Robust Conditional Wald Inference for Over-Identified IV††footnotetext: * Lee:
Princeton University and NBER (email: davidlee@princeton.edu); McCrary:
Columbia University and NBER (email: jmccrary@law.columbia.edu); Moreira: FGV
EPGE (email: mjmoreira@fgv.br); Porter: University of Wisconsin (email:
jrporter@ssc.wisc.edu); Yap: Princeton University (email: lyap@princeton.edu).

By David S. Lee, Justin McCrary, Marcelo J. Moreira,
Jack Porter, Luther Yap*

November 2023

For the over-identified linear instrumental variables model, researchers
commonly report the 2SLS estimate along with the robust standard error and
seek to conduct inference with these quantities. If errors are homoskedastic,
one can control the degree of inferential distortion using the first-stage $F$
critical values from Stock and Yogo (2005), or use the robust-to-weak
instruments Conditional Wald critical values of Moreira (2003). If errors are
non-homoskedastic, these methods do not apply. We derive the generalization of
Conditional Wald critical values that is robust to non-homoskedastic errors
(e.g., heteroskedasticity or clustered variance structures), which can also be
applied to nonlinear weakly-identified models (e.g. weakly-identified GMM).

Keywords: Instrumental Variables, Weak Instruments, $t$-ratio, First-stage
$F$-statistic, Conditional Wald

##  I Introduction

This paper considers inference in the over-identified linear instrumental
variables model and its generalization to weakly-identified models and GMM
more generally. The core problem of inference in the weak IV literature is
that when instruments are weak, conventional asymptotic approximations are
poor, causing standard inference procedures (like Wald or $t$-ratio-based
inference) to over-reject, even under the null. Indeed, Dufour (1997) pointed
out that any confidence set that is bounded with probability 1 (like the usual
$\hat{\beta}\pm 1.96\cdot\hat{se}(\hat{\beta})$) could have the potential to
cover the true parameter 0 percent of the time (i.e., zero percent confidence
level).

Moreira (2003) provided a generalized approach to constructing inference
procedures that addressed this weak instrument problem via data-dependent
critical values; this approach was first demonstrated in the case when errors
are homoskedastic. Moreira (2003) presented, for the hypothesis that the
parameter is equal to a particular value, conditional versions of the
"trinity" of test procedures: Likelihood Ratio (LR), Lagrange Multiplier (LM),
and Wald tests. The critical values for each of these test statistics were
functions of the observed data and the null hypothesis.

Since then, a number of efforts have generalized these tests to accommodate
non-homoskedastic settings, given the widespread preference of applied
researchers to remain somewhat agnostic about the properties of the errors in
the linear model.111See, for example, Andrews, Moreira and Stock (2004),
Kleibergen (2005), Andrews (2016). Curiously, while there have been efforts to
generalize the Conditional LR (CLR) and LM tests to general non-homoskedastic
errors, to the best of our knowledge, the extension of the Conditional Wald in
such a way has been neglected.

In this paper, we derive the extension of Conditional Wald to non-
homoskedastic settings. This effort delivers a Wald-based inference procedure
that is valid, similar, robust to arbitrarily weak instruments, robust to HAC
error structures, and applicable to more general weakly-identified settings
like GMM.

There are a number of practical reasons to revisit a testing procedure rooted
in a Wald approach. First, for the linear instrumental variables model,
applied research has revealed a preference for Wald-based inference. Most
typically, researchers compute and report the 2SLS estimator and robust
standard errors, regardless of concerns about instrument weakness. In
homoskedastic settings, researchers could pursue two different options for
conducting Wald-based inference using the computed $t$-ratio. Researchers can
either use the critical value function for Conditional Wald in Moreira (2003),
or they can use the first-stage $F$-statistic along with the critical value
tables in Stock and Yogo (2005) and the Bonferroni arguments used in Staiger
and Stock (1997). When the errors are non-homoskedastic – as is typically
allowed in modern empirical work – the values in Stock and Yogo (2005) tables
no longer reliably control size distortions, as pointed out in Andrews, Stock
and Sun (2019). The contribution of the current paper is to provide a method
for computing critical values for the $t$-ratio that will deliver valid,
robust inference, even in non-homoskedastic settings.222In this paper,
acceptance/rejection of the null is the result of comparing a single statistic
with a valid (in this case, data-dependent) critical value. A different, ”two-
step” inference approach where two different procedures (one ”robust to weak
instruments” and the other non-robust) are combined to form an overall valid
procedure (which can also accommodate non-homoskedastic settings) is proposed
by Andrews (2018).

A second reason to consider a robust-to-HAC version of Conditional Wald of
Moreira (2003) is that there are two recent studies pointing to power
advantages of Conditional Wald in the homoskedastic, over-identified setting
and in the non-homoskedastic just-identified setting. Van de Sijpe and
Windmeijer (2023) analyze power for the over-identified, homoskedastic case,
and provide simulation evidence that Conditional Wald using 2SLS tends to
produce shorter confidence set lengths, compared to CLR (Moreira (2003)). This
is a particularly striking finding, in light of papers that point to the near-
optimality, in terms of power, of CLR. Furthermore, Lee et al. (2023), in the
context of the just-identified (robust to HAC errors) IV model, show that two
different Wald-based – $VtF$ and Conditional Wald – confidence intervals
appear to be almost always shorter than that of Anderson and Rubin (1949), a
recommended benchmark in the literature. Thus, developing the Conditional Wald
robust to HAC errors is not only already aligned with practitioner practice,
but these recent studies suggest that it may even have power advantages in the
form of shorter confidence intervals.

Our motivation for deriving CW critical values is entirely practical and stems
from taking as given practitioners’ apparent preference for computing the 2SLS
point estimate and robust standard error (presuming non-homoskedasticity), and
finding critical values that lead to valid inference. Our approach is thus
different from identifying the optimal test after having defined a class of
procedures and an objective function. Nevertheless, the two studies mentioned
above do suggest the possibility that in terms of power and confidence
interval length, CW could fare well compared to existing alternatives for the
over-identified model.

