# Resistant Inference in Instrumental Variable Models111We are grateful to
Andreas Alfons, Isaiah Andrews, Chen Zhou, participants at the Nuffield
Econometrics Lunch Seminar, European Summer Meeting of the Econometric Society
(2023), International Association of Applied Econometrics conference (2023),
European Winter Meeting of the Econometric Society (2022) and the Netherlands
Econometric Study Group (2022) for helpful comments and suggestions.

Jens Klooster222Corresponding author. Department of Econometrics, Econometric
Institute, Erasmus University Rotterdam, The Netherlands. E-mail:
<EMAIL_ADDRESS>Mikhail Zhelonkin333Department of Econometrics,
Econometric Institute, Erasmus University Rotterdam, The Netherlands. E-mail:
<EMAIL_ADDRESS>
(March 25, 2024)

###### Abstract

The classical tests in the instrumental variable model can behave arbitrarily
if the data is contaminated. For instance, one outlying observation can be
enough to change the outcome of a test. We develop a framework to construct
testing procedures that are robust to weak instruments, outliers and heavy-
tailed errors in the instrumental variable model. The framework is constructed
upon M-estimators. By deriving the influence functions of the classical weak
instrument robust tests, such as the Anderson-Rubin test, K-test and the
conditional likelihood ratio (CLR) test, we prove their unbounded sensitivity
to infinitesimal contamination. Therefore, we construct contamination
resistant/robust alternatives. In particular, we show how to construct a
robust CLR statistic based on Mallows type M-estimators and show that its
asymptotic distribution is the same as that of the (classical) CLR statistic.
The theoretical results are corroborated by a simulation study. Finally, we
revisit three empirical studies affected by outliers and demonstrate how the
new robust tests can be used in practice.

Keywords:  Influence function; Local misspecification; Robust inference;
Outlier; Robust test; Weak instrument.

##  1 Introduction

The linear instrumental variable (IV) model is recognized as an important tool
that can be used to draw causal inferences in nonexperimental data (Angrist et
al., 1996). Its applicability spans over different fields and allows, for
example, to infer the causal effects of education on labor market earnings
(Angrist and Krueger, 1991), chemotherapy for advanced lung cancer in the
elderly (Earle et al., 2001) and foreign media on authoritarian regimes (Kern
and Hainmueller, 2009). In each of these examples, an endogeneity problem
occurs where the explanatory variable is correlated with the error term,
causing the Least Squares (LS) estimator to be biased. The researchers
introduce instrumental variables to resolve this problem. The instrumental
variables should be correlated with the (endogenous) explanatory variable and
uncorrelated with the error term. When instrumental variables are available,
then reliable estimation (and inference) is possible using IV estimators, such
as the Two-Stage Least Squares (2SLS) estimator.

In practice, it is difficult to find instruments that satisfy both conditions.
In particular, the instruments that researchers propose are oftentimes only
weakly correlated with the endogenous explanatory variable (Andrews et al.,
2019). When the instruments are weak, then classical IV estimators, such as
the 2SLS estimator, are biased and $t$-tests based on IV estimators can fail
to control the size of the tests and their associated confidence intervals are
invalid (Nelson and Startz, 1990; Bound et al., 1995). Therefore, it is common
to draw inference in the IV model using a two-step procedure. In the first
step, the strength of the instruments is tested by means of an $F$-test. When
the first-stage $F$ statistic is above a certain threshold (motivated by
Staiger and Stock 1997 a cutoff value of 10 is common), the instruments are
considered to be strong. When the instruments are strong, then estimation is
done with a 2SLS estimator and inference with a $t$-test. When the instruments
are weak, then weak instrument robust tests are used (Anderson and Rubin,
1949; Kleibergen, 2002; Moreira, 2003).

The implementation of the two-step procedure described above by practitioners
has typically been imperfect and it is advised to directly rely on weak
instrument robust tests, or, smoothly adjust 2SLS $t$-test inference based on
the first-stage $F$ (Andrews et al., 2019; Lee et al., 2022; Keane and Neal,
2023). In particular, in the just-identified model, i.e., with one endogenous
variable and one instrumental variable, the Anderson and Rubin (1949) (AR)
test is the uniformly most powerful among the class of unbiased tests
(Moreira, 2009). When there is one endogenous variable and multiple
instrumental variables, then the conditional likelihood ratio (CLR) test
introduced by Moreira (2003) enjoys good power properties in the linear
homoskedastic IV model (Andrews et al., 2006). As the CLR test is equal to the
AR test in the just-identified model it is generally advised to use the CLR
test.

Recently, Young (2022) pointed out that many empirical IV studies in economics
are affected by a few outliers or by small clusters of deviating observations
(see also Lal et al. 2023 in political science). In the two-step procedure,
the $F$-test is not robust against outliers (Ronchetti, 1982). Hence, even one
outlier is enough to inflate the first-stage $F$ leading to a false impression
that the instrument is strong, while it is weak (Klooster and Zhelonkin,
2024), which eventually results in incorrect inference in the second stage. On
the other hand, an outlier could also deflate the $F$-statistic, i.e., the
instrument is strong, but due to an outlier it seems weak. In that case, the
researcher might unnecessarily dismiss the instrument.

A possible solution could be to clean the data using outlier detection tools
and then apply the (classical) procedure. However, this data-analytic strategy
has been criticized in the literature, e.g., Welsh and Ronchetti (2002),
Maronna et al. (2019, Section 4.3), Heritier et al. (2009, Section 1.3). If
the data is cleaned, then one has to correct the inference for the data
cleaning step, otherwise the classical asymptotic theory becomes invalid (Chen
and Bien, 2020). A typical consequence is underestimation of the (asymptotic)
variance which leads to overrejection of the null hypothesis, see Klooster and
Zhelonkin (2024) for an example in the IV model. Moreover, in modern
applications involving complex structural models outlier detection and its
justification are themselves not trivial. Hence, a more viable strategy is to
directly use (outlier) robust methods for estimation and inference. In the
context of IV, several studies showed that classical IV estimators, such as
the 2SLS and Limited Information Maximum Likelihood (LIML) estimators, are not
robust to outliers and, consequently, introduced robust IV estimators (Cohen
Freue et al., 2013; Jiao, 2022; Sølvsten, 2020; Zhelonkin et al., 2012).
Klooster and Zhelonkin (2024) developed a robust version of the AR test.
However, a general treatment of the problem of (outlier) robust inference in
combination with weak instruments is missing.

In this article, we formally show that weak instrument robust tests such as
the AR, K (Kleibergen, 2002) and CLR tests are not robust to outliers. For
this reason, we develop a general framework that can be used to construct
robust versions of the AR, K and CLR tests. In particular, we show how to
construct an outlier robust CLR test that allows for reliable inference
(without having to rely on a first-stage pre-test) when the data might contain
some outliers. The robust CLR test provides reliable inference irrespective of
the strength of the instrumental variables and only loses a little bit of
power compared to the (classical) CLR test when the data does not contain any
outliers. Moreover, if the errors follow a heavy-tailed distribution, then our
tests are more powerful than the classical ones.

We formalize the outlier contamination using the Huber (1964) gross-error
model $F_{t}=(1-t)F+tG$, where $t$ is a (typically small) contamination
proportion. We are interested in drawing inference for the central model $F$,
but we assume that it only holds approximately and the data that we observe
comes from $F_{t}$. The model $G$ is assumed to be completely unknown and it
is the source of the contamination. The goal is to draw inference that is
valid at the central model $F$, but also remains stable and reliable when the
data is generated according to $F_{t}$. The (classical) AR, K and CLR tests
are valid when all the data is generated according to $F$, but, as we show
both analytically and numerically, when the data that we observe comes from
$F_{t}$, then they can be distorted. This setup is related to the literature
on sensitivity analysis and local misspecification, see Andrews et al. (2017,
2020), Athey and Imbens (2015), Bonhomme and Weidner (2022), Kitamura et al.
(2013) and references therein. The model $F_{t}$ can be viewed as locally
misspecified. If one assumes that the model $F$ holds exactly, then a
sensitivity analysis should complement the reported results. In Section 5, we
revisit studies by Alesina and Zhuravskaya (2011), Ananat (2011) and Angrist
and Krueger (1991) and show how our tests can be used to study the sensitivity
of the results to data contamination. Our approach allows to benefit from the
parametric structure, e.g., computational simplicity and interpretability,
while being resistant to small but harmful deviations from the assumed model
$F$.

When the assumed model $F$ does not even hold approximately, then the use of
nonparametric weak instrument robust inference (Andrews and Soares, 2007;
Andrews and Marmer, 2008) is advised. Note, however, that nonparametric
procedures are typically not designed to be robust to outliers. For example,
the sample mean is a nonparametric estimator of the expectation, but it is not
robust as one outlier can make it arbitrarily biased (see Huber and Ronchetti
2009, p. 6 for further discussion). Similarly, weak instrument robust quantile
methods introduced by Chernozhukov and Hansen (2008) and Jun (2008) are also
not designed to be robust against outliers.

The setup of the article is as follows. In Section 2, we introduce the model
and our notation. In Section 3, we introduce the general (robust) framework
that allows construction of weak instrument testing procedures that are robust
to outliers. In particular, in Section 3.6 we show how to construct a robust
version of the CLR test that can be readily used in practice. Then, in Section
4, we study the small sample properties of the robust CLR test and compare its
performance to the classical CLR test. In Section 5, we revisit three
empirical studies and show how the robust CLR test can be used in practice.
Finally, in Section 6 we conclude.

##  2 Instrumental Variables Model

We assume that the data is generated according to a linear instrumental
variable regression model. The model consists of a structural equation (1) and
a first-stage equation (2):

$\displaystyle y$ $\displaystyle=\beta x+w^{\top}\gamma_{1}+u,$ (1)
$\displaystyle x$ $\displaystyle=w^{\top}\gamma_{2}+z^{\top}\pi+v,$ (2)

where $y$ and $x$ are endogenous random variables, $z$ is a $k\times 1$ random
vector of instrumental variables and $w$ is a $p\times 1$ random vector of
control variables. We assume that $\gamma_{1},\gamma_{2}\in\mathbb{R}^{p}$ are
parameter vectors that both, if necessary, include an intercept,
$\pi\in\mathbb{R}^{k}$ and $\beta\in\mathbb{R}$. We assume that the errors
$(u,v)$ are mean zero, with covariance matrix
$\Sigma=\begin{pmatrix}\sigma_{u}^{2}&\sigma_{u,v}\\\
\sigma_{v,u}&\sigma_{v}^{2}\end{pmatrix}$. We assume that the instruments are
uncorrelated with the error terms, conditional on the control variables $w$.

We are interested in testing the hypothesis

$\displaystyle H_{0}\colon\beta=\beta_{0}\text{ against
}H_{1}\colon\beta\neq\beta_{0}$ (3)

in the linear IV model (1) - (2). We do so by constructing weak instrument
robust tests based on estimators in the reduced form model. The reduced form
equation can be obtained by substituting (2) into (1):

$\displaystyle y$ $\displaystyle=w^{\top}\gamma+z^{\top}\delta+\epsilon,$ (4)

with $\gamma=(\gamma_{1}+\gamma_{2}\beta)$, $\delta=\pi\beta$ and
$\epsilon=v\beta+u$. We refer to (2) - (4) as the reduced form model.

The tests we introduce in Section 3 are constructed upon estimators of the
parameters $\delta$ and $\pi$ in the reduced form model (2) - (4). To simplify
the presentation of our results, we assume that $\gamma_{1}=\gamma_{2}=0$ so
that $w$ drops out of the model. The results we present extend to the more
general case where $\gamma_{1}\neq 0$ and $\gamma_{2}\neq 0$. In particular,
when the estimators are regression equivariant (see Rousseeuw and Leroy 1987,
p. 116), then assuming $\gamma_{1}=\gamma_{2}=0$ is without loss of
generality. Thus, we focus on the simplified reduced form model

$\displaystyle y$ $\displaystyle=z^{\top}\delta+\epsilon,$ (5) $\displaystyle
x$ $\displaystyle=z^{\top}\pi+v.$ (6)

Define $\theta=(\delta^{\top},\pi^{\top})^{\top}$, then we assume that the
model (5) - (6) is governed by $F_{\theta}$. In practice, we assume that we
observe i.i.d. data $d_{i}=(y_{i},x_{i},z_{i}^{\top})^{\top}$, which are
random samples from $d=(y,x,z^{\top})^{\top}\sim F_{\theta}$. We define
$Y,X\in\mathbb{R}^{n}$, which are vectors with entries $y_{i}$ and $x_{i}$
respectively, and $Z\in\mathbb{R}^{n\times k}$, the matrix with rows
$z_{i}^{\top}$. Denote the partition of a vector $a\in\mathbb{R}^{2k}$ into
$k$ and $k$ components by $a^{\top}=(a_{(1)}^{\top},a_{(2)}^{\top})$ and
denote the corresponding partition of $2k\times 2k$ matrices by $A_{(ij)}$,
$i,j=1,2$.

