Wild Bootstrap Inference for Instrumental Variables Regressions with Weak and Few ClustersWe are grateful to Aureo de Paula, Firmin Doko Tchatoka, Kirill Evdokimov, Jungbin Hwang, Qingfeng Liu, Morten Ørregaard Nielsen, Ryo Okui, Kyungchul (Kevin) Song, Naoya Sueishi, Yoshimasa Uematsu, and
participants at the 2021 Annual Conference of the International Association for Applied Econometrics, the 2021 Asian Meeting of the Econometric Society, the 2021 China Meeting of the Econometric Society, the 2021 Australasian Meeting of the Econometric Society, the 2021 Econometric Society European Meeting, the 37th Canadian Econometric Study Group Meetings, the 16th International Symposium on Econometric Theory and Applications, UCONN Econometrics Seminar, and NUS Econometrics Seminar for their valuable comments. Special thanks to James MacKinnon for very insightful discussions. Wang acknowledges the financial support from NTU SUG Grant No.M4082262.SS0 and Singapore Ministry of Education Tier 1 grant RG53/20.
Zhang acknowledges the financial support from Singapore Ministry of Education Tier 2 grant under grant MOE2018-T2-2-169 and the Lee Kong Chian fellowship. Any possible errors are our own.

Wenjie Wang[Division of Economics, School of Social Sciences, Nanyang Technological University.
HSS-04-65, 14 Nanyang Drive, Singapore 637332.
E-mail address<EMAIL_ADDRESS>and Yichong Zhang[School of Economics, Singapore Management University. E-mail address<EMAIL_ADDRESS>
We study the wild bootstrap inference for instrumental variable regressions with a small number of large clusters. We first show that the wild bootstrap Wald test controls size asymptotically up to a small error as long as the parameters of endogenous variables are strongly identified in at least one of the clusters.
Second, we establish the conditions for the bootstrap tests to have power against local alternatives.We further develop a wild bootstrap Anderson-Rubin test for the full-vector inference and show that it controls size asymptotically even under weak identification in all clusters. We illustrate their good performance using simulations and provide an empirical application to a well-known dataset about US local labor markets.

Keywords: Wild Bootstrap, Weak Instrument, Clustered Data, Randomization Test.  JEL codes: C12, C26, C31

§ INTRODUCTION

The instrument variable (IV) regression is one of the five most commonly used causal inference methods identified by [14], and it is often applied with clustered data. For example, [74] analyzes 1,359 IV regressions in 31 papers published by the American Economic Association (AEA), out of which 24 papers account for clustering of observations.
Three issues arise when running IV regressions with clustered data.
First, the strength of IVs may be heterogeneous across clusters with  a few clusters providing the main identification power.
Indeed, [74] finds that in the average paper of his AEA samples, with the removal of just one cluster or observation, the first-stage $F$ can decrease by 28%,

and 38% of reported 0.05 significant two-stage least squares (TSLS) results can be rendered insignificant at that level.
Second, the number of clusters is small in many IV applications.
For instance, [3] cluster the standard errors at the country/polity level, resulting in 12-19 clusters, [34] cluster at the sectoral level with 16 sectors, and [66] clusters at the province and district level with 11 provinces and 30 districts, respectively. When the number of clusters is small, any inference procedures that require the number of clusters to diverge to infinity may be unreliable. Third, it is also possible that IVs are weak in all clusters, in which case researchers need to use weak-identification-robust inference methods [10].

Motivated by these issues, in this paper we study the inference for IV regressions with a small, and thus, fixed number of clusters.
We denote clusters in which the parameters of the endogenous variables are strongly identified as strong IV clusters.
First, we show that a wild bootstrap Wald test, with or without the cluster-robust covariance estimator (CCE), controls size asymptotically up to a small error, as long as there exists at least one strong IV cluster. Second, we show that the wild bootstrap tests have power against local alternatives at 10% and 5% significance levels when there are at least five and six strong IV clusters, respectively.

Third, in the common case of testing a single restriction,

we show the bootstrap Wald test with CCE is more powerful than that without CCE for distant local alternatives. Fourth, we develop the full-vector inference based on a wild bootstrap <cit.> test, which controls size asymptotically up to a small error regardless of instrument strength. Fifth, we show that in the common case with a single endogenous variable and a single IV, a wild bootstrap test based on the unstudentized Wald statistic (i.e., the one without CCE)
is asymptotically equivalent to a certain wild bootstrap AR test under both null and alternative, implying that in such a case it is fully robust to weak IV. Sixth, we show in the Online Supplement that bootstrapping weak-IV-robust tests other than the AR test (e.g., the Lagrange multiplier test or the conditional quasi-likelihood ratio test) controls asymptotic size when there is at least one strong IV cluster.

Our inference procedure is empirically relevant. First, we notice that besides the aforementioned examples, the numbers of clusters may also be rather small in studies that estimate the region-wise effects of certain intervention if the partition of clusters is at the state level.
We illustrate the usefulness of our methods in Section <ref> by applying them to the well-known dataset of [15] in the estimation of the effects of Chinese imports on local labor markets in three US Census Bureau-designated regions (South, Midwest, and West) with 11-16 clusters at the state level.  Second, our bootstrap inference is flexible with respect to IV strength:
the bootstrap Wald test allows for cluster-level heterogeneity in the first stage,

while its AR counterpart is fully robust to weak IVs. Figure 1 reports the estimated first-stage coefficients for each cluster (state) in [15]'s ([15]) dataset, which suggests that there exists substantial variation in the IV strength among states. Specifically, the first-stage coefficients of some states are quite large compared with the rest in the region (e.g., Idaho in the West region),
while some other states have coefficients that are rather close to zero, and thus, potentially subject to weak identification.

Some states even have opposite signs for their first-stage coefficients (e.g., Delaware, Maryland, Nebraska, and Washington).
However, there is no existing proven-valid inference method for IV regressions with few clusters, where IVs may be weak in some clusters.
Therefore, we believe our bootstrap methods enrich practitioners' toolbox by providing a reliable inference in this context.
Third, different from the analytical inference based on the widely used heteroskedasticity and autocorrelation consistent (HAC) estimators,
our approach is agnostic about the within-cluster (weak) dependence structure and thus avoids the use of tuning parameters to estimate the covariance matrix for dependent data.

State-level First-stage Coefficients in [15]

Note: This figure reports the estimated state-level first-stage coefficients for South (16 states), Midwest (12 states), and West (11 states), respectively, in [15]. The coefficients are obtained by running first-stage least squares regression for each state separately, controlling for the same exogenous variables as those specified in [15].

The contributions in the present paper relate to several strands of literature.
First, it is related to the literature on the cluster-robust inference.[See [18], [25], [47], [1], [37, 38, 39, 40], [57], [28], [54],
among others,
and [55]

for a recent survey.]

[28], [54], and [58] show the bootstrap validity under the asymptotic framework with a large number of clusters.
However, as emphasized by [44, 45],
[19], [20], [38, 39, 40], and  [21],
many empirical studies motivate an alternative framework in which the number of clusters is small, while the number of observations in each cluster is relatively large.

In such a case, we may consider applying the approaches developed by [17], [43],  <cit.>, and <cit.> for inference.

However, [17] and [43] require an (asymptotically) equal cluster-level sample size,[See <cit.> and <cit.> for details.]
while IM and CRS require strong identification in all clusters. In contrast, our bootstrap Wald tests are more flexible as it does not require an equal cluster size and only needs one strong IV cluster for size control and five to six for local power, thus allowing for substantial heterogeneity in IV strength.

To our knowledge, there is no alternative method in the literature that remains valid in such a context.
For full-vector inference, we further provide the bootstrap AR tests, which are fully robust to weak identification.