Section II establishes the notation we use for the standard linear IV model
with non-homoskedastic errors, Section III derives the critical values for
Robust Conditional Wald tests based on 2SLS, LIML, two-step, and CUE GMM
estimators, Section IV extends the test to nonlinear weakly-identified models
(e.g. GMM), and Section V concludes.

##  II The Linear IV Model

The standard linear IV model is represented by

$\displaystyle y_{1}$ $\displaystyle=$ $\displaystyle Y_{2}\beta+u$
$\displaystyle Y_{2}$ $\displaystyle=$ $\displaystyle Z\Pi+V_{2}$

where $y_{1}$ $\left(n\times 1\right)$ is the dependent variable, $Y_{2}$
$\left(n\times p\right)$ are the endogenous variable(s) of interest, and $Z$
$\left(n\times k\right)$ are the excluded instruments, while $u$
$\left(n\times 1\right)$ and $V_{2}$ $\left(n\times p\right)$ are the
unobserved structural-form errors. The single endogenous regressor case simply
corresponds to $p=1$. We will always take $k\geq p$ with $k=p$ corresponding
to the just-identified model and $k>p$ corresponding to the over-identified
model. The parameter of interest is $\beta$. It is straightforward to
accommodate additional covariates (including a constant), but we omit their
inclusion in the exposition below.

The reduced-form model is:

$\displaystyle y_{1}$ $\displaystyle=$ $\displaystyle Z\Pi\beta+v_{1}$
$\displaystyle Y_{2}$ $\displaystyle=$ $\displaystyle Z\Pi+V_{2},$

where $u\equiv v_{1}-V_{2}\beta$. It will be convenient to write the model in
a matrix form:

$Y=Z\Pi A+V,$

where $Y=\left[y_{1}:Y_{2}\right]$, $V=\left[v_{1}:V_{2}\right]$, and
$A=\left[\beta:I_{p}\right]$. We will use the notation $Y_{i}$, $V_{i}$,
$Z_{i}$, etc, to denote the $i$-th row of the corresponding matrix.

In Moreira (2003), the rows of $V$ were assumed to be i.i.d. This paper
relaxes this assumption for the derivation of the Conditional Wald test robust
to different DGPs. We are motivated by the observation that applied
researchers typically prefer not to make the assumption of homoskedasticity,
and often they are interested in a clustered error structure, for example.

##  III The Robust Conditional Wald Tests

In this section, we derive the Conditional Wald (CW) tests robust to HAC
errors for the linear model given in section II.

The Wald statistic is formed by three elements: a null value $\beta_{0}$, an
estimator $\widehat{\beta}_{n}$, and a robust asymptotic variance estimator
$\widehat{A.Var}$ for $\sqrt{n}\left(\widehat{\beta}_{n}-\beta_{0}\right)$:

$\widehat{\mathcal{W}}_{n}=n\left(\widehat{\beta}_{n}-\beta_{0}\right)^{\prime}\left[\widehat{A.Var}\right]^{-1}\left(\widehat{\beta}_{n}-\beta_{0}\right).$

In section III.A, we review the class of linear GMM estimators for $\beta$,
which includes common estimators like 2SLS, LIML, efficient two-step GMM, and
the CU (continuously updating) GMM estimator, all of which can be used for
constructing a Robust Conditional Wald test. In section III.B, we review the
robust variance estimators based on the asymptotic distribution of
$\sqrt{n}\left(\widehat{\beta}_{n}-\beta_{0}\right)$. With these components in
hand, we can form robust versions of the Wald statistic for various
estimators, $\widehat{\beta}_{n}$. Note that there are no new results in
Sections III.A and III.B, and there are many references that detail these
standard results (as one example, see Newey and McFadden (1994)). We review a
selected set of well-established facts about GMM to highlight that the Robust
Conditional Wald test we derive in III.C is not specific to the leading case
in applied work – 2SLS – and can be applied to tests based on other estimators
of the parameter of interest. Note that the multitude of different estimators
that could be employed arises in the over-identified case; in contrast, for
example, in the single instrument case, all of the estimators we discuss below
collapse to the standard IV estimator.

With that as context, an interested reader can skip to section III.C, in which
we show how to apply the conditional argument of Moreira (2003) to obtain a
critical value function for the Wald statistic that is robust to instrument
weakness. The critical value function is then used to form a robust similar
test, which can be inverted to generate confidence intervals for $\beta$.

###  III.A Estimators

Estimators like 2SLS or LIML can be viewed as particular GMM estimators based
on the linear moment condition:

$\overline{g}_{n}\left(\beta\right)=n^{-1}\sum\limits_{i=1}^{n}Z_{i}\left(y_{1i}-Y_{2i}^{\prime}\beta\right)=n^{-1}Z^{\prime}Yb,$

where $b=\left(1,-\beta^{\prime}\right)^{\prime}$. A GMM estimator for $\beta$
is the minimizer of the criterion

(1)
$\widehat{Q}_{n}\left(\beta\right)=\overline{g}_{n}\left(\beta\right)^{\prime}W_{n}\left(\beta\right)\overline{g}_{n}\left(\beta\right)$

where the weighting matrix may or may not depend on the unknown coefficient
$\beta$. Different choices of $W_{n}\left(\beta\right)$ will lead to different
estimators $\widehat{\beta}_{n}$.

For the 2SLS estimator, the weighting matrix does not depend on $\beta$:

$W_{n}{}^{-1}=\widehat{V}_{u}\cdot
n^{-1}\sum\limits_{i=1}^{n}Z_{i}Z_{i}^{\prime}=\widehat{V}_{u}\cdot
n^{-1}Z^{\prime}Z,$

where $\widehat{V}_{u}$ is an estimator of $V_{u}$, which is the variance of
$u$. Because $\widehat{Q}_{n}\left(\beta\right)$ is quadratic in $\beta$, it
is straightforward to show that the resulting estimator is 2SLS:

(2)
$\widehat{\beta}_{n}=\left[Y_{2}^{\prime}Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}Y_{2}\right]^{-1}Y_{2}^{\prime}Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}y_{1}.$

We can re-express the estimator as

$\widehat{\beta}_{n}=\left[\widehat{Y}_{2}^{\prime}\widehat{Y}_{2}\right]^{-1}\widehat{Y}_{2}^{\prime}y_{1},$

where $\widehat{Y}_{2}=NY_{2}$ and
$N=Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}$ is the usual projection matrix.
That is, in the first stage, we first regress $Y_{2}$ on $Z$ to obtain the
fitted values $\widehat{Y}_{2}$. In the second stage, we regress $y_{1}$ on
the fitted values $\widehat{Y}_{2}$.

