combination of three
instruments from the twelve transformed just-identifying ones, as long as all
indices $\left\\{1,2,3\right\\}$ are involved, as in for example
$\left\\{Z_{1},Z_{1|2},Z_{1|3}\right\\}$. The 2sls weights for
$\left\\{Z_{1},Z_{2},Z_{3}\right\\}$ and
$\left\\{Z_{1|2,3},Z_{2|1,3},Z_{3|1,2}\right\\}$ are here given by
$w=\left\\{0.757,0.126,0.117\right\\}$.

Table 2:  IV estimation results, Duranton et al. (2014, Table 5, column 2)

| Instruments |  |
---|---|---|---
| $Z_{1},Z_{2},Z_{3}$ | $Z_{1|2,3}$ | $Z_{2|1,3}$ | $Z_{3|1,2}$ | $Z_{1}$ | $Z_{2}$ | $Z_{3}$
log highway km | $\begin{array}[]{c}0.57\\\ \left(0.16\right)\end{array}$ | $\begin{array}[]{c}0.28\\\ \left(0.25\right)\end{array}$ | $\begin{array}[]{c}3.16\\\ \left(1.39\right)\end{array}$ | $\begin{array}[]{c}-0.32\\\ \left(0.86\right)\end{array}$ | $\begin{array}[]{c}0.55\\\ \left(0.17\right)\end{array}$ | $\begin{array}[]{c}1.09\\\ \left(0.26\right)\end{array}$ | $\begin{array}[]{c}0.13\\\ \left(0.38\right)\end{array}$
F | $90.30$ | $58.13$ | $6.97$ | $20.00$ | $154.5$ | $35.84$ | $15.97$
$J$-test p-value | $0.043$ |  |  |  |  |  |
|  | $Z_{1|2}$ | $Z_{1|3}$ | $Z_{2|1}$ | $Z_{2|3}$ | $Z_{3|1}$ | $Z_{3|2}$
log highway km |  | $\begin{array}[]{c}0.22\\\ \left(0.21\right)\end{array}$ | $\begin{array}[]{c}0.40\\\ \left(0.16\right)\end{array}$ | $\begin{array}[]{c}3.74\\\ \left(1.90\right)\end{array}$ | $\begin{array}[]{c}1.18\\\ \left(0.26\right)\end{array}$ | $\begin{array}[]{c}-0.61\\\ \left(1.11\right)\end{array}$ | $\begin{array}[]{c}-0.02\\\ \left(0.38\right)\end{array}$
F |  | $81.14$ | $122.45$ | $5.29$ | $34.31$ | $14.27$ | $31.07$
$\widehat{FAS}{}_{excl}$ | $\left[-0.32,0.28\right]$ |  |  |  |  |
$\widehat{FAS}{}_{exo}$ | $\left[0.13,1.09\right]$ |  |  |  |  |
$\widehat{FAS}$ | $\left[-0.61,1.18\right]$ |  |  |  |  |

Notes: Outcome variable “propensity to export weight”, $n=66$. Additional
controls “log employment” and “Market access (export)”. Heteroskedasticity
robust test statistics and (standard errors). $Z_{1}$ is instrument “Plan”,
$Z_{2}$ is “Railroads”, $Z_{3}$ is “Exploration”.

####  6.2.1 Instrument Selection

When a model is falsified, the $\widehat{FAS}$ gives the range of estimates of
all just-identified model specifications where the just-identifying
instruments are found to be relevant. If a relevant instrument is valid, then
the population FAS will contain $\beta$ under our stated assumptions. As such,
estimation of the FAS is not a selection tool. In the exclusion assumption
violation only setting, as $Z_{2|1,3}$ is found to be an irrelevant
instrument, MP however argued to automatically include $Z_{2}=$ Railroads, as
a control. If $Z_{1}$ and $Z_{3}$ are valid instruments and $Z_{2}$ violates
the exclusion assumption, then the resulting 2sls estimator would of course be
a consistent estimator of $\beta$, and the null hypothesis for the $J$-test
would be true. If $Z_{2}$ violates the exogeneity assumption instead, it
should clearly be omitted from the instrument set.. However, instrument
selection like that requires the assumption in this case that two instruments
are valid. Then the valid instruments can be found by finding the
specification for which the $J$-test does not reject. The MP approach for hard
thresholding with exclusion violation only is then the same as the approach of
Guo et al. (2018) and Windmeijer et al. (2021).