Second, [21] prove the validity of wild bootstrap with a few large clusters by innovatively connecting it with a randomization test with sign changes. Our results rely heavily on this technical breakthrough, but also complement  [21] in the following aspects.
First, [21] focus on the linear regression with exogenous regressors and then extend their analysis to
a score bootstrap for the GMM estimator.

Instead, we focus on extending the wild restricted efficient cluster (WREC) bootstrap, which is popular among empirical researchers, for general $k$-class IV estimators.
Therefore, our procedure cannot be formulated as a score bootstrap in the GMM setting.
Second, we study the local power for the Wald test both with and without CCE.
The power analysis of the Wald test with CCE is new to the literature and
technically involved because with a fixed number of clusters, the CCE has a random limit. We rely on the Sherman–Morrison–Woodbury formula for the matrix inverse to analyze the behaviors of the test statistic

and its wild bootstrap counterparts under local alternatives. Based on such a detailed asymptotic analysis, we establish the local power of the bootstrap Wald test studentized by CCE,
and show it is higher than that for the Wald test without CCE studentization for distant local alternatives in the empirically prevalent case of testing a single restriction.

Indeed, as the linear regression is a special case of the linear IV regression studied in this paper, we believe the same result holds for the local power of the Wald tests in [21]. Third, different from its unstudentized counterpart, the local power of the Wald test with CCE is established  without the assumption that the cluster-level first-stage coefficients have the same sign, which may not hold in some empirical studies (e.g., Figure <ref>).

Third, our paper is related to the literature of WREC bootstrap for IV regressions. See, for example,
[27], [31, 32], [67], and [53]. In particular, [31, 32] first propose the WREC bootstrap procedure for the TSLS-based Wald statistic and weak-IV-robust statistics in IV regressions with clustered data and perform extensive simulation experiments. Our key contribution to this strand of literature is to theoretically prove the validity of the (modified) WREC bootstrap under the new asymptotic framework in which IVs may be weak for some clusters and the number of clusters is fixed. In particular, we carefully re-design the first stage of the WREC procedure to adapt to the new framework,
and our analysis cover the general $k$-class IV estimators. In simulations, we show that the WREC bootstrap Wald tests with the limited information maximum likelihood (LIML) estimator or the modified LIML estimator proposed by <cit.> improve substantially over those with TSLS in the overidentified case. We also theoretically prove the power advantages of the Wald test with CCE compared with the one without, which justifies [32]'s ([31, 32]) proposal of using the bootstrap Wald test studentized by CCE under our alternative asymptotic framework.

Furthermore, our paper is related to the literature on weak identification,
in which various normal approximation-based inference approaches are available for nonhomoskedastic cases, among them [70],
[49], [5],
[7], [9], [8],
[60], and [6].

As <cit.> remark,
an important question concerns the quality of the normal approximations with nonhomoskedastic observations.

On the other hand, when implemented appropriately, alternative approaches such as
bootstrap may substantially improve the inference for IV regressions.[See, for example,
[26, 27], [62], [73], [32], and [74], among others.
In addition, we notice that [71] develops permutation versions of weak-IV-robust tests
with (non-clustered) heteroskedastic errors.]

We contribute to this literature by establishing formally that wild bootstrap-based AR tests control asymptotic size under both weak identification and a small number of clusters. However, contrary to the Wald tests, we find in simulations that the bootstrap AR test without CCE has better power properties than the one studentized by CCE. We formally establish the local power of the bootstrap AR test without CCE. Furthermore, we show that the bootstrap validity for other weak-IV-robust tests requires at least one strong IV clusters (Sections <ref>-<ref> in the Online Supplement), which provides theoretical support to the simulation results in [31, 32] documenting that with clustered observations, bootstrapping the AR tests results in better finite sample size control than bootstrapping alternative robust tests.

Finally, we note that although empirical applications often involve settings with substantial first-stage heterogeneity such as those shown in Figure <ref>, related econometric literature remains rather sparse.
[2] exploit such heterogeneity to improve the asymptotic mean squared error of IV estimators with independent and homoskedastic observations. Instead, we focus on developing inference methods that are robust to the first-stage heterogeneity for data with a small number of clusters, while allowing for (weak) within-cluster dependence and heteroskedasticity.

The remainder of this paper is organized as follows.
Section <ref> presents the setup and the bootstrap procedures.
Section <ref> presents main assumptions, while Sections <ref>-<ref> provide asymptotic results.

Simulations in Section <ref> suggest that our procedures have outstanding finite sample size control and, in line with our theoretical analysis, the bootstrap Wald test with CCE has power advantages compared with the other bootstrap tests. The empirical application is presented in Section <ref>. We conclude and provide practical recommendations in Section <ref>.

§ SETUP, ESTIMATION, AND INFERENCE

§.§ Setup

Throughout the paper, we consider the setup of a linear IV regression with clustered data,

\begin{eqnarray}\label{eq: IV-model}
y_{i,j} = X^{\top}_{i,j} \beta + W^{\top}_{i,j} \gamma + \eps_{i,j},
\end{eqnarray}

where the clusters are indexed by $j \in J = \{1, ..., q\}$ and units in the $j$-th cluster are indexed by $i \in I_{n,j}
= \{1, ..., n_j \}$. In (<ref>), we denote
$y_{i,j} \in \textbf{R}$, $X_{i,j} \in \textbf{R}^{d_x}$, and $W_{i,j} \in \textbf{R}^{d_w}$ as an outcome of interest, endogenous regressors, and exogenous regressors, respectively.
Furthermore, we let $Z_{i,j} \in \textbf{R}^{d_z}$
denote the IVs for $X_{i,j}$.

$\beta \in \textbf{R}^{d_x}$ and $\gamma \in \textbf{R}^{d_w}$ are unknown structural parameters.

We allow the parameter of interest $\beta$ to shift with respect to (w.r.t.) the sample size, which incorporates the analyses of size and local power in a concise manner: $\beta_n = \beta_0 + \mu_{\beta}/\sqrt{n}$, where $\mu_{\beta} \in \textbf{R}^{d_x}$ is the local parameter. We let $\lambda^{\top}_{\beta} \beta_0 = \lambda_0$, where $\lambda_{\beta} \in \textbf{R}^{d_x \times d_r}$, $\lambda_0 \in \textbf{R}^{d_r}$, and $d_r$ denotes the number of restrictions under the null hypothesis.
Define $\mu = \lambda_{\beta}^{\top} \mu_{\beta}$.
Then, the null and local alternative hypotheses studied in this paper can be written as

\begin{align}\label{eq: hypothesis}
\mathcal{H}_0: \mu = 0  \quad v.s. \quad \mathcal{H}_{1,n}:  \mu \neq 0.
\end{align}

§.§ K-Class IV Estimators

Throughout the paper, we consider the $k$-class IV estimators of the form:

\begin{align}\label{eq: k-class}
%    \left(\hat{\beta}^{\top}, \hat{\gamma}^{\top}\right)^{\top} = \left( \vec{X}^{\top} P_{\vec{Z}} \vec{X} - \hat{\mu} \vec{X}^{\top} M_{\vec{Z}}\vec{X} \right)^{-1} \left(\vec{X}^{\top} P_{\vec{Z}}Y - \hat{\mu} \vec{X}^{\top} M_{\vec{Z}}Y\right),
\left(\hat{\beta}^{\top}, \hat{\gamma}^{\top}\right)^{\top} = \left( \vec{X}^{\top} \vec{X} - \hat{\kappa} \vec{X}^{\top} M_{\vec{Z}}\vec{X} \right)^{-1} \left(\vec{X}^{\top} Y - \hat{\kappa} \vec{X}^{\top} M_{\vec{Z}}Y\right),
\end{align}

where $\vec{Z} = [Z:W]$, $\vec{X} = [X:W]$,
$Y$, $X$, $Z$, and $W$
are $n \times 1$, $n \times d_x$, $n \times d_z$, and $n \times d_w$-dimensional vectors and matrices formed
by $y_{i,j}$, $X^{\top}_{i,j}$, $Z^{\top}_{i,j}$, and $W^{\top}_{i,j}$, respectively. We also denote
$P_A = A(A^{\top} A)^{-1}A^{\top}$ and $M_A = I_n - P_A$, where $A$ is an $n$-dimensional matrix and $I_n$ is an $n$-dimensional identity matrix.