For the LIML estimator, the weight matrix is formed by using $\beta$ along
with residuals from OLS regressions of $y_{1}$ and $Y_{2}$ on $Z$. Let
$b=\left(1,-\beta^{\prime}\right)^{\prime}$,
$N=Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}$, and $M=I-N$. Then

$\widehat{V}_{u}\left(\beta\right)=n^{-1}\sum\limits_{i=1}^{n}\left(\widehat{v}_{1,i}-\widehat{V}_{2,i}^{\prime}\beta\right)^{2}=n^{-1}b^{\prime}Y^{\prime}MYb,$

where $\widehat{V}=MY=MV$. The GMM criterion is no longer quadratic in $\beta$
once we use

$W_{n}\left(\beta\right)^{-1}=\widehat{V}_{u}\left(\beta\right)\cdot
n^{-1}\sum\limits_{i=1}^{n}Z_{i}Z_{i}^{\prime}=\widehat{V}_{u}\left(\beta\right)\cdot
n^{-1}Z^{\prime}Z.$

Instead it is a ratio of quadratic forms:

$\widehat{Q}_{n}\left(\beta\right)=\frac{n^{-1}\sum\limits_{i=1}^{n}\left(y_{1i}-Y_{2i}^{\prime}\beta\right)Z_{i}\left(n^{-1}\sum\limits_{i=1}^{n}Z_{i}Z_{i}^{\prime}\right)^{-1}n^{-1}\sum\limits_{i=1}^{n}Z_{i}\left(y_{1i}-Y_{2i}^{\prime}\beta\right)}{n^{-1}\sum\limits_{i=1}^{n}\left(\widehat{V}_{1,i}-\widehat{V}_{2,i}^{\prime}\beta\right)^{2}}.$

The minimum of

$\widehat{Q}_{n}\left(\beta\right)=\frac{b^{\prime}Y^{\prime}NYb}{b^{\prime}Y^{\prime}MYb}$

is well-known to lead to the LIML estimator (see Davidson and MacKinnon (2021)
among others). This estimator is proportional to the eigenvector associated to
the smallest eigenvalue $\underline{\lambda}_{n}$of the characteristic
polynomial $\left|Y^{\prime}NY-\lambda.Y^{\prime}MY\right|=0$.

Once we consider different weighting functions $W_{n}\left(\beta\right)$, we
can wonder if there is the “best” possible choice. The answer depends if
errors are heteroskedastic, clustered, etc. Only in special cases, such as
with homoskedastic errors, are the 2SLS and LIML estimators “best.” To obtain
the weighting function that optimally accounts for heteroskedasticity,
clustering, serial correlation, and other departures from homoskedastic errors
with no serial correlation, one first considers the (infeasibly estimated)
variance of

$vec\left(n^{-1/2}\sum\limits_{i=1}^{n}Z_{i}V_{i}^{\prime}\right)=n^{-1/2}\sum\limits_{i=1}^{n}\left(V_{i}\otimes
Z_{i}\right).$

Under general conditions for the DGPs, the limiting variance exists and is
given by

$\Omega=\lim_{n}\,n^{-1}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}C\left(V_{i}\otimes
Z_{i},V_{j}\otimes Z_{j}\right).$

Since we do not observe the errors $V_{i}$, we can make this feasible by
replacing them with, as an example, the OLS residuals $\widehat{V}_{i}$. There
are different estimators for the variances and covariances above, each one of
them suited to different assumptions on the DGPs. For example, typically, one
uses the variance estimate of White (1980) for heteroskedastic errors that are
serially uncorrelated:

$\widehat{\Omega}_{n}=n^{-1}\sum\limits_{i=1}^{n}\left(\widehat{V}_{i}\otimes
Z_{i}\right)\left(\widehat{V}_{i}\otimes Z_{i}\right)^{\prime}.$

Henceforth, we will employ the broader notation $\widehat{\Omega}_{n}$ without
explicitly specifying its formulae for different departures from
homoskedasticity. Examples of robust variance estimators include White (1980)
for heteroskedasticity, Newey and West (1987) and Andrews (1991) for both
heteroskedasticity and autocorrelation (HAC), and Cameron, Gelbach and Miller
(2011) for clustered errors. See Andrews, Moreira and Stock (2004) for the IV
model.

The GMM criterion is then

$\displaystyle\widehat{Q}_{n}\left(\beta\right)$ $\displaystyle=$
$\displaystyle\left[n^{-1}\sum\limits_{i=1}^{n}\left(Y_{i}-X_{i}^{\prime}\beta\right)Z_{i}\right]W_{n}\left[n^{-1}\sum\limits_{i=1}^{n}Z_{i}\left(Y_{i}-X_{i}^{\prime}\beta\right)\right]\text{,
where}$ $\displaystyle W_{n}{}^{-1}$ $\displaystyle=$
$\displaystyle\left(\widetilde{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Omega}_{n}\left(\widetilde{b}_{n}\otimes
I_{k}\right)\text{ and
}\widetilde{b}_{n}=\left(1,-\widetilde{\beta}_{n}^{\prime}\right)^{\prime}\text{,}$

with $\widetilde{\beta}_{n}$ being a preliminary consistent estimator of
$\beta$. Again, this criterion

$\widehat{Q}_{n}\left(\beta\right)=b^{\prime}Y^{\prime}ZW_{n}Z^{\prime}Yb$

is quadratic in $\beta$ and we can easily find its closed-form solution:

(3)
$\widehat{\beta}_{n}=\left[Y_{2}^{\prime}ZW_{n}Z^{\prime}Y_{2}\right]^{-1}Y_{2}{}^{\prime}ZW_{n}Z^{\prime}y.$

which is the two-step GMM estimator. It simplifies to the 2SLS estimator if
$W_{n}^{-1}$ is proportional to $n^{-1}Z^{\prime}Z$. When the weight matrix
depends on $\beta$, we obtain

$\displaystyle\widehat{Q}_{n}\left(\beta\right)$ $\displaystyle=$
$\displaystyle
n^{-1}\sum\limits_{i=1}^{n}\left(y_{1i}-Y_{2i}^{\prime}\beta\right)Z_{i}W_{n}{}\left(\beta\right)n^{-1}\sum\limits_{i=1}^{n}Z_{i}\left(y_{1i}-Y_{2i}^{\prime}\beta\right)\text{,
where}$ $\displaystyle W_{n}{}\left(\beta\right)^{-1}$ $\displaystyle=$
$\displaystyle\left(b\otimes
I_{k}\right)^{\prime}\widehat{\Omega}_{n}\left(b\otimes I_{k}\right).$

which is the Continuously Updating (CU) GMM estimator, proposed by Hansen and
Singleton (1982).