Table 3:  Results for Estimation with Two Instruments | Instruments
---|---
| $Z_{1},Z_{2}$ | $Z_{1|3},Z_{2|3}$ | $Z_{1},Z_{3}$ | $Z_{1|2},Z_{3|2}$ | $Z_{2},Z_{3}$ | $Z_{2|1},Z_{3|1}$
log highway km | $\begin{array}[]{c}0.61\\\ \left(0.17\right)\end{array}$ | $\begin{array}[]{c}0.73\\\ \left(0.20\right)\end{array}$ | $\begin{array}[]{c}0.51\\\ \left(0.16\right)\end{array}$ | $\begin{array}[]{c}0.17\\\ \left(0.19\right)\end{array}$ | $\begin{array}[]{c}0.74\\\ \left(0.19\right)\end{array}$ | $\begin{array}[]{c}0.84\\\ \left(0.64\right)\end{array}$
F | $91.64$ | $81.13$ | $102.27$ | $65.34$ | $38.14$ | $11.87$
$J$-test p-value | $0.014$ | $0.020$ | $0.25$ | $0.51$ | $0.039$ | $0.019$
Notes: See notes to Table 2.

When considering the model specifications with two instruments only it seems
natural to consider all possible combinations allowing for the violation of
the exclusion or exogeneity assumption. The estimation results are presented
in Table 3. Although $Z_{2|1,3}$ and $Z_{2|1}$ are not relevant instruments,
we include them here as part of the pairs $\left\\{Z_{1},Z_{2}\right\\}$,
$\left\\{Z_{1|3},Z_{2|3}\right\\}$ and $\left\\{Z_{2|1},Z_{3|1}\right\\}$ as
there is enough information in the data to reject these specifications. We
find here that the $J$-statistics for the pairs $\left\\{Z_{1},Z_{3}\right\\}$
and $\left\\{Z_{1|2},Z_{3|2}\right\\}$ do not falsify the model
specifications, with the p-values respectively given by $0.25$ and $0.51$.
Hence there is no evidence against including Railroads as a control or
dropping it from the instrument set. The estimation results are quite
different in the two specifications, with an estimate of $0.51$ (se $0.16$)
and $F=102.3$ when $Z_{2}$ is dropped, and $0.17$ (se $0.19$) and $F=65.34$
when $Z_{2}$ is included as a control. Selecting the model with the larger
value of the $F$-statistic would then favor the model with $Z_{2}$ excluded.
When adopting the selection strategy of choosing the model with the smallest
value of the $J$-statistic, as in Andrews (1999), then the model with $Z_{2}$
included as a control would be selected, but there is no reason to specify
this model as the automatic default for a falsified model with irrelevant
instruments. And note that it could still be the case that $Z_{2}$ or
$Z_{2|3}$ is the only valid and relevant instrument, as the assumption that in
this case the majority of instruments is valid is not verifiable and a non-
rejection of the $J$-test does not imply that instruments are valid.

To illustrate this further, Tables 4 and 5 present the same estimation results
for the specification in Duranton et al. (2014) as presented in their Table 5,
column 4, and in MP Table I, column 4 and Table II. Although this
specification is not falsified by the $J$-test, MP do provide the estimate of
FASexcl. The results for the different falsification adaptive sets are here
given by $\widehat{FAS}{}_{excl}=\left[0.18,0.42\right]$,
$\widehat{FAS}{}_{exo}=\left[0.34,0.64\right]$ and
$\widehat{FAS}=\left[0.18,0.67\right]$. Railroads is again found to be an
irrelevant instrument in the sense that the F-statistic for $Z_{2|1,3}$ is
given by $1.27$ and for $Z_{2|1}$ is given by $0.16$. But $Z_{2}$ by itself
and $Z_{2|3}$ are found to be relevant instruments due to the correlation
structure. The sample partial correlation coefficients are given by
$\widehat{\rho}_{12}=0.63$, $\widehat{\rho}_{13}=0.38$ and
$\widehat{\rho}_{23}=0.12$.

