# An Identification-and Dimensionality-Robust Test for Instrumental Variables
Models

Manu Navjeevan Email<EMAIL_ADDRESS>The latest version of this paper
can be found here. I thank Denis Chetverikov for his guidance, numerous
discussions and permanent support. I am also grateful to the other members of
my dissertation committee, Andres Santos and Zhipeng Liao, and participants in
UCLA’s econometrics proseminar, Jinyong Hahn, Rosa Matzkin, Shuyang Sheng, and
Daniel Ober-Reynolds for a half-decade of useful comments. University of
California, Los Angeles

(Revised: )

###### Abstract

I propose a new identification-robust test for the structural parameter in a
heteroskedastic linear instrumental variables model. The proposed test
statistic is similar in spirit to a jackknife version of the K-statistic and
the resulting test has exact asymptotic size so long as an auxiliary parameter
can be consistently estimated. This is possible under approximate sparsity
even when the number of instruments is potentially much larger than the sample
size. As the number of instruments is allowed, but not required, to be large,
the limiting behavior of the test statistic is difficult to examine via
existing central limit theorems. Instead, I derive the asymptotic chi-squared
distribution of the test statistic using a direct Gaussian approximation
technique. To improve power against certain alternatives, I propose a simple
combination with the sup-score statistic of Belloni et al. (2012) based on a
thresholding rule. I demonstrate favorable size control andpower properties in
a simulation study and apply the new methods torevisit the effect of social
spillovers in movie consumption.

Keywords: Instrumental Variables, Weak Identification, High-DimensionalJEL
Codes: C12, C36, C55

##  1 Introduction

Consider a linear instrumental variables (IV) model

$y_{i}=x_{i}^{\prime}\beta+z_{1i}^{\prime}\Gamma+\epsilon_{i},\;\;\;\mathbb{E}[\epsilon_{i}|z_{i}]=0$
(1.1)

where $y_{i}\in\mathbb{R}$ is an outcome of interest and
$x_{i}\in\mathbb{R}^{d_{x}}$ is a vector of endogenous variables. The variable
$z_{i}=(z_{1i},z_{2i})^{\prime}\in\mathbb{R}^{d_{z}}$ represents a vector of
instrumental variables, of which a subvector of fixed dimension,
$z_{1i}^{\prime}\in\mathbb{R}^{d_{c}}$, is included in the structural equation
(1.1) as exogenous control. I assume that the researcher has access to $n$
independent observations of $(y_{i},x_{i}^{\prime},z_{i}^{\prime})^{\prime}$.
In this setting, I propose a new test for a two-sided restriction on the
structural parameter; $H_{0}\mathrel{\mathop{\ordinarycolon}}\beta=\beta_{0}$
versus $H_{1}\mathrel{\mathop{\ordinarycolon}}\beta\neq\beta_{0}$. The
proposed test has exact asymptotic size even when instruments are potentially
high-dimensional ($d_{z}\gg n$) and arbitrarily weak.When instruments are
suspected to be weak, researchers may want to test hypotheses about structural
parameters using testing procedures that are robust to identification
strength. These identification-robust testing procedures all require some
conditions on the rate of growth of the number of instruments, $d_{z}$, in
relation to the sample size, $n$. The tests of Anderson and Rubin (1949);
Staiger and Stock (1997); Moreira (2001, 2003), and Kleibergen (2002, 2005)
are shown by Andrews and Stock (2007) to control size under heteroskedasticity
when $d_{z}^{3}/n\to 0$. Recent tests developed in Mikusheva and Sun (2021);
Crudu et al. (2021); Matsushita and Otsu (2022), and Lim et al. (2022) allow
the number of instruments to be proportional to sample size,
$d_{z}/n\to\varrho\in[0,1)$, but require that the number of instruments be
large, $d_{z}\to\infty$. Moreover, while the tests proposed by Belloni et al.
(2012) and Mikusheva (2023) require only $\log^{M}(d_{z})/n\to 0$ for
someconstant $M>1$, they either rely on sample splitting or fail to
incorporate first-stage information, both of which may reduce power in
overidentified models.These conditions on the growth rate of $d_{z}$ can be
difficult to interpret and the variety of tests available under alternate
regimes may make it unclear to the researcher which test should be applied in
her particular setting.111For example, consider a setting similar to that of
Derenoncourt (2022) where the researcher has a dozen or so instruments and a
sample size of a few hundred. The number of instruments cubed is larger than
sample size, but asymptotic approximations based on $d_{z}\to\infty$ seem
unlikely to resemble the finite sample distribution.In contrast, the test
considered in this paper is applicable in any of the settings described above.
The proposed test statistic is similar in spirit to a jackknife version of the
K-statistic and makes use of an auxiliary conditional slope parameter to
construct first-stage estimates that are uncorrelated with the structural
errors under the null hypothesis. So long as this auxiliary parameter can be
consistently estimated, the proposed test statistic has a limiting chi-squared
distribution with degrees of freedom equal to the number of structural
parameters. The conditional slope parameter is simple to estimate with out-of-
the-box methods, and consistency is achievable under approximate sparsity even
when the number of instruments is much larger than the sample size. This
approximate sparsity assumption is trivially satisfied when the first- and
second-stage errors are homoskedastic.As the number of instruments is allowed
to be larger than the sample size, traditional central limit theorems are
insufficient to derive the limiting distribution of the test statistic.
Instead, the limiting behavior of the test statistic is examined directly with
a variation of Lindeberg’s interpolation method (Lindeberg, 1922). The direct
approach allows me to characterize the limiting behavior of the test statistic
in both high- and low-dimensional settings under a unified argument. I show
that, in local neighborhoods of the null, the distribution of the test
statistic can be uniformly approximated by the distribution of an analog
statistic that replaces each observation in the expression of the test
statistic with a Gaussian version that has the same mean and covariance matrix
as the original. These local neighborhoods of the null are characterized by a
local power index which I introduce in Section 3. An interesting feature of
the direct Gaussianapproximation argument is that, while the numerator and
denominator the jackknife K-statistic may both have nondegenerate
distributions under weak identification, neither needs to converge on its own
to a weak limit in order to characterize the limiting distribution of the
jackknife K-statistic.When there is a single endogenous variable, a leading
case in empirical applications, analysis of limiting behavior can be
considerably simplified by taking advantage of the particular form of the test
statistic.In this case I show that, under an additional regularity condition,
an infeasible version of the test that could be constructed if the auxiliary
parameter was known to the researcher is consistent whenever the local power
index diverges. When the local power index is bounded, I examine the limiting
power of the test by examining the behavior of the analog statistic. Under the
alternative hypothesis the analog statistic has a nearly non-central
$\chi^{2}$ distribution conditional on the first-stage estimates. The
noncentrality parameter is proportional to the correlation between the true
first-stage model and the first-stage estimates. Unfortunately, partialling
out the structural parameter may introduce bias into the first-stage estimates
under the alternate hypothesis. This issue is pointed out by Moreira (2001);
Andrews et al. (2006), and Andrews (2016) in the context of the non-jackknife
K-statistic.Against certain alternatives this bias can be particularly
pronounced and the additional regularity condition needed for the consistency
result may fail to hold.222This does not necessarily mean, however, that the
test is not consistent.To address this, I propose a simple combination of the
jackknife K-statistic with the sup-score statistic of Belloni et al. (2012)
based on a thresholding rule. As with the Anderson-Rubin statistic, while the
sup-score statistic does not yield efficient inference under strong
identification, it does not suffer from a loss of power against any particular
alternative when identification is weak (Andrews et al., 2006; Andrews, 2016).
The combination test decides whether the jackknife K-test or the sup-score
test should be run by comparing the value of a conditioning statistic to a
predetermined cutoff value. In the approximating Gaussian regime, this
conditioning statistic is marginally independent of both the jackknife
K-statistic and the sup-score statistic. This allows me to show that the
combination test controls size under the null without having to require that
the conditioning statistic converges in distribution to a stable limit. In a
simulation study, I find that taking this cutoff value to be the75th quantile
of the distribution of the conditioning statistic delivers a reasonable
balance of power against local and distant alternatives. Using results in
Chernozhukov et al. (2017) and Belloni et al. (2018) this quantile can be
simulated via a multiplier bootstrap procedure.When there are multiple
endogenous variables, I cannot take advantage of the simplified form of the
test statistic. Instead, I use a more involved interpolation argument that
relies on strengthened moment conditions. Under these strengthened conditions
I derive the limiting chi-squared distribution of the jackknife K-statistic in
the larger context and propose a generalization of the thresholding test to
improve power properties.I apply the proposed testing procedures to the data
of Gilchrist and Sands (2016) to generate weak instrument-robust confidence
intervals for the effect of social spillovers in movie consumption. Following
Belloni et al. (2012), the authors’ initial analysis uses conventional
heteroskedasticity-robust standard errors after estimating the first-stage via
post-LASSO. The validity of this analysis depends on the structural parameter
being strongly identified. Using a simple numerical demonstration, I argue
that the first-stage F-statistics reported by the authors may not be reliable
indicators of identification strength when LASSO is used to select
instruments. The identification-robust confidence intervals generated by
inverting the jackknife K-statistic are larger than those implied by the
initial analysis but do not rule out the authors’ point estimates. Moreover,
these confidence intervals are typically smaller than those implied by
inverting the sup-score statisticin specifications for which the sup-score
confidence interval is nonempty.Finally, I examine the applicability of the
theoretical results in this paper through a simulation study. Tests based on
the jackknife K-statistic are shown to have nearly exact size in a variety of
settings. While the jackknife K-statistic may have diminished power against
certain alternatives, this deficiency seems to be ameliorated by combining the
jackknife K-statistic with the sup-score statistic via the thresholding test.
Compared to the many-instrument tests of Mikusheva and Sun (2021) and
Matsushita and Otsu (2022), the tests proposed in this paper appear to have
favorable size control and power properties.The outline of this paper is as
follows. Section 2 formally defines the model considered and introduces the
jackknife K-statistic. Section 3 provides an overview of the Gaussian
approximation approach with a single endogenous variable and characterizes the
limiting behavior of the test statistic in this setting. Section 4 uses this
characterization to examine the power properties of the test and introduces
the combination test to address power deficiencies against certain
alternatives. Section 5 extends the analyses of Sections 3 and 4 to the case
of multiple endogenous variables and outlines the Gaussian approximation
argument in this setting. Section 6 contains the empirical application while
Section 7 provides evidence from simulation study. Proofs of the main results
are deferred to Appendices A, B and C.

#### Notation.

For any $n\in\mathbb{N}$ let $[n]$ denote the set $\\{1,\dots,n\\}$. I work
with a sequence of probability measures $P_{n}$ on the data
$\\{(y_{i},x_{i},z_{i})\mathrel{\mathop{\ordinarycolon}}i\in[n]\\}$. To
accommodate independent but not identically distributed observations, let
$\mathbb{E}_{n}[f_{i}]=n^{-1}\sumop\slimits@_{i=1}^{n}f_{i}$ denote the
empirical expectation and
$\bar{\mathbb}{E}[f]=\mathbb{E}_{n}[\mathbb{E}[f_{i}]]$ denote the average
expectation operator.

###  1.1 Prior Literature and Empirical Practice

When the first-stage F-statistic is small, standard asymptotic approximations
may fail to accurately describe the finite-sample behavior of IV estimates.
This was first pointed out by Nelson and Startz (1990) and Bound et al. (1995)
who consider the finite-sample behavior of two-stage least squares (2SLS) in
alternate settings where the IV is only weakly correlated with the endogenous
variable. In a seminal paper, Staiger and Stock (1997) capture this phenomena
in an asymptotic framework by considering a sequence of first-stage models
that shrink to zero with the sample size. Under this framework, standard IV
estimates are no longer consistent and inference procedures based on these
statistics fail to control size. Because of these results, there has been a
large interest in developing tests for the structural parameter that control
size regardless of identification strength.To test hypotheses about the
structural parameter when instruments are suspected to be weak, Staiger and
Stock (1997) propose the use of the Anderson-Rubin statistic, which does not
require any assumptions about identification strength to control size. Noting
that the Anderson-Rubin test is inefficient in overidentified models, Moreira
(2001) and Kleibergen (2002, 2005) propose the use of the (non-jackknife)
K-statistic, which has a limiting null distribution that does not depend on
the number of instruments. Compared to the Anderson-Rubin statistic, these
tests have improved power in local neighborhoods of the null but can perform
poorly against certain alternatives. To address this, Moreira (2003) and
Kleibergen (2005) suggest combinations of the K-statistic and Anderson-Rubin
statistic based on a conditioning statistic that is independent of them both
under the null. Andrews et al. (2006) characterize the power envelope in a
homoskedastic weakly identified IV model and show that the test based on
theconditional likelihood ratio statistic of Moreira (2003) has nearly optimal
power in this setting. When errors are heteroskedastic, Andrews (2016)
proposes alternate combinations of the K-statistic and the Anderson-Rubin
statistic based on a minimax regret criterion.These initial tests are
developed under asymptotic frameworks that treat the number of instruments as
fixed or growing slowly relative to the sample size (Han and Phillips, 2006;
Newey and Windmeijer, 2009; Andrews and Stock, 2007). However, with the
emergence of large datasets and more sophisticated research designs,
researchers may encounter scenarios where the number of instruments may not be
negligible as a ratio of sample size. A prominent example of this is in judge-
design settings where the number of instruments is equal to the number of
judges to whom an individual can be assigned to (Maestas et al., 2013; Sampat
and Williams, 2019; Dobbie et al., 2018). Since each judge can handle only a
finite number of cases the number of instruments is proportional to the sample
size. Moreover, to flexibly model the first-stage, researchers may generate a
large number of instruments by enriching a “small” initial set of instruments
via polynomial (or other) transformations. Angrist and Krueger (1991) famously
interact quarter-of-birth, state-of-birth, and year-of-birth dummies to
construct a total of 180 instruments.Belloni et al. (2012) show that, when
identification is strong, researchers can use a potentially high-dimensional,
$d_{z}\gg n$, set of first-stage instrument basis terms in conjunction with a
LASSO or post-LASSO estimate of the first-stage. This strategy has been
successfully employed in practice by Paravisini et al. (2014); Gilchrist and
Sands (2016); Derenoncourt (2022), and Jou and Morgan (2023).To address these
settings, there has been recent interest in developing weak instrument-robust
tests under asymptotic frameworks that do not require that the ratio of
instruments to sample size tends to zero. Crudu et al. (2021); Mikusheva and
Sun (2021), and Matsushita and Otsu (2022) take advantage of a new central
limit theorem for quadratic forms developed in Chao et al. (2012) and propose
weak identification-robust tests that are valid even when the number of
instruments is proportional to sample size; $d_{z}/n\to\varrho\in[0,1)$.
Following the many instruments asymptotic framework first introduced by Bekker
(1994), the analyses in these papers rely on the number of instruments
diverging. When the number of instruments is fixed or diverges slowly to
infinity, these asymptotic approximations may provide poor characterizations
of the proposed test statistics’ finite sample distribution.Limited
identification-robust testing procedures exist for the high-dimensional case,
$d_{z}\gg n$. To my knowledge, the only two options available are the sup-
score test of Belloni et al. (2012) and the split-sample optimal instrument AR
test developed in Mikusheva (2023). The sup-score test makes use of Gaussian
approximations for maxima of high-dimensional vectors developed in
Chernozhukov et al. (2013) but suffers from the same issue as the Anderson-
Rubin test in that its critical value is increasing with the number of
instruments. The spilt sample optimal instrument AR test splits the dataset
into two parts and uses one split to estimate an optimal instrument and the
other to test the null hypothesis. This may lead to a loss of power as only
half of the sample is being effectively used to test the null hypothesis.Weak
instrument-robust tests may be particularly interesting in high-dimensional
and heteroskedastic settings because of a lack of clarity on how to pretest
for identification strength. When the number of instruments is fixed and
errors are homoskedastic, Stock and Yogo (2005) propose pretesting for the
strength of identification via thte first-stage F-statistic. Based on their
results, common practice in empirical settings has been to use standard Wald
tests whenever the first-stage F-statistic exceeds 10. Lee et al. (2022) point
out this recommendation is not applicable in heteroskedastic models and update
the recommended F-statistic cutoff. To pretest for weak identification in the
many-instruments asymptotic framework, $d_{z}\to\infty$, Mikusheva and Sun
(2021) propose a new $\tilde{F}$-statistic and suggest using identification-
robust procedures when $\tilde{F}<4.14$. When the number of instruments is
larger than sample size thereis no accepted full-sample pretest for
identification strength.333Mikusheva (2023) suggests a split-sample pretest in
the same spirit as the split-sample optimal instrument AR test.In particular,
I argue in Section 6 that first-stage F-statistics resulting from first-stage
post-LASSO procedures can be misleading even if they are much larger the
standard cutoff of 10.

Asymptotic Regime | Main Tests
---|---
| Low-Dimensional:
---
$d_{z}^{3}/n\to 0$
| Anderson-Rubin
---
K/Lagrange Multiplier
Conditional Linear Combination
| Many-Instruments:
---
$d_{z}/n\to\phi\in[0,1)$
$d_{z}\to\infty$
| Jackknife-AR
---
Jackknife-LM
Conditional Linear Combination
| High-Dimensional:
---
$\log^{M}(d_{z}n)/n\to 0$
| Sup-Score Test
---
Split-Sample AR
Table 1.1: Existing Identification and Heteroskedasticity Robust Tests for
Linear IV models.