Specifically, we focus on four $k$-class estimators: (1) the TSLS estimator,
where $\hat{\kappa} = \hat{\kappa}_{tsls} = 1$, (2)
the LIML estimator, where

\begin{align*}
\hat{\kappa} = \hat{\kappa}_{liml} =
\min_{r} r^{\top} \vec{Y}^{\top} M_W \vec{Y} r / (r^{\top} \vec{Y}^{\top} M_{\vec{Z}}\vec{Y}r),
%\min_{r} r^{\top} \vec{Y}^{\top} M_W Z(Z^{\top} M_W Z)^{-1} Z^{\top} M_W \vec{Y} r / (r^{\top} \vec{Y}^{\top} M_{\vec{Z}}\vec{Y}r),
\;\;  \vec{Y} = [Y:X], \;\; \text{and} \;\; r=(1,-\beta^{\top} )^{\top},
\end{align*}

(3) the FULL estimator, where

$\hat{\kappa} = \hat{\kappa}_{full} = \hat{\kappa}_{liml} - C/(n-d_z-d_w)$ with some constant $C$,

and (4) the bias-adjusted TSLS (BA) estimator proposed by [63] and [68], where
$\hat{\kappa} = \hat{\kappa}_{ba} = n/(n-d_z+2)$.
Theoretically, we show that these four estimators are asymptotically equivalent. However, in simulations, we find that FULL has the best finite sample performance in the overidentified case.

§.§ Wild Bootstrap Inference

§.§.§ Inference Procedure for Wald Statistics

For inference, we construct Wald statistics based on the $k$-class estimator $\hat{\beta}_L$  defined in (<ref>) with $\hat k = \hat k_L$ for $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$. When the $d_r \times d_r$ weighting matrix $\hat{A}_{r}$  is asymptotically deterministic in the sense of Assumption <ref> below (such as $\hat{A}_r = I_{d_r}$, the $d_r \times d_r$ identity matrix),  we denote $T_n$ as the Wald statistic without CCE and define it as

\begin{eqnarray}\label{eq: Wald-unstudentized}
T_{n} = || \sqrt{n}(\lambda_{\beta}^{\top} \hat{\beta}_L - \lambda_0) ||_{\hat{A}_r},
\end{eqnarray}

where $||u||_A = \sqrt{u^{\top}Au}$ for a generic vector $u$ and a weighting matrix $A$.
When we use $\hat{A}_{r,CR}$, the inverse of the CCE defined in (<ref>) in Section <ref> as the weighting matrix, we denote $T_{CR,n}$ as the Wald statistic studentized by CCE and define it as

\begin{eqnarray}\label{eq: Wald-studentized}
T_{CR,n} = || \sqrt{n}(\lambda_{\beta}^{\top} \hat{\beta}_L - \lambda_0) ||_{\hat{A}_{r,CR}}.
\end{eqnarray}

In the case of testing the value of a single endogenous variable $\beta = \beta_0$, the test statistics reduce to
$T_n = |\sqrt{n}(\hat{\beta}_L - \beta_0)|$, and
$T_{CR,n} = |\sqrt{n}(\hat{\beta}_L - \beta_0)/\widehat{V}|$,
respectively, where $\widehat{V}$ denotes the usual cluster-robust variance estimator.

We reject the null hypothesis in (<ref>) at $\alpha$ significance level if the test statistic is greater than its corresponding critical value ($\hat c_n (1-\alpha)$ and $\hat c_{CR,n}(1-\alpha)$ defined below for $T_n$ and $T_{CR,n}$, respectively). We compute the critical values by a wild bootstrap procedure described below.

Step 1:

For $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$, compute the null-restricted residual

$$\hat{\eps}^r_{i,j} = y_{i,j} - X_{i,j}^{\top} \hat{\beta}^r_L - W_{i,j}^{\top} \hat{\gamma}^r_L,$$
where $\hat{\beta}^r_L$ and $\hat{\gamma}^r_L$ are null-restricted $k$-class IV estimators of $\beta$ and $\gamma$ from $(y_{i,j}, X^{\top}_{i,j}, W^{\top}_{i,j}, Z^{\top}_{i,j})^{\top}$,[The null-restricted $k$-class estimator is defined as

\begin{align*}
%%& \hat{\beta}^r_L = \hat{\beta}_L - \left(X^{\top}X -\hat{ \kappa}_L X^{\top}M_{\widetilde{Z}}X\right)^{-1} \lambda_{\beta}^{\top}
%%\left(\lambda_{\beta}^{\top}(X^{\top}X - \hat{ \kappa}_L X^{\top}M_{\widetilde{Z}}X)^{-1}
%%\lambda_{\beta}\right)^{-1} (\lambda_{\beta}^{\top}\hat{\beta}_L - \lambda_0) \quad \text{and} \\
& \hat{\beta}^r_L = \hat{\beta}_L - \left(X^{\top}P_{\widetilde{Z}}X -\hat{\mu}_L X^{\top}M_{\vec{Z}}X\right)^{-1}
\lambda_{\beta}
\left(\lambda_{\beta}^{\top}(X^{\top}P_{\widetilde{Z}}X - \hat{\mu}_L X^{\top}M_{\vec{Z}}X)^{-1}
\lambda_{\beta}\right)^{-1} (\lambda_{\beta}^{\top}\hat{\beta}_L - \lambda_0), \\
%\quad \text{and} \\
& \hat{\gamma}^r_L = (W^\top W)^{-1}W^\top(Y-X\hat{\beta}_L^r), \;\; \text{where} \;\; \hat{\mu}_L = \hat{\kappa}_L-1 \;\; \text{and} \;\; \tilde{Z} = M_W Z,
\end{align*}

for $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$;

e.g., see Appendix B of [67] for a general formula.]
and the unrestricted residual

\begin{align}\label{eq: unrest_residual}
\hat{\eps}_{i,j} = y_{i,j} - X_{i,j}^{\top} \hat{\beta}_L - W_{i,j}^{\top} \hat{\gamma}_L,
\end{align}

where $\hat{\beta}_L$ and $\hat{\gamma}_L$ are defined in (<ref>) with $\hat{\kappa}_L$.

Step 2:

Construct $\overline{Z}_{i,j}$ as

\begin{align*}
\overline{Z}_{i,j} = \left( \widetilde{Z}^{\top}_{i,j} 1\{j=1\}, ......, \widetilde{Z}^{\top}_{i,j} 1\{j=q\} \right)^{\top},
\end{align*}

where $\widetilde{Z}_{i,j}$ is the residual of regressing $Z_{i,j}$ on $W_{i,j}$ using the entire

Step 3: Compute the first-stage residual

\begin{align}\label{eq: boot-RE-first-stage}
\tilde{v}_{i,j} = X_{i,j} - \widetilde{\Pi}_{\overline{Z}}^\top \overline{Z}_{i,j} -  \widetilde{\Pi}_{w}^\top W_{i,j},
%- \hat{\eps}_{i,j} \hat{\pi}_e,
\end{align}

where $\widetilde{\Pi}_{\overline{Z}}$ and $\widetilde{\Pi}_{w}$ are the OLS coefficients of $\overline{Z}_{i,j}$ and $W_{i,j}$ from regressing $X_{i,j}$ on $(\overline{Z}^{\top}_{i,j}, W^{\top}_{i,j}, \hat{\eps}_{i,j})^{\top}$ using the entire sample.