Table 4:  IV estimation results, Duranton et al. (2014, Table 5, column 4)

| Instruments |  |
---|---|---|---
| $Z_{1},Z_{2},Z_{3}$ | $Z_{1|2,3}$ | $Z_{2|1,3}$ | $Z_{3|1,2}$ | $Z_{1}$ | $Z_{2}$ | $Z_{3}$
log highway km | $\begin{array}[]{c}0.39\\\ \left(0.12\right)\end{array}$ | $\begin{array}[]{c}0.18\\\ \left(0.21\right)\end{array}$ | $\begin{array}[]{c}3.66\\\ \left(4.16\right)\end{array}$ | $\begin{array}[]{c}0.42\\\ \left(0.52\right)\end{array}$ | $\begin{array}[]{c}0.38\\\ \left(0.13\right)\end{array}$ | $\begin{array}[]{c}0.64\\\ \left(0.22\right)\end{array}$ | $\begin{array}[]{c}0.34\\\ \left(0.19\right)\end{array}$
F | $84.71$ | $54.75$ | $1.27$ | $26.95$ | $129.7$ | $40.75$ | $23.79$
$J$-test p-value | $0.30$ |  |  |  |  |  |
|  | $Z_{1|2}$ | $Z_{1|3}$ | $Z_{2|1}$ | $Z_{2|3}$ | $Z_{3|1}$ | $Z_{3|2}$
log highway km |  | $\begin{array}[]{c}0.21\\\ \left(0.15\right)\end{array}$ | $\begin{array}[]{c}0.40\\\ \left(0.16\right)\end{array}$ | $\begin{array}[]{c}8.78\\\ \left(22.3\right)\end{array}$ | $\begin{array}[]{c}0.67\\\ \left(0.24\right)\end{array}$ | $\begin{array}[]{c}0.26\\\ \left(0.54\right)\end{array}$ | $\begin{array}[]{c}0.34\\\ \left(0.19\right)\end{array}$
F |  | $81.92$ | $98.57$ | $0.16$ | $44.07$ | $24.66$ | $39.41$
$\widehat{FAS}{}_{excl}$ | $\left[0.18,0.42\right]$ |  |  |  |  |
$\widehat{FAS}{}_{exo}$ | $\left[0.34,0.64\right]$ |  |  |  |  |
$\widehat{FAS}$ | $\left[0.18,0.67\right]$ |  |  |  |  |

Notes: Outcome variable “propensity to export weight”, $n=66$. Additional
controls “log employment”, “Market access (export)”, “log 1920 population”,
“log 1950 pop.”, “log 2000 pop.” and “log % manuf. emp.”. Heteroskedasticity
robust test statistics and (standard errors). $Z_{1}$ is instrument “Plan”,
$Z_{2}$ is “Railroads”, $Z_{3}$ is “Exploration”.

MP then again include Railroads by default as a control variable and presents
the 2sls estimation results for that particular specification. In Table 5 we
compare model specifications with $2$ instruments, but ignore here any
combination which would have an irrelevant instrument and therefore consider
only the pairs $\left\\{Z_{1},Z_{3}\right\\}$,
$\left\\{Z_{1|2},Z_{3|2}\right\\}$ and $\left\\{Z_{2},Z_{3}\right\\}$. As the
original specification was not falsified we don’t expect these specifications
to be falsified, as confirmed by the p-values of the $J$-statistics, which are
respectively $0.83$, $0.72$ and $0.25$. The model with largest value of the
$F$-statistic and smallest value of the $J$-statistic in this case is the
model with $Z_{2}$ excluded from the instrument set. The resulting 2sls
estimate of $0.38$ (se $0.12$) is very close to the one based on all three
instruments, $0.39$ (se $0.12$). The 2sls weights for the three instruments
$\left\\{Z_{1},Z_{2},Z_{3}\right\\}$ are given by
$w=\left\\{0.81,0.05,0.14\right\\}$, those for the two instruments
$\left\\{Z_{1},Z_{3}\right\\}$ are given by $w=\left\\{0.98,0.02\right\\}$.

Table 5:  Results for Estimation with Two Instruments | Instruments |
---|---|---
| $Z_{1},Z_{3}$ | $Z_{1|2},Z_{3|2}$ | $Z_{2},Z_{3}$
log highway km | $\begin{array}[]{c}0.38\\\ \left(0.12\right)\end{array}$ | $\begin{array}[]{c}0.23\\\ \left(0.14\right)\end{array}$ | $\begin{array}[]{c}0.51\\\ \left(0.16\right)\end{array}$
F | $112.28$ | $82.15$ | $46.65$
$J$-test p-value | $0.83$ | $0.72$ | $0.25$
Notes: See notes to Table 4.