##  3 Robust Inference

In this section, we introduce a general (robust) framework to construct weak
instrument robust testing procedures based on M-estimators (Huber, 1964). We
show that classical weak instrument robust tests, such as the AR, K and CLR
tests can be obtained by specifying the M-estimators to be the LS estimators.
To evaluate whether a statistic is robust, we use the influence function
(Hampel, 1974; Hampel et al., 1986), which we discuss in the next section.

###  3.1 Influence Function

To formally determine whether a statistic is robust, we analyze its influence
function. Let $T$ denote a (Fisher consistent) statistical functional, then
the influence function is defined as

$\displaystyle\operatorname*{IF}(d;T,F)=\lim_{t\downarrow
0}\frac{T\left\\{(1-t)F+t\Delta_{d}\right\\}-T(F)}{t},$ (7)

where $\Delta_{d}$ is a point mass at $d$. The heuristic interpretation of the
influence function is that it describes the effect of an infinitesimal
contamination at the point $d$ on the estimate, standardized by the mass of
the contamination (Hampel et al., 1986). If the statistical functional $T$ is
sufficiently regular, then a von Mises expansion (von Mises, 1947; Hampel,
1974) yields

$\displaystyle T(G)\approx T(F)+\int\operatorname*{IF}(d;T,F)d(G-F).$ (8)

When we consider the approximation (8) over a neighborhood of the model
$\mathcal{F}_{t}=\\{F_{t}=(1-t)F+tG\ |\ G\text{ an arbitrary
distribution}\\}$, we see that the influence function can be used to linearize
the asymptotic bias in a neighborhood of the ideal model $F$. In particular,
when we consider the maximum asymptotic bias, we obtain the following
relationship

$\displaystyle\sup_{G}||T(F_{t})-T(F)||\approx
t\sup_{d}||\operatorname*{IF}(d;T,F)||.$ (9)

Hence, when the influence function of a statistical functional $T$ is not
bounded, then the maximum bias in a neighborhood of $F$ can be infinite, even
when the contamination proportion $t$ is small. Therefore, for a statistical
functional to be (locally) robust, a bounded influence function is required.

###  3.2 M-estimators

We construct test statistics based on the M-estimator
$T(F_{n})=\left\\{\delta(F_{n})^{\top},\pi(F_{n})^{\top}\right\\}^{\top}$ of
$\theta$ defined by

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Psi\left\\{d_{i},T(F_{n})\right\\}=\frac{1}{n}\sum_{i=1}^{n}\begin{bmatrix}\psi\left\\{(y_{i},z_{i}),\delta(F_{n})\right\\}\\\
\psi\left\\{(x_{i},z_{i}),\pi(F_{n})\right\\}\end{bmatrix}=0,$ (10)

where $\psi$ is a (general) score function and $F_{n}$ denotes the empirical
distribution function. Note, letting $\psi\\{(a,b),c\\}=(a-b^{\top}c)b$,
results in the classical LS estimating equations, which are worked out in
detail in Example 1. Under general regularity conditions stated in Assumption
1 in the Appendix, if $(y_{i},x_{i},z_{i}^{\top})^{\top}$ are i.i.d. draws
from the distribution $F_{\theta}$, then M-estimators are consistent and
asymptotically normally distributed (Clarke, 1983, 1986), i.e., as
$n\rightarrow\infty$, then

$\displaystyle\sqrt{n}\left\\{T(F_{n})-\theta\right\\}=\sqrt{n}\left\\{\begin{matrix}\delta(F_{n})-\delta\\\
\pi(F_{n})-\pi\end{matrix}\right\\}\overset{d}{\to}\mathcal{N}\left[\begin{pmatrix}0\\\
0\end{pmatrix},\left\\{\begin{matrix}\Sigma_{\delta\delta}(F_{\theta})&\Sigma_{\delta\pi}(F_{\theta})\\\
\Sigma_{\pi\delta}(F_{\theta})&\Sigma_{\pi\pi}(F_{\theta})\end{matrix}\right\\}\right],$

where

$\displaystyle\Sigma_{\delta\delta}(F_{\theta})$
$\displaystyle=\int\operatorname*{IF}\left\\{d;\delta(\cdot),F_{\theta}\right\\}\operatorname*{IF}\left\\{d;\delta(\cdot),F_{\theta}\right\\}^{\top}dF_{\theta},$
$\displaystyle\Sigma_{\pi\pi}(F_{\theta})$
$\displaystyle=\int\operatorname*{IF}\left\\{d;\pi(\cdot),F_{\theta}\right\\}\operatorname*{IF}\left\\{d;\pi(\cdot),F_{\theta}\right\\}^{\top}dF_{\theta},$
$\displaystyle\Sigma_{\delta\pi}(F_{\theta})$
$\displaystyle=\int\operatorname*{IF}\left\\{d;\delta(\cdot),F_{\theta}\right\\}\operatorname*{IF}\left\\{d;\pi(\cdot),F_{\theta}\right\\}^{\top}dF_{\theta},$
$\displaystyle\Sigma_{\pi\delta}(F_{\theta})$
$\displaystyle=\int\operatorname*{IF}\left\\{d;\pi(\cdot),F_{\theta}\right\\}\operatorname*{IF}\left\\{d;\delta(\cdot),F_{\theta}\right\\}^{\top}dF_{\theta}.$

The influence functions of these estimators can be calculated using the
definition (7) (see Hampel et al. 1986, p. 101), and are given by

$\displaystyle\operatorname*{IF}\left\\{d_{i};\delta(\cdot),F_{\theta}\right\\}$
$\displaystyle=\operatorname*{IF}\left\\{d_{i};T(\cdot),F_{\theta}\right\\}_{(1)}=\left\\{M(F_{\theta})^{-1}\right\\}_{(11)}\psi\left\\{(y_{i},z_{i}),\delta\right\\},$
(11)
$\displaystyle\operatorname*{IF}\left\\{d_{i};\pi(\cdot),F_{\theta}\right\\}$
$\displaystyle=\operatorname*{IF}\left\\{d_{i};T(\cdot),F_{\theta}\right\\}_{(2)}=\left\\{M(F_{\theta})^{-1}\right\\}_{(22)}\psi\left\\{(x_{i},z_{i}),\pi\right\\},$
(12)

with
$M(F_{\theta})=-\int\frac{\partial\Psi}{\partial\theta}(d,\theta)dF_{\theta}$.
When we analyze the influence functions (11) - (12) we see that they are only
bounded when the function $\psi$ is bounded. Thus, the estimators
$\delta(F_{n})$ and $\pi(F_{n})$ are (locally) robust when $\psi$ is bounded.

As the LS estimators play an important role for the construction of the AR, K
and CLR statistics, we explicitly show how can they be considered as a special
case of the M-estimators in Example 1.

######  Example 1 (Least Squares).

When we use the following score function:

$\displaystyle\psi\colon\left(\mathbb{R}\times\mathbb{R}^{k}\right)\times\mathbb{R}^{k}\mapsto\mathbb{R}^{k}\colon\psi\\{(a,b),c\\}=(a-b^{\top}c)b,$

then $\delta(F_{n})=(Z^{\top}Z)^{-1}Z^{\top}Y$ and
$\pi(F_{n})=(Z^{\top}Z)^{-1}Z^{\top}X$ are the LS estimators that solve
estimating equations

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\begin{bmatrix}\left\\{y_{i}-z_{i}^{\top}\delta(F_{n})\right\\}z_{i}\\\
\left\\{x_{i}-z_{i}^{\top}\pi(F_{n})\right\\}z_{i}\end{bmatrix}=0.$

Their influence functions are

$\displaystyle\operatorname*{IF}\left\\{d_{i};\delta(\cdot),F_{\theta}\right\\}$
$\displaystyle=\left(\int
zz^{\top}dF_{\theta}\right)^{-1}\left(y_{i}-z_{i}^{\top}\delta\right)z_{i},$
(13)
$\displaystyle\operatorname*{IF}\left\\{d_{i};\pi(\cdot),F_{\theta}\right\\}$
$\displaystyle=\left(\int
zz^{\top}dF_{\theta}\right)^{-1}\left(x_{i}-z_{i}^{\top}\pi\right)z_{i}.$ (14)

The influence functions of the LS estimators are not bounded, so that one
outlying observation $d_{i}$ can arbitrarily bias the estimators
$\delta(F_{n})$ and $\pi(F_{n})$. Using the influence function, we can now
calculate the (co)variance matrices. For instance, assuming homoskedasticity,
we have

$\displaystyle\Sigma_{\delta\delta}(F_{\theta})$
$\displaystyle=\int\operatorname*{IF}\left\\{d;\delta(\cdot),F_{\theta}\right\\}\operatorname*{IF}\left\\{d;\delta(\cdot),F_{\theta}\right\\}^{\top}dF_{\theta}$
$\displaystyle=\int\left(\int
zz^{\top}dF_{\theta}\right)^{-1}\left(y-z^{\top}\delta\right)zz^{\top}\left(y-z^{\top}\delta\right)^{\top}\left(\int
zz^{\top}dF_{\theta}\right)^{-\top}dF_{\theta}$ $\displaystyle=\left(\int
zz^{\top}dF_{\theta}\right)^{-1}\int\left(y-z^{\top}\delta\right)^{2}dF_{\theta},$

which can be estimated by

${\Sigma}_{\delta\delta}(F_{n})=\left(\int
zz^{\top}dF_{n}\right)^{-1}\int\left\\{y-z^{\top}\delta(F_{n})\right\\}^{2}dF_{n}=\left(\frac{1}{n}Z^{\top}Z\right)^{-1}\sigma_{\epsilon}^{2}(F_{n}),$

where
$\sigma_{\epsilon}^{2}(F_{n})=\frac{1}{n}\sum_{i=1}^{n}\left\\{y_{i}-z_{i}^{\top}\delta(F_{n})\right\\}^{2}$.

###  3.3 The (Robust) Conditional Likelihood Ratio Test

Moreira (2003) shows that in the case of a scalar $\beta$, then the classical
CLR statistic can be decomposed into the classical AR, K, and a Wald
statistic. Therefore, to introduce a general (robust) version of the CLR
statistic, we start by introducing general (robust) versions of the AR, K and
Wald statistics. Then, we plug in the general (robust) AR, K and Wald
statistics into the decomposition to introduce a general (robust) CLR
statistic.