###  III.B Wald Test Statistics

Finally, the usual Wald test statistics are based on the standard asymptotic
approximation to the distribution of the GMM estimators. Under the true
parameter $\beta_{0}$,

$n^{1/2}\overline{g}_{n}\left(\beta_{0}\right)\rightarrow_{d}N\left(0,V_{0}\right)\text{,
where }V_{0}=\left(b_{0}\otimes I_{k}\right)^{\prime}\Omega\left(b_{0}\otimes
I_{k}\right)$

for $b_{0}=\left(1,-\beta_{0}^{\prime}\right)$. For convenience, we derive the
asymptotic distribution where we use the parameter $\beta_{0}$ in the
criterion function:

$\overline{Q}_{n}\left(\beta\right)=\overline{g}_{n}\left(\beta\right)^{\prime}W_{n}\left(\beta_{0}\right)\overline{g}_{n}\left(\beta\right).$

This setup allows for the possibility that the limiting behavior of
$W_{n}\left(\beta_{0}\right)$ is not necessarily proportional to $V_{0}^{-1}$.
Hence, $\widehat{\beta}_{n}$ is not necessarily optimal. Under the usual
asymptotics, the distribution of estimators which minimize
$\overline{Q}_{n}\left(\beta\right)$ is the same as if we had used
$\widehat{Q}_{n}\left(\beta\right)$ instead, where the weight uses a
preliminary estimator or uses $\beta$ itself (derivations of these results are
standard and can be found, e.g. in Newey and McFadden (1994)). As a result,
the 2SLS and LIML estimators are asymptotically equivalent, while the two-step
GMM and CUE estimators are asymptotically equivalent as well. We derive the
asymptotic distribution for the 2SLS and two-step GMM estimators from
equations (2) and (3) and, so, for the LIML and continuously updating
estimators as well.

For the 2SLS estimator, we can write

$n^{1/2}\left(\widehat{\beta}_{n}-\beta_{0}\right)=\left[n^{-1}Y_{2}^{\prime}Z\left(n^{-1}Z^{\prime}Z\right)^{-1}n^{-1}Z^{\prime}Y_{2}\right]^{-1}n^{-1}Y_{2}^{\prime}Z\left(n^{-1}Z^{\prime}Z\right)^{-1}n^{-1/2}Z^{\prime}Vb.$

Assuming that the following probability limits exist,

$plim\text{ }n^{-1}Y_{2}^{\prime}Z=E_{Y_{2}^{\prime}Z}\text{ and }plim\text{
}n^{-1}Z^{\prime}Z=E_{Z^{\prime}Z}\text{,}$

we then have
$n^{1/2}\left(\widehat{\beta}_{n}-\beta_{0}\right)\rightarrow_{d}N\left(0,B_{0}^{-1}A_{0}B_{0}^{-1}\right)$,
where

$\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle\left[E_{Y_{2}^{\prime}Z}E_{Z^{\prime}Z}^{-1}E_{Z^{\prime}Y_{2}}\right]^{-1}\text{
and}$ $\displaystyle A_{0}$ $\displaystyle=$ $\displaystyle
E_{Y_{2}^{\prime}Z}E_{Z^{\prime}Z}^{-1}\left(b_{0}\otimes
I_{k}\right)^{\prime}\Omega\left(b_{0}\otimes
I_{k}\right)E_{Z^{\prime}Z}^{-1}E_{Z^{\prime}Y_{2}}\text{.}$

We can find some consistent estimators for $A_{0}$ and $B_{0}$, and derive a
Wald statistic for the 2SLS and LIML estimators:

$\displaystyle\widehat{\mathcal{W}}_{n}^{\ast}$ $\displaystyle=$
$\displaystyle
n\left(\widehat{\beta}_{n}-\beta_{0}\right)^{\prime}\left[\widehat{B}_{n}^{-1}\widehat{A}_{n}\widehat{B}_{n}^{-1}\right]^{-1}\left(\widehat{\beta}_{n}-\beta_{0}\right)\text{,
where}$ $\displaystyle\widehat{B}_{n}$ $\displaystyle=$ $\displaystyle
n^{-1}Y_{2}^{\prime}Z\left(Z^{\prime}Z\right)^{-1}Z^{\prime}Y_{2}\text{ and}$
$\displaystyle\widehat{A}_{n}$ $\displaystyle=$ $\displaystyle
Y_{2}^{\prime}Z\left(Z^{\prime}Z\right)^{-1}\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Omega}_{n}\left(\widehat{b}_{n}\otimes
I_{k}\right)\left(Z^{\prime}Z\right)^{-1}Z^{\prime}Y_{2}\text{,}$

with $\widehat{b}_{n}=\left(1,-\widehat{\beta}_{n}^{\prime}\right)^{\prime}$
based on the respective 2SLS/LIML estimator.555We typically use
$\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Omega}_{n}\left(\widehat{b}_{n}\otimes
I_{k}\right)$ as a consistent estimator for $V_{0}$. However, other estimators
are possible, including $\left(b_{0}\otimes
I_{k}\right)^{\prime}\widehat{\Omega}_{n}\left(b_{0}\otimes I_{k}\right)$.