##  7 Conclusions

We have generalized the falsification adaptive set of Masten and Poirier
(2021) for the classical linear model with an endogenous variable, estimated
by the method of instrumental variables with multiple instruments. It is the
set of all just-identifying estimands where the just-identifying instrument is
relevant. It reflects the model uncertainty when the baseline model is
falsified, taking into account possible violations of both the exogeneity and
exclusion assumptions and where an invalid instrument can violate either the
exclusion or exogeneity assumption. It contains $\beta$ if there is at least
one valid and relevant instrument, and we recommend researchers to report
estimates of this set when their baseline model is falsified.

## Appendix

We show here that for the $k_{z}=2$ case we have that
$\widehat{\beta}_{2sls}=\tilde{\boldsymbol{w}}^{T}\left(\widehat{\beta}_{1},\widehat{\beta}_{1}^{*}\right)^{T}=\tilde{\boldsymbol{w}}^{T}\left(\widehat{\beta}_{2},\widehat{\beta}_{2}^{*}\right)^{T}$,
as observed in Section 6.1.

For general instruments
$\boldsymbol{Z}=\left[\boldsymbol{z}_{1}\,\,\boldsymbol{z}_{2}\right]$ the
2sls weights on the instrument specific estimators $\widehat{\beta}_{1}^{*}$
and $\widehat{\beta}_{2}^{*}$ was found in Section 4.2 to be given by

$w_{\ell}=\widehat{\pi}_{\ell}\boldsymbol{z}_{\ell}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1},$

for $\ell=1,2$, with $w_{1}+w_{2}=1$, and where $\widehat{\pi}_{\ell}$ is the
$\ell$-th element of
$\widehat{\boldsymbol{\pi}}=\left(\boldsymbol{Z}^{T}\boldsymbol{Z}\right)^{-1}\boldsymbol{Z}^{T}\boldsymbol{x}$,

$\widehat{\boldsymbol{\pi}}=\left(\begin{array}[]{c}\widehat{\pi}_{1}\\\
\widehat{\pi}_{2}\end{array}\right)=\left(\begin{array}[]{c}\left(\boldsymbol{z}_{1|2}^{T}\boldsymbol{z}_{1|2}\right)^{-1}\boldsymbol{z}_{1|2}^{T}\boldsymbol{x}\\\
\left(\boldsymbol{z}_{2|1}^{T}\boldsymbol{z}_{2|1}\right)^{-1}\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}\end{array}\right),$

where

$\boldsymbol{z}_{1|2}=\boldsymbol{M}_{z_{2}}\boldsymbol{z}_{1}=\boldsymbol{z}_{1}-\boldsymbol{z}_{2}\left(\boldsymbol{z}_{2}^{T}\boldsymbol{z}_{2}\right)^{-1}\boldsymbol{z}_{2}^{T}\boldsymbol{z}_{1}=\boldsymbol{z}_{1}-\boldsymbol{z}_{2}\widehat{\phi}_{21}$
$\boldsymbol{z}_{2|1}=\boldsymbol{M}_{z_{1}}\boldsymbol{z}_{2}=\boldsymbol{z}_{2}-\boldsymbol{z}_{1}\left(\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{1}\right)^{-1}\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{2}=\boldsymbol{z}_{2}-\boldsymbol{z}_{1}\widehat{\phi}_{12},$

with
$\widehat{\phi}_{21}=\left(\boldsymbol{z}_{2}^{T}\boldsymbol{z}_{2}\right)^{-1}\boldsymbol{z}_{2}^{T}\boldsymbol{z}_{1}$
and
$\widehat{\phi}_{12}=\left(\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{1}\right)^{-1}\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{2}.$

Consider the instruments
$\tilde{\boldsymbol{Z}}_{1}=\left[\boldsymbol{z}_{1|2}\,\,\boldsymbol{z}_{1}\right]$,
with instruments specific estimators respectively $\widehat{\beta}_{1}$ and
$\widehat{\beta}_{1}^{*}$. Denote the associated 2sls weights
$\tilde{\boldsymbol{w}}_{1}$, with first element given by