Let $\left\\{\delta(F_{n})^{\top},\pi(F_{n})^{\top}\right\\}^{\top}$ denote
M-estimators of the parameters $(\delta^{\top},\pi^{\top})^{\top}$ as defined
in (10). Then, given a hypothesized $\beta_{0}$ value, we define (see also
Andrews et al. 2019),

$\displaystyle g(F_{n},\beta_{0})$
$\displaystyle=\delta(F_{n})-\pi(F_{n})\beta_{0},$ $\displaystyle
D(F_{n},\beta_{0})$
$\displaystyle=\pi(F_{n})-\left\\{\Sigma_{\pi\delta}(F_{\theta})-\Sigma_{\pi\pi}(F_{\theta})\beta_{0}\right\\}\Omega(F_{\theta},\beta_{0})^{-1}g(F_{n},\beta_{0}),$

where
$\Omega(F_{\theta},\beta_{0})=\Sigma_{\delta\delta}(F_{\theta})-\beta_{0}\left\\{\Sigma_{\delta\pi}(F_{\theta})+\Sigma_{\pi\delta}(F_{\theta})\right\\}+\beta_{0}^{2}\Sigma_{\pi\pi}(F_{\theta})$.
The general (robust) AR, K and Wald statistics are defined as:

$\displaystyle RAR(F_{n},\beta_{0})$
$\displaystyle=g(F_{n},\beta_{0})^{\top}\Omega(F_{\theta},\beta_{0})^{-1}g(F_{n},\beta_{0}),$
(15) $\displaystyle RK(F_{n},\beta_{0})$
$\displaystyle=g(F_{n},\beta_{0})^{\top}D(F_{n},\beta_{0})\left\\{D(F_{n},\beta_{0})^{\top}\Omega(F_{\theta},\beta_{0})D(F_{n},\beta_{0})\right\\}^{-1}D(F_{n},\beta_{0})^{\top}g(F_{n},\beta_{0}),$
(16) $\displaystyle RW(F_{n},\beta_{0})$
$\displaystyle=D(F_{n},\beta_{0})^{\top}\Lambda(F_{\theta},\beta_{0})^{-1}D(F_{n},\beta_{0}),$
(17)

where
$\Lambda(F_{\theta},\beta_{0})=\Sigma_{\pi\pi}(F_{\theta})-\left\\{\Sigma_{\pi\delta}(F_{\theta})-\Sigma_{\pi\pi}(F_{\theta})\beta_{0}\right\\}\Omega(F_{\theta},\beta_{0})^{-1}\left\\{\Sigma_{\delta\pi}(F_{\theta})-\Sigma_{\pi\pi}(F_{\theta})\beta_{0}\right\\}.$
The general (robust) CLR statistic can then be written as:

$\displaystyle RCLR(F_{n},\beta_{0})=$
$\displaystyle\frac{1}{2}\bigg{[}RAR(F_{n},\beta_{0})-RW(F_{n},\beta_{0})\ +$
(18) $\displaystyle\
\sqrt{\left\\{RAR(F_{n},\beta_{0})-RW(F_{n},\beta_{0})\right\\}^{2}+4RW(F_{n},\beta_{0})\cdot
RK(F_{n},\beta_{0})}\bigg{]}.$

In Example 2, we show how plugging in the LS estimators from Example 1 results
in the classical AR, K, Wald and CLR statistics.

######  Example 2 (Least Squares (continued)).

When $\delta(F_{n})$ and $\pi(F_{n})$ are the LS estimators, and using the
estimators $\Omega(F_{n},\beta_{0})$ and $\Lambda(F_{n},\beta_{0})$, then we
have

$\displaystyle g(F_{n},\beta_{0})$
$\displaystyle=(Z^{\top}Z)^{-1}Z^{\top}(Y-\beta_{0}X),$ $\displaystyle
D(F_{n},\beta_{0})$
$\displaystyle=(Z^{\top}Z)^{-1}Z^{\top}\left\\{X-\frac{\sigma_{u,v}(F_{n})}{\sigma_{u}^{2}(F_{n})}\left(Y-X\beta_{0}\right)\right\\}.$

Plugging these into the statistics (15), (16) and (17) yields the classical
AR, K and Wald statistics (up to a factor $n$):

$\displaystyle nAR(F_{n},\beta_{0})$
$\displaystyle=\frac{(Y-\beta_{0}X)^{\top}Z(Z^{\top}Z)^{-1}Z^{\top}(Y-\beta_{0}X)}{\sigma_{u}^{2}(F_{n})},$
$\displaystyle nK(F_{n},\beta_{0})$
$\displaystyle=\frac{1}{\sigma_{u}^{2}(F_{n})}(Y-X\beta_{0})^{\top}P_{ZD(F_{n},\beta_{0})}(Y-X\beta_{0}),$
$\displaystyle nW(F_{n},\beta_{0})$
$\displaystyle=\left\\{X-\frac{\sigma_{u,v}(F_{n})}{\sigma_{u}^{2}(F_{n})}\left(Y-X\beta_{0}\right)\right\\}^{\top}\left[\left\\{\sigma_{v}^{2}(F_{n})-\frac{\sigma_{u,v}(F_{n})}{\sigma_{u}^{2}(F_{n})}\right\\}Z^{\top}Z\right]^{-1}\times$
$\displaystyle\qquad\left\\{X-\frac{\sigma_{u,v}(F_{n})}{\sigma_{u}^{2}(F_{n})}\left(Y-X\beta_{0}\right)\right\\},$

with
$P_{ZD(F_{n},\beta_{0})}=ZD(F_{n},\beta_{0})\left\\{D(F_{n},\beta_{0})^{\top}Z^{\top}ZD(F_{n},\beta_{0})\right\\}^{-1}D(F_{n},\beta_{0})^{\top}Z^{\top}$
and
$\sigma_{u}^{2}(F_{n})=\sigma_{\epsilon}^{2}(F_{n})-2\beta_{0}\sigma_{\epsilon,v}(F_{n})+\beta_{0}^{2}\sigma_{v}^{2}(F_{n})$.
Plugging these statistics into the decomposition (18) yields the CLR statistic
introduced by Moreira (2003).

Remark We are not the first to use the decomposition (18) to construct a
generalization of the CLR statistic. For example, Kleibergen (2005) uses the
decomposition to introduce a generalized method of moments (GMM) version of
the CLR statistic based on GMM versions of the AR (Stock and Wright, 2000) and
K statistic. Magnusson (2010) uses the decomposition to introduce a CLR test
for inference in weakly identified limited dependent variable models. Their
goal is to provide tests that can be used for inference in weakly identified
models beyond the linear IV model. In this article, we restrict our focus to
the linear IV model and leave connections to more general versions open for
future research. Note, Ronchetti and Trojani (2001) study the robustness
properties of GMM estimators, but they do not consider weak identification.

###  3.4 Robustness Properties

To determine the robustness properties of the general (robust) CLR statistic
we derive its influence function. It turns out that directly applying the
influence function defined in (7) to the statistical functional
$RCLR(\cdot,\beta_{0})$ is not meaningful as the influence function is always
zero. This happens, because the statistical functional $RCLR(\cdot,\beta_{0})$
is not Fisher consistent. A functional $T$ is Fisher consistent when
$T(F_{\theta})=\theta$ for all $\theta$, while $RCLR(F_{\theta},\beta)=0$.
Therefore, to still obtain a meaningful result, similar as in Hampel et al.
(1986, p. 348), we derive the influence function of the square root of the
statistic. Note, generalizations of the influence function towards functionals
that are not Fisher consistent have been introduced as well (Rousseeuw and
Ronchetti, 1981).

In Proposition 1, we derive the influence function of the general (robust) CLR
statistic, conditional on $D(F_{n},\beta_{0})=\tilde{D}$, denoted by
$RCLR(\cdot,\beta_{0}|\tilde{D})$.

######  Proposition 1.

Under the null hypothesis $\beta=\beta_{0}$, and the regularity conditions in
Assumption 1, the influence function of the general (robust) CLR statistic,
conditional on $D(F_{n},\beta_{0})=\tilde{D}$, is

$\displaystyle\operatorname*{IF}\left\\{d_{i};\sqrt{RCLR(\cdot,\beta_{0}|\tilde{D})},F_{\theta}\right\\}=\begin{cases}\operatorname*{IF}\left\\{d_{i};\sqrt{RAR(\cdot,\beta_{0})},F_{\theta}\right\\},&\text{if
}\tilde{D}=0,\\\
\operatorname*{IF}\left\\{d_{i};\sqrt{RK(\cdot,\beta_{0}|\tilde{D})},F_{\theta}\right\\},&\text{if
}\tilde{D}\neq 0.\end{cases}$

###### Proof.

See Appendix. ∎

Proposition 1 shows that the influence function of the RCLR statistic,
conditional on $D(F_{n},\beta_{0})=\tilde{D}$, depends on the influence
functions of the RAR statistic and RK statistic. This corresponds to the
theory, as $D(F_{n},\beta_{0})=0$ suggests that true value of $\pi=0$ as
$D(F_{\theta},\beta)=\pi$. When $D(F_{n},\beta_{0})=0$, it suggests that the
instruments are completely irrelevant. In this case, the RCLR statistic is the
same as the RAR statistic and therefore has the same influence function as the
RAR statistic. When $\tilde{D}\neq 0$, it suggests that $\pi\neq 0$. Moreira
(2003) shows that when $\pi\neq 0$, then the CLR statistic converges towards
the $K$ statistic as the amount of data goes towards infinity. For this
reason, we see the same behavior in the IF of the RCLR statistic as it is
equal to the IF of the RK statistic when $\tilde{D}\neq 0$.

We continue with deriving the influence functions of the RAR, RW and RK
statistics.

######  Proposition 2.

Under the null hypothesis $\beta=\beta_{0}$, and the regularity conditions in
Assumption 1, the influence functions of the general (robust) AR, W and K
statistics are

$\displaystyle\operatorname*{IF}\\{d_{i};\sqrt{RAR(\cdot,\beta_{0})},F_{\theta}\\}$
$\displaystyle=\sqrt{\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\Omega(F_{\theta},\beta_{0})^{-1}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}},$
(19)
$\displaystyle\operatorname*{IF}\\{d_{i};\sqrt{RK(\cdot,\beta_{0})},F_{\theta}\\}$
$\displaystyle=\sqrt{\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\pi\left(\pi^{\top}\Omega(F_{\theta},\beta_{0})\pi\right)^{-1}\pi^{\top}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}},$
(20)
$\displaystyle\operatorname*{IF}\\{d_{i};\sqrt{RW(\cdot,\beta_{0})},F_{\theta}\\}$
$\displaystyle=\sqrt{\operatorname*{IF}\\{d_{i};D(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\Lambda(F_{\theta},\beta_{0})^{-1}\operatorname*{IF}\\{d_{i};D(\cdot,\beta_{0}),F_{\theta}\\}},$
(21)

where

$\displaystyle\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}$
$\displaystyle=\operatorname*{IF}\\{d_{i};\delta(\cdot),F_{\theta}\\}-\beta_{0}\operatorname*{IF}\\{d_{i};\pi(\cdot),F_{\theta}\\}$
$\displaystyle\operatorname*{IF}\\{d_{i};D(\cdot,\beta_{0}),F_{\theta}\\}$
$\displaystyle=\operatorname*{IF}\\{d_{i};\pi(\cdot),F_{\theta}\\}-\left\\{\Sigma_{\pi\delta}(F_{\theta})-\Sigma_{\pi\pi}(F_{\theta})\beta_{0}\right\\}\Omega(F_{\theta},\beta_{0})^{-1}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}.$

The influence function of the general (robust) K statistic, conditional on
$D(F_{n},\beta_{0})=\tilde{D}$, is:

$\displaystyle\operatorname*{IF}\\{d_{i};$
$\displaystyle\sqrt{RK(\cdot,\beta_{0}|\tilde{D})},F_{\theta}\\}=\sqrt{\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\tilde{D}\left\\{\tilde{D}^{\top}\Omega(F_{\theta},\beta_{0})\tilde{D}\right\\}^{-1}\tilde{D}^{\top}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}}.$

###### Proof.

See Appendix. ∎

Proposition 2 shows that the influence functions of the RAR and RK statistics
depend directly on the influence function of the functional
$g(\cdot,\beta_{0})$. The influence function of the functional
$g(\cdot,\beta_{0})$ depends on the influence functions of the functionals
$\delta(\cdot)$ and $\pi(\cdot)$.

####  3.4.1 Robustness Issues With the Conditional Likelihood Ratio Test

In this section, we study the robustness properties of the classical CLR
statistic by analyzing its influence function. Example 2 showed us how the
classical CLR statistic can be viewed as a special case where LS estimators
are used. Therefore, we can use Propositions 1 and 2 to analyze the influence
function of the classical CLR statistic. As in Example 2, we denote the
classical AR, K, Wald and CLR statistics as
$AR(F_{n},\beta_{0}),K(F_{n},\beta_{0}),W(F_{n},\beta_{0})$ and
$CLR(F_{n},\beta_{0})$, respectively.