Likewise, for the two-step GMM estimator, we find that
$n^{1/2}\left(\widehat{\beta}_{n}-\beta_{0}\right)\rightarrow_{d}N\left(0,B_{0}^{-1}\right)$,
where

$B_{0}=E_{Y_{2}^{\prime}Z}\left[\left(b_{0}\otimes
I_{k}\right)^{\prime}\Omega\left(b_{0}\otimes
I_{k}\right)\right]^{-1}E_{Z^{\prime}Y_{2}}.$

We can find a consistent estimator for $B_{0}$ and derive a Wald statistic for
the two-step GMM and continuously updating estimator:

$\displaystyle\widehat{\mathcal{W}}_{n}^{o}$ $\displaystyle=$ $\displaystyle
n\left(\widehat{\beta}_{n}-\beta_{0}\right)^{\prime}\widehat{B}_{n}\left(\widehat{\beta}_{n}-\beta_{0}\right)\text{,
where}$ $\displaystyle\widehat{B}_{n}$ $\displaystyle=$ $\displaystyle
n^{-1}Y_{2}^{\prime}Z\left[\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Omega}_{n}\left(\widehat{b}_{n}\otimes
I_{k}\right)\right]^{-1}n^{-1}Z^{\prime}Y_{2}.$

with $\widehat{b}_{n}=\left(1,-\widehat{\beta}_{n}^{\prime}\right)^{\prime}$
based on the GMM/CU estimators.666We can also use here either the null value
$\beta_{0}$ or the preliminary estimator $\widetilde{\beta}_{n}$ for the
variance estimator.

###  III.C Valid Critical Value Functions

As emphasized in Dufour (1997), since the nuisance parameter representing the
strength of the first stage may be arbitrarily close to zero, then the usual
constant critical values cannot be valid; indeed, Dufour (1997) points out
that any valid confidence set in this context must be unbounded with positive
probability, which clearly cannot be the case with a constant critical value
for any of the Wald statistics mentioned above. To derive valid critical
values, using the conditioning strategy of Moreira (2003), we begin by
defining the quantity

$R=\left(Z^{\prime}Z\right)^{-1/2}Z^{\prime}Y=\left[R_{1}:R_{2}\right],$

where the $k$-dimensional vector $R_{1}$ is the first column of $R$ and the
$k\times p$-matrix $R_{2}$ is the last $p$ columns of $R$. The standardization
avoids multiplication by the sample size $n$. The asymptotic variance of
$vec(R)$ is

$\Sigma=\left(I_{p+1}\otimes\left(E_{Z^{\prime}Z}\right)^{-1/2}\right)\Omega\left(I_{p+1}\otimes\left(E_{Z^{\prime}Z}\right)^{-1/2}\right)=\left[\begin{array}[]{cc}\Sigma_{11}&\Sigma_{12}\\\
\Sigma_{21}&\Sigma_{22}\end{array}\right],$

where the matrix $\Sigma$ is being partitioned by submatrices of columns/rows
of dimensions $1$ and $p$. Analogously, we can use the estimator

$\widehat{\Sigma}_{n}=\left(I_{p+1}\otimes\left(n^{-1}Z^{\prime}Z\right)^{-1/2}\right)\widehat{\Omega}_{n}\left(I_{p+1}\otimes\left(n^{-1}Z^{\prime}Z\right)^{-1/2}\right)=\left[\begin{array}[]{cc}\widehat{\Sigma}_{11,n}&\widehat{\Sigma}_{12,n}\\\
\widehat{\Sigma}_{21,n}&\widehat{\Sigma}_{22,n}\end{array}\right].$

Up to a scale of the sample size $n$, the GMM criterion is

$\widehat{Q}_{n}\left(\beta\right)=b^{\prime}R^{\prime}\left(n^{-1}Z^{\prime}Z\right)^{1/2}W_{n}\left(\beta\right)\left(n^{-1}Z^{\prime}Z\right)^{1/2}Rb=b^{\prime}R^{\prime}\overline{W}_{n}\left(\beta\right)Rb,$

where
$\overline{W}_{n}\left(\beta\right)=\left(n^{-1}Z^{\prime}Z\right)^{1/2}W_{n}\left(\beta\right)\left(n^{-1}Z^{\prime}Z\right)^{1/2}$.
It is clear that only $R$ and the weight function
$\overline{W}_{n}\left(\beta\right)$ fully determine the estimator
$\widehat{\beta}_{n}$. To illustrate this connection, recall that the 2SLS
estimator results if $W_{n}^{-1}$ is proportional to $n^{-1}Z^{\prime}Z$. For
such a weight, we have $\overline{W}_{n}\left(\beta\right)=I_{k}$, and we
trivially have the 2SLS being dependent only on $R$. Indeed, the 2SLS
estimator can be written as

$\widehat{\beta}=\left(R_{2}^{\prime}R_{2}\right)^{-1}R_{2}^{\prime}R_{1}.$

The same holds for the other estimators as well. For example, we take the LIML
estimator. When
$W_{n}\left(\beta\right)^{-1}=\widehat{V}_{u}\left(\beta\right)\cdot
n^{-1}Z^{\prime}Z$, we have
$\overline{W}_{n}\left(\beta\right)=\widehat{V}_{u}\left(\beta\right)^{-1}I_{k}$.
The LIML estimator solves

$\widehat{Q}_{n}\left(\beta\right)=\frac{b^{\prime}R^{\prime}Rb}{b^{\prime}\widehat{\Phi}_{n}b}\text{,
where }\widehat{\Phi}_{n}=n^{-1}Y^{\prime}MY.$

Having found that the estimators are completely determined by the standardized
reduced-form coefficients $R$ and the function
$\overline{W}_{n}\left(\beta\right)$, we can turn our attention to the Wald
statistics.