I contribute to these literatures by proposing a new identification-robust
test for the structural parameter that can work in potentially high-
dimensional settings ($d_{z}\gg n$) without requiring that the number of
instruments diverges. The testing procedures in this paper may be particularly
applicable in intermediate cases where the number of instruments cubed may not
be negligible relative to sample size but it is unclear whether asymptotic
approximations based on $d_{z}\to\infty$ will accurately describe finite
sample behavior. Examples of such intermediate cases include the analyses of
Derenoncourt (2022), where $d_{z}=9$ and $n=239$, Paravisini et al. (2014),
where $d_{z}=10$ and $n=5{,}995$, and Gilchrist and Sands (2016), where
$d_{z}=52$ and $n=1{,}671$.In addition to the literature on weak-instrument
robust testing, I contribute to a growing literature on direct gaussian
approximation and interpolation techniques (Chatterjee, 2006, 2010; Pouzo,
2015; Chernozhukov et al., 2013, 2017; Celentano et al., 2020). These
techniques have proven useful to approximate the behaviors of statistics in a
variety of nonstandard settings, such as high-dimensional random vectors or
spectral analysis of random matrices. Prior analysis of statistics via
interpolation techniques has relied on the boundedness of the derivatives of
these statistics with respect to individual observations. This condition does
not hold in my setting as the derivative of the jackknife K-statistic with
respect to terms in the denominator may be unbounded. This poses a number of
technical challenges for my interpolation argument that must be overcome in
order to characterize the limiting behavior of the jackknife K-statistic,
particularly when $d_{x}>1$. I contribute to this literature by proposing
modifications of the original Lindeberg (1922) interpolation technique that
can accommodate statistics with unbounded derivatives.

##  2 Model and Setup

Though the analysis below allows for exogenous regressors, to simplify the
exposition I follow Mikusheva and Sun (2021) and assume that they have already
been partialed out of both the outcome, $y_{i}$, and the endogenous
regressors, $x_{i}$. As the controls are assumed to be of fixed dimension,
this is without loss of generality.111For discussion refer to Appendix E.
Along with the structural equation in (1.1), the IV model can then be written
with the first stage as a system of simultaneous equations:

$\begin{split}y_{i}&=x_{i}^{\prime}\beta+\varepsilon_{i}\\\
x_{i}&={}_{i}+v_{i}\end{split}$ (2.1)

The researcher observes the outcome $y_{i}\in\mathbb{R}$, the endogenous
variable $x_{i}\in\mathbb{R}^{d_{x}}$, and the instruments
$z_{i}\in\mathbb{R}^{d_{z}}$ but neither the structural error
$\varepsilon_{i}\in\mathbb{R}$ nor the first-stage errors
$v_{i}\in\mathbb{R}^{d_{x}}$. The structural error is assumed to be
conditional-mean independent of the instruments,
$\mathbb{E}[\varepsilon_{i}|z_{i}]=0$. I denote $\mathbb{E}[x_{i}|z_{i}]$ as
${}_{i}\coloneqq\mathbb{E}[x_{i}|z_{i}]$ and make no assumptions about the
functional form of the conditional expectation so the instruments are allowed
to affect the endogenous variable in a nonlinear fashion.The random variables
$\\{(z_{i},\varepsilon_{i},v_{i})\\}_{i=1}^{n}$ are assumed to be independent
across observations. Observations need not be identically distributed but the
errors are assumed to have a common covariance structure conditional on the
instruments $z_{i}$:

$\operatorname{Var}((\varepsilon_{i},v_{i})^{\prime}|z_{i})\coloneqq\Omega(z_{i})=\begin{pmatrix}\sigma^{2}_{\epsilon\epsilon}(z_{i})&{}_{v\epsilon}(z_{i})\\\
{}_{\epsilon
v}(z_{i})&{}_{vv}(z_{i})\end{pmatrix}\in\mathbb{R}^{(1+d_{x})\times(1+d_{x})}$

As $\Omega(z_{i})$ is otherwise left unrestricted, the errors are allowed to
be heteroskedastic. All results in this paper hold conditionally on a
realization of the instruments
$\bm{z}\mathrel{\mathop{\ordinarycolon}}=(z_{1}^{\prime},\dots,z_{n}^{\prime})\in\mathbb{R}^{n\,\times\,d_{z}}$
so from this point forth they are treated as fixed and all expectations can be
understood as conditional on the instruments.Under this setup, the researcher
wishes to test a two-sided restriction on the structural parameter:

$H_{0}\mathrel{\mathop{\ordinarycolon}}\beta=\beta_{0}\hbox{ }\text{ vs.
}\hbox{ }H_{1}\mathrel{\mathop{\ordinarycolon}}\beta\neq\beta_{0}$

I am interested in constructing powerful tests for this null-alternate pair
that are asymptotically valid under arbitrarily weak identification and with
minimal restrictions on the number of instruments $d_{z}$. To this end, define
the null errors $\varepsilon_{i}(\beta_{0})\coloneqq
y_{i}-x_{i}^{\prime}\beta_{0}$. Using these, I construct a variable, $r_{i}$,
that is a “partialed-out” version of the endogenous variable satisfying
$\operatorname{Cov}(r_{i},\epsilon_{i}(\beta_{0}))=0$:

$\displaystyle r_{i}\coloneqq
x_{i}-\rho(z_{i})\epsilon_{i}(\beta_{0}),\;\;\;\rho(z_{i})$
$\displaystyle\coloneqq\frac{\operatorname{Cov}(\epsilon_{i}(\beta_{0}),x_{i})}{\operatorname{Var}(\epsilon_{i}(\beta_{0}))}\in\mathbb{R}^{d_{x}}$
$\displaystyle\;=\frac{{}_{v\epsilon}(z_{i})+{}_{vv}(z_{i})(\beta-\beta_{0})}{(1,\beta-\beta_{0})^{\prime}\Omega(z_{i})^{\prime}(1,\beta-\beta_{0})}.$

Each element of the nuisance parameter $\rho(z_{i})$, $\rho_{\ell}(z_{i})$ for
$\ell=1,\dots,d_{x}$, can be interpreted as the (conditional) slope
coefficient from a simple linear regression of $x_{\ell i}$ on
$\epsilon_{i}(\beta_{0})$. Thus, if $\rho_{\ell}(\cdot)$ falls in some
function class it can be estimated directly under $H_{0}$ by solving empirical
analogs of:222Under $H_{1}$, $\rho_{\ell}(z_{i})$ can be estimated directly by
solving empirical analogs of
$\rho_{\ell}(z_{i})=\arg\min_{\phi\in\Phi}\mathbb{E}[(x_{\ell
i}-\eta_{i}(\beta_{0})\phi(z_{i}))^{2}]$ where
$\eta_{i}(\beta_{0})=\epsilon_{i}(\beta_{0})-\mathbb{E}[\epsilon_{i}(\beta_{0})|z_{i}]$.
This requires an initial estimate of
$\mathbb{E}[\epsilon_{i}(\beta_{0})|z_{i}]$, however.

$\rho_{\ell}(z_{i})=\arg\min_{\varphi\in\Phi}\bar{\mathbb}{E}[(x_{\ell
i}-\epsilon_{i}(\beta_{0})\varphi(z_{i}))^{2}].$

I will largely work under the assumption that $\rho(z_{i})$ has an
approximately sparse representation in some (growing) basis
$b(z_{i})\coloneqq(b_{1}(z_{i}),\dots,b_{d_{b}}(z_{i}))^{\prime}\in\mathbb{R}^{d_{b}}$,
that is $\rho_{\ell}(z_{i})=b(z_{i})^{\prime}\phi_{\ell}+\xi_{\ell i}$ where
$\xi_{\ell i}$ represents an approximation error that tends to zero with the
sample size and $\phi_{\ell}$ is sparse in the sense that many of its
coefficients are zero. This allows for nesting of the low-dimensional case,
where the number of instruments is fixed, and the high dimensional case, where
the number of instruments is potentially much larger than the sample size,
under a unified estimation procedure. Under homoskedasticity,
$\rho_{\ell}(z_{i})$ is a constant function and thus has a spare
representation in any basis that contains a constant term.As in Chernozhukov
et al. (2022), the parameter $\phi_{\ell}$ can be estimated via LASSO:

$\hat{\phi}_{\ell}=\arg\min_{\phi\in\mathbb{R}^{d_{b}}}\mathbb{E}_{n}[(x_{\ell
i}-\epsilon_{i}(\beta_{0})b(z_{i})^{\prime}\phi)^{2}]+\lambda\|\phi\|_{1},$
(2.2)

or via post-LASSO, refitting an unpenalized version of (2.2) using only the
basis terms associated with nonzero coeffecients in the inital LASSO
regression. The estimating procedure in (2.2) is a simple $\ell_{1}$-penalized
regression of $x_{\ell i}$ against $\epsilon_{i}(\beta_{0})b(z_{i})$. It can
be easily implemented using out-of-the-box software available on most
platforms. Under standard conditions, this leads to a consistent estimate of
$\rho_{\ell}(z_{i})$ as long as the sparsity condition
$s^{2}\log^{M}(d_{b}n)/n\to 0$ where $s$ is the number of nonzero elements of
$\phi_{\ell}$ and $M$ is a positive constant that depends on the moment bounds
imposed. The estimation procedure is discussed in more detail in Section 3.2.
With $\hat{\rho}(z_{i})\coloneqq b(z_{i})^{\prime}\hat{\phi}_{\ell}$, I
construct the estimated version of $r_{\ell i}$, $\hat{r}_{\ell i}\coloneqq
x_{i}-\hat{\rho}(z_{i})\epsilon_{i}(\beta_{0})$ for each $\ell\in[d_{x}]$.

###  2.1 Test Statistic

The test statistic is based on an arbitrary jackknife-linear estimate of the
first stage,

$\mathaccent 0{\symlargesymbols}062{}_{\ell i}=\sumop\slimits@_{j\neq
i}h_{ij}\hat{r}_{\ell j},\;\;\ell\in[d_{x}]$

for some “hat” matrix $H=[h_{ij}]\in\mathbb{R}^{n\times n}$. The phrase “hat
matrix” is borrowed from ordinary least squares (OLS) where the projection
matrix, $\bm{z}(\bm{z}^{\prime}\bm{z})^{-1}\bm{z}^{\prime}$, is sometimes
referred to as the hat matrix in the sense that
$\hat{x}=\bm{z}(\bm{z}^{\prime}\bm{z})^{-1}\bm{z}^{\prime}x$. In practice, the
hat matrix, $H$, can be any matrix that depends only on $\bm{z}$. It is
important to note that while $\mathaccent 0{\symlargesymbols}062{}_{\ell i}$
does not depend on the observation $r_{\ell i}$, it may depend on $z_{i}$
through the hat matrix $H$. This gives the test power against alternatives
where $\mathbb{E}[\epsilon_{i}(\beta_{0})z_{i}]\neq 0$.For technical reasons,
I will assume that $h_{ii}=0$ for each $i\in[n]$ so that $\mathaccent
0{\symlargesymbols}062{}_{\ell i}$ can be written as $\mathaccent
0{\symlargesymbols}062{}_{\ell i}=\sumop\slimits@_{j=1}^{n}h_{ij}r_{\ell
j}$.Formally, the only structure I require on the hat matrix $H$ is a
balanced-design condition described in Section 3. However, for reasons
explained in Section 4 it may be optimal to introduce some regularization in
estimating the first-stage models $\mathaccent 0{\symlargesymbols}062{}_{\ell
i}$ so I suggest using the deleted diagonal ridge-regression hat matrix
$H(\lambda^{\star})$:

$[H(\lambda^{\star})]_{ij}=\begin{cases}[\bm{z}(\bm{z}^{\prime}\bm{z}+\lambda^{\star}I_{d_{z}})^{-1}\bm{z}^{\prime}]_{ij}&\text{if
}i\neq j\\\ 0&\text{otherwise }\end{cases}$ (2.3)

where, following recommendations in Harrell (2015) and van Wieringen (2023),
the penalty parameter $\lambda^{\star}$ is set so that the effective degrees
of freedom of the resulting hat matrix is no more than a fraction of the
sample size:

$\lambda^{\star}=\inf\\{\lambda\geq
0\mathrel{\mathop{\ordinarycolon}}\text{trace}(\bm{z}(\bm{z}^{\prime}\bm{z}+\lambda
I_{d_{z}})^{-1}\bm{z}^{\prime})\leq n/5\\}$

The ridge hat matrix has the benefit of being well defined even when the
number of instruments is larger than the sample size.I stress, though, that
the researcher may use any other hat matrix that she believes will lead to
plausible first-stage estimates. Other possible choices of hat matrix include
the jackknife OLS hat matrix of Angrist et al. (1999), the deleted diagonal
projection matrix introduced in Chao et al. (2012) and successfully used in
Kline et al. (2020); Crudu et al. (2021); Mikusheva and Sun (2021), and
Matsushita and Otsu (2022), or hat matrices based on selecting instruments via
some preliminary unsupervised technique such as principal component analysis
(PCA). Remark 3.1 below discusses how the balanced-design condition may be
verified for arbitrary choices of hat matrices.For each $i=1,\dots,n$, define
$\mathaccent 0{\symlargesymbols}062{}_{i}=(\mathaccent
0{\symlargesymbols}062{}_{1i},\dots,\mathaccent
0{\symlargesymbols}062{}_{d_{x}i})\in\mathbb{R}^{d_{x}}$ and $\mathaccent
0{\symlargesymbols}062{}_{\epsilon i}=\epsilon_{i}(\beta_{0})\mathaccent
0{\symlargesymbols}062{}_{i}$. Collect these in the matrices

$\begin{split}\varepsilon(\beta_{0})&=\big{(}\varepsilon_{1}(\beta_{0}),\dots,\varepsilon_{n}(\beta_{0})\big{)}^{\prime}\in\mathbb{R}^{n}\\\
\mathaccent 0{\symlargesymbols}062{\Pi}&=\big{(}\mathaccent
0{\symlargesymbols}062{}_{1}^{\prime},\dots,\mathaccent
0{\symlargesymbols}062{}_{n}^{\prime}\big{)}^{\prime}\in\mathbb{R}^{n\,\times\,d_{x}}\\\
\mathaccent 0{\symlargesymbols}062{}_{\varepsilon}&=\big{(}\mathaccent
0{\symlargesymbols}062{}_{\epsilon 1}^{\prime},\dots,\mathaccent
0{\symlargesymbols}062{}_{\epsilon
n}^{\prime}\big{)}^{\prime}\in\mathbb{R}^{n\,\times\,d_{x}}\end{split}$ (2.4)

The jackknife K-statistic can then be defined

$\displaystyle\text{\it JK}(\beta_{0})=\epsilon(\beta_{0})^{\prime}\mathaccent
0{\symlargesymbols}062{\Pi}\big{(}\mathaccent
0{\symlargesymbols}062{}_{\epsilon}^{\prime}\mathaccent
0{\symlargesymbols}062{}_{\epsilon}\big{)}^{-1}\mathaccent
0{\symlargesymbols}062{}^{\prime}\epsilon(\beta_{0})\times\bm{1}{\\{\lambda_{\text{min}}(\mathaccent
0{\symlargesymbols}062{}_{\epsilon}^{\prime}\mathaccent
0{\symlargesymbols}062{}_{\epsilon})>0\\}}$ (2.5)

I will show that, under appropriate conditions on the hat matrix, $H$, and
moment bounds, I will show that the limiting distribution of $\text{\it
JK}(\beta_{0})$ under $H_{0}$ is $\chi^{2}_{d_{x}}$. For exposition, I will
largely focus on the case where $d_{x}=1$, in which case the form of the test
statistic simplifies to $\text{\it
JK}(\beta_{0})=\big{(}\sumop\slimits@_{i=1}^{n}\epsilon_{i}(\beta_{0})\mathaccent
0{\symlargesymbols}062{}_{i}\big{)}^{2}/\sumop\slimits@_{i=1}^{n}\epsilon_{i}^{2}(\beta_{0})\mathaccent
0{\symlargesymbols}062{}_{i}^{2}$. The extension to $d_{x}>1$ is not immediate
but is possible under strengthened moment conditions and is explored in
Section 5.

##  3 Limiting Behavior with a Single Endogenous Variable

The limiting behavior of the test statistic is analyzed via a direct Gaussian
approximation technique. When there is a single endogenous variable this
approach can be considerably simplified. In this section, I detail the
approach and take advantage of the simplified analysis to characterize the
limiting behavior of the test statistic under local alternatives to $H_{0}$.
This direct approach has the advantage of not relying on any particular
central limit theorem, which allows a great deal of flexibility in the choice
of hat matrix $H$.For each $i\in[n]$, let
$(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})^{\prime}$ be jointly Gaussian
random variables generated (i) independently of each other and the data and
(ii) with the same mean and covariance matrix as
$(\epsilon_{i}(\beta_{0}),r_{i})^{\prime}$. In addition, define
$\tilde{\Pi}_{i}\coloneqq\sumop\slimits@_{j\neq i}h_{ij}\tilde{r}_{j}$.The
goal will be to show that the quantiles of $\text{\it JK}(\beta_{0})$ can be
approximated by corresponding quantiles of the Gaussian statistic,

$\text{\it
JK}_{G}(\beta_{0})\coloneqq\frac{(\sumop\slimits@_{i=1}^{n}\tilde{\epsilon}_{i}(\beta_{0})\tilde{\Pi}_{i})^{2}}{\sumop\slimits@_{i=1}^{n}\mathbb{E}[\epsilon_{i}^{2}(\beta_{0})]\tilde{\Pi}_{i}^{2}}$
(3.1)

Since uncorrelated jointly Gaussian random variables are independent, under
$H_{0}$ the vector
$(\tilde{\epsilon}_{1}(\beta_{0}),\dots,\tilde{\epsilon}_{n}(\beta_{0}))^{\prime}$
is mean zero and independent of
$(\tilde{r}_{1},\dots,\tilde{r}_{n})^{\prime}$. The null distribution of
$\text{\it JK}_{G}(\beta_{0})$ conditional on any realization of
$(\tilde{r}_{1},\dots,\tilde{r}_{n})^{\prime}$ is then $\chi^{2}_{1}$ and so
its unconditional null distribution is also $\chi^{2}_{1}$.