$\tilde{w}_{1,1}=\tilde{\pi}_{1,1}\boldsymbol{z}_{1|2}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{\tilde{Z}_{1}}\boldsymbol{x}\right)^{-1}=\tilde{\pi}_{1,1}\boldsymbol{z}_{1|2}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1}$

with the second equality following from the fact that $\boldsymbol{z}_{1|2}$
is a linear combination of $\boldsymbol{z}_{1}$ and $\boldsymbol{z}_{2}$.
$\tilde{\pi}_{1,1}$ is the first element of
$\tilde{\boldsymbol{\pi}}_{1}=\left(\tilde{\boldsymbol{Z}}_{1}^{T}\tilde{\boldsymbol{Z}}_{1}\right)^{-1}\tilde{\boldsymbol{Z}}_{1}^{T}\boldsymbol{x}$.

We have that

$\tilde{\boldsymbol{z}}_{1|2}=\boldsymbol{M}_{z_{1}}\boldsymbol{M}_{z_{2}}\boldsymbol{z}_{1}=-\boldsymbol{z}_{2|1}\widehat{\phi}_{21}$

and so

$\tilde{\pi}_{1,1}=-\frac{\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}}{\boldsymbol{z}_{2|1}^{T}\boldsymbol{z}_{2|1}\widehat{\phi}_{21}}=-\frac{\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}}{\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{2}\left(1-\widehat{\phi}_{12}\widehat{\phi}_{21}\right)},$

as

$\boldsymbol{z}_{2|1}^{T}\boldsymbol{z}_{2|1}\widehat{\phi}_{21}=\left(\boldsymbol{z}_{2}^{T}\boldsymbol{z}_{2}-\boldsymbol{z}_{2}^{T}\boldsymbol{z}_{1}\widehat{\phi}_{12}\right)\widehat{\phi}_{21}=\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{2}\left(1-\widehat{\phi}_{12}\widehat{\phi}_{21}\right),$

and so

$\tilde{\pi}_{1,1}\boldsymbol{z}_{1|2}^{T}\boldsymbol{x}=-\frac{\boldsymbol{x}^{T}\boldsymbol{z}_{2|1}\boldsymbol{z}_{1|2}^{T}\boldsymbol{x}}{\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{2}\left(1-\widehat{\phi}_{12}\widehat{\phi}_{21}\right)}.$

Next consider the instruments
$\tilde{\boldsymbol{Z}}_{2}=\left[\boldsymbol{z}_{2|1}\,\,\boldsymbol{z}_{2}\right]$,
with instruments specific estimators respectively $\widehat{\beta}_{2}$ and
$\widehat{\beta}_{2}^{*}$. Denote the associated 2sls weights
$\tilde{\boldsymbol{w}}_{2}$, with first element given by

$\tilde{w}_{2,1}=\tilde{\pi}_{2,1}\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{\tilde{Z}_{2}}\boldsymbol{x}\right)^{-1}=\tilde{\pi}_{2,1}\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}\left(\boldsymbol{x}^{T}\boldsymbol{P}_{Z}\boldsymbol{x}\right)^{-1},$

with $\tilde{\pi}_{2,1}$ the first element of
$\tilde{\boldsymbol{\pi}}_{2}=\left(\tilde{\boldsymbol{Z}}_{2}^{T}\tilde{\boldsymbol{Z}}_{2}\right)^{-1}\tilde{\boldsymbol{Z}}_{2}^{T}\boldsymbol{x}$.
By the derivations as above we get,

$\tilde{\pi}_{2,1}\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}=-\frac{\boldsymbol{x}^{T}\boldsymbol{z}_{1|2}\boldsymbol{z}_{2|1}^{T}\boldsymbol{x}}{\boldsymbol{z}_{1}^{T}\boldsymbol{z}_{2}\left(1-\widehat{\phi}_{12}\widehat{\phi}_{21}\right)},$

and so $\tilde{w}_{1,1}=\tilde{w}_{2,1}$. As
$\tilde{w}_{1,2}=1-\tilde{w}_{1,1}$ and $\tilde{w}_{2,2}=1-\tilde{w}_{2,1}$,
it follows that
$\tilde{\boldsymbol{w}}_{1}=\tilde{\boldsymbol{w}}_{2}=\tilde{\boldsymbol{w}}$.

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