Step 4: Let $\textbf{G} = \{-1, 1\}^q$ and for any $g = (g_1, ..., g_q) \in \textbf{G}$ generate

\begin{eqnarray*}\label{eq: boot-algo-unstud-2}
X_{i,j}^*(g) = \widetilde{\Pi}_{\overline{Z}}^\top\overline{Z}_{i,j} +  \widetilde{\Pi}_{w}^\top W_{i,j} + g_j \tilde{v}_{i,j}, \quad y_{i,j}^*(g) = X_{i,j}^{*\top}(g) \hat{\beta}^r_L + W_{i,j}^{\top} \hat{\gamma}^r_L + g_j \hat{\eps}^r_{i,j}.
\end{eqnarray*}

For each $g=(g_1, ..., g_q) \in \textbf{G}$, compute $\hat{\beta}^*_{L,g}$ and $\hat{\gamma}^*_{L,g}$,
the analogues of the estimators $\hat{\beta}_L$ and $\hat{\gamma}_L$
using $\left( y_{i,j}^{*}(g), X_{i,j}^{*\top}(g) \right)^{\top}$ in place of $\left( y_{i,j}, X_{i,j}^{\top} \right)^{\top}$ and the same $(Z^{\top}_{i,j}, W^{\top}_{i,j})^{\top}$.

Compute the bootstrap analogue of the Wald statistics:[Let $X^*(g)$ and $Y^*(g)$ be the $n \times d_x$ matrix constructed using $X_{i,j}^*(g)$ and the $n \times 1$ vector constructed using $Y_{i,j}^*(g)$, respectively. For $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$ and $g \in \textbf{G}$,

\begin{align*}
\hat{\beta}^*_{L,g}= \left(X^{*\top}(g) P_{\widetilde{Z}} X^*(g) - \hat{\mu}^*_{L,g} X^{*\top}(g) M_{\vec{Z}} X^*(g) \right)^{-1}\left(X^{*\top}(g) P_{\widetilde{Z}} Y^*(g) - \hat{\mu}^*_{L,g} X^{*\top}(g) M_{\vec{Z}} Y^*(g)\right), \;\; \text{where} \;\; \hat{\mu}^*_{L,g} = \hat{\kappa}^*_{L,g}-1.
\end{align*}

\begin{align}\label{eq: boot-algo-unstud-3}
T^*_{n}(g) = || \sqrt{n} (\lambda_{\beta}^{\top} \hat{\beta}^*_{L,g} - \lambda_0) ||_{\hat{A}_{r}}, \quad T^*_{CR,n}(g) = || \sqrt{n} (\lambda_{\beta}^{\top} \hat{\beta}^*_{L,g} - \lambda_0) ||_{\hat{A}_{r,CR,g}^*},
%T^*_n(\lambda, g) &=& T^*_{n}(\lambda, g) / \sqrt{ c' \widehat{V}^*_n(\hat{\beta}^*_g) c},
\end{align}

where $\hat{A}^*_{r,CR,g}$ is the bootstrap counterpart of $\hat A_{r,CR}$ defined in (<ref>).

Step 5:
Step 5We compute the $1-\alpha$ quantiles of $\{ T^*_{n}(g): g \in \textbf{G} \}$ and $\{ T^*_{CR,n}(g): g \in \textbf{G} \}$:

\begin{align*}
%\label{eq: boot-unstud-cv}
& \hat{c}_{n}(1-\alpha) = \inf
\left\{ x \in \textbf{R}: \frac{1}{|\textbf{G}|} \sum_{g \in \textbf{G}}1 \{ T^*_{n}(g) \leq x \} \geq 1-\alpha
\right\},\\
& \hat{c}_{CR,n}(1-\alpha) = \inf
\left\{ x \in \textbf{R}: \frac{1}{|\textbf{G}|} \sum_{g \in \textbf{G}}1 \{ T^*_{CR,n}(g) \leq x \} \geq 1-\alpha
\right\},
\end{align*}

where $1\{E\}$ equals one whenever the event $E$ is true and equals zero otherwise and $|\textbf{G}| = 2^q$.

The bootstrap test for $\mathcal{H}_0$ rejects whenever $T_{CR,n}$ exceeds the critical value $\hat{c}_{CR,n}(1-\alpha)$ and $T_{n}$ exceeds $\hat{c}_{n}(1-\alpha)$ for Wald statistics with and without CCE, respectively.

Three remarks are in order. First, Step 1 imposes null when computing the residuals in the structural equation (<ref>), which is advocated by [18],
[27], [55], and [21], among others. Second, the estimators $\widetilde{\Pi}_{\overline{Z}}$ and $\widetilde{\Pi}_w$ in Step 3 are similar to the efficient reduced-form estimators in the WREC bootstrap procedures

which have superior finite sample performance for IV regressions, even when the instruments are rather weak.

In this paper, we focus on extending the WREC procedure because (1) we find the resulting bootstrap also has excellent finite sample performance for IV regressions with a small number of clusters, and (2) we want to be consistent with the suggestions in the literature.
Third, to adapt to the new framework, we carefully modify the original WREC procedure and use the fully interacted $\widetilde{Z}_{i,j}$ in Step 3, which is crucial to guarantee that the bootstrap Jacobian matrix $    \widehat{Q}^*_{\widetilde{Z}X}(g) :=
\frac{1}{n}\sum_{j \in J}\sum_{i \in I_{n,j}} \tilde{Z}_{i,j}X_{i,j}^{*\top}(g)$ is asymptotically equivalent to the original Jacobian $\widehat{Q}_{\widetilde{Z}X}$ under cluster-level heterogeneity in IV strength.
In the whole procedure, the fully interacted IVs (i.e., $\overline{Z}_{i,j}$) are only needed to construct $X_{i,j}^*(g)$,
and we still use the original uninteracted IVs (i.e., $Z_{i,j}$) when computing $(\hat{\beta}_L^{\top}, \hat{\gamma}_L^{\top})^{\top}$ in (<ref>) and their null-restricted and bootstrap counterparts (i.e., $(\hat{\beta}^{r\top}_L, \hat{\gamma}_L^{r\top})^{\top}$ in Step 1 and $(\hat{\beta}^{*\top}_{L,g}, \hat{\gamma}^{*\top}_{L,g})^{\top}$ in Step 4).

In addition, when regressing $X_{i,j}$ on $(\overline{Z}^{\top}_{i,j}, W^{\top}_{i,j}, \hat{\eps}_{i,j})^{\top}$ in Step 3, we need to use the unrestricted residuals $\hat{\eps}_{i,j}$ instead of the null-restricted residuals $\hat{\eps}_{i,j}^r$.
This modification is required to establish the asymptotic power results.