###  3.1 Interpolation Approach

Error arising from estimation of $\rho(z_{i})$ prevents immediate comparison
of the distribution of $\text{\it JK}(\beta_{0})$ to the distribution of
$\text{\it JK}_{G}(\beta_{0})$. As such, I begin by considering the
distribution of an infeasible statistic, $\text{\it JK}_{I}(\beta_{0})$, which
could be constructed if $\rho(z_{i})$ were known to the researcher:

$\text{\it
JK}_{I}(\beta_{0})\coloneqq\frac{\big{(}\sumop\slimits@_{i=1}^{n}\epsilon_{i}(\beta_{0})\mathaccent
0{\symlargesymbols}062{}_{i}^{I}\big{)}^{2}}{\sumop\slimits@_{i=1}^{n}\epsilon_{i}^{2}(\beta_{0})\big{(}\mathaccent
0{\symlargesymbols}062{}_{i}^{I}\big{)}^{2}}\times\bm{1}{\Big{\\{}\sumop\slimits@_{i=1}^{n}\epsilon_{i}^{2}(\beta_{0})\big{(}\mathaccent
0{\symlargesymbols}062{}_{i}^{I}\big{)}^{2}>0\Big{\\}}}$

where $\mathaccent 0{\symlargesymbols}062{}^{I}_{i}=\sumop\slimits@_{j\neq
i}h_{ij}r_{j}$. To show that the distribution of $\text{\it
JK}_{I}(\beta_{0})$ can be approximated by the distribution of $\text{\it
JK}_{G}(\beta_{0})$, I adapt Lindeberg’s interpolation method, first
introduced by Lindeberg (1922) in an elegant proof of the central limit
theorem. This method consists of one-by-one replacment of the terms
$(\epsilon_{i}(\beta_{0}),r_{i})$ in the expression of $\text{\it
JK}_{I}(\beta_{0})$ with their Gaussian analogs,
$(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})$, and bounding the resulting
one-step deviations.Applying the interpolation method directly on the
statistics $\text{\it JK}_{I}(\beta_{0})$ and $\text{\it JK}_{G}(\beta_{0})$,
however, is not tractable as it requires bounding expectations of derivatives
with respect to terms in the denominator. When identification is weak, the
denominators of $\text{\it JK}_{I}(\beta_{0})$ and $\text{\it
JK}_{G}(\beta_{0})$ may both be arbitrarily close to zero with positive
probability. Derivatives with respect to terms in the denominators thus may
not have finite expectations.Instead, I consider a different approach. For a
scaling factor $s_{n}$, introduced below, define the scaled numerators and
denominators

$\displaystyle N$
$\displaystyle\coloneqq\Big{(}\frac{s_{n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\epsilon_{i}(\beta_{0})\mathaccent
0{\symlargesymbols}062{}_{i}^{I}\Big{)}^{2}$
$\displaystyle\coloneqq\Big{(}\frac{s_{n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\tilde{\epsilon}_{i}(\beta_{0})\tilde{\Pi}_{i}\Big{)}^{2}$
$\displaystyle D$
$\displaystyle\coloneqq\frac{s_{n}^{2}}{n}\sumop\slimits@_{i=1}^{n}\epsilon_{i}^{2}(\beta_{0})\big{(}\mathaccent
0{\symlargesymbols}062{}_{i}^{I}\big{)}^{2}$
$\displaystyle\coloneqq\frac{s_{n}^{2}}{n}\sumop\slimits@_{i=1}^{n}\mathbb{E}[\epsilon_{i}^{2}(\beta_{0})]\big{(}\tilde{\Pi}_{i}\big{)}^{2}$

and for any $a\geq 0$, define the decomposed statistics

$\displaystyle\text{\it JK}^{a}_{I}(\beta_{0})$ $\displaystyle\coloneqq N-aD$
$\displaystyle\text{\it JK}^{a}_{G}(\beta_{0})$
$\displaystyle\coloneqq\tilde{N}-a\tilde{D}$

Since $D=0$ implies $N=0$ and since $\tilde{D}\neq 0$ almost surely, the
events $(\\{\text{\it JK}_{I}(\beta_{0})\leq a\\},\\{\text{\it
JK}_{G}(\beta_{0})\leq a\\})$ are almost surely equivalent to the events
$(\\{\text{\it JK}_{I}^{a}(\beta_{0})\leq 0\\},\\{\text{\it
JK}_{G}^{a}(\beta_{0})\leq 0\\})$. The decomposed statistics no longer have
denominators to be dealt with and are tractable for the interpolation
argument. I show for any $\varphi(\cdot)\in C_{b}^{3}(\mathbb{R})$, the space
of all thrice continuously differentiable functions with bounded derivatives
up to the third order, that there is a fixed constant $M>0$ such that

$\begin{split}|\mathbb{E}[\varphi(\text{\it JK}^{a}_{I})-\varphi(\text{\it
JK}^{a}_{G})]|\leq\frac{M(a^{3}\vee
1)}{\sqrt{n}}(L_{2}(\varphi)+L_{3}(\varphi))\end{split}$ (3.2)

where $L_{2}(\varphi)\coloneqq\sup_{x}|\varphi^{\prime\prime}(x)|$ and
$L_{3}(\varphi)\coloneqq\sup_{x}|\varphi^{\prime\prime\prime}(x)|$. By taking
$\varphi(\cdot)$ to be a sequence of functions approximating the indicator
function, $\bm{1}\\{x\leq 0\\}$, the result in (3.2) can be used to show that
the cumulative distribution function (CDF) of the infeasible statistic
$\text{\it JK}_{I}(\beta_{0})$ can be approximated by the CDF of the Gaussian
statistic $\text{\it JK}_{G}(\beta_{0})$ at each point $a\in\mathbb{R}$. A
Glivenko-Cantelli type argument is then be applied to show the approximation
holds uniformly over all points on the real line.The Lindeberg interpolation
argument on the decomposed test statistics makes use of the fact that the
numerator and denominator of the Gaussian test statistic are functions of
quadratic forms in the random vectors
$\epsilon(\beta_{0})\coloneqq(\epsilon_{1}(\beta_{0}),\dots,\epsilon_{n}(\beta_{0}))^{\prime}$
and $r\coloneqq(r_{1},\dots,r_{n})^{\prime}$.111See Pouzo (2015) for another
example of the Lindeberg interpolation method applied to approximate
thedistribution of quadratic forms.Moving from approximation of expectations
of smooth functions to approximation of the CDF relies on a particular
anticoncentration bound on $\tilde{D}$. I show that that this bound can be
established under either weak or strong identification. This allows for the
limiting null distribution of the test statistic under various identification
regimes to be derived via a unifying argument. Additionally, even though
$(N,D,\tilde{N},\tilde{D})$ may all have nonnegligible distributions when
identification is weak, the interpolation argument does not require any of
these to individually converge in distribution or probability anywhere stable.
This allows for a wide range of possible hat matrices $H$ to be used in
constructing the first stage estimates, $(\mathaccent
0{\symlargesymbols}062{}_{1},\dots,\mathaccent 0{\symlargesymbols}062{}_{n})$.
In particular, no assumption need be made on the number of instruments used to
construct $H$ nor any requirement imposed that the first-stage estimates
$(\mathaccent 0{\symlargesymbols}062{}_{1},\dots,\mathaccent
0{\symlargesymbols}062{}_{n})$ are consistent.I now detail the assumptions
needed for the argument.Define
$\eta_{i}\coloneqq(\beta-\beta_{0})v_{i}+\epsilon_{i}$ and $\zeta_{i}\coloneqq
v_{i}-\rho(z_{i})\eta_{i}$, noting
$\eta_{i}=\epsilon_{i}(\beta_{0})-\mathbb{E}[\epsilon_{i}(\beta_{0})]$ and
$\zeta_{i}=r_{i}-\mathbb{E}[r_{i}]$.In what comes below $c>1$ can be
considered an arbitrary constant that may be updated upon each use but that
does not depend on sample size $n$.

######  Assumption 3.1 (Moment Conditions).

There is a fixed constant $c>1$ such that (i)
$\\{|{}_{i}|+|(\beta-\beta_{0})|+|\rho(z_{i})|\\}\leq c$, and (ii) for any
$l,k\in\mathbb{N}\cup\\{0\\}$ such that $l+k\leq 6$,
$c^{-1}\leq\mathbb{E}[|\eta_{i}|^{l}|\zeta_{i}|^{k}]\leq c$.

######  Assumption 3.2 (Balanced Design).

(i) For $s_{n}^{-2}=\max_{i}\mathbb{E}[(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2}]$ the following is bounded away from
zero,
$c^{-1}\leq\mathbb{E}[\frac{s_{n}^{2}}{n}\sumop\slimits@_{i=1}^{n}(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2}]$; (ii)
$\max_{i}s_{n}^{2}\sumop\slimits@_{j\neq i}h_{ji}^{2}\leq c$; and (iii) the
following ratio is bounded away from zero:
$\frac{\sumop\slimits@_{k=2}^{n}\lambda_{k}^{2}(HH^{\prime})}{\sumop\slimits@_{k=1}^{n}\lambda_{k}^{2}(HH^{\prime})}\geq
c^{-1}$ where $\lambda_{k}(HH^{\prime})$ represents the $k^{\text{th}}$
largest eigenvalue of the matrix $HH^{\prime}$.

Assumptions 3.1 and 3.2 allow characterization of the null distribution of
$\text{\it JK}(\beta_{0})$.Assumption 3.1 imposes light moment conditions on
the random variables $\eta_{i}$ and $\zeta_{i}$, which in turn imply
restrictions on $\epsilon_{i}(\beta_{0})$ and $r_{i}$. In particular,
Assumption 3.1(i) imposes that $\epsilon_{i}(\beta_{0})$ and $r_{i}$ have
finite means while Assumption 3.1(ii) bounds, both from above and away from
zero, the first through sixth central moments of the random
variables.Assumption 3.2(i) requires that the average second moment of the
infeasible first-stage estimators be on the same order as the maximum first-
stage estimator second moment. This is imposed mainly to rule out hat matrices
that are all zeroes or nearly all zeros so that the effective number of
observations used to test the null is growing with the sample size. Remark 3.1
below discusses how this assumption and Assumption 3.2(ii) may be verified in
practice. Remark 3.2 compares this balanced design assumption to that in the
many-instruments literature (Crudu et al., 2021; Mikusheva and Sun, 2021;
Matsushita and Otsu, 2022; Lim et al., 2022), noting that their balanced
design neither implies nor is implied by the one in this paper.Assumption
3.2(ii) requires that the maximum leverage of any observation be bounded. When
$H$ is symmetric, it is automatically satisfied under Assumption 3.1(i) and
the definition of $s_{n}$.222To see this, notice that
$s_{n}^{-2}=\max_{i}\mathbb{E}[(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2}]\geq\max_{i}\operatorname{Var}(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})=\max_{i}\sumop\slimits@_{j\neq
i}h_{ij}^{2}\operatorname{Var}(r_{j})$. By Assumption 3.1,
$\operatorname{Var}(r_{j})$ is bounded from below by $c^{-1}$. Inverting this
chain of inequalities yields that $s_{n}^{2}\sumop\slimits@_{j\neq
i}h_{ij}^{2}$ is bounded from above uniformly over all $i\in[n]$.The scaling
factor $s_{n}$ captures both the “size” of the elements in the hat matrix $H$
and the strength of identification. If elements of the hat matrix are on the
same order as a constant, one would expect $s_{n}=O(n^{-1})$ under strong
identification (${}_{i}\propto 1$) while $s_{n}=O(n^{-1/2})$ under weak
identification (${}_{i}\lesssim n^{-1/2}$).Assumption 3.2(iii) can be viewed
as a technical requirement that there be more than one “effective” instrument
in the hat matrix.333In the case of a standard projection matrix (no deleted
diagonal), Assumption 3.2(iii) would be satisfied whenever
$\text{rank}(z(z^{\prime}z)^{-1}z)>1$. This condition can be easily verified
in practice by examining the eigenvalues of $HH^{\prime}$.In addition to
characterizing the limiting distribution of $\text{\it JK}(\beta_{0})$ under
$H_{0}$, I also examine the behavior of $\text{\it JK}(\beta_{0})$ in local
neighborhoods of the null. These local neighborhoods are characterized by the
local power index $P$, defined below, as well as an additional regularity
condition that restricts the size of $\mathbb{E}[\epsilon_{i}(\beta_{0})]$
relative to $\mathbb{E}[r_{i}]$.

$P\coloneqq(\beta-\beta_{0})^{2}\mathbb{E}\bigg{[}\bigg{(}\frac{s_{n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}{}_{i}\mathaccent
0{\symlargesymbols}062{}_{i}^{I}\bigg{)}^{2}\bigg{]}$

######  Assumption 3.3 (Local Identification).

(i) The local power index $P$ is bounded, $P\leq c$; and (ii)
$\max_{i}\mathbb{E}[(s_{n}\sumop\slimits@_{j\neq
i}h_{ji}\epsilon_{j}(\beta_{0}))^{2}]\leq c$.

Under $H_{0}$, Assumption 3.3 is trivially satisfied since
$(\beta-\beta_{0})=0$ and $\sumop\slimits@_{j\neq i}s_{n}^{2}h_{ji}^{2}\leq
c$. The local power index is the second moment of the scaled numerator, $N$
and is a measure of the association between the true first stage i and the
first-stage estimates $\mathaccent 0{\symlargesymbols}062{}_{i}$. In Section
4, I discuss how the strength of this association is related to the power of
the test under local alternatives. Proposition 3.1 below shows that when
Assumption 3.3(ii) holds, $P\to\infty$ implies that the test based on the
infeasible statistic $\text{\it JK}_{I}(\beta_{0})$ is consistent.Assumption
3.3(ii) is an additional technical condition that requires that the maximum
value of $\mathbb{E}[(\sumop\slimits@_{j\neq
i}h_{ji}\epsilon_{j}(\beta_{0}))^{2}]$ be on the same or lesser order than the
maximum value of $\mathbb{E}[(\sumop\slimits@_{j\neq i}h_{ij}r_{j})^{2}]$.
Using the moment bounds in Assumption 3.1 and Assumption 3.2(ii) one can
verify that Assumption 3.3(ii) is equivalent to the existence of constants
$C_{1},C_{2}>0$ such that

$\displaystyle\max_{i}\big{(}\sumop\slimits@_{j\neq
i}h_{ji}\mathbb{E}[\epsilon_{j}(\beta_{0})]\big{)}^{2}$ $\displaystyle\leq
C_{1}\max_{i}\mathbb{E}\big{[}(\sumop\slimits@_{j\neq
i}h_{ij}r_{j})^{2}\big{]}+C_{2}$
$\displaystyle=C_{1}\max_{i}\bigg{\\{}\sumop\slimits@_{j\neq
i}h_{ij}^{2}\operatorname{Var}(r_{j})+\big{(}\sumop\slimits@_{j\neq
i}h_{ij}\mathbb{E}[r_{j}]\big{)}^{2}\bigg{\\}}+C_{2}$

for all $i\in[n]$. It is always satisfied whenever
$\mathbb{E}[\epsilon_{i}(\beta_{0})]={}_{i}(\beta-\beta_{0})$ is in a
$\sqrt{n}$-neighborhood of zero in the sense that
$|{}_{i}(\beta-\beta_{0})|\leq C/\sqrt{n}$ for all $i\in[n]$ and some constant
$C$.In general, Assumption 3.3(ii) can be roughly interpreted as requiring the
local neighborhoods of $H_{0}$ considered to be those in which the means of
$(\epsilon_{1}(\beta_{0}),\dots,\epsilon_{n}(\beta_{0}))$ are of the same or
lesser order than the means of $(r_{1},\dots,r_{n})$.Under Assumptions 3.1,
3.2 and 3.3, I establish a main technical lemma stating that the CDF of the
infeasible statistic, $\text{\it JK}_{I}(\beta_{0})$, can be uniformly
approximated by the CDF of the Gaussian statistic, $\text{\it
JK}_{G}(\beta_{0})$. This result does not require $\text{\it
JK}_{G}(\beta_{0})$ to have a fixed limiting distribution.

######  Lemma 3.1 (Infeasible Uniform Approximation).

Suppose that Assumptions 3.1, 3.2 and 3.3 hold. Then,

$\sup_{a\in\mathbb{R}}\big{|}\mathrm{Pr}(\text{\it JK}_{I}(\beta_{0})\leq
a)-\mathrm{Pr}(\text{\it JK}_{G}(\beta_{0})\leq a)\big{|}\to 0$

I additionally show that the test based on the $\text{\it JK}_{I}(\beta_{0})$
statistic is consistent whenever the power index diverges, $P\to\infty$, and
Assumption 3.3(ii) holds.

######  Proposition 3.1 (Consistency).

Suppose that Assumptions 3.1, 3.2, and 3.3(ii) hold. Then if $P\to\infty$ the
test based on $\text{\it JK}_{I}(\beta_{0})$ is consistent; i.e for any fixed
$a\in\mathbb{R}$, $\mathrm{Pr}(\text{\it JK}_{I}(\beta_{0})\leq a)\to 0$.

The dependence of the consistency result on Assumption 3.3(ii) is a nontrivial
restriction because of the bias taken on in constructing $r_{i}$. In
particular, against certain alternatives it is possible that
$\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{i}^{I}]=0$ for all $i\in[n]$
even under strong identification. This is an extreme case, however. In
general, bias in $\mathbb{E}[r_{i}]$ does not imply a violation of Assumption
3.3(ii), which requires only that the _size_ of $\mathbb{E}[r_{i}]$ be of a
weakly greater order than that of $\mathbb{E}[\epsilon_{i}(\beta_{0})]$.
Moreover, as discussed in Remark 3.5, Proposition 3.1 does not necessarily
rule out consistency when $P\to\infty$ but Assumption 3.3(ii)
fails.Regardless, bias taken on in constructing $r_{i}$ has consequences for
the power of the test in finite samples. This is particularly true when the
mean of $r_{i}$ is of a lesser order than that of $\epsilon_{i}(\beta_{0})$ as
will be discussed in Section 4. To rectify this deficiency in tests based on
the jackknife K-statistic, I suggest a thresholding test that decides whether
to use the jackknife K-statistic or the sup-score Belloni et al. (2012)
statistic based on the value of the conditioning statistic. This conditioning
statistic, in turn, is based on a test statistic for the null hypothesis that
$\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{i}^{I}]=0$ for all
$i\in[n]$.