The Wald statistic for the 2SLS/LIML estimators has the form

$\widehat{\mathcal{W}}_{n}^{\ast}=\left(\widehat{\beta}_{n}-\beta_{0}\right)^{\prime}\left[\left(R_{2}^{\prime}R_{2}\right)^{-1}R_{2}^{\prime}\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Sigma}_{n}\left(\widehat{b}_{n}\otimes
I_{k}\right)R_{2}\left(R_{2}^{\prime}R_{2}\right)^{-1}\right]^{-1}\left(\widehat{\beta}_{n}-\beta_{0}\right).$

Hence, it is a function of $R$ and $\widehat{\Sigma}_{n}$ (or, for LIML,
$\widehat{\Phi}_{n}$). Likewise, the Wald statistic for the two-step GMM and
CU estimators can be written as

$\widehat{\mathcal{W}}_{n}^{o}=\left(\widehat{\beta}_{n}-\beta_{0}\right)^{\prime}\left[R_{2}^{\prime}\left[\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Sigma}_{n}\left(\widehat{b}_{n}\otimes
I_{k}\right)\right]^{-1}R_{2}\right]\left(\widehat{\beta}_{n}-\beta_{0}\right),$

which again depends only on $R$ and $\widehat{\Sigma}_{n}$ (as long as the
preliminary estimator $\widetilde{\beta}_{n}$ depends only on $R$ and
$\widehat{\Sigma}_{n}$ as well, such as the 2SLS estimator). In short, the
Wald statistics associated with any of the estimators we have discussed above
are functions of $R$, $\widehat{\Sigma}_{n}$, and $\widehat{\Phi}_{n}$ as
shown above.

We now apply the conditioning approach of Moreira (2003), beginning by finding
a useful transformation of $R$:

$R_{0}=RB_{0}=\left[R_{u}:R_{2}\right]\text{, where
}B_{0}=\left[\begin{array}[]{cc}1&0_{1\times p}\\\
-\beta_{0}&I_{p}\end{array}\right].$

That is, $R_{u}=R_{1}-R_{2}\beta_{0}$. Note that the asymptotic variance of
$R_{0}$ is given by

$\Sigma_{0}=\left(B_{0}^{\prime}\otimes I_{k}\right)\Sigma\left(B_{0}\otimes
I_{k}\right)=\left[\begin{array}[]{cc}\Sigma_{uu}&\Sigma_{u2}\\\
\Sigma_{2u}&\Sigma_{22}\end{array}\right].$

This quantity can of course be consistently estimated as well (regardless of
identification of $\beta$):

$\widehat{\Sigma}_{0,n}=\left(B_{0}^{\prime}\otimes
I_{k}\right)\widehat{\Sigma}_{n}\left(B_{0}\otimes
I_{k}\right)=\left[\begin{array}[]{cc}\widehat{\Sigma}_{uu,n}&\widehat{\Sigma}_{u2,n}\\\
\widehat{\Sigma}_{2u,n}&\widehat{\Sigma}_{22,n}\end{array}\right].$

Consider a transformation of $R$.

$\widehat{D}=vec\left(R_{2}\right)-\widehat{\Sigma}_{2u,n}\widehat{\Sigma}_{uu,n}^{-1}R_{u}.$

Given $\widehat{\Sigma}_{n}$, there is a one-to-one transformation between the
pair $R$ and $R_{2}$ and the pair $R_{u}$ and $\widehat{D}$. Since we have
established that all of the Wald statistics above can be written as functions
of $R,\widehat{\Sigma}_{n},$ and $\widehat{\Phi}_{n}$, this means that they
can also be written as functions of $R_{u},\widehat{D},\widehat{\Sigma}_{n},$
and $\widehat{\Phi}_{n}$. Importantly, adopting the appropriate assumptions
relevant for HAC (e.g. see Kleibergen (2005) or Andrews (2016)), it can be
shown that
$\left(R_{u},\widehat{D}\right)\rightarrow_{d}\left(\mathcal{R}_{u},\mathcal{D}\right)$,
where $\mathcal{R}_{u}$ and $\mathcal{D}$ are asymptotically normal and
independent, with $\mathcal{R}_{u}$ being mean zero with a variance matrix
that can be consistently estimated under the null – that is,
$\mathcal{R}_{u}\sim N\left(0,\Sigma_{uu}\right)$ under the null. As Moreira
(2003) shows, this allows one to establish the distribution of test statistics
even in the presence of the unknown nuisance parameter (the mean of $R_{2}$),
since the distribution of $\mathcal{R}_{u}$ conditional on $\mathcal{D}$ is
the same as the marginal distribution.777We will not standardize here the
$R_{u}$ and $D$ statistics. However, we could have worked with their
respective standardized versions, $S=\left[\left(b_{0}^{\prime}\otimes
I_{k}\right)\Sigma\left(b_{0}\otimes I_{k}\right)\right]^{-1/2}Rb_{0}$ and
$T=\left[\left(A_{0}^{\prime}\otimes I_{k}\right)\Sigma^{-1}\left(A_{0}\otimes
I_{k}\right)\right]^{-1/2}\left(A_{0}^{\prime}\otimes
I_{k}\right)\Sigma^{-1}vec\left(R\right)$, as in Moreira and Moreira (2019).

We can write all Wald statistics as

$\widehat{\mathcal{W}}_{n}=\psi\left(R_{u},\widehat{D},\widehat{\Sigma}_{n},\widehat{\Phi}_{n}\right)$

(where we explicitly state the distribution of $R_{u}$ depends on the sample
size $n$). Its asymptotic behavior is given by

$\mathcal{W}_{n}=\psi\left(\mathcal{R}_{u},\mathcal{D},\Sigma,\Phi\right).$

where $\Phi=plim$ $\widehat{\Phi}_{n}=plim$ $n^{-1}V^{\prime}MV$ (if the
process is ergodic, $\Phi$ is just the variance of the reduced-form errors
$V$). We then find the $1-\alpha$ quantile, say,
$c_{\alpha}\left(d,\Sigma,\Phi\right)$ of the null asymptotic distribution of

$\psi\left(\mathcal{R}_{u},d,\Sigma,\Phi\right)\text{, where
}\mathcal{R}_{u}\sim N\left(0,\Sigma_{uu}\right).$

The final conditional test rejects the null when

$\widehat{\mathcal{W}}_{n}=\psi\left(R_{u},\widehat{D},\widehat{\Sigma}_{n},\widehat{\Phi}_{n}\right)>c_{\alpha}\left(\widehat{D},\widehat{\Sigma}_{n},\widehat{\Phi}_{n}\right).$