§.§.§ Inference Procedure for Weak-instrument-robust Statistics

In this section, we describe a full-vector wild bootstrap procedure for Anderson-Rubin (AR) type weak-IV-robust statistics with or without CCE.  Recall that $\beta_n = \beta_0 + \mu_{\beta}/\sqrt{n}$.
Under the null, we have $\mu_\beta  = 0$, or equivalently, $\beta_n =\beta_0$. First, define the AR statistic without CCE as

\begin{eqnarray*}
AR_{n} &=& \big\Vert \sqrt{n} \widehat{f} \big\Vert_{\hat{A}_z}, \quad
\widehat{f}  =  n^{-1} \sum_{j \in J} \sum_{i \in I_{n,j}} f_{i,j},
\end{eqnarray*}

$\hat{A}_z$ is a $d_z \times d_z$ weighting matrix with an asymptotically deterministic limit as specified in Assumption <ref> below,

$f_{i,j} = \widetilde{Z}_{i,j}\bar{\eps}^r_{i,j}$,
$\bar{\eps}^r_{i,j} = y_{i,j} - X^{\top}_{i,j} \beta_0 - W_{i,j}^{\top} \bar{\gamma}^r $,

and $\bar{\gamma}^r$ is the null-restricted ordinary least squares (OLS) estimator of $\gamma$:

$$\bar{\gamma}^r = \left( \sum_{j \in J} \sum_{i \in I_{n,j}} W_{i,j} W_{i,j}^{\top} \right)^{-1}\sum_{j \in J} \sum_{i \in I_{n,j}} W_{i,j}
(y_{i,j} - X^{\top}_{i,j} \beta_0).$$

Second, we also define the AR statistic with the (null-imposed) CCE as

\begin{eqnarray*}
%\label{eq: AR}
%AR_{CR,n} &=& n \widehat{f}^{\top} \widehat{\Omega}^{-1}\widehat{f},
AR_{CR,n} &=& \big\Vert \sqrt{n} \widehat{f} \big\Vert_{\hat{A}_{CR}},
\quad
\hat{A}_{CR} =  \left( n^{-1} \sum_{j \in J} \sum_{i \in I_{n,j}} \sum_{k \in I_{n,j} } f_{i,j} f_{k,j}^{\top} \right)^{-1}.
\end{eqnarray*}

Our wild bootstrap procedure for the AR statistics
is defined as follows.

Step 1: Compute the null-restricted residual $\bar{\eps}^r_{i,j} = y_{i,j} - X_{i,j}^{\top} \beta_0 - W_{i,j}^{\top} \bar{\gamma}^r.$

Step 2: Let $\textbf{G} = \{-1, 1\}^q$ and, for any $g = (g_1, ..., g_q) \in \textbf{G}$, define

\begin{eqnarray*}
%\label{eq: boot-algo-robust-2}
\widehat{f}^*_g = n^{-1} \sum_{j \in J} \sum_{i \in I_{n,j}} f^*_{i,j}(g_j),\quad
%\hat{\delta}^*_g = \widehat{Q}^{-1}_{\widetilde{Z}\widetilde{Z}} \widehat{f}^*_g, \quad
\text{and} \quad f^*_{i,j}(g_j) = \widetilde{Z}_{i,j}\eps^*_{i,j}(g_j),
%\widehat{\Omega}^*_{CR,g} &=& n^{-1} \sum_{j \in J} \sum_{i \in I_{n,j}} \sum_{k \in I_{n,j}}
%f^*_{i,j}(g_j) f^*_{k,j}(g_j)^{\top}, \notag \\
%\widehat{\Omega}^*_{R,g} &=& n^{-1} \sum_{j \in J} \sum_{i \in I_{n,j}} \sum_{k \in I_{n,j}}
%\widetilde{Z}_{i,j} \widetilde{Z}_{k,j}^{\top} \hat{u}^*_{i,j} (g) \hat{u}^*_{k,j} (g),
\end{eqnarray*}

$\eps^*_{i,j}(g_j) = g_j \bar{\eps}^r_{i,j}$.

Compute the bootstrap statistics:

\begin{align*}
AR^*_{n}(g) = \big\Vert \sqrt{n} \widehat{f}^*_g \big\Vert_{\hat{A}_{z}} \quad \text{and} \quad
AR^*_{CR,n}(g) = \big\Vert \sqrt{n} \widehat{f}^*_g \big\Vert_{\hat{A}_{CR}}. \quad
\end{align*}

Step 3:
Let $\hat{c}_{AR,n}(1-\alpha)$ and $\hat{c}_{AR,CR,n}(1-\alpha)$
denote the $(1-\alpha)$-th quantile of $\{AR_{n}^*(g)\}_{g \in \textbf{G}}$ and $\{AR^*_{CR,n}(g)\}_{g \in \textbf{G}}$, respectively.

We note that unlike the $T_{CR,n}$-based Wald test in Section <ref>, there is no need to bootstrap the CCE for the $AR_{CR,n}$ test even though $\hat{A}_{CR}$ also admits a random limit. This is because $\hat{A}_{CR}$, unlike $\hat{A}_{r,CR}$ defined in (<ref>), is invariant to the sign changes.

§ MAIN ASSUMPTIONS AND SEVERAL EXAMPLES

In this section, we introduce the assumptions that will be used in our analysis of the wild bootstrap tests under a small number of clusters in Sections <ref>-<ref>. For the rest of the paper, we define $\widetilde{Z}_{i,j}$ as the residual of regressing  $Z_{i,j}$ on $W_{i,j}$ using the entire sample, and that for any random vectors $U_{i,j}$ and $V_{i,j}$, $\widehat{Q}_{UV,j} = \frac{1}{n_j}\sum_{i \in I_{n,j}}U_{i,j}V_{i,j}^\top$ and $\widehat{Q}_{UV} = \frac{1}{n}\sum_{j \in J}\sum_{i \in I_{n,j}}U_{i,j}V_{i,j}^\top$. Further define $\widehat{Q} = \widehat{Q}_{\widetilde{Z}X}^{\top} \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}
\widehat{Q}_{\widetilde{Z}X}$
and $Q$ as the probability limits of $\widehat{Q}$.

The following statements hold:
(i) For each $j \in J$, either (1) $\widehat{Q}_{\widetilde{Z}W,j} = 0$, or (2) $\widehat{Q}_{\widetilde{Z}W,j} = o_p(1)$ and $\frac{1}{\sqrt{n}} \sum_{j \in J}\sum_{i \in I_{n,j}} W_{i,j}\eps_{i,j} = O_p(1)$.

(ii) There exists a collection of independent random variables $\{ \mathcal{Z}_j : j \in J \}$,
where $\mathcal{Z}_j \sim N(0, \Sigma_j)$ with positive definite $\Sigma_j$ for all $j \in J$, such that

\begin{eqnarray*}
\left\{
\frac{1}{\sqrt{n_j}} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j}
: j \in J \right\} \xrightarrow{\enskip d \enskip}
\left\{
\mathcal{Z}_{j}
: j \in J \right\}.
\end{eqnarray*}

(iii) For each $j \in J$, $n_j/n \rightarrow \xi_j >0$.

(iv) $\frac{1}{n}\sum_{j \in J} \sum_{i \in I_{n,j}}W_{i,j}W_{i,j}^\top$ is invertible.

Several remarks are in order. First, we have $\widehat{Q}_{\widetilde{Z}W,j} = 0$ if $W_{i,j}$ contains the interactions between baseline exogenous regressors and cluster dummies or $Z_{i,j}$ is constructed as the residual from the cluster-level projection of original IVs onto the linear space spanned by $W_{i,j}$.[Specifically, suppose $\mathscr{Z}_{i,j}$ are the base IVs. We can construct $Z_{i,j}$ as $Z_{i,j} = \mathscr{Z}_{i,j} - \hat{\chi}_j^\top W_{i,j}$, where

$\hat{\chi}_j = \widehat{Q}_{WW,j}\widehat{Q}_{WW,j}^{-}\widehat{Q}_{WW,j}^{-}\widehat{Q}_{W\mathscr{Z},j}$,

and $A^{-}$ denotes the pseudo inverse of the positive semidefinite matrix $A$. It is possible to show that $\widetilde{Z}_{i,j} = Z_{i,j}$ and $\frac{1}{n_j}\sum_{i \in I_{n,j}}\widetilde{Z}_{i,j}W_{i,j}^\top = 0$.]
In this case, Assumption <ref>(i) holds automatically and our wild bootstrap procedures allow the joint distributions of $(Z_{i,j},W_{i,j})_{i \in I_{n,j}}$ to be heterogeneous across clusters. In Section <ref>, we find good finite sample performance of our method when the distributions of $(Z_{i,j},W_{i,j})_{i \in I_{n,j}}$ are different across clusters.