###  3.2 Limiting Behavior of Test Statistic

The final step in characterizing the limiting behavior of the feasible test
statistic is to show that the difference between the infeasible and feasible
statistics is negligible. I begin with a technical lemma stating that the
difference between $\text{\it JK}(\beta_{0})$ and $\text{\it
JK}_{I}(\beta_{0})$ is asymptotically negligible whenever the differences
between the scaled numerators and the scaled denominators are asymptotically
negligible. Define these differences:

N
$\displaystyle\coloneqq\frac{s_{n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\epsilon_{i}(\beta_{0})(\mathaccent
0{\symlargesymbols}062{}_{i}-\mathaccent 0{\symlargesymbols}062{}_{i}^{I})$ D
$\displaystyle\coloneqq\frac{s_{n}^{2}}{n}\sumop\slimits@_{i=1}^{n}\epsilon_{i}^{2}(\beta_{0})(\mathaccent
0{\symlargesymbols}062{}_{i}^{2}-(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2})$

######  Lemma 3.2.

Suppose Assumptions 3.1, 3.2 and 3.3 hold and
$({}_{N},{}_{D})^{\prime}\to_{p}0$. Then $|\text{\it JK}(\beta_{0})-\text{\it
JK}_{I}(\beta_{0})|\to_{p}0$.

While Lemma 3.2 is a simple statement, it is not obvious. In particular,
showing that the difference between the infeasible and feasible statistics is
negligible requires showing that $1/(D+{}_{D})$ is bounded in probability,
where $D$ represents the scaled denominator of $\text{\it JK}_{I}(\beta_{0})$.
In a standard analysis, this would be done by arguing that $D$ converges in
distribution to a stable limit and then applying the continuous mapping
theorem.444This is the approach taken by Kleibergen (2002, 2005) This approach
is not applicable here as neither the scaled numerator nor the scaled
denominator has a limiting distribution.Instead, I directly show that
$1/(D+{}_{D})$ is bounded in probability by showing
$\mathrm{Pr}(D\leq\delta_{n})\to 0$ for any sequence $\delta_{n}\to 0$. This
is done by first establishing that quantiles of $D$ can be approximated by
quantiles of $\tilde{D}$, the scaled denominator of $\text{\it
JK}_{G}(\beta_{0})$. If the variance of $\tilde{D}$ is bounded away from zero,
its density can also be bounded with new bounds on Gaussian quadratic form
densities from Götze et al. (2019), which yields the result. Otherwise, if
$\operatorname{Var}(\tilde{D})\to 0$, the result holds by an application of
Chebyshev’s inequality and $\mathbb{E}[D]>c^{-1}$ from Assumption 3.2(i).This
particular anticoncentration bound for $\tilde{D}$ is also important in the
proof of Lemma 3.1 to establish anticoncentration for the decomposed Gaussian
test statistic.Lemma 3.2 allows the researcher to use alternate choices of
estimators for $\rho(z_{i})$, so long as they can verify that
$({}_{N},{}_{D})^{\prime}\to_{p}0$. Below, I verify that this condition can be
satisfied for the $\ell_{1}$-penalized estimation procedure proposed in (2.2).
This requires a strengthened moment condition on $\eta_{i}$. Given a random
variable $X$ and $\upsilon>0$ the Orlicz (quasi-)norm is defined

$\|X\|_{\psi_{\upsilon}}\coloneqq\inf\\{t>0\mathrel{\mathop{\ordinarycolon}}\mathbb{E}\exp(|X|^{\upsilon}/t^{\upsilon})\leq
2\\}$

Random variables with a finite Orlicz norm for some
$\upsilon\in(0,1]\cup\\{2\\}$ are termed $\alpha$-sub-exponential random
variables (Gotze et al., 2021; Sambale, 2022). This class encompasses a wide
range of potential distributions including all bounded and sub-Gaussian random
random variables (with $\upsilon=2$), all sub-exponential random variables
such as Poisson or noncentral $\chi^{2}$ random variables (with $\upsilon=1$),
as well as random variables with “fatter” tails such as Weibull distributed
random variables with shape parameter $\upsilon\in(0,1]$.

######  Assumption 3.4 (Estimation Error).

(i) There is a fixed constant $\upsilon\in(0,1]\cup\\{2\\}$ such that
$\|\eta_{i}\|_{\psi_{\upsilon}}\leq c$; (ii) The basis terms $b(z_{i})$ are
bounded, $\|b(z_{i})\|_{\infty}\leq C$ for all $i=1,\dots,n$; (iii) the
approximation error satisfies
$(\mathbb{E}_{n}[\xi_{i}^{2}])^{1/2}=o(n^{-1/2})$; (iv) the researcher has
access to an estimator $\mathaccent 0{\symlargesymbols}062{\phi}$ of $\phi$
that satisfies $\log(d_{b}n)^{2/(\upsilon\wedge 1)}\|\mathaccent
0{\symlargesymbols}062{\phi}-\phi\|_{1}\to_{p}0$; (v) the following moment
bounds hold

  1. (va)

$\max_{1\leq\ell\leq
d_{b}}\big{|}\mathbb{E}\big{[}\frac{s_{n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\sumop\slimits@_{j\neq
i}h_{ij}\epsilon_{i}(\beta_{0})b_{\ell}(z_{j})\epsilon_{j}(\beta_{0})\big{]}\big{|}\leq
c$

  2. (vb)

$\max_{\begin{subarray}{c}1\leq i\leq n\\\ 1\leq\ell\leq
d_{b}\end{subarray}}|\mathbb{E}[s_{n}\sumop\slimits@_{j\neq
i}h_{ij}b_{\ell}(z_{j})\epsilon_{j}(\beta_{0})]|\leq c.$

Assumption 3.4(i) strengthens the moment condition on $\eta_{i}$ to require
that $\eta_{i}$ be in the class of $\alpha$-sub-exponential random variables.
While this condition is more restrictive than the moment condition in
Assumption 3.1, as discussed above, it still allows for a wide range of
potential distributions. Assumption 3.4(ii) is a standard condition in
$\ell_{1}$-penalized estimation. At the cost of extra notation, it can be
relaxed and the sup-norm of the basis terms can be allowed to grow slowly with
the sample size to accommodate bases such as normalized b-splines or wavelets.
Assumption 3.4(iii) is a bound on the rate of decay ofthe approximation error,
similar to the approximate sparsity condition of Belloni et al.
(2012).Assumption 3.4(iv) is a high-level condition on the rate of consistency
of the parameter estimate $\hat{\phi}$ in the $\ell_{1}$ norm. This can be
verified under approximate sparsity for both the LASSO estimator in (2.2) or
post-LASSO procedures based on refitting an unpenalized version of (2.2) only
using the basis terms selected in a LASSO first stage. See Belloni et al.
(2012); van der Greer (2016); Tan (2017), and Chetverikov and Sørensen (2021)
for references under various choices of penalty parameter. This condition
allows for the dimensionality of the basis terms, $d_{b}$, to grow near
exponentially as a function of the sample size. Following the analysis of Tan
(2017) one can see that, under appropriate choice of penalty parameter, this
may be satisfied as long as $s^{2}\log^{2(\upsilon+1)/\upsilon}(d_{b}n)/n\to
0$, where the sparsity index $s$ denotes the number of nonzero elements of
$\phi$.Assumption 3.4(v) is a strengthening of the definition of local
neighborhoods and can be interpreted similarly to Assumption 3.3(ii). Since
the moment conditions in Assumption 3.4(va,vb) hold with
$b_{\ell}(z_{j})\epsilon_{j}(\beta_{0})$ replaced with $r_{j}$, Assumption
3.4(v) can be interpreted as requiring that
$|\mathbb{E}[\sumop\slimits@_{j\neq
i}h_{ij}b_{\ell}(z_{j})\epsilon_{j}(\beta_{0})]|$ is on the same order as
$|\mathbb{E}[\sumop\slimits@_{j\neq i}h_{ij}r_{j}]|$ for all $i=1,\dots,n$ and
$\ell=1,\dots,d_{b}$. As with Assumption 3.3(ii), it is trivially satisfied
under $H_{0}$ or, using the fact that $\max_{i}\sumop\slimits@_{j\neq
i}s_{n}^{2}h_{ij}^{2}\leq c$, whenever
$\mathbb{E}[\epsilon_{i}(\beta_{0})]={}_{i}(\beta-\beta_{0})$ is in a
$\sqrt{n}$-neighborhood of zero.Under Assumptions 3.1, 3.2, 3.3 and 3.4, I
establish that the difference between the infeasible and feasible statistics
can be treated as negligible when the estimation procedure proposed in (2.2)
is used.

######  Lemma 3.3.

Suppose that Assumptions 3.1, 3.3, 3.2 and 3.4 hold. Then
$({}_{N},{}_{D})^{\prime}\to_{p}0$.

Lemmas 3.1, 3.2 and 3.3 are combined for the main result, local approximation
of the distribution of the feasible test statistic, $\text{\it
JK}(\beta_{0})$, by the distribution of the Gaussian statistic, $\text{\it
JK}_{G}(\beta_{0})$. An immediate corollary is that the limiting null
distribution of $\text{\it JK}(\beta_{0})$ is $\chi^{2}_{1}$.

######  Theorem 3.1 (Uniform Approximation).

Suppose that Assumptions 3.1, 3.3, 3.2 and 3.4 hold. Then

$\sup_{a\in\mathbb{R}}\big{|}\mathrm{Pr}(\text{\it JK}(\beta_{0})\leq
a)-\mathrm{Pr}(\text{\it JK}_{G}(\beta_{0})\leq a)\big{|}\to 0$

######  Corollary 3.1 (Size Control).

Suppose that Assumptions 3.1, 3.2 and 3.4 hold. Then, under $H_{0}$,
$\text{\it JK}(\beta_{0})\rightsquigarrow\chi^{2}_{1}$.

If the limiting $\text{\it JK}_{G}(\beta_{0})$ had a fixed distribution under
$H_{1}$,Theorem 3.1 would follow immediately from Lemmas 3.1, 3.2 and 3.3, and
an application of Slutsky’s lemma. However, under $H_{1}$, there is nothing
preventing the distribution of $\text{\it JK}_{G}(\beta_{0})$ changing with
the sample size. Instead I establish Theorem 3.1 directly using the fact that
both $\text{\it JK}(\beta_{0})$ and $\text{\it JK}_{G}(\beta_{0})$ are bounded
in probability and that $\text{\it JK}_{G}(\beta_{0})$ has a density that is
bounded uniformly over $n$.While $\text{\it JK}_{G}(\beta_{0})$ does not have
a fixed distribution, examining its behavior is still tractable and allows for
insight into the power properties of the jackknife K-test. In the next
section, I use this result to analyze the local power of the proposed test. To
improve power against certain alternatives, I suggest a combination with the
sup-score statistic of Belloni et al. (2012).

######  Remark 3.1.

A sufficient condition for Assumption 3.2(i) is that there is some fixed
quantile $q\in(0,100)$ such that $(cq)^{-1}\leq\frac{q^{\text{\tiny
th}}\text{-quantile of }\mathbb{E}[(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2}]}{\max_{i}\mathbb{E}[(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2}]}$. In practice this can be verified by
checking that there is some quantile $q$ such that both

$\frac{q^{\text{\tiny th}}\text{-quantile of }\sumop\slimits@_{j\neq
i}h_{ij}^{2}}{\max_{i}\sumop\slimits@_{j\neq i}h_{ij}^{2}}\hbox{ }\text{ and
}\hbox{ }\frac{q^{\text{\tiny th}}\text{-quantile of }(\sumop\slimits@_{j\neq
i}h_{ij}\hat{r}_{j})^{2}}{\max_{i}(\sumop\slimits@_{j\neq
i}h_{ij}\hat{r}_{j})^{2}}$ (3.3)

are bounded away from zero. Similarly, Assumption 3.2(ii) can be verified by
checking that $\max_{i}\sumop\slimits@_{j\neq
i}h_{ji}^{2}/\max_{i}\sumop\slimits@_{j\neq i}h_{ij}^{2}$ is bounded from
above.

######  Remark 3.2.

The balanced-design condition in Assumption 3.2(i) is neither weaker nor
stronger than that in the many instruments literature (Crudu et al., 2021;
Mikusheva and Sun, 2021; Matsushita and Otsu, 2022; Lim et al., 2022). These
papers require that the projection matrix
$P=\bm{z}(\bm{z}^{\prime}\bm{z})^{-1}\bm{z}^{\prime}$ satisfies
$[P]_{ii}\leq\delta\leq 1$ for some value $\delta$ and all $i\in[n]$. Since
$P$ is idempotent, $[P]_{ii}=1$ for some $i\in[n]$ implies that $[P]_{ij}=0$
for $j\neq i$.555Since $P$ is idempotent,
$[P]_{ii}=\sumop\slimits@_{i=1}^{n}[P]_{ij}^{2}=[P]_{ii}^{2}+\sumop\slimits@_{j\neq
i}[P]_{ij}^{2}$. This would not violate Assumption 3.2 if one were to take $H$
such that $h_{ij}=[P]_{ij}\bm{1}\\{i\neq j\\}$; $\mathbb{E}[(\mathaccent
0{\symlargesymbols}062{}_{i}^{I})^{2}]=0$ is allowed for a constant share of
$i\in[n]$. Conversely, if the instruments are fixed or growslowly, it is
possible to construct a projection matrix $P$of rank $d_{z}$ where $[P]_{ii}$
is bounded away from one for all $i\in[n]$, but “most” of the rows are zero. I
view this as a theoretical edge case, however, that seems unlikely to result
from real data.

######  Remark 3.3.

The Lindeberg interpolation method allows me to give a nearly uniform explicit
bound on the Gaussian approximation error. In particular, using the bound in
(3.2), I show that for any fixed value $\Delta>0$;

$\displaystyle\sup_{a\leq\Delta}\big{|}\mathrm{Pr}(\text{\it
JK}_{I}(\beta_{0})\leq a)-\mathrm{Pr}(\text{\it JK}_{G}(\beta_{0})\leq
a)\big{|}\leq Cn^{-2/13}$

where $C$ is a constant that depends only on $(c,\Delta)$. Lemma 3.1 makes use
of the fact that the limiting statistic $\text{\it JK}_{G}(\beta_{0})$ is
bounded in probability and extends this result to show that the approximation
error tends to zero uniformly over the real line. While it does not account
for estimation error in $\hat{\rho}(\cdot)$, obtaining an explicit bound
reflects an improvement over the original analyses of K-statistics in
Kleibergen (2002, 2005). These original studies rely on continuous mapping
theorems to obtain the limiting chi-squared distributions, making the rate of
decay of the approximation error difficult to analyze.

######  Remark 3.4.

The interpolation argument relies on the fact that the first and second
moments of $(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})$ are the same as
the first and second moments of $(\epsilon_{i}(\beta_{0}),r_{i})$ to match the
first and moments of one-step deviations with Gaussian analogs. Without the
jackknife form of $\mathaccent 0{\symlargesymbols}062{}_{i}^{I}$, these one
step deviations would additionally contain cross-terms such as
$h_{ii}r_{i}\epsilon_{i}(\beta_{0})$, for $i\in[n]$. While the first moment of
this cross-term is matched by the first moment of the Gaussian analog,
$h_{ii}\tilde{\epsilon}_{i}(\beta_{0})\tilde{r}_{i}$, the second moment is not
matched. This is manageable, however, so long as the terms $h_{ii}$ are
“small.” An example of when the $h_{ii}$ terms are small is when $H$ is taken
to be the OLS projection matrix, $H=\bm{z}(\bm{z}^{\prime}\bm{z})^{-1}\bm{z}$,
and the number of instruments satisfies $d_{z}^{3}/n\to 0$. See Appendices A
and E for details.

######  Remark 3.5.

Proposition 3.1 does not necessarily rule out that a test based on $\text{\it
JK}_{I}(\beta_{0})$ is consistent when $P\to\infty$ but Assumption 3.3(ii)
fails to hold.The proof of Proposition 3.1 relies on showing that, when
$P\to\infty$ and Assumption 3.3(ii) holds, $\mathbb{E}[|N|]\to\infty$ while
$\operatorname{Var}(|N|)$ and $\mathbb{E}[D]$ are bounded. These facts can be
combined to show that $\mathrm{Pr}(N^{2}-aD\leq 0)\to 0$ for any fixed
$a\in\mathbb{R}$. When Assumption 3.3(ii) fails, $P\to\infty$ may imply that
$\operatorname{Var}(|N|)\to\infty$ as well, making the limiting behavior of
the test difficult to analyze. There is reason to believe that this issue can
be overcome, Andrews et al. (2004) show that the K-statistic of Kleibergen
(2002) is consistent against fixed alternatives under strong identification.
However, a full consistency result is not pursued here and left to future
work.

######  Remark 3.6.

Approximate sparsity of $\rho(z_{i})$ may be a particularly
palatableassumption in cases where the instrument set is generated by
functions of a smaller initial set of instruments, as in Angrist and Krueger
(1991); Paravisini et al. (2014); Gilchrist and Sands (2016), and Derenoncourt
(2022). In these cases, the dimensionality of the basis, $d_{b}$, may not need
to be much larger than the dimensionality of the instruments, $d_{z}$, to
provide a good approximation of $\rho(z_{i})$.Interestingly, if taking
$b(z_{i})=z_{i}$ provides a good approximation of $\rho(z_{i})$, the Tan
(2017) result suggests that consistency of $\hat{\rho}(\cdot)$ is achievable
under $d_{z}^{2}\log^{2(\upsilon+1)/\upsilon}(d_{z}n)/n\to 0$ even if $\phi$
is fully dense. This requirement is weaker than the $d_{z}^{3}/n\to 0$
requirement of the standard K-statistic.

##  4 Improving Power against Certain Alternatives

Using the characterization of the limiting behavior of the test statistic
derived in Section 3, I analyze the local power properties of the test.
Unfortunately, against certain alternatives the test statistic may have
trivial power, a deficiency shared with the K-statistics of Kleibergen (2002,
2005). To combat this, I propose a simple combination with the sup-score
statistic of Belloni et al. (2012) based on a thresholding rule.