The previous section showed that the influence function of the general
(robust) CLR statistic ultimately depends on the influence functions of the
estimators $\delta(F_{n})$ and $\pi(F_{n})$ it is constructed upon. The
classical CLR statistic is constructed upon LS estimators. The influence
functions of the LS estimators were derived in Example 1, see (13) and (14).
These influence functions are not bounded, so that one outlying observation
$d_{i}=(y_{i},x_{i},z_{i}^{\top})^{\top}$ can arbitrarly bias the estimators
$\delta(F_{n})$ and $\pi(F_{n})$. As the influence functions of
$\delta(\cdot)$ and $\pi(\cdot)$ are not bounded, the influence function of
$g(\cdot,\beta_{0})$ is not bounded, so that the influence functions
$\operatorname*{IF}\\{d_{i};\sqrt{AR(\cdot,\beta_{0})},F_{\theta}\\}$ and
$\operatorname*{IF}\\{d_{i};\sqrt{K(\cdot,\beta_{0}|\tilde{D})},F_{\theta}\\}$
are not bounded, and finally the influence function
$\operatorname*{IF}\\{d_{i};\sqrt{CLR(\cdot,\beta_{0}|\tilde{D})},F_{\theta}\\}$
is not bounded. Therefore, we can conclude that the classical CLR statistic is
not (locally) robust.

Intuitively, the classical CLR statistic inherits the robustness properties of
the estimators it is constructed upon. As the LS estimators are not robust,
the classical CLR statistic is also not robust. However, from Propositions 1
and 2 it follows that if the influence function of the estimators would be
bounded, then the influence function of the general (robust) CLR statistic
would be bounded. From Section 3.2, it follows that, when the function $\psi$
is bounded the influence function of the M-estimator is bounded. Therefore,
the general (robust) CLR statistic introduced in (18) is robust whenever the
M-estimator it is constructed upon has a bounded (general) score function
$\psi$. An example of an M-estimator with a bounded influence function is the
M-estimator of Mallows type. In Section 3.6, we show how these types of
estimators can be used to construct a robust CLR statistic that can be used in
practice.

###  3.5 Asymptotic Distributions

In this section, we derive the asymptotic distribution of the general (robust)
AR, K and CLR statistics defined in (15), (16) and (18), and show how to
construct tests based on these statistics. We start with a lemma that plays a
key role in deriving all of the asymptotic distributions of the statistics.

######  Lemma 1.

Under the null hypothesis $\beta=\beta_{0}$, and the regularity conditions of
Assumption 1, then as $n\rightarrow\infty$,

$\displaystyle\sqrt{n}\left\\{\begin{matrix}g(F_{n},\beta_{0})\\\
D(F_{n},\beta_{0})\end{matrix}\right\\}\overset{d}{\to}\mathcal{N}\left[\begin{pmatrix}0\\\
\pi\end{pmatrix},\left\\{\begin{matrix}\Omega(F_{\theta},\beta_{0})&0\\\
0&\Lambda(F_{\theta},\beta_{0})\end{matrix}\right\\}\right].$

###### Proof.

See Appendix. ∎

From Lemma 1, we see that under the null hypothesis $\beta=\beta_{0}$, as
$n\rightarrow\infty$,

$\displaystyle nRAR(F_{n},\beta_{0})\overset{d}{\to}\chi_{k}^{2}.$ (22)

Moreover, under the null hypothesis $\beta=\beta_{0}$ and $\pi=0$, as
$n\rightarrow\infty$,

$\displaystyle nRW(F_{n},\beta_{0})\overset{d}{\to}\chi_{k}^{2}.$

Based on (22), we can construct the general (robust) AR test as:

$\displaystyle\phi_{RAR}(\beta_{0})=1\\{nRAR(F_{n},\beta_{0})>\chi_{k,1-\alpha}^{2}\\},$

where $\chi_{k,1-\alpha}^{2}$ denotes the $1-\alpha$ quantile of a
$\chi_{k}^{2}$ distribution.

Next, in Proposition 3, we derive the asymptotic distribution of the RK
statistic.

######  Proposition 3.

Under the null hypothesis $\beta=\beta_{0}$, and the regularity conditions in
Assumption 1, then as $n\rightarrow\infty$,

$\displaystyle nRK(F_{n},\beta_{0})\overset{d}{\to}\chi_{1}^{2}.$ (23)

###### Proof.

See Appendix. ∎

Based on Proposition 3, we can introduce the general (robust) K test as:

$\displaystyle\phi_{RK}(\beta_{0})=1\\{nRK(F_{n},\beta_{0})>\chi_{1,1-\alpha}^{2}\\}.$

Finally, we derive the asymptotic distribution of the RCLR statistic.

######  Proposition 4.

Under the null hypothesis $\beta=\beta_{0}$, and the regularity conditions in
Assumption 1, it holds that, conditional on $D(F_{n},\beta_{0})=\tilde{D}$ and
with $\tilde{W}=\tilde{D}^{\top}\Lambda(F_{\theta},\beta_{0})^{-1}\tilde{D}$,
then as $n\rightarrow\infty$,

$\displaystyle
nRCLR(F_{n},\beta_{0})\overset{d}{\to}\frac{1}{2}\left\\{\chi_{k-1}^{2}+\chi_{1}^{2}-\tilde{W}+\sqrt{(\chi_{k-1}^{2}+\chi_{1}^{2}+\tilde{W})^{2}-4\tilde{W}\chi_{k-1}^{2}}\right\\},$
(24)

where $\chi_{k-1}^{2}$ and $\chi_{1}^{2}$ are independent chi-squared
distributed random variables with $k-1$ and $1$ degrees of freedom.

###### Proof.

See Appendix. ∎

Using Proposition 4 we can introduce the robust CLR test as follows:

$\displaystyle\phi_{RCLR}(\beta_{0})=1[nRCLR(F_{n},\beta_{0})>c_{\alpha}\\{\sqrt{n}D(F_{n},\beta_{0})\\}],$

where $c_{\alpha}\\{\sqrt{n}D(F_{n},\beta_{0})\\}$ denotes the conditional
$1-\alpha$ quantile of the asymptotic distribution given in (24). The critical
values and confidence sets can be computed using simulation and test inversion
(Moreira, 2003; Andrews et al., 2019). That is, we find all the values of
$\beta_{0}$ for which the data does not reject the null hypothesis. The
confidence set is then $\left\\{\beta_{0}\ |\ \beta_{0}\in\mathbb{R}\text{ and
}\phi_{RCLR}(\beta_{0})=0\right\\}$.

###  3.6 A Robust Conditional Likelihood Ratio Test of Mallows Type

In this section, we show how to construct a robust CLR statistic based on
Mallows type M-estimators. These estimators solve the following estimating
equations:

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\omega(z_{i})\rho\left\\{\frac{y_{i}-z_{i}^{\top}\delta(F_{n})}{\sigma_{\epsilon}(F_{n})}\right\\}z_{i}=0,$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\omega(z_{i})\rho\left\\{\frac{x_{i}-z_{i}^{\top}\pi(F_{n})}{\sigma_{v}(F_{n})}\right\\}z_{i}=0,$

where we use $\omega(z_{i})=\sqrt{1-z_{i}^{\top}(Z^{\top}Z)^{-1}z_{i}}$ as a
weight function, and $\rho\colon\mathbb{R}\mapsto\mathbb{R}$ is the Huber
function:

$\displaystyle\rho(x)=\begin{cases}x,&\text{ if }|x|\leq 1.345,\\\
\text{sgn}(x)\cdot 1.345,&\text{ if }|x|>1.345.\end{cases}$

The estimates $\sigma_{\epsilon}(F_{n})$ and $\sigma_{v}(F_{n})$ are robust
estimates of the scales. In our case, for the practical implementation in
Sections 4 and 5, we use the function rlm from the R package MASS (Venables
and Ripley, 2002). We use the default robust scale estimator provided, which
is based on a re-scaled median absolute deviation of the residuals. In
Sections 4 and 5, for simplicity, we do the estimation equation per equation
and not simulteneously.

Now we only need to obtain estimates for the (co)variances belonging to these
estimators. Using Equations (11) and (12), we derive the following influence
functions for the Mallows type M-estimators:

$\displaystyle\operatorname*{IF}\left\\{d_{i};\delta(\cdot),F_{\theta}\right\\}$
$\displaystyle=\left\\{\int\frac{w(z)}{\sigma_{\epsilon}}\rho^{\prime}\left(\frac{y-z^{\top}\delta}{\sigma_{\epsilon}}\right)zz^{\top}dF_{\theta}\right\\}^{-1}\omega(z_{i})\rho\left(\frac{y_{i}-z_{i}^{\top}\delta}{\sigma_{\epsilon}}\right)z_{i},$
$\displaystyle\operatorname*{IF}\left\\{d_{i};\pi(\cdot),F_{\theta}\right\\}$
$\displaystyle=\left\\{\int\frac{w(z)}{\sigma_{v}}\rho^{\prime}\left(\frac{x-z^{\top}\pi}{\sigma_{v}}\right)zz^{\top}dF_{\theta}\right\\}^{-1}\omega(z_{i})\rho\left(\frac{x_{i}-z_{i}^{\top}\pi}{\sigma_{v}}\right)z_{i},$

where

$\displaystyle\rho^{\prime}(x)=\begin{cases}1,&\text{ if }|x|\leq 1.345,\\\
0,&\text{ if }|x|>1.345.\end{cases}$

Using these influence functions, we can calculate the (co)variance matrices.
For instance,
$\Sigma_{\delta\delta}(F_{\theta})=\int\operatorname*{IF}\\{d;\delta(\cdot),F_{\theta}\\}\operatorname*{IF}\\{d;\delta(\cdot),F_{\theta}\\}^{\top}dF_{\theta}=M_{\delta\delta}(F_{\theta})^{-1}Q_{\delta\delta}(F_{\theta})M_{\delta\delta}(F_{\theta})^{-\top}$,
where

$\displaystyle M_{\delta\delta}(F_{\theta})$
$\displaystyle=\int\frac{w(z)}{\sigma_{\epsilon}}\rho^{\prime}\left(\frac{y-z^{\top}\delta}{\sigma_{\epsilon}}\right)zz^{\top}dF_{\theta},$
$\displaystyle Q_{\delta\delta}(F_{\theta})$
$\displaystyle=\int\omega(z)^{2}\rho\left(\frac{y-z^{\top}\delta}{\sigma_{\epsilon}}\right)^{2}zz^{\top}dF_{\theta}.$

Evaluating $M_{\delta\delta}(\cdot)$ and $Q_{\delta\delta}(\cdot)$ at the
empirical distribution $F_{n}$ yields empirical estimate
$\Sigma_{\delta\delta}(F_{n})$ of the variance matrix
$\Sigma_{\delta\delta}(F_{\theta})$.

With these estimates, we can construct the robust CLR test (and also the
robust AR and K tests). We study the performance of this test in different
contaminated scenarios in Section 4. Note, this is just one possible way to
construct a robust CLR test. Other promising estimators that can be used to
construct a robust CLR statistic, are MM-estimators (Yohai, 1987) and robust
SUR estimators based on MM-estimators (Saraceno et al., 2021).

##  4 Simulation Study

In this section, we numerically evaluate the robustness properties of the
robust CLR test introduced in Section 3.6 compared to the classical CLR test.
We consider four different scenarios: a baseline scenario without
contamination, two scenarios with contamination by a single outlier and
scenario with “distributional” contamination in the error terms. These
contaminated scenarios are generated by slightly altering the baseline
scenario. The exact details of these different contaminations are given below.
We start by introducing the scenario without contamination.

Let $\iota_{3}$ denote a 3-dimensional vector of ones. We generate data from
the following model:

$\displaystyle x$ $\displaystyle=w+\pi z^{\top}\iota_{3}+v,$ (25)
$\displaystyle y$ $\displaystyle=\beta x+2w+u.$ (26)

We generate three exogeneous instruments $z=(z_{1},z_{2},z_{3})^{\top}$ and
one exogeneous control variable $w$. We draw $z_{1},z_{2},z_{3}$ and $w$ from
independent standard normal distributions. The errors $u$ and $v$ are drawn
from a bivariate normal distribution with variances equal to $1$ and
correlation $\rho=0.5$. We consider two different cases for the parameter
$\pi\in\\{0.1,1\\}$. When $\pi=0.1$ we are in the weak instrument case and
when $\pi=1$ we are in the strong instrument case. The sample size is $n=250$
and we repeat the study $10000$ times. In the simulation, we test
$H_{0}\colon\beta=0$ at a 5% significance level.