Second, when some clusters contain a small number of observations, including the full set of interactions between baseline exogenous variables and cluster dummies might cause numerical instability.[This is just a finite sample issue as asymptotically, we assume that the cluster-level sample sizes diverges to infinity while the number of clusters remains fixed, which implies the dimension of the full interactions is also fixed.] In this case, researchers may prefer to use baseline exogenous regressors without interacting them with cluster dummies. To accommodate such a practice, we give another set of conditions in Assumption <ref>(i)(2). Specifically, we have $\widehat{Q}_{\widetilde{Z}W,j} = o_p(1)$ if

\begin{eqnarray}\label{eq: ZW_proj}
\frac{1}{n_j} \sum_{i \in I_{n,j}} \Big\Vert W_{i,j}^{\top} \left( \widehat{\Gamma}_n - \widehat{\Gamma}_{n,j} \right) \Big\Vert^2
\convP  0,
\end{eqnarray}

where $\widehat{\Gamma}_n$ and $\widehat{\Gamma}_{n,j}$ are the $d_w \times d_z$ matrices that satisfy the following orthogonality conditions:

$\sum_{j \in J} \sum_{i \in I_{n,j}}W_{i,j}(Z_{i,j} - \widehat{\Gamma}_n^\top W_{i,j})^\top =0$, and $\sum_{i \in I_{n,j}}W_{i,j}(Z_{i,j} - \widehat{\Gamma}_{n,j}^\top W_{i,j})^\top =0.$

<cit.> impose the same condition as (<ref>) and point out that one sufficient but not necessary condition for (<ref>) is that the distributions of $(Z^{\top}_{i,j}, W^{\top}_{i,j})_{i \in I_{n,j}}$ are the same across clusters. In our empirical application in Section <ref>, we compute $\widehat{Q}_{\widetilde{Z}W,j}$ for each cluster and find that they are all close to zero. We refer to Tables <ref>–<ref> in the Online Supplement for more details.[It is also possible to rigorously test $Q_{\widetilde ZW,j} = 0$ for $j \in J$ if we can consistently estimate the covariance matrix of $\widehat Q_{\widetilde ZW,j}$ under a more detailed within-cluster dependence structure.]

Moreover, the condition that

$$\frac{1}{\sqrt{n}} \sum_{j \in J}\sum_{i \in I_{n,j}} W_{i,j}\eps_{i,j} = O_p(1)$$

is similar to Assumption 1(i) in [21]. If $\eps_{i,j}$ follows an error-component model, i.e.,

\begin{align}\label{eq: error-component}
\eps_{i,j} = \eta_j + e_{i,j},
\end{align}

where $\eta_j$ is a cluster-wide shock for cluster $j$, $e_{i,j}$ is an idiosyncratic shock for observation $i$, and the two shocks are independent, then we suggest including cluster dummies in $W_{i,j}$. In such a regression, the cluster-wide shock $\eta_j$ can be viewed as a coefficient for the cluster dummy and the regression is observationally equivalent to the one with $e_{ij}$ as the error, i.e.,

\begin{align*}
y_{i,j} = X_{i,j}^\top \beta + W_{i,j}^\top \gamma + e_{i,j}.
\end{align*}

Hence, as long as $W_{i,j}$ is exogenous and the within-cluster dependence is sufficiently weak, we have
$\frac{1}{\sqrt{n}} \sum_{j \in J}\sum_{i \in I_{n,j}} W_{i,j}e_{i,j} = O_p(1).$
We emphasize again that if a full set of interactions between baseline exogenous regressors and cluster dummies are used as $W_{ij}$, then Assumption <ref>(i)(1) holds automatically and we do not need Assumption <ref>(i)(2).

Third, Assumption <ref>(ii) is reasonable because $\widetilde{Z}_{i,j}$ is exogenous. It is satisfied whenever the within-cluster dependence is sufficiently weak to permit the application of a suitable central limit theorem (CLT) and the data are independent across clusters. Assumption <ref>(iii) gives the restriction on cluster sizes, and Assumption <ref>(iv) ensures $\widetilde Z_{i,j}$ is uniquely defined.

The following statements hold:
(i) The quantities $\widehat{Q}_{\widetilde{Z}X,j}$, $\widehat{Q}_{\widetilde{Z}\widetilde{Z},j}$, $\widehat{Q}_{\widetilde{Z}X}$, and $\widehat{Q}_{\widetilde{Z}\widetilde{Z}}$
converge in probability to deterministic matrices, which are denoted as $Q_{\widetilde{Z}X,j}$, $Q_{\widetilde{Z}\widetilde{Z},j}$, $Q_{\widetilde{Z}X}$, and $Q_{\widetilde{Z}\widetilde{Z}}$, respectively.

(ii) The matrices $Q_{\widetilde{Z}\widetilde{Z},j}$ is invertible for $j \in J$.

(iii) For all $j \in J$, $\widehat{Q}_{XX,j} = O_p(1)$,
$\widehat{Q}_{X\eps,j} = O_p(1)$, and $\widehat{Q}_{\dot{\eps}\dot{\eps},j} \geq c>0$ for constant $c$ with probability approaching one,
where $\dot{\eps}_{i,j}$ is the residual from the cluster-level projection of $\eps_{i,j}$ on $W_{i,j}$.

Assumption <ref> is required for the our analysis of the Wald test, but not for the AR test. Specifically, Assumption <ref>(i) holds if the dependence of units within clusters is weak enough to render some type of LLN to hold. Assumption <ref>(ii) is standard in the literature and holds regardless of the IV strength. However, it rules out the case that the IVs $Z_{i,j}$ defined in Section <ref>
are constructed as the interaction between baseline IVs and cluster dummies, as discussed in Example <ref> below.

We conclude this section with several examples. Specifically, Assumptions <ref> and <ref> hold in Examples <ref> and <ref>, and we explain why they may not hold in Examples <ref>–<ref>. For Example <ref>, Assumption <ref> holds while Assumption <ref> does not. Consequently, the wild bootstrap AR tests are still applicable while the Wald tests are not.

[Cluster Fixed Effects]

\begin{align}
y_{i,j} = X_{i,j}^\top \beta + B_{i,j}^\top \theta_{j} + \eta_{j} + e_{i,j},
\label{eq:ex1}
\end{align}

where $y_{i,j}$ is the outcome variable, $X_{i,j}$ contains the endogenous variables, $\eta_{j}$ denotes cluster fixed effects, $B_{i,j}$ contains the individual-level baseline exogenous variables, $e_{i,j}$ is the individual-level idiosyncratic error,
and $\theta_j$ are allowed to be heterogeneous across clusters. Further denote $W_{i,j}$ as the full interaction between $\widetilde B_{i,j} = (1,B_{i,j}^\top)^\top$ and cluster dummies, i.e.,

\begin{align}
W_{i,j}    = (\widetilde B_{i,j}^\top 1\{j=1\},\widetilde B_{i,j}^\top 1\{j=2\},\cdots, \widetilde B_{i,j}^\top 1\{j=q\})^\top.
\end{align}

Then, (<ref>) can be rewritten as (<ref>) with $\eps_{i,j} = \eta_j + e_{i,j}$.