###  4.1 Local Power Properties

In local neighborhoods of $H_{0}$, as defined in Assumptions 3.3 and 3.4,
Theorem 3.1 implies that the limiting behavior of $\text{\it JK}(\beta_{0})$
can be analyzed by examining the behavior of the Gaussian analog statistic,
$\text{\it JK}_{G}(\beta_{0})$. Conditional on the vector
$\tilde{r}=(\tilde{r}_{1},\dots,\tilde{r}_{n})$, the distribution of
$\text{\it JK}_{G}(\beta_{0})$ is nearly non-central $\chi^{2}_{1}$ with
noncentrality parameter $\mu(\tilde{r})$, $\text{\it
JK}_{G}(\beta_{0})|\tilde{r}\sim
A^{2}(\tilde{r})\cdot\chi^{2}_{1}(\mu(\tilde{r}))$:

$\displaystyle A(\tilde{r})$
$\displaystyle=\frac{\sumop\slimits@_{i=1}^{n}\operatorname{Var}(\eta_{i})\tilde{\Pi}_{i}^{2}}{\sumop\slimits@_{i=1}^{n}\\{{}_{i}^{2}(\beta-\beta_{0})^{2}+\operatorname{Var}(\eta_{i})\\}\tilde{\Pi}_{i}^{2}}$
$\displaystyle\mu^{2}(\tilde{r})$
$\displaystyle=(\beta-\beta_{0})^{2}\frac{\big{(}\sumop\slimits@_{i=1}^{n}{}_{i}\tilde{\Pi}_{i}\big{)}^{2}}{\sumop\slimits@_{i=1}^{n}\\{{}_{i}^{2}(\beta-\beta_{0})^{2}+\operatorname{Var}(\eta_{i})\\}\tilde{\Pi}_{i}^{2}}.$

Under local alternatives, the terms ${}_{i}^{2}(\beta-\beta_{0})^{2}\to 0$ so
that $A(\tilde{r})\to 1$ and
$|\mu^{2}(\tilde{r})-\mu_{\infty}^{2}(\tilde{r})|\to 0$, where

$\mu^{2}_{\infty}(\tilde{r})=(\beta-\beta_{0})^{2}\frac{\big{(}\sumop\slimits@_{i=1}^{n}{}_{i}\tilde{\Pi}_{i})^{2}}{\sumop\slimits@_{i=1}^{n}\operatorname{Var}(\eta_{i})\tilde{\Pi}_{i}^{2}}.$
(4.1)

The numerator of $\mu^{2}_{\infty}(\tilde{r})$ suggests that power is
maximized when the first-stage estimate $\tilde{\Pi}_{i}$ is close to the true
first stage value i. Indeed, when errors are homoskedastic
$\mu_{\infty}^{2}(\tilde{r})$ is maximized by setting $\tilde{\Pi}_{i}={}_{i}$
reflecting the classical result of Chamberlain (1987). The denominator of
$\mu^{2}_{\infty}(\tilde{r})$ suggests that having first-stage estimates
$\tilde{\Pi}_{i}$ with low second moments may increase power. This guides the
recommendation for the use of $\ell_{2}$-regularization in constructing the
hat matrix, $H$.Unfortunately, estimators of i based on
$r_{i}=x_{i}-\rho(z_{i})\epsilon_{i}(\beta_{0})$ may not be close to i under
$H_{1}$. This is because the mean of $r_{i}$ will in general differ from i

$\mathbb{E}[r_{i}]={}_{i}-\rho(z_{i}){}_{i}(\beta-\beta_{0})$

This deficiency is inherited from the similarity of the $\text{\it
JK}(\beta_{0})$ statistic to the K-statistic.As pointed out by Moreira (2001),
this need not be an issue as long as there is a fixed constant $C\neq 0$ such
that $\mathbb{E}[r_{i}]=C{}_{i}$ for all $i\in[n]$. However, in general, this
will introduce bias into the first-stage estimates $\mathaccent
0{\symlargesymbols}062{}_{i}$ under $H_{1}$. The power implications of this
bias are particularly pronounced when $\rho(z_{i})$ is a constant
$(\beta-\beta_{0})=1/\rho(z_{i})$. In this case, $\mathbb{E}[r_{i}]$, and thus
$\mathbb{E}[\tilde{\Pi}_{i}]$, will equal zero for each $i\in[n]$, and the
$\text{\it JK}(\beta_{0})$ statistic will select a direction completely at
random to direct power into.111Andrews et al. (2006) and Andrews (2016) point
out this deficiency in the context of the K-statistics of Kleibergen (2002,
2005).

###  4.2 A Simple Combination Test

To combat this loss of power for tests based on the K-statistic, a common
strategy is to combine the K-statistic with the Anderson-Rubin statistic based
on a conditioning statistic. While the Anderson-Rubin statistic does not have
optimal power on its own, it has the benefit of directing power equally in all
directions avoiding the pitfalls of the K-statistic which lacks power in
certain directions. Prominent examples of such tests are the conditional
likelihood ratio test of Moreira (2003), the GMM-M test of Kleibergen (2005),
and the minimax regret tests of Andrews (2016). These combinations make use of
the fact that the Anderson-Rubin statistic is asymptotically independent of
both the K-statistic and the conditioning statistic.Unfortunately, the
asymptotic validity of these tests under heteroskedasticity is based on the
assumption that $d_{z}^{3}/n\to 0$, which may not reasonably describe many
settings discussed above. Instead, to improve the power of tests based on the
jackknife K-statistic, I consider a simple combination with the sup-score
statistic of Belloni et al. (2012). The test based on the sup-score statistic
(4.2) is similar in spirit to the Anderson-Rubin test but controls size even
when $d_{z}$ grows near exponentially as a function of the sample size.

$S(\beta_{0})\coloneqq\sup_{1\leq\ell\leq
d_{z}}\bigg{|}\frac{\sumop\slimits@_{i=1}^{n}\epsilon_{i}(\beta_{0})z_{\ell
i}}{(\sumop\slimits@_{i=1}^{n}z_{\ell i}^{2})^{1/2}}\bigg{|}$ (4.2)

A size $\theta\in(0,1)$ test based on the sup-score statistic rejects whenever
$S(\beta_{0})>c^{S}_{1-\theta}$ where, for $e_{1},\dots,e_{n}$ iid standard
normal and generated independently of the data, $c^{S}_{1-\theta}$ is the
simulated multiplier bootstrap critical value:

$c^{S}_{1-\theta}\coloneqq(1-\theta)\text{ quantile of }\sup_{1\leq\ell\leq
d_{z}}\bigg{|}\frac{\sumop\slimits@_{i=1}^{n}e_{i}\epsilon_{i}(\beta_{0})z_{\ell
i}}{(\sumop\slimits@_{i=1}^{n}z_{\ell i}^{2})^{1/2}}\bigg{|}\text{ conditional
on }\\{(y_{i},x_{i},z_{i})\\}_{i=1}^{n}.$

As with the Anderson-Rubin test, tests based on the sup-score statistic may
have suboptimal power properties in overidentified models as it does not
incorporate first-stage information. However, the sup-score statistic does
retain the benefit of directing power evenly in all directions, avoiding
pitfalls of tests based on $\text{\it JK}(\beta_{0})$ against certain
alternatives.The combination test will be based on an attempt to detect
whether the alternative $\beta$ is such that $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{i}^{I}]=0$ for all $i=1,\dots,n$. When this is the
case, the test based on $\text{\it JK}(\beta_{0})$ will choose a direction
completely at random to direct power into. It would then be optimal for the
researcher to test the null hypothesis using the sup-score statistic.
Detection of whether $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{i}^{I}]=0$ is based on the conditioning statistic:

$\begin{split}C=\max_{1\leq i\leq n}\bigg{|}\frac{\sumop\slimits@_{j\neq
i}h_{ij}\hat{r}_{j}}{(\sumop\slimits@_{j\neq
i}h_{ij}^{2})^{1/2}}\bigg{|}.\end{split}$ (4.3)

Under the assumption that $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{i}^{I}]=0$ for all $i\in[n]$, quantiles of the
conditioning statistic can be simulated analogously to the sup-score critical
value.For a new set of $e_{1},\dots,e_{n}$ iid standard normal and generated
independently of the data, and for any $\theta\in(0,1)$, define the
conditional quantile

$c^{C}_{1-\theta}\coloneqq(1-\theta)\text{ quantile of }\max_{1\leq i\leq
n}\bigg{|}\frac{\sumop\slimits@_{j\neq
i}e_{i}h_{ij}\hat{r}_{j}}{(\sumop\slimits@_{j\neq
i}h_{ij}^{2})^{1/2}}\bigg{|}\text{ conditional on
}\\{(y_{i},x_{i},z_{i})\\}_{i=1}^{n}$ (4.4)

Depending on the value of the conditioning statistic, the thresholding test
decides whether the test based on $\text{\it JK}(\beta_{0})$ or one based on
$S(\beta_{0})$ should be run.

$\begin{split}T(\beta_{0};\tau)=\begin{cases}\bm{1}\\{\text{\it
JK}(\beta_{0})>\chi^{2}_{1;1-\alpha}\\}&\text{if }C\geq\tau\\\
\bm{1}\\{S(\beta_{0})>\;c_{1-\alpha}^{S}\\}&\text{if }C<\tau\\\
\end{cases}\end{split}$ (4.5)

for some cutoff $\tau$, which I take in the simulation study and empirical
exercise to be the 75th quantile of the distribution of $C$ under the
assumption that $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{i}^{I}]=0,\forall i\in[n]$.To show that the
thresholding test controls size, I compare the rejection probability to that
of a Gaussian analog. In addition to $\text{\it JK}_{G}(\beta_{0})$, defined
in (3.1), define the Gaussian analogs of $\text{\it S}(\beta_{0})$ and the
conditioning statistic $C$:

$\displaystyle S_{G}(\beta_{0})$ $\displaystyle\coloneqq\sup_{1\leq\ell\leq
d_{z}}\bigg{|}\frac{\sumop\slimits@_{i=1}^{n}\tilde{\epsilon}_{i}(\beta_{0})z_{\ell
i}}{(\sumop\slimits@_{i=1}^{n}z_{\ell i}^{2})^{1/2}}\bigg{|}$ $\displaystyle
C_{G}$ $\displaystyle\coloneqq\sup_{1\leq i\leq
n}\bigg{|}\frac{\sumop\slimits@_{j\neq
i}h_{ij}\tilde{r}_{j}}{(\sumop\slimits@_{j\neq i}h_{ij}^{2})^{1/2}}\bigg{|}$

where, as in Section 3,
$(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})^{\prime}$ are generated
independently of each other and the data following a Gaussian distribution
with the same mean and covariance matrix as $(\epsilon_{i}(\beta_{0}),r_{i})$.
Since $\operatorname{Cov}(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})=0$
under $H_{0}$, the statistics $C_{G}$ and $S_{G}(\beta_{0})$ are independent
under the null. Similarly, the null distribution of $\text{\it
JK}_{G}(\beta_{0})$ is the same conditional on any realization of
$(\tilde{r}_{1},\dots,r_{n})$; it is also independent of $C_{G}$ under the
null. The Gaussian analog thresholding test decides whether the researcher
should run a test based on $S_{G}(\beta_{0})$ or $\text{\it
JK}_{G}(\beta_{0})$ depending on the value of $C_{G}$ as in (4.5).The test
statistics $\text{\it JK}_{G}(\beta_{0})$ and $S_{G}(\beta_{0})$ are only
marginally independent of the conditioning statistic $C_{G}$ under the null.
This limits the ways in which the test statistics can be combined using the
conditioning statistic while still controlling size. This marginal
independence in the Gaussian limit is enough, however, for the asymptotic
validity of the thresholding test, $T(\beta_{0};\tau)$. To establish that the
behavior of the pairs $(C,\text{\it JK}(\beta_{0}))$ and $(C,S(\beta_{0}))$
can be approximated by the behavior of $(C_{G},\text{\it JK}_{G}(\beta_{0}))$
and $(C_{G},S_{G}(\beta_{0}))$, respectively, I rely on the following
assumption:

######  Assumption 4.1 (Combination Conditions).

Assume that (i) there is a $\upsilon\in(0,1]\cup\\{2\\}$ such that
$\|\zeta_{i}\|_{\psi_{\upsilon}}\leq c$; (ii)
$\max_{i,j}|\frac{h_{ij}}{(\mathbb{E}_{n}[h_{ij}^{2}])^{1/2}}|+\max_{l,i}|\frac{z_{li}}{(\mathbb{E}_{n}[z_{li}^{2}])^{1/2}}|\leq
c$; and (iii) $\log^{7+4/\upsilon}(d_{z}n)/n\to 0$.

Assumption 4.1(i) is a strengthening of the moment bound on $r_{i}$ similar to
that of Assumption 3.4(i). As discussed, while more restrictive than the
condition in Assumption 3.1, this still allows for a wide range of potential
distributions for $r_{i}$. Assumption 4.1(ii) requires that the number of
observations used to test $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{i}]=0$ via the conditioning statistic and the number
of observations used to test the null hypothesis via the sup-score test are
both growing with the sample size. It can be verified by looking at the hat
matrix $H$ and the instruments. Finally, Assumption 4.1(iii) is a light
requirement on the number of instruments $d_{z}$ needed for the validity of
the sup-score test. It allows the number of instruments to grow near
exponentially as a function of sample size.

######  Theorem 4.1.

Suppose Assumptions 3.1, 3.2, 3.3, 3.4 and 4.1 hold. Then,

  1. 1.

the test based on $T(\beta_{0};\tau)$ has asymptotic size $\alpha$ for any
choice of cutoff $\tau$, and

  2. 2.

if $\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{i}^{I}]=0$ for all
$i\in[n]$, there exist sequences $\delta_{n}\searrow 0$ and $\beta_{n}\searrow
0$ such that with probability at least $1-\delta_{n}$,

$\sup_{\theta\in(0,1)}\big{|}\mathrm{Pr}_{e}(C\leq
c^{C}_{1-\theta})-(1-\theta)\big{|}\leq\beta_{n},$

where $\mathrm{Pr}_{e}(\cdot)$ denotes the probability with respect to only
the variables $e_{1},\dots,e_{n}$.

The first part of Theorem 4.1 establishes the asymptotic validity of the
thresholding test $T(\beta_{0};\tau)$ for any choice of cutoff $\tau$. The
proof of this statement follows the logic outlined above. The second part of
Theorem 4.1 establishes the validity of the multiplier bootstrap procedure to
approximate quantiles of the conditioning statistic. It follows directly from
results in Belloni et al. (2018) after verifying that the conditions needed
for error taken on from estimation of $\rho(z_{i})$ can treated as negligible
under Assumption 3.4.In Section 7, I investigate the power properties of the
thresholding test via simulation study. I find that combining the $\text{\it
JK}(\beta_{0})$ statistic with the sup-score statistic based on $C$ improves
power against distant alternatives and helps alleviate a power decline
suffered by the $\text{\it JK}(\beta_{0})$ statistic against a particular set
of alternatives.

######  Remark 4.1.

As mentioned by Andrews (2016) in the context of the standard K-statistic,
this attempt to rectify the power deficiency via this particular conditioning
statistic is not perfect. In particular, under heteroskedasticity, the means
of the partialed-out endogenous variables, $\mathbb{E}[r_{i}]$, may not be
scaled versions of the true first stages. However, as long as
$\mathbb{E}[r_{i}]\neq 0$, one can still expect $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{i}^{I}]=\sumop\slimits@_{j\neq
i}h_{ij}{}_{i}+(\beta-\beta_{0})\sumop\slimits@_{j\neq
i}h_{ij}\rho(z_{i}){}_{i}$ to be related to the true fist stage i and for the
test to have nontrivial power. Moreover, in light of the dependence of the
consistency result in Proposition 3.1 on Assumption 3.3(ii), in the case where
$\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{i}]=0$ for all $i\in[n]$ it
may be particularly important to avoid using the jackknife K-statistic to test
$H_{0}$.

##  5 Analysis with Multiple Endogenous Variables

To analyze the limiting behavior of the test statistic when $d_{x}>1$, I
follow the basic idea of Section 3, which is to show that quantiles of the
jackknife K-statistic can be approximated by analogous quantiles of the
Gaussian statistic:

$\text{\it
JK}_{G}(\beta_{0})\coloneqq\tilde{\epsilon}(\beta_{0})\tilde{\Pi}(\tilde{\Pi}_{\epsilon}^{\prime}\tilde{\Pi}_{\epsilon})^{-1}\tilde{\Pi}^{\prime}\tilde{\epsilon}(\beta_{0});$

where $(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})^{\prime}$ are Gaussian
with the same mean and covariance matrix as
$(\epsilon_{i}(\beta_{0}),r_{i})^{\prime}$ and for $\tilde{\Pi}_{\ell
i}=\sumop\slimits@_{j\neq i}h_{ij}\tilde{r}_{\ell j}$ define
$\tilde{\Pi}_{i}\coloneqq(\tilde{\Pi}_{1i},\dots,\tilde{\Pi}_{d_{x}i})^{\prime}\in\mathbb{R}^{d_{x}}$,
$\tilde{\Pi}_{\epsilon
i}\coloneqq(\mathbb{E}[\epsilon_{i}^{2}(\beta_{0})])^{1/2}\tilde{\Pi}_{i}$,
and

$\displaystyle\tilde{\epsilon}(\beta_{0})$
$\displaystyle\coloneqq(\tilde{\epsilon}_{1}(\beta_{0}),\dots,\tilde{\epsilon}_{n}(\beta_{0}))^{\prime}\in\mathbb{R}^{n}$
$\displaystyle\coloneqq(\tilde{\Pi}_{1},\dots,\tilde{\Pi}_{n})^{\prime}\in\mathbb{R}^{n\times
d_{x}}$ $\displaystyle\tilde{\Pi}_{\epsilon}$
$\displaystyle\coloneqq(\tilde{\Pi}_{\epsilon 1},\dots,\tilde{\Pi}_{\epsilon
n})^{\prime}\in\mathbb{R}^{n\times d_{x}}$

As in Section 3, notice that, since uncorrelated random variables are
independent, under $H_{0}$ the vector $\tilde{\epsilon}(\beta_{0})$ is mean
zero and independent of $(\tilde{\Pi},\tilde{\Pi}_{\epsilon})$. Conditional on
any realization of $(\tilde{\Pi},\tilde{\Pi}_{\epsilon})$ the $\text{\it
JK}_{G}(\beta_{0})$ statistic then follows a $\chi^{2}_{d_{x}}$ distribution,
and thus, its unconditional distribution is also $\chi^{2}_{d_{x}}$.In
addition to characterizing the local behavior of $\text{\it JK}(\beta_{0})$
with multiple endogenous variables, I show that the thresholding test of
Section 4.2 can be applied with multiple endogenous variables with a
generalized conditioning statistic.