###  4.1 Results

In Figure 1, we show the power curves of the RCLR and CLR tests in a scenario
without contamination. In this case, we see that both the RCLR and CLR tests
are size correct in both the weak and strong instrument cases. Furthermore, we
see that the CLR test is more powerful than the RCLR test in both the weak and
strong instrument cases. However, we see that the loss of power of the RCLR
test is not very large implying that the extra robustness comes at a low cost.

Figure 1: Power curves of robust CLR and CLR tests that test
$H_{0}\colon\beta=0$ for various values of $\beta$ with three instruments,
$\rho=0.5$ and no contamination.

In Figure 2, we show the power curves of the RCLR and CLR tests in a scenario
with contamination by a large outlier in the $y$ variable. The data is first
generated according to the baseline scenario and then an outlier is
introduced. The outlier is generated by changing the first observation to
$(y_{1},x_{1},z_{11},z_{12},z_{13},w_{1})=(20,x_{1},z_{11},z_{12},z_{13},w_{1})$,
which means we only change the data point $y_{1}$ to $20$. In this case, we
see that the RCLR test remains almost as powerful as in the case without
contamination in both the weak and strong instrument cases. This happens,
because the outlier is successfully downweighted by the robust M-estimators.
In contrast, the CLR test loses power compared to the uncontaminated case and
is now less powerful compared to the RCLR test. The loss of power happens,
because the outlier increases the variance estimates of the classical test.
This increased variance results in fewer rejections of the null, lowering the
power. Overall, the CLR test remains reliable as we can see that it is still
size correct. This happens because the (vertical) outlier does not bias the LS
estimators $\delta(F_{n})$ and $\pi(F_{n})$. We would see a similar pattern if
the outlier was in any other variable, for example the $x$ or $w$ variable,
where the CLR test loses power but remains reliable. Therefore, we can ask the
question whether it is necessary to use a robust CLR test. The simple answer
is yes, as we demonstrate in the following contamination scenario.

Figure 2: Power curves of robust CLR and CLR tests that test
$H_{0}\colon\beta=0$ for various values of $\beta$ with three instruments,
$\rho=0.5$ and a large outlier in the $y$ variable.

In Figure 3, we show the power curves of the RCLR and CLR tests in a scenario
with contamination by a large outlier in the $y$ and $z$ variable. This
outlier was generated by replacing the first observation with
$(y_{1},x_{1},z_{11},z_{12},z_{13},w_{1})=(20,x_{1},5,z_{12},z_{13},w_{1})$.
We see that the power curves of the robust CLR test remain size correct in
both weak and strong instrument cases and only lose a little bit of power
compared to the scenario without contamination. This shows that the robust CLR
test effectively downweights the outlier. However, this is not the case for
the CLR test. We see that in both the strong and weak instrument cases the CLR
test is not size correct anymore and the minimum is shifted to the left. To
understand how this happens, for simplicity, we consider the case with only
one instrument. In this case the CLR statistic behaves the same as the AR
statistic. The AR statistic is minimized when
$g(F_{n})=\delta(F_{n})-\pi(F_{n})\beta_{0}=0$. The outlier we constructed can
positively bias the estimator $\delta(F_{n})$ when a nonrobust estimator is
used. Hence, under the null, we obtain
$\delta(F_{n})-\pi(F_{n})\beta_{0}=\delta(F_{n})\approx\delta+b=\pi\beta+b$,
where $b>0$ denotes a positive bias term. Solving $\pi\beta+b=0$ for $\beta$
gives $\beta=-b/\pi$. This formula shows that if there is a positive bias $b$,
then the minimum will not be at $\beta=0$ implying there must an overrejection
at $\beta=0$. Furthermore, as in our case the bias is positive and $\pi>0$, we
obtain a negative value for $\beta$ so that the power curves shift to the
left. At last, when the instrument is weak and $\pi\approx 0$, then the
$\beta=-b/\pi$ term will be large showing that a (small) bias caused by
outliers can be more problematic when the instrument is weak compared to the
strong instrument case. This last finding also holds more generally for biases
that are not necessarily due to outliers (Small and Rosenbaum, 2008).

Figure 3: Power curves of robust CLR and CLR tests that test
$H_{0}\colon\beta=0$ for various values of $\beta$ with three instruments,
$\rho=0.5$ and a large outlier in the $y$ variable combined with an outlier in
the $z$ variable.

In Figure 4, we show the power curves of the RCLR and CLR tests in a scenario
with contamination in the error terms. In this case, the first $50$ error
terms $(u_{i},v_{i}),i=1,\dots,50$, are drawn from a mean zero bivariate
$t(3)$-distribution with correlation $\rho=0.5$ instead of a mean zero
bivariate normal distribution. In this case, we see that the RCLR test is
strictly more powerful than the CLR test in both the weak and strong
instrument settings. This happens, because the error terms drawn from the
$t(3)$-distribution sometimes introduce larger outlying values that increase
the variance estimates the CLR statistic relies on. This results in fewer
rejections of the null hypothesis when $\beta\neq\beta_{0}$ lowering the
power. The RCLR effectively downweights these outlying values, resulting in
better variance estimates, which results in a higher power.

Figure 4: Power curves of robust CLR and CLR tests that test
$H_{0}\colon\beta=0$ for various values of $\beta$ with three instruments,
$\rho=0.5$ and distributional contamination in the error term.

##  5 Empirical Examples

In this section, we show how the robust CLR test introduced in Section 3.6 can
be used in practice by revisiting three empirical studies. First, we consider
the data and several specifications in Alesina and Zhuravskaya (2011) who
examine the effect of segregation on the quality of government. Second, we
revisit the main specifications considered in Ananat (2011) where the effect
of (racial) segregation on urban poverty and inequality is studied. Finally,
we revisit the Staiger and Stock (1997) specifications for the Angrist and
Krueger (1991) data where the effect of education on labor market earnings is
studied.

###  5.1 Alesina and Zhuravskaya (2011)

Alesina and Zhuravskaya (2011) study the effect of segregation on the quality
of government in a cross section of countries. They find that ethnically and
linguistically segregated countries have a lower quality of government.
Furthermore, they find that there is no relationship between religious
segregation and governance. To address endogeneity concerns caused by mobility
and endogeneous internal borders, an instrument is constructed for
segregation. For more information, data and the construction of the instrument
we refer to Alesina and Zhuravskaya (2011).

In Section 5D Alesina and Zhuravskaya (2011) mention that they carefully
examined whether a handful of influential observations drive their results. By
exluding influential observations and recalculating their statistics they
conclude that this is not the case. However, Alesina and Zhuravskaya (2011)
mention that in the specifications of Panel D in Table 7 removing two
influential observations leads to the first-stage $F$-statistic dropping from
17.22 to 7.82 making inference based on 2SLS estimator unreliable (Staiger and
Stock, 1997). Alesina and Zhuravskaya (2011) solve this by also removing the
most influential observation in the first stage from the data so that the
instrument becomes “strong enough”. The practice of manually removing outliers
from the data and then relying on classical statistical methods is not
recommended for several reasons. The asymptotic distribution of the estimators
and test statistics are unknown as they become conditional on the outlier
detection procedure(s). The variability may be underestimated, which can lead
to an overrejection of the null hypothesis. Lastly, some outliers might be
difficult to detect due to masking effects. We refer to Section 4.3 in Maronna
et al. (2019) and Section 1.3 in Heritier et al. (2009) for a more detailed
discussion on these issues. Therefore, it is advisable to rely on robust
estimation and testing procedures from the start. It seems that in case of
Alesina and Zhuravskaya (2011) that the use of the 2SLS estimator might be
questionable due to the outliers and the weak instrument. Therefore, we
believe that it is desirable to re-evaluate the robustness of the results and
the stability of the conclusions. We apply our test to six specifications
(“Voice”, “Political stability”, “Government effectiveness”, “Regulatory
Quality”, “Rule of law” and “Control of corruption”) of Panel D in Table 7 of
Alesina and Zhuravskaya (2011). As there is only one instrument, we calculate
the $95\%$ confidence sets of the robust AR statistic and the (classical) AR
statistic. The results are reported in Table 1.

Table 1: Results using data from Alesina and Zhuravskaya (2011).
Specifications correspond to the specifications in Panel D of Table 7 in
Alesina and Zhuravskaya (2011). Confidence sets are given for the parameter
belonging to endogeneous regressor $(x)$ “Segregation” for six different
specifications $(y)$. The RAR confidence sets are calculated based on Mallows
Type estimators as in Section 3.6.

| Language
---|---
Specification |  I |  II |  III |  IV |  V |  VI
|  Voice |  Political stability |  Government effectiveness |  Regulatory quality | Rule of law |  Control of corruption
$95\%$ AR confidence set |  $[-6.04,-0.93]$ |  $[-5.73,-1.01]$ |  $[-3.70,0.79]$ |  $[-4.86,2.20]$ |  $[-3.62,0.52]$ |  $[-3.48,1.81]$
$95\%$ RAR confidence set |  $[-9.67,-0.65]$ |  $[-8.68,-0.63]$ |  $[-3.86,1.37]$ |  $[-2.40,5.58]$ |  $[-3.31,1.64]$ |  $[-4.16,2.21]$
All control variables |  Yes |  Yes |  Yes |  Yes |  Yes |  Yes
No. of observations |  92 |  92 |  92 |  92 |  92 |  92
First-stage $F$ |  $17.22$ |  $17.22$ |  $17.22$ |  $17.22$ |  $17.22$ |  $17.22$

From Table 1, we can see that in specifications IV and V that the confidence
set of the robust AR test is shifted compared to the AR confidence set. This
suggests that outliers did have an effect as we would expect the confidence
set of the robust AR test to be wider than the confidence set of the AR test
when there are no outliers, but not shifted. The shift of the confidence set
suggests that the LS estimators the AR test is constructed upon are biased due
to the outlier(s). Therefore, the confidence sets based on the robust AR are
more reliable. In specifications I, II, III and VI, we see that the robust AR
confidence sets are wider, but not (much) shifted, compared to the AR
confidence set. This suggests that outliers did not have a large effect.
Overall, the outliers do not seem to be very problematic as in all
specifications the final decision whether to reject or not reject the null
hypothesis $H_{0}\colon\beta=0$ remains the same.

###  5.2 Ananat (2011)

Ananat (2011) studies the effect of racial segregation on urban poverty and
inequality. To overcome endogeneity issues, a railroad division index is used
to instrument for racial segregation. Using this instrumental variable, Ananat
(2011) shows that segregation increases metropolitan rates of black poverty
and overall black-white income disparities, while decreasing rates of white
poverty and inequality within the white population. For more information, data
and the construction of the instrument we refer to Ananat (2011).

Klooster and Zhelonkin (2024) show that an outlier in the control variable
used in the main results of Ananat (2011) inflates the first-stage
$F$-statistic from 1.83 to 19.32. As the outlier was not taken into account in
the original study, it was assumed that the instrument was strong.
Consequently, estimation was done with a 2SLS estimator and inference with a
$t$-test. Due to the outlier (and the weak instrument) inference based on the
2SLS estimator might be unreliable. Therefore, to re-evaluate the robustness
of the results, we apply our test to the four main specifications (“Gini index
whites”, “Gini index blacks”, “Poverty rate whites”, “Poverty rate blacks”),
which can be found in columns (3) and (4) of Table 2 in Ananat (2011). As
there is only one instrument, we calculate the $95\%$ confidence sets of the
robust AR statistic and the (classical) AR statistic. The results are reported
in Table 2.

Table 2: Results using data from Ananat (2011). Specifications correspond to the specifications in columns (3) and (4) in Table 2 in Ananat (2011). Confidence sets are given for the parameter belonging to endogeneous regressor $(x)$ “Segregation” for four different specifications $(y)$. The RAR confidence sets are calculated based on Mallows Type estimators as in Section 3.6. Specification |  I |  II |  III |  IV
---|---|---|---|---
|  Gini index whites |  Gini index blacks |  Poverty rate whites |  Poverty rate blacks
95% AR confidence set |  $(-0.64,-0.18)$ |  $(0.22,2.15)$ |  $(-0.38,-0.09)$ |  $(0.00,0.48)$
95% RAR confidence set |  $(-\infty,-0.15)$ $\cup(2.60,\infty)$ |  $(-\infty,-6.21)$ $\cup(0.23,\infty)$ |  $(-\infty,-0.09)$ $\cup(1.45,\infty)$ |  $(-\infty,0.77)$ $\cup(2.30,\infty)$
First-stage $F$ |  $19.32$ |  $19.32$ |  $19.32$ |  $19.32$
No. of observations |  121 |  121 |  121 |  121

When we analyze Table 2, we find large differences between the robust and
classical confidence sets. When there are no outliers in the data, we would
expect that the robust confidence sets are only a bit wider than the classical
confidence sets. However, in this case, the classical confidence sets are
bounded convex sets, while the robust confidence sets are unbounded sets. The
shape of the robust confidence sets do correctly suggest that the instrument
is weak. In this example, the outlier does have a large effect and we
recommend using the robust confidence sets for reliable inference.