As $W_{i,j}$ contains full interaction between $\widetilde B_{i,j}$ and cluster dummies, $\widetilde Z_{i,j}$ is numerically equivalent to the residual from the cluster-level projection of $Z_{i,j}$ on  $\widetilde B_{i,j}$, which implies $\widehat{Q}_{\widetilde Z W,j}=0$ automatically and Assumption <ref>(i) holds.  We further have $\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j} \eta_j = 0$ so that Assumption <ref>(ii) reduces to

\begin{eqnarray*}
\left\{
\frac{1}{\sqrt{n_j}} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}e_{i,j}
: j \in J \right\} \xrightarrow{\enskip d \enskip}
\left\{
\mathcal{Z}_{j}
: j \in J \right\},
\end{eqnarray*}

which holds if $Z_{i,j}$ and $B_{i,j}$ are exogenous and the within-cluster dependence is sufficiently weak.

[Heterogeneous IV Strength across Clusters]
We allow for cluster-level heterogeneity with regard to IV strength. Consider the following first stage regression:

\begin{align*}
X_{i,j} =     Z_{i,j}^\top \Pi_{z,j,n} + W_{i,j}^\top \Pi_{w,j,n} + V_{i,j},
\end{align*}

where $\Pi_{z,j,n}$ and $\Pi_{w,j,n}$ are the coefficients of $Z_{i,j}$ and $W_{i,j}$, respectively, via the cluster-level population projection of $X_{i,j}$ on $Z_{i,j}$ and $W_{i,j}$, for each $j \in J$.[We note $\Pi_{z,j,n}$ and $\Pi_{w,j,n}$ depend on the sample size because the underlying distribution is indexed by $n$.]
Then, our model (<ref>) allows for both $\Pi_{z,j,n}$ and $\Pi_{w,j,n}$ to vary across clusters.
In particular, we allow for some of $\Pi_{z,j,n}$ to decay to or be zero for the bootstrap Wald tests and all $\Pi_{z,j,n}$ to decay to or be zero for the bootstrap AR tests. We will come back to this point in Sections <ref>-<ref> with more details.

[Heterogeneous Slope for the Endogenous Variable]
Similar to <cit.>, we cannot allow for $\beta$ to be heterogeneous across clusters. As a stylized example, let

\begin{align}
y_{i,j} = X_{i,j}^\top \beta_j + W_{i,j}^\top \gamma_j +e_{i,j},
\label{eq:ex2}
\end{align}

where $W_{i,j}$ is just the cluster dummies. For $\beta$ equal to a suitable weighted average of $\beta_j$'s, we may rewrite (<ref>) as (<ref>) with $\eps_{i,j} = X_{i,j}^\top(\beta_j - \beta) + e_{i,j}$, which implies

\begin{align*}
\frac{1}{\sqrt{n_j}} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} =  \frac{1}{\sqrt{n_j}} \sum_{i \in I_{n,j}} (Z_{i,j} - \bar Z_j) (X_{i,j}^\top(\beta_j - \beta) + e_{i,j} ).
\end{align*}

We then see that Assumption <ref>(ii) is violated unless $\beta_j = \beta$.

The wild bootstrap tests are unable to allow for the case with cluster-level IVs, which usually occur in difference-in-difference analysis with endogenous treatments.
In this setting, the treatment status and group assignment are interpreted as our $X_{i,j}$ and $Z_{i,j}$, respectively, and they are different due to imperfect compliance.
In addition, when the assignment is at the cluster level (i.e., some clusters are assigned to the treatment group while the others are assigned to the control group), the value of $Z_{i,j}$
is invariant within each cluster. If we use the full interaction of baseline covariates and cluster dummies as our $W_{i,j}$, then our Assumption <ref>(ii) is violated because $\Sigma_j$ can be degenerate.

Following the suggestion by <cit.>, it may be possible to avoid such an issue by clustering more coarsely.

[Interacting IVs with Cluster Dummies] We require the IVs $Z_{i,j}$ defined in Section <ref>
to be the baseline instruments which are not interacted with cluster dummies.
To see this, we consider the simplest case that $W_{i,j}$ only contains the intercept and $Z_{i,j}$ is constructed as the interactions of a scalar baseline IV $\mathscr{Z}_{i,j}$ with cluster dummies. Then, for the last cluster (the $q$-th cluster), we have

\begin{align*}
\frac{1}{\sqrt{n_q}}\sum_{i \in I_{n,q}} \widetilde{Z}_{i,q}\eps_{i,q} = \frac{1}{\sqrt{n_q}}\sum_{i \in I_{n,q}}(-\xi_1 \overline{Z}_1 \eps_{i,q},\cdots, -\xi_{q-1} \overline{Z}_{q-1} \eps_{i,q},(\mathscr{Z}_{i,q} - \xi_{q} \overline{Z}_{q})\eps_{i,q})^\top,
\end{align*}

where $\overline{Z}_j = \frac{1}{n_j}\sum_{i \in I_{n,j}}\mathscr{Z}_{i,j}$. Clearly, $\frac{1}{\sqrt{n_q}}\sum_{i \in I_{n,q}} \widetilde{Z}_{i,q}\eps_{i,q}$ is linearly dependent, which implies $\Sigma_q$ is degenerated, and thus, Assumption <ref>(ii) is violated. Similar results hold for the other clusters.

We notice that [2] interact the baseline IVs with subgroup dummies such as those for states, gender, or race. However, [2] use an analytical covariance matrix estimator for independent and conditionally homoskedastic observations for inference. In contrast, as we allow for heteroskedasiticity and are agnostic about the within-cluster dependence, it is difficult, if not impossible, to derive a consistent estimator of the covariance matrix in our setting without imposing additional restrictions. Instead, following the lead of [21], we rely on the connection between the wild bootstrap and the randomization test to avoid the consistent estimation of the covariance matrix. Assumption <ref>(ii) is crucial for such a connection to hold.

[Cluster-level Endogenous Variables]
If $X_{i,j}$ is a cluster-level variable (say, $X_j$),
then the resulting within-cluster limiting Jacobian matrix $Q_{\widetilde{Z}X,j}$ may be random and potentially correlated with the within-cluster score component $\mathcal{Z}_j$
as $X_{j}$ is endogenous, which violates Assumption <ref>(i). We notice that similar issues can arise for the approaches by [17], [43], IM, and CRS. On the other hand, our wild bootstrap AR tests ($AR_n$ and $AR_{CR,n}$) only requires Assumption <ref> but not Assumption <ref>, and thus, remain valid in this case.

§ ASYMPTOTIC RESULTS FOR THE WALD TESTS WITHOUT CCE

For the Wald test, we further assume the following assumption.

(i) $Q_{\widetilde{Z}X}$ is of full column rank.

(ii) One of the following two conditions holds:
(1) $d_x = 1$,

or (2) there exists a scalar $a_j$ for each $j \in J$ such that  $Q_{\widetilde{Z}X,j} = a_j Q_{\widetilde{Z}X}$.