###  5.1 Modified Interpolation Approach

As with a single endogenous variable, error taken on from the estimation of
$\rho(z_{i})$ prevents immediate comparison of $\text{\it JK}(\beta_{0})$ to
$\text{\it JK}_{G}(\beta_{0})$. Instead as an intermediate step consider
showing that the quantiles of $\text{\it JK}_{I}(\beta_{0})$ can be
approximated by corresponding quantiles of $\text{\it JK}_{G}(\beta_{0})$
where $\text{\it JK}_{I}(\beta_{0})$ is an infeasible statistic:

$\text{\it JK}_{I}(\beta_{0})\coloneqq\epsilon(\beta_{0})(\mathaccent
0{\symlargesymbols}062{}^{I})((\mathaccent
0{\symlargesymbols}062{}_{\epsilon}^{I})^{\prime}(\mathaccent
0{\symlargesymbols}062{}_{\epsilon}^{I}))^{-1}(\mathaccent
0{\symlargesymbols}062{}^{I})^{\prime}\epsilon(\beta_{0}),$

for $\mathaccent 0{\symlargesymbols}062{}^{I}$ and $\mathaccent
0{\symlargesymbols}062{}_{\epsilon}^{I}$ defined the same way as $\mathaccent
0{\symlargesymbols}062{\Pi}$ and $\mathaccent
0{\symlargesymbols}062{}_{\epsilon}$ in (2.4), respectively, but using the
true values $(r_{1},\dots,r_{n})^{\prime}$ in place of their estimates
$(\hat{r}_{1},\dots,\hat{r}_{n})^{\prime}$.When there are multiple endogenous
variables, $d_{x}>1$, I cannot take advantage of the simplified form of the
test statistic to establish this approximation as in Section 3. Instead I deal
directly with the test statistics themselves. Consider functions
$\varphi_{\gamma}(\cdot)\in C_{b}^{3}(\mathbb{R})$ that approximate the
indicators $\bm{1}\\{\cdot\leq a\\}$, where $a\in\mathbb{R}$ is arbitrary and
$\gamma$ is a scaling factor inversely proportional to the quality of the
approximation but positively proportional to the derivatives of
$\varphi_{\gamma}$. The goal is to show, for a sequence $\gamma_{n}$ tending
to zero, that

$\mathbb{E}[\varphi_{\gamma_{n}}(\text{\it
JK}_{I}(\beta_{0}))-\varphi_{\gamma_{n}}(\text{\it JK}_{G}(\beta_{0}))]\to 0$
(5.1)

The classical interpolation argument of Lindeberg (1922) would attempt to show
(5.1) by one-by-one replacement of each pair,
$(\epsilon_{i}(\beta_{0}),r_{i})^{\prime}$, in the expression of
$\varphi_{\gamma_{n}}(\text{\it JK}_{I}(\beta_{0}))$ with its Gaussian analog,
$(\tilde{\epsilon}_{i}(\beta_{0}),\tilde{r}_{i})^{\prime}$, and bounding of
the size of each of these deviations. As mentioned in Section 3, the problem
arises as the derivative of the test statistic, $\text{\it
JK}_{I}(\beta_{0})$, with respect to terms in the denominator matrix,
$\mathaccent 0{\symlargesymbols}062{}_{\epsilon}^{\prime}\mathaccent
0{\symlargesymbols}062{}_{\epsilon}$, may be as large as the inverse of the
minimum eigenvalue of the denominator matrix. When identification is
sufficiently weak, the denominator matrix will have a nonnegligible
distribution and the inverse of its minimum eigenvalue may not have finite
moments.To get around this, I modify the argument by considering a “data-
dependent” choice of approximation parameter $\gamma_{n}$. This choice of
approximation parameter inversely scales with the determinant of the
denominator matrix and thus, since the determinant is the product of the
eigenvalues, inversely scales with the minimum eigenvalue.111The determinant
has the benefit of being a smooth function of elements of the matrix. This
makes it nicer to work with than the minimum eigenvalue itself, which loses
differentiability when the dimension of its eigenspace is larger than one.
Geometrically, this approach can be thought of as “stretching out” the
function $\varphi_{\gamma_{n}}(\cdot)$ in directions where the minimum
eigenvalue of the denominator matrix is close to zero. Since the overall
derivatives of $\varphi_{\gamma_{n}}(\text{\it JK}_{I}(\beta_{0}))$ with
respect to $(\epsilon_{i}(\beta_{0}),r_{i})^{\prime}$ depend on the product of
derivatives with respect to the test statistic andderivatives of
$\varphi_{\gamma_{n}}(\cdot)$, which scale inversely with the approximation
parameter, this adjustment of the approximation parameter allows control of
the overall derivative. Details of this approach can be found in Appendix
D.This approach relies on stronger moment conditions, which I detail below.
These strengthened moment conditions are needed mainly needed to bound moments
of the determinant of the denominator matrix. For all $\ell=1,\dots,d_{x}$ let
$\zeta_{\ell i}\coloneqq v_{i}-\rho_{\ell}(z_{i})\eta_{i}$, noting that
$\zeta_{\ell i}=r_{\ell i}-\mathbb{E}[r_{\ell i}]$. Recall also the definition
of $\eta_{i}=\epsilon_{i}-v_{i}^{\prime}(\beta-\beta_{0})$, which is equal to
$\epsilon_{i}(\beta_{0})-\mathbb{E}[\epsilon_{i}(\beta_{0})]$.

######  Assumption 5.1 (Moment Conditions).

Assume (i) there are constants $c>1$ and $\upsilon\in(0,1]\cup\\{2\\}$ such
that $\|\epsilon_{i}\|_{\psi_{a}}\leq c$ and $\|\zeta_{\ell
i}\|_{\psi_{\upsilon}}\leq c$, and (ii)
$c^{-1}\leq\lambda_{\min}(\mathbb{E}[\eta_{i}\eta_{i}^{\prime}])\leq\lambda_{\max}(\mathbb{E}[\eta_{i}\eta_{i}^{\prime}])\leq
c$.

######  Assumption 5.2 (Balanced Design).

(i) For any $\ell=1,\dots,d_{x}$ let $s_{\ell,n}^{-2}=\max_{1\leq i\leq
n}\mathbb{E}[(\mathaccent 0{\symlargesymbols}062{}_{\ell i}^{I})^{2}]$; then,
the minimum eigenvalue of the following matrix is bounded away from zero:

$c^{-1}\leq\lambda_{\min}\mathbb{E}\begin{pmatrix}\frac{s_{\ell,n}s_{k,n}}{n}\sumop\slimits@_{i=1}^{n}(\mathaccent
0{\symlargesymbols}062{}_{\ell i}^{I})(\mathaccent
0{\symlargesymbols}062{}_{ki}^{I})\end{pmatrix}_{\begin{subarray}{c}1\leq\ell\leq
d_{x}\\\ 1\leq k\leq d_{x}\end{subarray}}$

(ii) $\max_{i}s_{n}\sumop\slimits@_{j\neq i}h_{ji}^{2}\leq c$; and(iii) the
following ratio is bounded away from zero:
$\frac{\sumop\slimits@_{k=2}^{n}\lambda_{k}^{2}(HH^{\prime})}{\sumop\slimits@_{k=1}^{n}\lambda_{k}^{2}(HH^{\prime})}\geq
c^{-1}$ where $\lambda_{k}(HH^{\prime})$ represents the $k^{\text{th}}$
largest eigenvalue of the matrix $HH^{\prime}$.

Assumption 5.1(i) strengthens Assumption 3.1 to require that the random
variables $(\eta_{i},\zeta_{i})$, and thus, by extension,
$(\epsilon_{i}(\beta_{0}),r_{i})$ are $\upsilon$-sub-exponential. As discussed
below Assumption 3.4 this is more restrictive than the finite sixth moments
needed to establish Lemma 3.1 but still allows for a wide range of possible
distributions. Assumption 5.1(ii) is a light regularity condition requiring
that the random variables $(\eta_{1i},\dots,\eta_{d_{x}i})$ be linearly
independent.Assumption 5.2(i) is a natural extension of Assumption 3.2(i) to
the setting where $d_{x}>1$. It requires that the average second moment of any
linear combination of the first-stage estimates is proportional to the maximum
second moment of the same linear combination. Assumption 5.2(ii,iii) are the
same conditions as Assumption 3.2(ii,iii) and can again be implicitly thought
of as requiring that the maximum leverage of any one observation be bounded
and there be than two effective instruments in the hat matrix. Assumption 5.2
thus reduces to Assumption 3.2 when $d_{x}=1$.

######  Assumption 5.3 (Local Identification).

(i) The local power index is bounded $P\leq c$ for

$P=\sumop\slimits@_{\ell=1}^{d_{x}}\mathbb{E}\bigg{[}\bigg{(}\frac{s_{\ell,n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\mathaccent
0{\symlargesymbols}062{}_{\ell
i}^{I}{}_{i}^{\prime}(\beta-\beta_{0})\bigg{)}^{2}\bigg{]}$

(ii) $\mathbb{E}[(s_{n,\ell}\sumop\slimits@_{j\neq
i}h_{ji}\epsilon_{j}(\beta_{0}))^{2}]\leq c$ for all $\ell=1,\dots,d_{x}$.

######  Lemma 5.1 (Infeasible Uniform Approximation).

Suppose that Assumptions 5.1, 5.2 and 5.3 hold. Then

$\sup_{a\in\mathbb{R}}|\mathrm{Pr}(\text{\it JK}_{I}(\beta_{0})\leq
a)-\mathrm{Pr}(\text{\it JK}_{G}(\beta_{0}))\leq a)|\to 0$

###  5.2 Limiting Behavior of Test Statistic

Having derived the limiting behavior of the infeasible statistic, I next
present a high-level condition under which estimation error taken on from
estimation of $\rho(z_{i})$ can be treated as negligible. I then verify this
high-level condition for the $\ell_{1}$-regularized estimators proposed in
(2.2). For any $\ell=1,\dots,d_{x}$ define the scaled differences

N,ℓ
$\displaystyle\coloneqq\frac{s_{\ell,n}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\epsilon_{i}(\beta_{0})(\mathaccent
0{\symlargesymbols}062{}_{\ell,i}-\mathaccent
0{\symlargesymbols}062{}_{\ell,i}^{I})$ D,ℓ
$\displaystyle\coloneqq\frac{s_{\ell,n}^{2}}{n}\sumop\slimits@_{i=1}^{n}\epsilon_{i}^{2}(\beta_{0})(\mathaccent
0{\symlargesymbols}062{}_{\ell,i}^{2}-(\mathaccent
0{\symlargesymbols}062{}_{\ell,i}^{I})^{2})$

As long as these scaled differences tend to zero, Lemma 5.2 shows that the
difference between the feasible and infeasible test statistics converges to
zero:

######  Lemma 5.2.

Suppose that Assumptions 5.1, 5.2 and 5.3 hold and that
$({}_{N,\ell},{}_{D,\ell})\to_{p}0$ for all $\ell=1,\dots,d_{x}$. Then
$|\text{\it JK}(\beta_{0})-\text{\it JK}_{I}(\beta_{0})|\to_{p}0$.

As with Lemma 3.2, while Lemma 5.2 is a simple statement, it is not immediate.
In particular, establishing Lemma 5.2 requires showing that
$\lambda_{\max}(D^{-1})$ is bounded in probability, where $D$ represents a
scaled version of the denominator matrix. This requires some work as the
scaled denominator matrix is not required to converge in distribution to a
stable limit. Instead I directly show that $\lambda_{\max}(D^{-1})$ is bounded
in probability by showing that
$\mathrm{Pr}(\lambda_{\min}(D)\leq\delta_{n})\to 0$ for any sequence
$\delta_{n}\to 0$.To do this, I first demonstrate that it is sufficient to
show that $\mathrm{Pr}(a^{\prime}Da\leq\delta_{n})\to 0$ for any
$\delta_{n}\to 0$ and fixed
$a\in\mathcal{S}^{d_{x}-1}=\\{v\in\mathbb{R}^{d_{x}}\mathrel{\mathop{\ordinarycolon}}\|v\|=1\\}$.
I then establish the claim for an arbitrary choice of $a$. As in Lemma 3.2 I
do this by comparing the scaled quadratic form of the denominator matrix to a
Gaussian analog and then establishing the corresponding result for the
Gaussian analog. This corresponding result is again also useful for
establishing the validity of the interpolation approach with a dynamic choice
of approximation parameter.I state conditions under which
$({}_{N,\ell},{}_{D,\ell})\to_{p}0$ holds for the $\ell_{1}$-regularized
estimation procedure proposed in (2.2). These conditions are equivalent to
those in Assumption 3.4 but hold for each the $d_{x}$ estimation procedures.

######  Assumption 5.4 (Estimation Error).

(i) The basis terms $b(z_{i})$ are bounded, $\|b(z_{i})\|_{\infty}\leq C$ for
all $i=1,\dots,n$; (ii) the approximation error satisfies
$(\mathbb{E}_{n}[\xi_{\ell i}^{2}])^{1/2}=o(n^{-1/2})$; (iii) the researcher
has access to estimators $\mathaccent 0{\symlargesymbols}062{\phi}_{\ell}$ of
$\phi_{\ell}$ that satisfy $\log(d_{b}n)^{2/(\upsilon\wedge 1)}\|\mathaccent
0{\symlargesymbols}062{\phi}_{\ell}-\phi_{\ell}\|_{1}\to_{p}0$ for each
$\ell\in[d_{x}]$; and (iv) locally identified in the sense that

  1. (iva)

$\max_{\begin{subarray}{c}1\leq\ell\leq d_{x}\\\ 1\leq k\leq
d_{b}\end{subarray}}\big{|}\mathbb{E}\big{[}\frac{s_{n,\ell}}{\sqrt{n}}\sumop\slimits@_{i=1}^{n}\sumop\slimits@_{j\neq
i}h_{ij}\epsilon_{i}(\beta_{0})b_{k}(z_{j})\epsilon_{j}(\beta_{0})\big{]}\big{|}\leq
c$

  2. (ivb)

$\max_{\begin{subarray}{c}1\leq i\leq n\\\ 1\leq\ell\leq
d_{b}\end{subarray}}|\mathbb{E}[s_{n,\ell}\sumop\slimits@_{j\neq
i}h_{ij}b_{\ell}(z_{j})\epsilon_{j}(\beta_{0})]|\leq c.$

Under Assumption 5.4 the conditions of Lemma 5.2 are satisfied. If these
conditions are satisfied, Lemmas 5.1 and 5.2 can be combined to analyze the
behavior of $\text{\it JK}(\beta_{0})$ statistics in local neighborhoods of
the null.

######  Theorem 5.1 (Uniform Approximation).

Suppose that Assumptions 5.1, 5.2, 5.3 and 5.4 hold. Then,

$\sup_{a\in\mathbb{R}}|\mathrm{Pr}(\text{\it JK}(\beta_{0})\leq
a)-\mathrm{Pr}(\text{\it JK}_{G}(\beta_{0})\leq a)|\to 0$

In particular, under $H_{0}$, $\text{\it
JK}(\beta_{0})\rightsquigarrow\chi^{2}_{d_{x}}$.

As in Lemma 3.1, the result in Theorem 5.1 does not require $\text{\it
JK}_{G}(\beta_{0})$ to have a stable limiting distribution under $H_{1}$.

###  5.3 Improving Power against Certain Alternatives

As discussed in Section 4.1, tests based on the jackknife K-statistic may
suffer from suboptimal power properties. These properties are particularly bad
whenever $\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{\ell i}]=0$ for
some $\ell\in[d_{x}]$ and all $i\in[n]$. To improve power in this direction, I
propose a generalization of the thresholding test in Section 4.2 based on the
conditioning statistic $C$

$\displaystyle C\coloneqq\min_{1\leq\ell\leq d_{x}}\max_{1\leq i\leq
n}\bigg{|}\frac{\sumop\slimits@_{j\neq i}h_{ij}\hat{r}_{\ell
j}}{(\sumop\slimits@_{j\neq i}h_{ij}^{2})^{1/2}}\bigg{|}$ (5.2)

The conditioning statistic $C$ attempts to detect whether, for _some_
$\ell\in[d_{x}]$, $\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{\ell
i}^{I}]=0$ for all $i\in[n]$. Under the assumption that
$\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{\ell i}^{I}]=0,\forall
i\in[n],\ell\in[d_{x}]$, quantiles of $C$ can be simulated by multiplier
bootstrap. Let $e_{1},\dots,e_{n}$ be generated iid standard normal
independent of the data and for any $\theta\in(0,1)$, define the conditional
bootstrap quantile:

$c^{C}_{1-\theta}\coloneqq(1-\theta)\text{ quantile of }\min_{1\leq\ell\leq
d_{x}}\max_{1\leq i\leq n}\bigg{|}\frac{\sumop\slimits@_{j\neq
i}e_{j}h_{ij}\hat{r}_{j}}{(\sumop\slimits@_{j\neq
i}h_{ij}^{2})^{1/2}}\bigg{|}\text{ conditional on
}\\{(y_{i},x_{i},z_{i})\\}_{i=1}^{n}$

Based on the value of the conditioning statistic the researcher can decide
whether to run a test based on $\text{\it JK}(\beta_{0})$ or a test based on
the sup-score statistic $S(\beta_{0})$.