###  5.3 Angrist and Krueger (1991)

Angrist and Krueger (1991) study the effect of education on labor market
earnings. To adress endogeneity issues, quarter of birth instruments are
constructed for education. Using these instrumental variables they find a
positive relationship between years of education and labor market earnings. We
revisit four specifications presented in Table 2 of Staiger and Stock (1997)
based on the 1930 - 1939 cohort. For more information, data and the
construction of the instrument, we refer to Staiger and Stock (1997) and
Angrist and Krueger (1991).

Bound et al. (1995) showed that the relationship between the instruments and
the endogeneous regressor is quite weak in certain specifications of Angrist
and Krueger (1991). Furthermore, more recently, Sølvsten (2020) shows that the
LIML residuals of the structural equation of a certain specification in
Angrist and Krueger (1991) is distributed roughly like a normal distribution
at the center with outlying errors that closely follow a $t(3)$-distribution
(reminiscent of the “distributional” contamination scenario in Section 4). Due
to the weak instruments (and possible outliers), inference based on the 2SLS
estimator used by Angrist and Krueger (1991) might be unreliable. Moreover,
due to outliers, weak instrument robust tests might be corrupted and/or
inefficient. Therefore, to re-evaluate the robustness of the results we
replicate the results reported in Panel A of Table 2 in Staiger and Stock
(1997). For each specification, we give the $95\%$ confidence set of the CLR
and robust CLR, and the first-stage $F$-statistic. The results are reported in
Table 3.

When we compare the $95\%$ confidence sets of the CLR and RCLR statistics, we
note that the RCLR confidence sets are smaller than the CLR confidence sets in
every specification. This happens because the RCLR statistic effectively
downweights outlying values in the residuals. Similar as in the
“distributional” contamination scenario in Section 4 this results in better
variance estimates and hence tighter confidence sets.

Table 3: Results for Angrist and Krueger (1991) data. Specifications as in
Table 2 of Staiger and Stock (1997), except for specification III (see the
note below). Confidence sets are given for the parameter belonging to the
endogeneous regressor $(x)$ “Years of schooling” for four different
specifications that all use the same dependent variable $(y)$ “log weekly
wages”. The RCLR confidence sets are calculated based on Mallows type
estimators as in Section 3.6. QOB, YOB and SOB stand for quarter, year and
state of birth.

Specification |  I |  II |  III∗ |  IV
---|---|---|---|---
$95\%$ CLR confidence set |  $[0.042,0.136]$ |  $[0.026,0.116]$ | $[-0.064,0.278]$ | $[-0.068,0.265]$
$95\%$ RCLR confidence set | $[0.047,0.122]$ | $[0.032,0.100]$ | $[-0.038,0.189]$ | $[-0.047,0.179]$
First-stage F |  $30.53$ |  $4.74$ |  $2.43$ |  $1.87$
controls $(w)$ |  |  |  |
Base controls |  Yes |  Yes |  Yes |  Yes
SOB |  No |  No |  Yes |  Yes
Age, Age2 |  No |  No |  No |  Yes
Instruments $(z)$ |  |  |  |
QOB |  Yes |  Yes |  Yes |  Yes
QOB*YOB |  No |  Yes |  Yes |  Yes
QOB*SOB |  No |  No |  Yes |  Yes
No. of instruments |  3 |  30 |  180 |  178
Observations |  329,509 |  329,509 |  329,509 |  329,509

  * •

*This specification slightly different from specification III of Table 2 in Staiger and Stock (1997). Instead of using Age and Age2 as control variables, we use the SOB controls instead. This was done as we encountered some small numerical difficulties when replicating the original specification leading to unusual confidence sets for both the CLR and RCLR tests.

##  6 Conclusion

In this article, we proposed a general framework to construct weak instrument
robust testing procedures that are also robust to outliers in the linear
instrumental variable model. The framework is constructed upon M-estimators
and we showed that the classical weak instrument robust tests, such as the AR,
K and CLR tests, can be obtained by specifying the M-estimators to be the LS
estimators. We formally showed that influence functions of the classical test
statistics are not bounded and hence not (locally) robust. As the classical
test statistics are not robust, we showed how to construct a robust CLR test
based on a Mallows type M-estimator. By means of a simulation study, we
documented good performance of our robust CLR test in different contamination
scenarios. Finally, we illustrated how the robust CLR test can be used in
practice by revisiting three different empirical studies.

## Appendix

######  Assumption 1 (Adapted from Heritier and Ronchetti (1994)).

Let $F$ be any arbitrary distribution on $\mathbb{R}^{2(p+k)}$ and define

$\displaystyle K_{F}(\theta)$ $\displaystyle=\int\Psi(d,\theta)dF,$

with
$\Psi(d,\theta)=\begin{bmatrix}\psi\\{(y,z),\delta\\}^{\top}&\psi\\{(x,z),\pi\\}^{\top}\end{bmatrix}^{\top}$.

  1. 1.

$K_{F}(\theta)$ exists at least on a (nondegenerate) open set $\mathcal{O}$.

  2. 2.

There exists $\theta^{*}\in\mathcal{O}$ satisfying $K_{F}(\theta^{*})=0$.

  3. 3.

$\int\Psi(d,\theta)dF_{\theta}=0$ (Fisher consistency).

  4. 4.

$\Psi(d,\theta)$ is a $2(p+k)\times 1$ vector function that is continuous and
bounded on $\mathcal{D}\times\Theta$, where $\Theta$ is some nondegenerate
compact interval containing $\theta^{*}$.

  5. 5.

$\Psi(d,\theta)$ is locally Lipschitz in $\theta$ about $\theta^{*}$, i.e.,
there exists a constant $\alpha$ such that

$\displaystyle||\Psi(d,\theta)-\Psi(d,\theta^{*})||\leq\alpha||\theta-\theta^{*}||$

uniformly in $d\in\mathcal{D}$ and for all $\theta$ in a neighbourhood of
$\theta^{*}$.

  6. 6.

The generalized Jacobian $\partial K_{F}(\theta)$ is of maximal rank at
$\theta=\theta^{*}$.

  7. 7.

Given $\Delta>0$, there exists $\epsilon>0$ such that for all distributions in
a $\epsilon$ neighborhood of $F$,
$\sup_{\theta\in\Theta}||K_{G}(\theta)-K_{F}(\theta)||>\delta$ and $\partial
K_{G}(\theta)\subset\partial K_{F}(\theta)+\Delta B$, uniformly in
$\theta\in\Theta$, where $B$ is the unit ball of $2(p+k)\times 2(p+k)$
matrices.

  8. 8.

$K_{F}(\theta)$ has at least a continuous derivative
$(\partial/\partial\theta)K_{F}(\theta)$ at $\theta=\theta^{*}$.

  9. 9.

$(\partial\Psi/\partial\theta)(d,\theta)$ exists for $\theta=\theta^{*}$ and
almost everywhere in a neighborhood of $\theta^{*}$. Moreover, for all
$\theta$ in this neighborhood, there exists an integrable function $v(d)$ such
that $||(\partial\Psi/\partial\theta)(d,\theta)||\leq v(d)$ a.e.

### Proof of Lemma 1

###### Proof.

Assumptions 1.4 - 1.8 ensure Fréchet differentiability (Clarke, 1986), so that
we can write

$\displaystyle
T(F_{n})-\theta=\int\operatorname*{IF}\left\\{d,T(\cdot),F_{\theta}\right\\}dF_{\theta}+\mathsf{o}_{\mathsf{p}}(||F_{n}-F_{\theta}||_{\infty}).$

Therefore, we have

$\displaystyle\sqrt{n}\begin{Bmatrix}\delta(F_{n})-\delta\\\
\pi(F_{n})-\pi\end{Bmatrix}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\begin{bmatrix}\operatorname*{IF}\\{d_{i};\delta(\cdot),F_{\theta}\\}\\\
\operatorname*{IF}\\{d_{i};\pi(\cdot),F_{\theta}\\}\end{bmatrix}+\mathsf{o}_{\mathsf{p}}(\sqrt{n}||F_{n}-F_{\theta}||_{\infty}).$

We can use the fact that
$\sqrt{n}||F_{n}-F||_{\infty}=\mathsf{O}_{\mathsf{p}}(1)$ (Dvoretzky et al.,
1956), and apply the Lindeberg-Feller Theorem to obtain, as
$n\rightarrow\infty$,

$\displaystyle\sqrt{n}\begin{Bmatrix}\delta(F_{n})\\\
\pi(F_{n})\end{Bmatrix}\overset{d}{\to}\mathcal{N}\left[\begin{pmatrix}\delta\\\
\pi\end{pmatrix},\begin{Bmatrix}\Sigma_{\delta\delta}(F_{\theta})&\Sigma_{\delta\pi}(F_{\theta})\\\
\Sigma_{\pi\delta}(F_{\theta})&\Sigma_{\pi\pi}(F_{\theta})\end{Bmatrix}\right].$

Note,

$\displaystyle\begin{bmatrix}I_{k}&-\left\\{\Sigma_{\pi\delta}(F_{\theta})-\Sigma_{\pi\pi}(F_{\theta})\beta_{0}\right\\}\Omega(F_{\theta},\beta_{0})^{-1}\\\
-\beta_{0}I_{k}&\left\\{\Sigma_{\pi\delta}(F_{\theta})-\Sigma_{\pi\pi}(F_{\theta})\beta_{0}\right\\}\Omega(F_{\theta},\beta_{0})^{-1}\beta_{0}+I_{k}\end{bmatrix}^{\top}\begin{Bmatrix}\delta(F_{n})\\\
\pi(F_{n})\end{Bmatrix}=\begin{Bmatrix}g(F_{n})\\\ D(F_{n})\end{Bmatrix},$

we conclude that, as $n\rightarrow\infty$,

$\displaystyle\sqrt{n}\begin{Bmatrix}g(F_{n},\beta_{0})\\\
D(F_{n},\beta_{0})\end{Bmatrix}\overset{d}{\to}\mathcal{N}\left[\begin{pmatrix}0\\\
\pi\end{pmatrix},\begin{Bmatrix}\Omega(F_{\theta},\beta_{0})&0\\\
0&\Lambda(F_{\theta},\beta_{0})\end{Bmatrix}\right].$

∎

### Proof of Proposition 1

###### Proof.

For simplicity, in this proof we surpress the dependency on $\beta_{0}$ and
$\tilde{D}$ in all the test statistics. Let $G$ be a general distribution
function and define
$\tilde{W}=\tilde{D}^{\top}\Lambda(F_{\theta},\beta_{0})^{-1}\tilde{D}$. The
functional form of the RCLR statistic, conditional on $\tilde{D}$, is

$\displaystyle
RCLR(G)=\frac{1}{2}\left[RAR(G)-\tilde{W}+\sqrt{\left\\{RAR(G)-\tilde{W}\right\\}^{2}+4\tilde{W}\cdot
RK(G)}\right].$

To simplify the notation, we write

$\displaystyle A(G)=\left\\{RAR(G)-\tilde{W}\right\\}^{2}+4\tilde{W}\cdot
RK(G),$

so that

$\displaystyle
RCLR(G)=\frac{1}{2}\left\\{RAR(G)-\tilde{W}+\sqrt{A(G)}\right\\}.$

We start with the case $\tilde{D}\neq 0$. In this case it must hold that
$\tilde{W}>0$ and as $A(F_{\theta})=\tilde{W}^{2}$, we have
$RCLR(F_{\theta})=0$. To calculate the first derivative, note that