##  IV Generalization to weakly-identified models (including GMM)

Summarizing the setup in Andrews (2016), it is assumed that there is a
sequence of models $F_{n}\left(\theta,\gamma\right)$, which is indexed by the
sample size $n$. To illustrate the extension, we focus on a parameter of
interest $\theta\in\Theta$, and presume there is an $l\times 1$ consistently
estimable nuisance parameter $\gamma\in\Gamma$. The objective is to test the
null hypothesis $\theta=\theta_{0}$, presuming the availability of three
quantities: 1) a standardized sample moment vector (or distance function)
evaluated at the null, $h_{n}\left(\theta_{0}\right)$888For the linear model,
we can take either the (standardized) moment condition
$h_{n}\left(\beta\right)=\left(Z^{\prime}Z\right)^{-1/2}Z^{\prime}\left(y_{1}-Y_{2}\beta\right)$
or the distance function
$h\left(\Pi,\beta\right)=\left(Z^{\prime}Z\right)^{-1}Z^{\prime}Y-\left[\Pi\beta:\Pi\right]$.;
2) a sample gradient of $h_{n}\left(\theta\right)$ with respect to $\theta$
evaluated at the null, $\Delta h_{n}\left(\theta_{0}\right)$; and 3) the
consistent estimate $\hat{\gamma}$ for $\gamma$.

The main assumptions in Andrews (2016) are that for any true value
$\left(\theta,\gamma\right)\in\Theta\times\Gamma$:

$\left(\begin{array}[]{c}h_{n}\left(\theta_{0}\right)\\\ \Delta
h_{n}\left(\theta_{0}\right)\end{array}\right)\overset{d}{\rightarrow}\left(\begin{array}[]{c}h\left(\theta_{0}\right)\\\
\Delta h\left(\theta_{0}\right)\end{array}\right)$

and

$\left(\begin{array}[]{c}h\left(\theta_{0}\right)\\\ vec\left(\Delta
h\left(\theta_{0}\right)\right)\end{array}\right)\sim
N\left(\left(\begin{array}[]{c}m\left(\theta_{0}\right)\\\
vec\left(\mu\right)\end{array}\right),\Sigma_{0}\right)\text{, where
}\Sigma_{0}=\left(\begin{array}[]{cc}\Sigma_{hh}&\Sigma_{h\theta}\\\
\Sigma_{\theta h}&\Sigma_{\theta\theta}\end{array}\right)$

and $\hat{\gamma}\overset{p}{\rightarrow}\gamma$. It is further assumed that
$\Sigma_{h\theta}$ and $\Sigma_{hh}$ are continuous in $\gamma$, and hence
consistently estimable.

The mean $m\left(\theta_{0}\right)$ belongs to a set
$M\left(\mu,\gamma\right)\subseteq R^{k}$, with $\mu\in\mathcal{M}$, and is
defined so that when $\theta=\theta_{0}$, $m\left(\theta_{0}\right)=0$. The
goal is to test the null hypothesis
$\left(m\left(\theta_{0}\right),\mu\right)=\left(0,\mu\right)$ against the
alternative
$\left(m\left(\theta_{0}\right),\mu\right)=\left(\mathcal{M}\backslash\left\\{0\right\\},\mu\right)$
, for any unknown value of $\mu$.

We can once again consider the $k\times 1$ quantity

$\mathcal{D}=vec\left(\Delta h\left(\theta_{0}\right)\right)-\Sigma_{\theta
h}\Sigma_{hh}^{-1}h\left(\theta_{0}\right)$

which, by construction is independent of $h\left(\theta_{0}\right)$. The Wald
statistic based on the 2SLS estimator for the linear model simplifies to

$\displaystyle\widehat{\mathcal{W}}_{n}$ $\displaystyle=$ $\displaystyle
R_{u}^{\prime}R_{2}\left[R_{2}^{\prime}\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Sigma}_{0,n}\left(\widehat{b}_{n}\otimes
I_{k}\right)R_{2}\right]^{-1}R_{2}^{\prime}R_{u}\text{, where}$
$\displaystyle\widehat{b}_{n}$ $\displaystyle=$
$\displaystyle\left(1,-R_{u}^{\prime}R_{2}\left(R_{2}^{\prime}R_{2}\right)^{-1}\right)^{\prime}.$

We can thus define a nonlinear analog as

$\displaystyle\widehat{\mathcal{W}}_{n}$ $\displaystyle=$ $\displaystyle
h_{n}^{\prime}\Delta h_{n}\left[\Delta
h_{n}^{\prime}\left(\widehat{b}_{n}\otimes
I_{k}\right)^{\prime}\widehat{\Sigma}_{n}\left(\widehat{b}_{n}\otimes
I_{k}\right)\Delta h_{n}\right]^{-1}\Delta h_{n}^{\prime}h_{n}\text{, where}$
$\displaystyle\widehat{b}_{n}$ $\displaystyle=$
$\displaystyle\left(1,-h_{n}^{\prime}\Delta h_{n}\left(\Delta
h_{n}^{\prime}\Delta h_{n}\right)^{-1}\right)^{\prime}.$

(where we have suppressed the dependence on $\theta_{0}$), which will converge
in distribution to

$\mathcal{W}\equiv\left(h^{\prime}\Delta h\right)\left[\Delta
h^{\prime}\left(b\otimes I_{k}\right)^{\prime}\Sigma\left(b\otimes
I_{k}\right)\Delta h\right]^{-1}\left(\Delta h^{\prime}h\right)$

After substituting in $\Delta h=\mathcal{D}+\Sigma_{\theta
h}\Sigma_{hh}^{-1}h$, then it is easy to compute the $\left(1-\alpha\right)$th
conditional quantile defined by

$\Pr\left[\mathcal{W}>c\left(d,\Sigma;\alpha\right)|\mathcal{D}=d\right]=\alpha$

The test is straightforward to implement as follows: reject the hypothesis if
and only if

$\widehat{\mathcal{W}}_{n}>c\left(\widehat{D}_{n},\widehat{\Sigma}_{n};\alpha\right)\text{,
where }\widehat{D}_{n}=vec\left(\Delta
h_{n}\left(\theta_{0}\right)\right)-\widehat{\Sigma}_{\theta
h}\widehat{\Sigma}_{hh}^{-1}h_{n}\left(\theta_{0}\right).$

This test will have the property, under the null, that

$\lim\Pr\left[\widehat{\mathcal{W}}_{n}>c\left(\widehat{D}_{n},\widehat{\Sigma}_{n};\alpha\right)|\widehat{D}_{n}=d\right]=\alpha$

for all values of $d$ and hence

$\lim\Pr\left[\widehat{\mathcal{W}}_{n}>c\left(\widehat{D}_{n},\widehat{\Sigma}_{n};\alpha\right)\right]=\alpha$

as desired.