$\displaystyle T(\beta_{0};\tau)\coloneqq\begin{cases}\bm{1}\\{\text{\it
JK}(\beta_{0})>\chi^{2}_{d_{x};1-\alpha}\\}&\text{if }C>\tau\\\
\bm{1}\\{S(\beta_{0})>c^{S}_{1-\alpha}\\}&\text{if }C\leq\tau\end{cases}$
(5.3)

As with Theorem 4.1, I show the asymptotic validity of the thresholding test
by first establishing that quantiles of $(\text{\it JK}(\beta_{0}),C)$ and
$(S(\beta_{0}),C)$ can jointly be approximated by Gaussian analogs and then
using the marginal independence of the Gaussian analog testing and
conditioning statistics under the null; $(\text{\it JK}(\beta_{0})\perp C)$
and $(S\beta_{0})\perp C)$ under $H_{0}$.

######  Theorem 5.2.

Suppose that Assumptions 4.1(ii,iii), 5.1, 5.2, and 5.4 hold. Then,

  1. 1.

the test based on $T(\beta_{0};\tau)$ has asymptotic size $\alpha$ for any
choice of cutoff $\tau$, and

  2. 2.

if $\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{\ell i}^{I}]=0$ for all
$i\in[n]$ and $\ell\in[d_{x}]$, there exist sequences $\delta_{n}\searrow 0$
and $\beta_{n}\searrow 0$ such that with probability at least $1-\delta_{n}$,

$\sup_{\theta\in(0,1)}|\mathrm{Pr}_{e}(C\leq
c_{1-\theta}^{C})-(1-\theta)|\leq\beta_{n}$

where $\mathrm{Pr}_{e}(\cdot)$ denotes the probability with respect to only
the variables $e_{1},\dots,e_{n}$.

The first part of Theorem 5.2 establishes the validity of the test based on
the thresholding statistic for any choice of cutoff $\tau$. In practice, I
recommend taking the cutoff, $\tau$, to be the 75th quantile of the
distribution of $C$ under the assumption that $\mathbb{E}[\mathaccent
0{\symlargesymbols}062{}_{\ell i}^{I}]=0$ for all $\ell\in[d_{x}]$ and
$i\in[n]$. The second part of Theorem 5.2 establishes that this quantile can
be simulated via the multiplier bootstrap procedure described above.

##  6 Empirical Application

I apply the testing procedures proposed in this paper to the data of Gilchrist
and Sands (2016), who seek to determine the effect of social spillovers in
movie consumption. The sample consists of all 1,671 opening weekend days111An
opening weekend day is a Friday, Saturday, or Sunday of opening weekend.
between January 1, 2002 and January 1, 2012. For each opening weekend, the
authors observe gross ticket sales for all movies wide released in theaters in
the United States.222A wide released movie is any movie that ever shows on 600
or more screens. The data are obtained through Box Office Mojo, a subsidiary
of the Internet Movie Database (IMDb). To focus on movies in theaters long
enough for social spillovers to be a relevant factor, the authors consider
only movies that remain in theaters for at least six weeks.The outcome
variables of interest are gross ticket sales of movies that opened in a given
weekend in the second through sixth weeks of their run, while the endogenous
variable is the gross ticket sales of a movie in its opening weekend. To
control for seasonal periodicity in both the supply of and demand for movies,
a vector of date controls are included. Formally, Gilchrist and Sands (2016)
are interested in the parameters $\beta_{w}$, $w=2,\dots,7$ from the linear IV
model(s):

$\text{Sales}_{wi}^{\perp}=\beta_{w}\text{Sales}^{\perp}_{1i}+\epsilon_{wi}$
(6.1)

where, for $i=1,\dots,6$, $\text{Sales}_{wi}^{\perp}$ represents gross
national ticket sales, after the partialing out of date controls and a
constant, $7w$ days after day $i$, of movies that opened on the opening
weekend of $i$. The variable
$\text{Sales}_{7i}^{\perp}=\sumop\slimits@_{w=1}^{6}\text{Sales}_{wi}^{\perp}$
denotes the cumulative national ticket sales from the second through sixth
running weekends of movies who opened in weekend $i$, after the partialing out
of date controls and a constant.The parameter $\beta_{w}$ represents the
social spillover effect of strong opening weekend sales on sales in later
weeks; more people seeing a movie on its opening weekend will mean more people
telling their friends about the movie potentially leading to larger sales
later on.Because movies with high first-week sales may have high sales in
succeeding weeks for reasons other than word of mouth spillover effects (e.g
the movie may receive positive critical reviews prerelease or be part of a
previously successful franchise), the parameter $\beta_{w}$ cannot be
plausibly recovered from ordinary least squares regression of
$\text{Sales}_{wi}^{\perp}$ on $\text{Sales}_{1i}^{\perp}$. To identify the
structural parameter, Gilchrist and Sands (2016) employ a vector of nationally
aggregated weather measures. These weather measures reflect the proportion of
movie theaters experiencing a particular type of weather on a particular
weekend. The measures include the proportion of movie theaters experiencing
maximum temperatures in $5^{\circ}$ Fahrenheit bins on the interval
$[10^{\circ},100^{\circ}]$, the proportion of movie theaters experiencing
precipitation levels in 0.25 inch per hour increments on the interval
$[0,1.5]$, and the proportions of theaters experiencing any type of snow and
of theaters experiencing any type of rain.The nationally aggregated weather
conditions on opening weekend days serve as plausibly exogenous instrumental
variables, affecting ticket sales in later weeks only through their effect on
opening-weekend-day sales. Same-day weather conditions may also have an effect
on movie ticket sales: when the weather is particularly nice, people may be
more inclined to engage in outdoor activities while in poorer, weather people
may choose to stay indoors and see a new movie. Putting together the
nationally aggregated weather measures leaves Gilchrist and Sands (2016) with
a vector of 52 instrumental variables. After the partialing out a constant and
the date controls, four of these are linearly dependent. I discard these and
work with the remaining 48 partialed-out instruments in my analysis.To handle
the large number of instruments, the authors follow Belloni et al. (2012) and
employ a post-LASSO estimate of the first stage. In their main specifications,
they set the first-stage penalty parameter so that the number of instrument
selected is one, two, or three. The resulting first-stage F-statistics using
the selected instrument(s), 38.80, 25.86, and 20.95, respectively, seem to
indicate strong identification.333Typical empirical practice is to use the
Wald test when the first stage F-statistic is larger than 10. However, the
first-stage F-statistic on the full set of instrumental variables is only
3.80. Moreover, since the LASSO objective is an $\ell_{1}$ penalized version
of the OLS loss, using the variables selected by LASSO may mechanically lead
to higher F-statistics even if the underlying relationship between the
instruments and the endogenous variables is weak.Figure 6.1 provides evidence
from a simple simulation experiment to demonstrate this. For the simulation
experiment I generate an iid sample of 10 instrumental variables,
$\\{Z_{1i},\dots,Z_{10i}\\}_{i=1}^{n}$ from a normal distribution with a
Toeplitz covariance structure, $\operatorname{Cov}(Z_{\ell
i},Z_{ki})=(1.1)^{-|j-k|}$, $1\leq j,k\leq 10$. The endogenous variable is
generated to only have a weak relationship with the instruments
<EMAIL_ADDRESS>Z_{\ell
i}+v_{i}$, where the first-stage errors $v_{i}$ are independent standard
normals. From this initial set of 10 instrumental variables I generate an
additional 55 technical instruments by squaring and taking all interactions
between variables in the initial set. These generated instruments are
correlated with the initial instruments but do not directly enter the first
stage.I then set the LASSO penalty so that only a certain number of
instruments are chosen and report the resulting average first stage
F-statistics over one thousand simulations. As seen in Figure 6.1, these
first-stage F-statistics increase significantly as the number of selected
instruments decreases. While the “true” F-statistic, computed with only the 10
initial instruments directly relevant for the first stage, is only 5.234, the
average F-statistic on the selected variables can be larger than 40. The
persistence of this pattern between sample sizes $n=500$ and $n=1000$ suggests
that this is not a small-sample issue and that pretesting for weak
identifications based on post-LASSO F-statistics may be problematic generally.
Figure 6.2 shows how the first stage F-statistic changes with the number of
LASSO-selected variables in the Gilchrist and Sands (2016) data. The pattern
is similar to that seen in the Figure 6.1 simulation experiment.

Figure 6.1: Results from Simulation Experiment. The endogenous variable is
generated to be weakly related to a set of ten initial instruments. I take
quadratic powers and interactions of these ten initial instruments to create
an additional 55 technical instruments that do not directly enter the first
stage. The LASSO penalty is then set to select a certain number of variables
and I report the resulting average post-LASSO F-statistics over 1000
simulations. The average F-statistic using only the relevant ten initial
instruments is 5.234 for both $n=500$ and $n=1000$. Figure 6.2: First-Stage
F-statistic as Function of Number of LASSO-Selected Variables in the Data of
Gilchrist and Sands (2016). When selecting variables using a cross-validated
choice of LASSO penalty parameter, the first-stage F-statistic is 6.42.

Given a lack of clarity on the strength of identification, I seek to validate
the results of Gilchrist and Sands (2016) using the weak identification
testing procedures proposed in this paper. The setting is particularly
suitable for weak IV testing using the jackknife K-statistic. With 48
instruments and a sample size of 1671, $d_{z}^{3}=110{,}592\gg n$, making the
tests of Moreira (2003, 2009); Kleibergen (2005), and Andrews (2016)
inapplicable. On the other hand, it is unclear whether asymptotic
approximations based on $d_{z}\to\infty$ will accurately describe the finite-
sample distribution of test statistics with 48 instruments. Moreover, since
fluctuations in movie theater attendance seem to be largely driven by either
particularly cold or particularly hot weather (see Figure 4 in Gilchrist and
Sands (2016)), the nuisance parameter $\rho(z_{i})$ is plausibly approximately
sparse.Table 6.1 compares the $95\%$ confidence intervals for
$\beta_{1},\dots,\beta_{7}$ generated by the jackknife K test to the
confidence intervals generated by the sup-score test of Belloni et al. (2012)
and the jackknife LM test (JLM) test of Matsushita and Otsu (2022). I form
these confidence intervals by running the tests for each $\beta_{0}$ on a 300
point grid between zero and two and inverting the results; a point $\beta_{0}$
is included in the $95\%$ confidence interval if the test fails to reject the
null that $\beta_{w}=\beta_{0}$ at level $\alpha=0.05$.For the $\text{\it
JK}(\beta_{0})$ statistic I use the choice of hat matrix in (2.3) and estimate
the auxiliary parameter $\rho(z_{i})$ as in (2.2). The penalty parameter
$\lambda$ is chosen with leave-one-out cross-validation using the cv.glmnet
command from the glmnet package in R (R Core Team, 2021; Friedman et al.,
2010). The critical value for the sup-score statistic $S(\beta_{0})$ is
simulated using 2,500 bootstrap draws. Confidence intervals based on the
combination test, $T(\beta_{0};\tau)$, are not directly reported as the
pretesting procedure based on simulating the 75th quantile of $C$ as in (4.4)
always suggests using the $\text{\it JK}(\beta_{0})$ statistic.For reference,
I also provide point estimates and standard errors for
$\beta_{1},\dots,\beta_{7}$ from Gilchrist and Sands (2016), Table 2. To
facilitate comparison, these point estimates and standard errors come from a
specification that uses all the instruments in the first stage of a 2SLS
procedure. While the Gilchrist and Sands (2016) point estimates are always in
the 95% confidence intervals generated by the $\text{\it JK}(\beta_{0})$ and
JLM tests, the confidence intervals from the identification-robust procedures
are significantly wider than those generated with the 2SLS standard
errors.Interestingly, the confidence intervals from inverting the jackknife
K-test tend to be quite similar to the confidence intervals from the JLM test.
This is surprising given the distinct forms of the $\text{\it JK}(\beta_{0})$
and the JLM test statistics.For the parameters
$\beta_{2},\beta_{4},\beta_{5},$ and $\beta_{6}$, the confidence intervals
generated by the sup-score statistic are empty while the sup-score confidence
interval for $\beta_{2}$ is nearly empty. This is also the case when using the
jackknife AR-statistic of Crudu et al. (2021) and Mikusheva and Sun (2021),
whose confidence intervals are not reported as they are always empty. With 48
instruments and a single parameter the linear IV model in (6.1) is
overidentified and as such the empty confidence intervals could be interpreted
as evidence of model misspecification. For the parameter $\beta_{7}$ the
confidence interval generated by inverting the sup-score statistic is not
empty and is instead 36% larger than the $\text{\it JK}(\beta_{0})$ confidence
interval and 41% larger than the JLM confidence interval. This result suggests
that the jackknife K tests and JLM tests may have better power properties than
the sup-score test in this setting.

Parameter | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{5}$ | $\beta_{6}$ | $\beta_{7}$
---|---|---|---|---|---|---
$\underset{\text{(s.e.)}}{\text{Estimate}}$ | $\underset{(0.024)}{0.475}$ | $\underset{(0.023)}{0.269}$ | $\underset{(0.017)}{0.164}$ | $\underset{(0.013)}{0.121}$ | $\underset{(0.010)}{0.093}$ | $\underset{(0.074)}{1.222}$
$\text{\it JK}(\beta_{0})$ | $[0.436,0.557]$ | $[0.227,0.334]$ | $[0.134,0.214]$ | $[0.100,0.167]$ | $[0.080,0.134]$ | $[1.003,1.391]$
$S(\beta_{0})$ | $\emptyset$ | $[0.294,0.334]$ | $[0.087,0.094]$ | $\emptyset$ | $\emptyset$ | $[0.990,1.518]$
JLM | $[0.436,0.557]$ | $[0.227,0.334]$ | $[0.134,0.214]$ | $[0.107,0.167]$ | $[0.087,0.134]$ | $[1.010,1.384]$

Table 6.1: 95% Confidence Intervals based on inverting various test
statistics. Instrument set used is the same as the Gilchrist and Sands (2016)
instrument set less four collinear instruments; $d_{z}=48$ with $n=1,671$.
Thresholding test confidence intervals are not reported as they coincide with
confidence intervals for $\text{\it JK}(\beta_{0})$.

Tables 6.2 and 6.3 repeat the analysis of Table 6.1 but with alternative
instrument sets. The confidence intervals of Table 6.2 use only $5^{\circ}$
Fahrenheit temperature bins ($d_{z}=36$) while the confidence intervals of
Table 6.3 include all the instruments used in Table 6.1 and all interactions
between the $5^{\circ}$ Fahrenheit temperature bins and the other weather
measures for a total of 524 instruments.444The instrument set of Table 6.3
does not include interactions between temperature bins nor interactions
between other weather measures. For the most part, the confidence intervals
generated by inverting the jackknife K-statistic are similar across Tables
6.1-6.3. The confidence intervals for the jackknife LM statistic however,
become much narrower when using the largest set of instruments is used. Thisis
interesting as the results from the $\text{\it JK}(\beta_{0})$ test as well as
the power analysis in Section 4 seem to suggest that use of the extra
isntruments does not lead to better first-stage estimates. Interestingly, the
JLM confidence intervals in for $\beta_{6},\beta_{7}$ in Table 6.3 do not
contain the point estimates for $\beta_{6}$ and $\beta_{7}$ from Gilchrist and
Sands (2016). As with Table 6.1, Tables 6.2 and 6.3 do not report confidence
intervals from $T(\beta_{0};\tau)$ as these always agree with the $\text{\it
JK}(\beta_{0})$ confidence intervals and do not report jackknife AR confidence
intervals as these are always empty.

Parameter | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{5}$ | $\beta_{6}$ | $\beta_{7}$
---|---|---|---|---|---|---
$\underset{\text{(s.e.)}}{\text{Estimate}}$ | $\underset{(0.024)}{0.475}$ | $\underset{(0.023)}{0.269}$ | $\underset{(0.017)}{0.164}$ | $\underset{(0.013)}{0.121}$ | $\underset{(0.010)}{0.093}$ | $\underset{(0.074)}{1.222}$
$\text{\it JK}(\beta_{0})$ | $[0.449,0.597]$ | $[0.255,0.389]$ | $[0.148,0.248]$ | $[0.114,0.194]$ | $[0.094,0.154]$ | $[1.086,1.555]$
$S(\beta_{0})$ | $\emptyset$ | $[0.302,0.329]$ | $\emptyset$ | $\emptyset$ | $\emptyset$ | $\emptyset$
JLM | $[0.449,0.597]$ | $[0.255,0.389]$ | $[0.154,0.248]$ | $[0.114,0.194]$ | $[0.094,0.154]$ | $[1.092,1.555]$

Table 6.2: 95% Confidence Intervals based on inverting various test
statistics. Instrument set used includes only temperatures measures;
$d_{z}=36$, with $n=1,671$. Thresholding test condidence intervals are not
reported as they coincide with confidence intervals for $\text{\it
JK}(\beta_{0})$.

Parameter | $\beta_{2}$ | $\beta_{3}$ | $\beta_{4}$ | $\beta_{5}$ | $\beta_{6}$ | $\beta_{7}$
---|---|---|---|---|---|---
$\underset{\text{(s.e.)}}{\text{Estimate}}$ | $\underset{(0.024)}{0.475}$ | $\underset{(0.023)}{0.269}$ | $\underset{(0.017)}{0.164}$ | $\underset{(0.013)}{0.121}$ | $\underset{(0.010)}{0.093}$ | $\underset{(0.074)}{1.222}$
$\text{\it JK}(\beta_{0})$ | $[0.443,0.604]$ | $[0.215,0.342]$ | $[0.094,0.228]$ | $[0.087,0.154]$ | $[0.054,0.121]$ | $[0.916,1.435]$
$S(\beta_{0})$ | $[0.416,0.477]$ | $\emptyset$ | $\emptyset$ | $[0.034,0.121]$ | $[0.121,0.208]$ | $[0.918,1.562]$
JLM | $[0.463,0.497]$ | $[0.268,0.282]$ | $[0.161,0.174]$ | $[0.101,0.107]$ | $[0.063,0.084]$ | $[1.059,1.137]$

Table 6.3: 95% Confidence Intervals based on inverting various test
statistics. Instrument set used includes the original instrument set along
with interactions of the temperature measures set with all other aggregated
weather measures; $d_{z}=524$, with $n=1,671$. Thresholding test confidence
intervals are not reported as they coincide with confidence intervals for
$\text{\it JK}(\beta_{0})$.