$\displaystyle\frac{\partial}{\partial
t}RCLR(F_{t})\Big{|}_{t=0}=\frac{1}{2}\left\\{\frac{\partial}{\partial
t}RAR(F_{t})\Big{|}_{t=0}+\frac{1}{2\sqrt{A(F_{\theta})}}\cdot\frac{\partial}{\partial
t}A(F_{t})\Big{|}_{t=0}\right\\}.$

We know that $\frac{\partial}{\partial t}RAR(F_{t})\big{|}_{t=0}$ = 0, and

$\displaystyle\frac{\partial}{\partial t}A(F_{t})\Big{|}_{t=0}$
$\displaystyle=2\left\\{RAR(F_{\theta})-\tilde{W}\right\\}\frac{\partial}{\partial
t}RAR(F_{t})\Big{|}_{t=0}+4\tilde{W}\frac{\partial}{\partial
t}RK(F_{t})\Big{|}_{t=0}$ $\displaystyle=0,$

as $\frac{\partial}{\partial t}RAR(F_{t})\big{|}_{t=0}=0$, and
$\frac{\partial}{\partial t}RK(F_{t})\big{|}_{t=0}=0$. Furthermore, as
$\tilde{W}>0$, we have

$\displaystyle
A(F_{\theta})=\\{RAR(F_{\theta})-\tilde{W}\\}^{2}+4\tilde{W}RK(F_{\theta})=(0-\tilde{W})^{2}+0=\tilde{W}^{2}.$

Hence, $A(F_{\theta})>0$ so that we are not dividing by zero. It thus follows
that

$\displaystyle\frac{\partial}{\partial t}RCLR(F_{t})\Big{|}_{t=0}=0.$

Next, we calculate the second derivative. We have

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}RCLR(F_{t})\Big{|}_{t=0}=\frac{1}{2}\left\\{\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\Big{|}_{t=0}+\frac{1}{2\tilde{W}}\frac{\partial^{2}}{\partial
t^{2}}A(F_{t})\Big{|}_{t=0}\right\\},$

where we used the results that $\frac{\partial}{\partial
t}A(F_{t})\big{|}_{t=0}=0$ and $A(F_{\theta})=\tilde{W}^{2}$. We continue and
obtain

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}A(F_{t})\Big{|}_{t=0}$
$\displaystyle=2\left\\{\frac{\partial}{\partial
t}RAR(F_{t})\Big{|}_{t=0}\right\\}^{2}-2\tilde{W}\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\Big{|}_{t=0}+4\tilde{W}\frac{\partial^{2}}{\partial
t^{2}}RK(F_{t})\Big{|}_{t=0}$
$\displaystyle=-2\tilde{W}\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\Big{|}_{t=0}+4\tilde{W}\frac{\partial^{2}}{\partial
t^{2}}RK(F_{t})\Big{|}_{t=0}.$

Substituting this back into the previous equation, it follows that

$\displaystyle\frac{\partial^{2}}{\partial t^{2}}RCLR(F_{t})\Big{|}_{t=0}$
$\displaystyle=\frac{1}{2}\left\\{\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\Big{|}_{t=0}+\frac{4\tilde{W}\frac{\partial^{2}}{\partial
t^{2}}RK(F_{t})\Big{|}_{t=0}-2\tilde{W}\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\Big{|}_{t=0}}{2\tilde{W}}\right\\}$
$\displaystyle=\frac{\partial^{2}}{\partial t^{2}}RK(F_{t})\Big{|}_{t=0}.$

Therefore, using L’Hôpital’s rule twice, the influence function of the CLR
statistic, given $\tilde{W}>0$, is

$\displaystyle\operatorname*{IF}\\{d_{i};\sqrt{RCLR},F\\}$
$\displaystyle=\lim_{t\rightarrow
0}\left\\{\sqrt{RCLR(F_{t})}-\sqrt{RCLR(F)}\right\\}/t$
$\displaystyle=\left\\{\lim_{t\rightarrow 0}RCLR(F_{t})/t^{2}\right\\}^{1/2}$
$\displaystyle=\left\\{\frac{1}{2}\frac{\partial^{2}}{\partial
t^{2}}RCLR(F_{t})\Big{|}_{t=0}\right\\}^{1/2}$
$\displaystyle=\left\\{\frac{1}{2}\frac{\partial^{2}}{\partial
t^{2}}RK(F_{t})\Big{|}_{t=0}\right\\}^{1/2}$
$\displaystyle=\operatorname*{IF}(d_{i};\sqrt{RK},F).$

Next, we assume $\tilde{D}=0$. In this case $\tilde{W}=0$, so that

$\displaystyle RCLR(G)=RAR(G).$

Hence,

$\displaystyle\operatorname*{IF}(d_{i};\sqrt{RCLR},F)=\operatorname*{IF}(d_{i};\sqrt{RAR},F).$

∎

### Proof of Proposition 2

###### Proof.

To simplify the notation, we suppress the dependency on $\beta_{0}$ for the
RAR and RK statistics. We have $\frac{\partial}{\partial
t}RAR(F_{t})\big{|}_{t=0}=2\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\Omega(F_{\theta},\beta_{0})^{-1}g(F_{\theta},\beta_{0})=0$,
as $g(F_{\theta},\beta_{0})=0$ due to Fisher consistency. The second
derivative gives

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\Big{|}_{t=0}=2\operatorname*{IF}\\{d;g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\Omega(\beta_{0})^{-1}\operatorname*{IF}\\{d;g(\cdot,\beta_{0}\\},F_{\theta}).$

Using L’Hôpital’s rule twice, we obtain

$\displaystyle\operatorname*{IF}(d_{i};\sqrt{RAR},F_{\theta})$
$\displaystyle=\lim_{t\rightarrow
0}\left\\{\sqrt{RAR(F_{t})}-\sqrt{RAR(F)}\right\\}/t$
$\displaystyle=\left\\{\lim_{t\rightarrow 0}RAR(F_{t})/t^{2}\right\\}^{1/2}$
$\displaystyle=\left\\{\frac{1}{2}\frac{\partial^{2}}{\partial
t^{2}}RAR(F_{t})\big{|}_{t=0}\right\\}^{1/2}$
$\displaystyle=\sqrt{\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\Omega(F_{\theta},\beta_{0})^{-1}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}}.$

Similarly, for the RK statistic we have

$\displaystyle\frac{\partial}{\partial t}RK(F_{t})\Big{|}_{t=0}=\ $
$\displaystyle
2\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\pi\left\\{\pi^{\top}\Omega(F_{\theta},\beta_{0})\pi\right\\}^{-1}\pi^{\top}g(F_{\theta},\beta_{0})$
$\displaystyle=$ $\displaystyle\ 0$

as $g(F_{\theta},\beta_{0})=0$. The second derivative gives

$\displaystyle\frac{\partial^{2}}{\partial
t^{2}}RK(F_{t})\Big{|}_{t=0}=2\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\pi\left\\{\pi^{\top}\Omega(\beta_{0})\pi\right\\}^{-1}\pi^{\top}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}.$

Again, using L’Hopitals Rule twice, we obtain

$\displaystyle\operatorname*{IF}(d_{i};\sqrt{RK},F)$
$\displaystyle=\lim_{t\rightarrow
0}\left\\{\sqrt{RK(F_{t})}-\sqrt{RK(F)}\right\\}/t$
$\displaystyle=\left\\{\lim_{t\rightarrow 0}RK(F_{t})/t^{2}\right\\}^{1/2}$
$\displaystyle=\left\\{\frac{1}{2}\frac{\partial^{2}}{\partial
t^{2}}RK(F_{t})\big{|}_{t=0}\right\\}^{1/2}$
$\displaystyle=\sqrt{\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}^{\top}\pi\left\\{\pi^{\top}\Omega(\beta_{0})\pi\right\\}^{-1}\pi^{\top}\operatorname*{IF}\\{d_{i};g(\cdot,\beta_{0}),F_{\theta}\\}}$

We omit the proof of the IF of the RW statistic, as it follows the exact same
steps as the IF of the RAR statistic. We also omit the proof of the influence
function of the K statistic, conditional on $D(F_{n},\beta_{0})=\tilde{D}$, as
this also follows the exact same steps as the IF of the RAR statistic. ∎

### Proof of Proposition 3

###### Proof.

In this proof we follow similar arguments as in Proof 1 of Kleibergen (2005).
From Lemma 1, we know that, as $n\rightarrow\infty$,

$\displaystyle\sqrt{n}g(F_{n},\beta_{0})\overset{d}{\to}N\sim\mathcal{N}\\{0,\Omega(F_{\theta},\beta_{0})\\}.$

We denote the asymptotic distribution of $D(F_{n},\beta_{0})$ by $D$, i.e.,
$\sqrt{n}D(F_{n},\beta_{0})\overset{d}{\to}D\sim\mathcal{N}\\{\pi,\Lambda(\beta_{0})\\}$.
Then, as $n\rightarrow\infty$,
$\sqrt{n}D(F_{n},\beta_{0})^{\top}\sqrt{n}g(F_{n},\beta_{0})\overset{d}{\to}D^{\top}N$.
The conditional distribution of $D^{\top}N$ given $D$ reads

$\displaystyle
D^{\top}N\big{|}D\sim\mathcal{N}\\{0,D^{\top}\Omega(F_{\theta},\beta_{0})D\\}.$

From Lemma 1, we know that $D$ is independent of $N$ as they are jointly
normally distributed and uncorrelated. Therefore, we obtain an unconditional
result by normalizing the expression by
$\\{D^{\top}\Omega(F_{\theta},\beta_{0})D\\}^{-1/2}$. Thus, as
$n\rightarrow\infty$,

$\displaystyle\\{D(F_{n},\beta_{0})^{\top}\Omega(F_{\theta},\beta_{0})D(F_{n},\beta_{0})\\}^{-1/2}D(F_{n},\beta_{0})\sqrt{n}g(F_{n},\beta_{0})$
$\displaystyle\quad\quad\quad\quad=\\{\sqrt{n}D(F_{n},\beta_{0})^{\top}\Omega(F_{\theta},\beta_{0})\sqrt{n}D(F_{n},\beta_{0})\\}^{-1/2}\sqrt{n}D(F_{n},\beta_{0})\sqrt{n}g(F_{n},\beta_{0})$
$\displaystyle\quad\quad\quad\quad\overset{d}{\to}\\{D^{\top}\Omega(F_{\theta},\beta_{0})D\\}^{-1/2}D^{\top}N\sim\mathcal{N}(0,1).$

Hence, as $n\rightarrow\infty$,
$nRK(F_{n},\beta_{0})\overset{d}{\to}\chi_{1}^{2}$. ∎

### Proof of Proposition 4

###### Proof.

Define $U(F_{n})=\Omega(F_{\theta},\beta_{0})^{-1/2}g(F_{n},\beta_{0})$ and
$R(F_{n})=\Omega(F_{\theta},\beta_{0})^{1/2}D(F_{n},\beta_{0})$. Note, by
Lemma 1, as $n\rightarrow\infty$,
$\sqrt{n}\begin{Bmatrix}U(F_{n})&R(F_{n})\end{Bmatrix}^{\top}\overset{p}{\rightarrow}\begin{pmatrix}U&R\end{pmatrix}^{\top}$,
where $U$ and $R$ are independent normal distributions.

We have $RAR(F_{n},\beta_{0})=U(F_{n})^{\top}U(F_{n})$ and
$RK(F_{n},\beta_{0})=U(F_{n})^{\top}P_{R(F_{n})}U(F_{n})$, where
$P_{R(F_{n})}=R(F_{n})\left\\{R(F_{n})^{\top}R(F_{n})\right\\}^{-1}R(F_{n})^{\top}$.
We can write $RAR(F_{n},\beta_{0})=RK(F_{n},\beta_{0})+RJ(F_{n},\beta_{0})$,
with

$\displaystyle
RJ(F_{n},\beta_{0})=U(F_{n})^{\top}\left\\{I_{k}-P_{R(F_{n})}\right\\}U(F_{n}).$

It holds that
$\left\\{I_{k}-P_{R(F_{n})}\right\\}^{-1/2}\sqrt{n}U(F_{n})\overset{p}{\to}\left(I_{k}-P_{R}\right)U$,
with $U\sim\mathcal{N}(0,I_{k})$. The conditional distribution of
$\left(I_{k}-P_{R}\right)^{-1/2}U$ given
$\tilde{R}=\Omega(F_{\theta},\beta_{0})^{1/2}\tilde{D}$ reads

$\displaystyle\left(I_{k}-P_{\tilde{R}}\right)^{-1/2}U\big{|}\tilde{R}\sim\mathcal{N}(0,I_{k}-P_{\tilde{R}}).$

We know that $\tilde{R}$ is independent of $N$ by Lemma 1. Therefore, this
result also holds unconditionally. Furthermore, as the rank of
$I_{k}-P_{\tilde{R}}$ is $tr(I_{k}-P_{\tilde{R}})=k-1$, we have

$\displaystyle nRJ(F_{n})\overset{d}{\to}\chi_{k-1}^{2}.$

The asymptotic distribution of $nRK(F_{n},\beta_{0})$ and
$nRJ(F_{n},\beta_{0})$ are independent, as $nRJ(F_{n},\beta_{0})$ projects on
the orthogonal complement of $R(F_{n})$.