##  V Conclusion

We are motivated by providing an inference method for researchers interested
in the over-identified linear instrumental variables model, and who have a
preference for using the 2SLS estimator $\hat{\beta}$ for inference, and who
do not wish to rely on the assumption of homoskedasticity. If errors are
assumed to be homoskedastic, one can use the results of Staiger and Stock
(1997) and Stock and Yogo (2005) to control the amount of distortion in
inference. As noted in Andrews, Stock and Sun (2019), the tables in Stock and
Yogo (2005) do not apply to non-homoskedastic settings. Andrews (2018)
provides a conservative two-step procedure that builds on Stock and Yogo
(2005) for more general DGPs.

To accommodate practitioners’ preference for using the 2SLS estimator and
conventional robust standard errors, we present the robust Conditional Wald
(data-dependent) critical values for the Wald statistics robust to
heteroskedastic, autocorrelated, and/or clustered errors, which turns out to
be a relatively straightforward extension of the Conditional Wald test of
Moreira (2003); its derivation has been neglected in the weak-IV literature,
which has provided a number of other non-Wald procedures that are both robust
to non-homoskedastic errors and to arbitrarily weak instruments.

Using existing results from the weak IV literature, we also generalize the
procedure to apply to the more general nonlinear models that are typically
estimated via minimum distance or GMM. We can explore several Wald statistics
within the nonlinear setup as well, contingent on the weights employed in the
criterion function. The final conditional test would substitute the
conventional critical value with a conditional quantile.

## References

  * (1)
  * Anderson and Rubin (1949) Anderson, T. W., and H. Rubin. 1949\. “Estimation of the Parameters of a Single Equation in a Complete System of Stochastic Equations.” Annals of Mathematical Statistics, 20: 46–63.
  * Andrews (1991) Andrews, Donald W. K. 1991\. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica, 59: 817–858.
  * Andrews, Moreira and Stock (2004) Andrews, D. W. K., M. J. Moreira, and J. H. Stock. 2004\. “Optimal Invariant Similar Tests for Instrumental Variables Regression.” NBER Working Paper t0299.
  * Andrews (2016) Andrews, Isaiah. 2016\. “Conditional Linear Combination Tests for Weakly Identified Models.” Econometrica, 84: 2155–2182.
  * Andrews (2018) Andrews, Isaiah. 2018\. “Valid Two-Step Identification-Robust Confidence Sets for GMM.” The Review of Economics and Statistics, 100(2): 337–348.
  * Andrews, Stock and Sun (2019) Andrews, Isaiah, James H. Stock, and Liyang Sun. 2019\. “Weak Instruments in Instrumental Variables Regression: Theory and Practice.” Annual Review of Economics, 11: 727–753.
  * Cameron, Gelbach and Miller (2011) Cameron, A. Colin, Jonah B. Gelbach, and Douglas L. Miller. 2011\. “Robust Inference With Multiway Clustering.” Journal of Business Economics and Statistics, 77: 238–249.
  * Davidson and MacKinnon (2021) Davidson, Russell, and James G. MacKinnon. 2021\. Estimation and Inference in Econometrics. New York: Oxford University Press.
  * Dufour (1997) Dufour, J-M. 1997\. “Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models.” Econometrica, 65: 1365–1388.
  * Hansen and Singleton (1982) Hansen, Lars P., and Kenneth J. Singleton. 1982\. “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models.” Econometrica, 50(5): 1269–1286.
  * Kleibergen (2005) Kleibergen, F. 2005\. “Testing Parameters in GMM without Assuming that they are Identified.” Econometrica, 73: 1103–1123.
  * Lee et al. (2023) Lee, David S, Justin McCrary, Marcelo J Moreira, Jack R Porter, and Luther Yap. 2023\. “What to do when you can’t use ’1.96’ Confidence Intervals for IV.” National Bureau of Economic Research Working Paper 31893.
  * Moreira and Moreira (2019) Moreira, H., and M. J. Moreira. 2019\. “Optimal Two-Sided Tests for Instrumental Variables Regression with Heteroskedastic and Autocorrelated Errors.” Journal of Econometrics, 213: 398–433.
  * Moreira (2003) Moreira, M. J. 2003\. “A Conditional Likelihood Ratio Test for Structural Models.” Econometrica, 71: 1027–1048.
  * Newey and McFadden (1994) Newey, Whitney, and Daniel L. McFadden. 1994\. “Large Sample Estimation and Hypothesis Testing.” In Handbook of EconometricsVol. 4, , ed. Robert F. Engle and Daniel L. McFadden, Chapter 36, 2111–2245. Amsterdam:Elsevier Science.
  * Newey and West (1987) Newey, Whitney K., and Kenneth D. West. 1987\. “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica, 55: 703–708.
  * Staiger and Stock (1997) Staiger, D., and J. H. Stock. 1997\. “Instrumental Variables Regression with Weak Instruments.” Econometrica, 65: 557–586.
  * Stock and Yogo (2005) Stock, James H., and Motohiro Yogo. 2005\. “Testing for Weak Instruments in Linear IV Regression.” In Identification and Inference in Econometric Models: Essays in Honor of Thomas J. Rothenberg, ed. Donald W.K. Andrews and James H. Stock, Chapter 5, 80–108. Cambridge University Press.
  * Van de Sijpe and Windmeijer (2023) Van de Sijpe, Nicolas, and Frank Windmeijer. 2023\. “On the power of the conditional likelihood ratio and related tests for weak-instrument robust inference.” Journal of Econometrics, 1: 82–104.
  * White (1980) White, Halbert. 1980\. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.” Econometrica, 48: 817–838.