##  7 Simulation Study

In this simulation study, I examine the performance of tests based on the
$\text{\it JK}(\beta_{0})$ statistic and compare it with that of other tests
that may be used in settings where the number of instruments is nonnegligible
as a fraction of sample size. I consider a reduced-form data-generating
process (DGP) similar to that of Matsushita and Otsu (2022). The outcome
variable, $y_{i}$, and endogenous variable, $x_{i}$, are generated according
to

$\begin{split}y_{i}&=x_{i}+\epsilon_{i}\\\ x_{i}&={}_{i}+v_{i}\end{split}$
(7.1)

where
${}_{i}=\frac{1}{r_{n}}\sumop\slimits@_{k=1}^{5}\frac{3}{4}\bar{z}_{ki}+\frac{1}{4}\bar{z}_{ki}^{2}+\frac{1}{4}\bar{z}_{ki}^{3}$
is a transformation of an initial set of instruments
$\bar{z}_{i}\in\mathbb{R}^{10}$ generated as described below. The value of
$r_{n}$ varies depending on the strength of identification considered; for
strong identification, $r_{n}=1$, while under weak identification,
$r_{n}=1/\sqrt{n}.$To model heteroskedasticity, the errors $(e_{i},v_{i})$ are
generated
$\epsilon_{i}=(1+\varrho_{1}(\bar{z}_{1i}^{2}+\bar{z}_{2i}^{2}+\bar{z}_{2i}\bar{z}_{3i}))e_{1i}$,
and $v_{i}=\varrho_{2}(1+\bar{z}_{1i})\epsilon_{i}+(1-\varrho_{2})^{2}e_{2i}$
where $e_{1i}$ and $e_{2i}$ are generated independently of each other and
other variables in the model according to a Laplace distribution with location
parameter $\mu=0$ and scale parameter $b=1$.111The Laplace distribution is
often referred to as a “double exponential” distribution. If $X_{1}$ and
$X_{2}$ are independently distributed according $\text{Exponential}(1)$, then
$Y=X_{1}-X_{2}$ has a Laplace distribution with parameters $\mu=0$ and $b=1$.
If $X$ has a Laplace distribution with parameters $\mu=0$ and $b=1$, then
$|X|\sim\text{Exponential}(1)$. Since the limiting $\chi^{2}$ distribution of
the jackknife K-statistic is exact when the errors are jointly Gaussian and
$\rho(z_{i})$ is known, I purposefully avoid normally distributed errors to
investigate the quality of asymptotic approximations to the finite-sample
behavior of the test. The parameters $\varrho_{1}$ and $\varrho_{2}$ control
the degree of heteroskedasticity and endogeneity, respectively.

DGP | Testing Procedure
---|---
|  |  |  |  |  |  |  |  |  |
$n$ | $d_{z}$ | $\varrho_{1}$ | $\varrho_{2}$ | $\text{\it JK}(\beta_{0})$ | $S(\beta_{0})$ | $T(\beta_{0};\tau_{0.3})$ | $T(\beta_{0};\tau_{0.75})$ | A.Rbn. | JAR | JLM
200 | 10 | 0.2 | 0.3 | 0.0516 | 0.0352 | 0.0406 | 0.0406 | 0.0296 | 0.0766 | 0.0502
|  | 0.2 | 0.6 | 0.0542 | 0.0306 | 0.0442 | 0.0384 | 0.0258 | 0.0748 | 0.0400
|  | 0.5 | 0.3 | 0.0470 | 0.0338 | 0.0416 | 0.0418 | 0.0238 | 0.0784 | 0.0460
|  | 0.5 | 0.6 | 0.0506 | 0.0350 | 0.0416 | 0.0390 | 0.0280 | 0.0676 | 0.0384
|  |  |  |  |  |  |  |  |  |
| 30 | 0.2 | 0.3 | 0.0570 | 0.0124 | 0.0422 | 0.0200 | 0.0088 | 0.1000 | 0.0382
|  | 0.2 | 0.6 | 0.0564 | 0.0126 | 0.0408 | 0.0208 | 0.0124 | 0.0962 | 0.0322
|  | 0.5 | 0.3 | 0.0498 | 0.0100 | 0.0366 | 0.0190 | 0.0096 | 0.1090 | 0.0318
|  | 0.5 | 0.6 | 0.0562 | 0.0118 | 0.0420 | 0.0216 | 0.0088 | 0.1104 | 0.0292
|  |  |  |  |  |  |  |  |  |
| 65 | 0.2 | 0.3 | 0.0542 | 0.0316 | 0.0428 | 0.0370 | 0.0314 | 0.0764 | 0.0420
|  | 0.2 | 0.6 | 0.0532 | 0.0366 | 0.0418 | 0.0398 | 0.0250 | 0.0780 | 0.0376
|  | 0.5 | 0.3 | 0.0474 | 0.0308 | 0.0388 | 0.0362 | 0.0244 | 0.0748 | 0.0354
|  | 0.5 | 0.6 | 0.0484 | 0.0324 | 0.0366 | 0.0388 | 0.0282 | 0.0708 | 0.0402
|  |  |  |  |  |  |  |  |  |
500 | 10 | 0.2 | 0.3 | 0.0590 | 0.0468 | 0.0478 | 0.0516 | 0.0376 | 0.0652 | 0.0452
|  | 0.2 | 0.6 | 0.0530 | 0.0420 | 0.0460 | 0.0466 | 0.0366 | 0.0692 | 0.0434
|  | 0.5 | 0.3 | 0.0496 | 0.0370 | 0.0408 | 0.0368 | 0.0338 | 0.0710 | 0.0464
|  | 0.5 | 0.6 | 0.0512 | 0.0426 | 0.0456 | 0.0438 | 0.0334 | 0.0696 | 0.0404
|  |  |  |  |  |  |  |  |  |
| 30 | 0.2 | 0.3 | 0.0522 | 0.0202 | 0.0386 | 0.0278 | 0.0238 | 0.0818 | 0.0322
|  | 0.2 | 0.6 | 0.0558 | 0.0208 | 0.0408 | 0.0310 | 0.0266 | 0.0888 | 0.0342
|  | 0.5 | 0.3 | 0.0554 | 0.0178 | 0.0392 | 0.0280 | 0.0174 | 0.0940 | 0.0272
|  | 0.5 | 0.6 | 0.0570 | 0.0156 | 0.0426 | 0.0236 | 0.0206 | 0.0984 | 0.0280
|  |  |  |  |  |  |  |  |  |
| 65 | 0.2 | 0.3 | 0.0542 | 0.0372 | 0.0434 | 0.0432 | 0.0384 | 0.0754 | 0.0464
|  | 0.2 | 0.6 | 0.0584 | 0.0442 | 0.0482 | 0.0470 | 0.0334 | 0.0676 | 0.0438
|  | 0.5 | 0.3 | 0.0614 | 0.0460 | 0.0504 | 0.0496 | 0.0316 | 0.0708 | 0.0434
|  | 0.5 | 0.6 | 0.0526 | 0.0378 | 0.0434 | 0.0420 | 0.0298 | 0.0692 | 0.0358
Table 7.1: Simulated Size of Identification and Heteroskedasticity Robust Tests under Weak Identification. Each DGP is simulated 5000 times. Critical values of the sup-score statistic and quantiles of the conditioning statistic are calculated using 1000 multiplier bootstrap simulations. DGP | Testing Procedure
---|---
|  |  |  |  |  |  |  |  |  |
$n$ | $d_{z}$ | $\varrho_{1}$ | $\varrho_{2}$ | $\text{\it JK}(\beta_{0})$ | $S(\beta_{0})$ | $T(\beta_{0};\tau_{0.3})$ | $T(\beta_{0};\tau_{0.75})$ | A.Rbn. | JAR | JLM
200 | 10 | 0.2 | 0.3 | 0.0474 | 0.0420 | 0.0474 | 0.0468 | 0.0308 | 0.0728 | 0.0424
|  | 0.2 | 0.6 | 0.0512 | 0.0386 | 0.0512 | 0.0506 | 0.0304 | 0.0764 | 0.0378
|  | 0.5 | 0.3 | 0.0416 | 0.0318 | 0.0414 | 0.0414 | 0.0248 | 0.0794 | 0.0428
|  | 0.5 | 0.6 | 0.0446 | 0.0342 | 0.0446 | 0.0442 | 0.0244 | 0.0806 | 0.0384
|  |  |  |  |  |  |  |  |  |
| 30 | 0.2 | 0.3 | 0.0482 | 0.0122 | 0.0448 | 0.0264 | 0.0110 | 0.1048 | 0.0370
|  | 0.2 | 0.6 | 0.0498 | 0.0120 | 0.0480 | 0.0312 | 0.0118 | 0.0980 | 0.0378
|  | 0.5 | 0.3 | 0.0456 | 0.0126 | 0.0410 | 0.0262 | 0.0082 | 0.1146 | 0.0268
|  | 0.5 | 0.6 | 0.0482 | 0.0110 | 0.0474 | 0.0308 | 0.0094 | 0.1090 | 0.0302
|  |  |  |  |  |  |  |  |  |
| 65 | 0.2 | 0.3 | 0.0528 | 0.0380 | 0.0526 | 0.0510 | 0.0276 | 0.0696 | 0.0460
|  | 0.2 | 0.6 | 0.0464 | 0.0360 | 0.0464 | 0.0468 | 0.0302 | 0.0728 | 0.0416
|  | 0.5 | 0.3 | 0.0482 | 0.0298 | 0.0480 | 0.0466 | 0.0246 | 0.0738 | 0.0412
|  | 0.5 | 0.6 | 0.0396 | 0.0320 | 0.0390 | 0.0386 | 0.0258 | 0.0748 | 0.0356
|  |  |  |  |  |  |  |  |  |
500 | 10 | 0.2 | 0.3 | 0.0524 | 0.0444 | 0.0524 | 0.0524 | 0.0394 | 0.0684 | 0.0472
|  | 0.2 | 0.6 | 0.0476 | 0.0430 | 0.0476 | 0.0476 | 0.0400 | 0.0644 | 0.0490
|  | 0.5 | 0.3 | 0.0434 | 0.0410 | 0.0434 | 0.0434 | 0.0340 | 0.0702 | 0.0404
|  | 0.5 | 0.6 | 0.0448 | 0.0382 | 0.0448 | 0.0448 | 0.0350 | 0.0736 | 0.0432
|  |  |  |  |  |  |  |  |  |
| 30 | 0.2 | 0.3 | 0.0502 | 0.0214 | 0.0502 | 0.0498 | 0.0240 | 0.0854 | 0.0368
|  | 0.2 | 0.6 | 0.0522 | 0.0208 | 0.0522 | 0.0524 | 0.0224 | 0.0858 | 0.0392
|  | 0.5 | 0.3 | 0.0456 | 0.0202 | 0.0456 | 0.0434 | 0.0220 | 0.0918 | 0.0264
|  | 0.5 | 0.6 | 0.0500 | 0.0186 | 0.0500 | 0.0498 | 0.0204 | 0.0924 | 0.0268
|  |  |  |  |  |  |  |  |  |
| 65 | 0.2 | 0.3 | 0.0490 | 0.0426 | 0.0490 | 0.0490 | 0.0350 | 0.0742 | 0.0472
|  | 0.2 | 0.6 | 0.0522 | 0.0458 | 0.0522 | 0.0522 | 0.0436 | 0.0652 | 0.0442
|  | 0.5 | 0.3 | 0.0542 | 0.0476 | 0.0542 | 0.0542 | 0.0294 | 0.0712 | 0.0446
|  | 0.5 | 0.6 | 0.0438 | 0.0420 | 0.0438 | 0.0438 | 0.0306 | 0.0666 | 0.0500
Table 7.2: Simulated Size of Identification and Heteroskedasticity Robust
Tests under Strong Identification. Each DGP is simulated 5000 times. Critical
values of the sup-score statistic and quantiles of the conditioning statistic
are calculated using 1000 multiplier bootstrap simulations.

In addition to considering the behavior of tests under both weak and strong
identification, I examine the size of the test under three different
instrument regimes. In all three regimes, I begin with an initial set of
instruments $\bar{z}_{i}=(\bar{z}_{1i},\dots,\bar{z}_{10i})^{\prime}$
generated independently across indices according to a multivariate Gaussian
distribution with Toeplitz covariance structure,
$\operatorname{Cov}(\bar{z}_{\ell i},\bar{z}_{ki})=2^{-|\ell-k|}$. In the
first regime, the full set instruments $z_{i}$ is taken to be equal to
$\bar{z}_{i}$ so that $d_{z}=10$. In the second regime, the full set of
instruments $z_{i}$ additionally includes all quadratic and cubic terms,
$(z_{\ell i}^{2},z_{\ell i}^{3})$, $\ell=1,\dots,10$ so that in total
$d_{z}=30$. In the third regime, the full set of instrument includes the
initial set of instruments, $\bar{z}_{i}$, and all quadratic terms (10
additional terms) and interactions of the initial set of instruments
($\binom{10}{2}=45$additional terms), so that in total $d_{z}=65$. Under each
regime, the full set of instruments is passed to the test statistics with no
indication about which instruments correspond to the initial set, and thus no
indication about which instruments are relevant to the DGP.

Figure 7.1: Calibrated Local Power Curves under Intermediate Identification
Strength and 65 Instruments. Sample size is 500 and rejection probability is
calculated on a grid of 100 $(\beta_{0}-\beta)$ points between -4 and 4. At
each point the DGP is simulated 2000 times.

I compare the simulated size of the jackknife K test and to the performance of
the sup-score test, $S(\beta_{0})$, of Belloni et al. (2012), the thresholding
test introduced in Section 4.2, the standard Anderson-Rubin (A.Rbn.) test of
Anderson and Rubin (1949) and Staiger and Stock (1997), the jackknife AR test
(JAR) of Crudu et al. (2021) and Mikusheva and Sun (2021), and the jackknife
LM test (JLM) of Matsushita and Otsu (2022). To estimate the parameter
$\rho(z_{i})$, I implement the $\ell_{1}$-penalized procedure of (2.2) via the
glmnet package in R (Friedman et al., 2010). The penalty parameter $\lambda$
is selected via tenfold cross-validation. I use the full vector of instruments
as the basis to approximate $\rho(z_{i})$. For the jackknife AR test I use
cross-fit estimates of test statistic variances proposed and shown to improve
power by Mikusheva and Sun (2021).Critical values of the sup-score and
conditioning statistic are simulated with the procedures described in Section
4 with 1000 bootstrap replications. For the combination test cutoff, I
consider two different quantiles of the conditioning statistic under the
assumption that $\mathbb{E}[\mathaccent 0{\symlargesymbols}062{}_{i}^{I}]=0$
for all $i\in[n]$; $\tau_{0.3}$ corresponding to the 30th quantile and
$\tau_{0.75}$ corresponding to the 75th quantile.Tables 7.1 and 7.2 report the
simulated size for all tests under weak and strong identification,
respectively. One can see that the $\text{\it JK}(\beta_{0})$ statistic has
nearly exact size in almost all the setups considered. In contrast, the
jackknife AR test seems to overreject in nearly all the simulation setups
considered. This is also the case in the simulation study of Matsushita and
Otsu (2022) and so may be an artifact of the similarity of my simulation
design to theirs.The sup-score, jackknife AR, and jackknife LM test all seem
to have particularly poor performance under both weak and strong
identification when $d_{z}=30$. This is the setup with the most correlation
between the instruments. While tests based on the jackknife AR statistic can
have a simulated size that is nearly double the nominal size in this setting,
both the sup-score and and jackknife LM tests appear to be conservative. The
size of the sup-score test is always under $0.025$ while the size of the JLM
test can be under half of the nominal size. Notably, the size properties of
the sup-score test do seem to improve under both weak and strong
identification when the sample size increases from $n=200$ to $n=500$. This is
in line with theoretical results showing that the sup-score test has exact
asymptotic size under standard conditions. In contrast, the size properties of
the jackknife LM test do not seem to improve when the sample size increases
and indeed worsen for three out of the four DGPs considered under both weak
and strong identification. This suggests that the requirement of
$d_{z}\to\infty$ may be important for the quality of finite-sample
approximation by its limiting distribution.The thresholding test seems to
control size in all the setups considered. However, under weak identification
the thresholding test appears to inherit the conservative nature of the sup-
score test, even in the “large” sample size regime of $n=500$. This is not the
case under strong identification, suggesting that the thresholding-test is
choosing to run tests based on the $\text{\it JK}(\beta_{0})$ with high
probability in this regime. This behavior is similar to the conditional
combination tests of Moreira (2003); Andrews (2016) which weigh the
K-statistic more under strong identification. This behavior is optimal as the
K-statistic yields effecient inference when the data is informative about the
structural parameter (Andrews et al., 2004, 2006). When errors are
homoskedastic and the number of instruments is fixed, the jackknife
K-statistic can also be shown to yield effecient inference under strong
identification.

Figure 7.2: Calibrated Local Power Curves under Intermediate identification
Strength and 75 Instruments. Sample size is 500 and rejection power is
calculated on a grid of 100 $(\beta_{0}-\beta)$ points between -4 and 4. At
each point the DGP is simulated 2000 times.

Figure 7.1 plots calibrated local power curves under an intermediate-strength
identification where the first stage is in a $n^{-1/3}$ neighborhood of zero,
$d_{z}=65$, $\varrho_{1}\in\\{0.2,0.5\\}$ and $\varrho_{2}\in\\{0.3,0.6\\}$.
The critical value of each test is set to simulated 95th quantile of the
distribution of the corresponding test-statistic under $H_{0}$. I compare the
calibrated local power curves of the $\text{\it JK}(\beta_{0})$ test, the
combination test with cutoff $\tau_{0.75}$, the jackknife AR test, the
Jackknife LM test, and the sup-score test. The jackknife K-test appears to
have stronger power than the jackknife AR, jackknife LM, and sup-score tests
in local neighborhoods of the null as well as for negative values of
$(\beta_{0}-\beta)$. For values of $(\beta_{0}-\beta)$ larger than 1.5, tests
based on the jackknife K-statistic appear to suffer from a loss of power as
described inSection 4. This power decline appears to be largerly ameliorated
by combining the jackknife K-statistic with the sup-score statistic and the