Now the result follows. ∎

## References

  * Alesina and Zhuravskaya (2011) Alesina, A. and Zhuravskaya, E. (2011), “Segregation and the Quality of Government in a Cross Section of Countries,” American Economic Review, 101, 1872–1911.
  * Ananat (2011) Ananat, E. O. (2011), “The Wrong Side(s) of the Tracks: The Causal Effects of Racial Segregation on Urban Poverty and Inequality,” American Economic Journal: Applied Economics, 3, 34–66.
  * Anderson and Rubin (1949) Anderson, T. W. and Rubin, H. (1949), “Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations,” The Annals of Mathematical Statistics, 20, 46–63.
  * Andrews and Marmer (2008) Andrews, D. W. and Marmer, V. (2008), “Exactly Distribution-Free Inference in Instrumental Variables Regression With Possibly Weak Instruments,” Journal of Econometrics, 142, 183–200.
  * Andrews et al. (2006) Andrews, D. W., Moreira, M. J., and Stock, J. H. (2006), “Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression,” Econometrica, 74, 715–752.
  * Andrews and Soares (2007) Andrews, D. W. and Soares, G. (2007), “Rank Tests for Instrumental Variables Regression With Weak Instruments,” Econometric Theory, 23, 1033–1082.
  * Andrews et al. (2017) Andrews, I., Gentzkow, M., and Shapiro, J. M. (2017), “Measuring the Sensitivity of Parameter Estimates to Estimation Moments,” Quarterly Journal of Economics, 132, 1553–1592.
  * Andrews et al. (2020) — (2020), “On the Informativeness of Descriptive Statistics for Structural Estimates,” Econometrica, 88, 2231–2258.
  * Andrews et al. (2019) Andrews, I., Stock, J. H., and Sun, L. (2019), “Weak Instruments in Instrumental Variables Regression: Theory and Practice,” Annual Review of Economics, 11, 727–753.
  * Angrist et al. (1996) Angrist, J. D., Imbens, G. W., and Rubin, D. B. (1996), “Identification of Causal Effects Using Instrumental Variables,” Journal of the American Statistical Association, 91, 444–455.
  * Angrist and Krueger (1991) Angrist, J. D. and Krueger, A. B. (1991), “Does Compulsory School Attendance Affect Schooling and Earnings?” The Quarterly Journal of Economics, 106, 979–1014.
  * Athey and Imbens (2015) Athey, S. and Imbens, G. W. (2015), “A Measure of Robustness to Misspecification,” American Economic Review, 105, 476–480.
  * Bonhomme and Weidner (2022) Bonhomme, S. and Weidner, M. (2022), “Minimizing Sensitivity to Model Misspecification,” Quantitative Economics, 13, 907–954.
  * Bound et al. (1995) Bound, J., Jaeger, D. A., and Baker, R. M. (1995), “Problems With Instrumental Variables Estimation When the Correlation Between the Instruments and the Endogeneous Explanatory Variable Is Weak,” Journal of the American Statistical Association, 90, 443–450.
  * Chen and Bien (2020) Chen, S. and Bien, J. (2020), “Valid Inference Corrected for Outlier Removal,” Journal of Computational and Graphical Statistics, 29, 323–334.
  * Chernozhukov and Hansen (2008) Chernozhukov, V. and Hansen, C. (2008), “Instrumental Variable Quantile Regression: A Robust Inference Approach,” Journal of Econometrics, 142, 379–398.
  * Clarke (1983) Clarke, B. R. (1983), “Uniqueness and Fréchet Differentiability of Functional Solutions to Maximum Likelihood Type Equations,” Annals of Statistics, 1196–1205.
  * Clarke (1986) — (1986), “Nonsmooth Analysis and Fréchet Differentiability of M-functionals,” Probability Theory and Related Fields, 73, 197–209.
  * Cohen Freue et al. (2013) Cohen Freue, G., Ortiz-Molina, H., and Zamar, R. H. (2013), “A Natural Robustification of the Ordinary Instrumental Variables Estimator,” Biometrics, 69, 641–650.
  * Dvoretzky et al. (1956) Dvoretzky, A., Kiefer, J., and Wolfowitz, J. (1956), “Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator,” The Annals of Mathematical Statistics, 642–669.
  * Earle et al. (2001) Earle, C. C., Tsai, J. S., Gelber, R. D., Weinstein, M. C., Neumann, P. J., and Weeks, J. C. (2001), “Effectiveness of Chemotherapy for Advanced Lung Cancer in the Elderly: Instrumental Variable and Propensity Analysis,” Journal of Clinical Oncology, 19, 1064–1070.
  * Hampel (1974) Hampel, F. R. (1974), “The Influence Curve and Its Role in Robust Estimation,” Journal of the American Statistical Association, 69, 383–393.
  * Hampel et al. (1986) Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. (1986), Robust Statistics: The Approach Based on Influence Functions, New York: John Wiley & Sons.
  * Heritier et al. (2009) Heritier, S., Cantoni, E., Copt, S., and Victoria-Feser, M.-P. (2009), Robust Methods in Biostatistics, Chichester, UK: John Wiley & Sons.
  * Heritier and Ronchetti (1994) Heritier, S. and Ronchetti, E. (1994), “Robust Bounded-Influence Tests in General Parametric Models,” Journal of the American Statistical Association, 89, 897–904.
  * Huber (1964) Huber, P. J. (1964), “Robust Estimation of a Location Parameter,” The Annals of Mathematical Statistics, 35, 73–101.
  * Huber and Ronchetti (2009) Huber, P. J. and Ronchetti, E. M. (2009), Robust Statistics, New York: John Wiley & Sons, 2nd ed.
  * Jiao (2022) Jiao, X. (2022), “A Simple Robust Procedure in Instrumental Variables Regression,” Technical Report.
  * Jun (2008) Jun, S. J. (2008), “Weak Identification Robust Tests in an Instrumental Quantile Model,” Journal of Econometrics, 144, 118–138.
  * Keane and Neal (2023) Keane, M. and Neal, T. (2023), “Instrument Strength in IV Estimation and Inference: A Guide to Theory and Practice,” Journal of Econometrics, 235, 1625–1653.
  * Kern and Hainmueller (2009) Kern, H. L. and Hainmueller, J. (2009), “Opium for the Masses: How Foreign Media Can Stabilize Authoritarian Regimes,” Political Analysis, 17, 377–399.
  * Kitamura et al. (2013) Kitamura, Y., Otsu, T., and Evdokimov, K. (2013), “Robustness, Infinitesimal Neighborhoods, and Moment Restrictions,” Econometrica, 81, 1185–1201.
  * Kleibergen (2002) Kleibergen, F. (2002), “Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression,” Econometrica, 70, 1781–1803.
  * Kleibergen (2005) — (2005), “Testing Parameters in GMM Without Assuming That They Are Identified,” Econometrica, 73, 1103–1123.
  * Klooster and Zhelonkin (2024) Klooster, J. and Zhelonkin, M. (2024), “Outlier Robust Inference in the Instrumental Variable Model With Applications to Causal Effects,” Journal of Applied Econometrics, 39, 86–106.
  * Lal et al. (2023) Lal, A., Lockhart, M., Xu, Y., and Zu, Z. (2023), “How Much Should We Trust Instrumental Variable Estimates in Political Science? Practical Advice Based on Over 60 Replicated Studies,” Political Analysis (forthcoming).
  * Lee et al. (2022) Lee, D. S., McCrary, J., Moreira, M. J., and Porter, J. (2022), “Valid t-Ratio Inference for IV,” American Economic Review, 112, 3260–90.
  * Magnusson (2010) Magnusson, L. M. (2010), “Inference in Limited Dependent Variable Models Robust to Weak Identification,” The Econometrics Journal, 13, S56–S79.
  * Maronna et al. (2019) Maronna, R. A., Martin, R. D., Yohai, V. J., and Salibián-Barrera, M. (2019), Robust Statistics: Theory and Methods (With R), New York: Wiley, 2nd ed.
  * Moreira (2003) Moreira, M. J. (2003), “A Conditional Likelihood Ratio Test for Structural Models,” Econometrica, 71, 1027–1048.
  * Moreira (2009) — (2009), “Tests With Correct Size When Instruments Can Be Arbitrarily Weak,” Journal of Econometrics, 152, 131–140.
  * Nelson and Startz (1990) Nelson, C. and Startz, R. (1990), “The Distribution of the Instrumental Variables Estimator and Its t-Ratio When the Instrument Is a Poor One,” The Journal of Business, 63, 5125––5140.
  * Ronchetti (1982) Ronchetti, E. (1982), “Robust Testing in Linear Models: The Infinitesimal Approach,” Ph.D. thesis, ETH Zürich.
  * Ronchetti and Trojani (2001) Ronchetti, E. and Trojani, F. (2001), “Robust Inference With GMM Estimators,” Journal of Econometrics, 101, 37–69.
  * Rousseeuw and Leroy (1987) Rousseeuw, P. J. and Leroy, A. M. (1987), Robust Regression and Outlier Detection, John Wiley & Sons.
  * Rousseeuw and Ronchetti (1981) Rousseeuw, P. J. and Ronchetti, E. (1981), “Influence Curves of General Statistics,” Journal of Computational and Applied Mathematics, 7, 161–166.
  * Saraceno et al. (2021) Saraceno, G., Alqallaf, F., and Agostinelli, C. (2021), “A Robust Seemingly Unrelated Regressions for Row-Wise and Cell-Wise Contamination,” Technical Report.
  * Small and Rosenbaum (2008) Small, D. S. and Rosenbaum, P. R. (2008), “War and Wages: The Strength of Instrumental Variables and Their Sensitivity to Unobserved Biases,” Journal of the American Statistical Association, 103, 924–933.
  * Sølvsten (2020) Sølvsten, M. (2020), “Robust Estimation With Many Instruments,” Journal of Econometrics, 214, 495–512.
  * Staiger and Stock (1997) Staiger, D. and Stock, J. H. (1997), “Instrumental Variables Regression With Weak Instruments,” Econometrica, 65, 557–586.
  * Stock and Wright (2000) Stock, J. H. and Wright, J. H. (2000), “GMM With Weak Identification,” Econometrica, 68, 1055–1096.
  * Venables and Ripley (2002) Venables, W. N. and Ripley, B. D. (2002), Modern Applied Statistics with S, New York: Springer-Verslag, 4th ed.
  * von Mises (1947) von Mises, R. (1947), “On the Asymptotic Distribution of Differentiable Statistical Functions,” The Annals of Mathematical Statistics, 18, 309–348.
  * Welsh and Ronchetti (2002) Welsh, A. H. and Ronchetti, E. (2002), “A Journey in Single Steps: Robust One-Step M-Estimation in Linear Regression,” Journal of Statistical Planning and Inference, 103, 287–310.
  * Yohai (1987) Yohai, V. J. (1987), “High Breakdown-Point and High Efficiency Robust Estimates for Regression,” Annals of Statistics, 15, 642–656.
  * Young (2022) Young, A. (2022), “Consistency Without Inference: Instrumental Variables in Practical Application,” European Economic Review, 147, 104–112.
  * Zhelonkin et al. (2012) Zhelonkin, M., Genton, M. G., and Ronchetti, E. (2012), “On the Robustness of Two-Stage Estimators,” Statistics & Probability Letters, 82, 726–732.