Several remarks are in order. First, Assumption <ref>(i) requires (overall) strong identification for $\beta_n$.
Second, Assumption <ref>(ii)(1) states that if there is only one endogenous variable, no further restrictions are required. A single endogenous variable is the leading case in empirical applications involving IV regressions.
In fact, 101 out of 230 specifications in [10]'s ([10]) sample and 1,087 out of 1,359 in [74]'s ([74]) sample has one endogenous regressor and one IV. [52] find that among 123 papers published in AER between 2013 and 2019 that include IV regressions, 61 employ single instrumental variable (just-identified) regressions. They point out that the single-IV case “includes applications such as randomized trials with imperfect compliance (estimation of LATE, [46]), fuzzy regression discontinuity designs
(see discussion in [51]), and fuzzy regression kink designs (see discussion in
[11] also point out that “most studies using IV (including [12] and [13]) report just-identified IV estimates computed with a single instrument."[In our empirical application, we revisit the influential study by [15],

which also has only one IV.]
In addition, Assumption <ref>(ii)(1) further allows for the case of single endogenous regressor and multiple IVs.  Third, Assumption <ref>(ii)(2) is needed if we have multiple endogenous variables. The condition is similar to that in <cit.>, which restricts the type of heterogeneity of the within-cluster Jacobian matrices.
However, it is still weaker than restrictions assumed in the literature for cluster-robust Wald tests with few clusters. For example,
[17] and [43] provide asymptotic approximations that are based on $t$ and $F$ distributions for the Wald statistics with CCE.
The conditions in their papers require the within-cluster Jacobian matrices to have the same limit for all clusters (i.e., Assumption <ref>(ii)(2) to hold with $a_j=1$ for all $j \in J$).[For example, see <cit.> and <cit.> for details.]
They also impose that the cluster sizes are approximately equal for all clusters and the cluster-level scores in Assumption <ref>(ii) have the same normal limiting distribution for all clusters, which are not necessary for the wild bootstrap. Finally, Assumption <ref> will not be needed for the bootstrap AR tests in Section <ref>, as they require neither strong identification nor homogeneity conditions.

To further clarify our setting, we can relate the Jacobian matrices with the first-stage projection coefficient.

Specifically, recall $\Pi_{z,j,n}$ is the coefficient of $Z_{i,j}$ via the cluster-level population projection of $X_{i,j}$ on $Z_{i,j}$ and $W_{i,j}$ as defined in Example <ref>.
Then, we have $\lim_{n \rightarrow \infty}\Pi_{z,j,n} = \Pi_{z,j} := Q_{\widetilde{Z}\widetilde{Z},j}^{-1} Q_{\widetilde{Z}X,j}$ under our framework. Also define $\Pi_z = Q_{\widetilde{Z}\widetilde{Z}}^{-1} Q_{\widetilde{Z}X}$.
Assumption <ref>(i) ensures that overall we have strong identification as $Q_{\widetilde{Z}X}$ (and $\Pi_z$) is of full column rank.
Furthermore, we call the clusters in which $Q_{\widetilde{Z}X,j}$ (and $\Pi_{z,j}$) are of full column rank the strong IV clusters, i.e., $\beta_n$ is strongly identified in these clusters.
On the other hand, strong identification for $\beta_n$ is not ensured in the rest of the clusters. We can draw three observations from Assumption <ref>. First, given the number of clusters is fixed, only one strong IV cluster is needed for Assumption <ref>(i) to hold. Second, Assumption <ref>(ii)(2) implies that when $a_j \neq 0$, $Q_{\widetilde{Z}X,j}$ (and $\Pi_{z,j}$) is of full column rank, so that the $j$-th cluster is a strong IV cluster. Third, Assumption <ref>(ii)(2) excludes the case that $Q_{\widetilde{Z}X,j}$ is of a reduced rank but is not a zero matrix.
It is possible to select out the clusters with Jacobian matrices of reduced rank
[65, 50, 24]. To formally extend these rank tests to the case with a fixed number of large clusters would merit a separate paper and is thus left  as a topic for future research.

Suppose  $|| \hat{A}_r - A_r ||_{op} = o_p(1)$, where
$A_r$ is a $d_r \times d_r$ symmetric deterministic weighting matrix such that $0 < c \leq \lambda_{min}(A_r) \leq \lambda_{max}(A_r) \leq C < \infty$ for some constants $c$ and $C$, $\lambda_{\min}(A)$ and  $\lambda_{\max}(A)$ are the minimum and maximum eigenvalues of the generic matrix $A$, and
$||A||_{op}$ denotes the operator norm of the matrix $A$.

Assumption <ref> requires that the weighting matrix $\hat{A}_r$ in (<ref>)
has a deterministic limit.  It rules out the case that $\hat{A}_r$ equals the inverse of CCE, which has a random limit under a small number of clusters. We will discuss the bootstrap Wald test with CCE in Section <ref>.

Suppose that Assumptions <ref>-<ref> hold. Then under $\mathcal{H}_0$, for all four estimation methods (namely, TSLS, LIML, FULL, and BA),

\begin{eqnarray*}
\alpha - \frac{1}{2^{q-1}} \leq \liminf_{n \rightarrow \infty} \mathbb{P} \{ T_{n} > \hat{c}_{n}(1-\alpha) \}
\leq \limsup_{n \rightarrow \infty} \mathbb{P} \{ T_{n} > \hat{c}_{n}(1-\alpha) \} \leq
\alpha + \frac{1}{2^{q-1}}.
\end{eqnarray*}

Two remarks are in order. First, Theorem <ref> states that as long as there exists at least one strong IV cluster, the $T_n$-based wild bootstrap test has limiting null rejection probability no greater than $\alpha+1/2^{q-1}$ and no smaller than $\alpha - 1/2^{q-1}$. For the test to have power against local alternatives, we later show that at least 5 or 6 strong IV clusters are needed. The error $1/2^{q-1}$ can be viewed as the upper bound for the asymptotic size distortion, which vanishes exponentially with the total number of clusters rather than the number of strong IV clusters. Intuitively, although the weak IV clusters do not contribute to the identification of $\beta_n$,
the scores of such clusters still contribute to the limiting distributions of the IV estimators, which in turn determines the total number of sign changes in the bootstrap Wald statistics. We note that  $1/2^{q-1}$ equals $1.56\%$ and $0.2\%$ when $q=7$ and $10$, respectively. If such an distortion is still of concern, researchers can replace $\alpha$ in our context by $\alpha - 1/2^{q-1}$ to ensure null rejection rate.

Second, it is well known that estimators such as LIML and FULL have reduced finite sample bias relative to TSLS in the over-identified case, especially when the IVs are not strong. Since the validity of the randomization with sign changes requires a distributional symmetry around zero, the LIML and FULL-based bootstrap Wald tests may therefore achieve better finite sample size control than that based on TSLS. This is confirmed by the simulation experiments in Section <ref>.[To theoretically document the asymptotic bias due to the dimensionality of IVs, one needs to consider an alternative framework in which the number of clusters is fixed but the number of IVs tends to infinity, following the literature on many/many weak instruments [16, 23, 59]. We leave this direction of investigation for future research.]

We next examine the power of the
wild bootstrap test against local alternatives.

Suppose that Assumptions <ref>-<ref> hold. Further suppose that there exists a subset $J_s$ of $J$ such that $a_j>0$ for each $j \in J_s$, $a_j = 0$ for $j \in J \backslash J_s$,	and $\lceil |\textbf{G}|(1-\alpha) \rceil \leq |\textbf{G}|-2^{q-q_s+1}$, where $|\textbf{G}| = 2^q$, $q_s = |J_s|$, and $a_j$ is defined in Assumption <ref>.[When $d_x= 1$, we define $a_j = Q^{-1}Q_{\widetilde{Z}X}^\top Q_{\widetilde{Z}\widetilde{Z}}^{-1} Q_{\widetilde{Z}X,j}$.]
Then under $\mathcal{H}_{1,n}$, for all four estimation methods (namely, TSLS, LIML, FULL, and BA),

\begin{eqnarray*}
\lim_{|| \mu ||_2 \rightarrow \infty} \liminf_{n \rightarrow \infty} \mathbb{P} \{ T_{n} > \hat{c}_{n}(1-\alpha) \} =1.
\end{eqnarray*}

Three remarks are in order.
First, to establish the power of the $T_n$-based wild bootstrap test against $n^{-1/2}$-local alternatives,
we need
homogeneity of the signs of Jacobians for the strong IV clusters (i.e., $a_j>0$ for each $j \in J_s$).
For example, in the case with a single IV, it requires $\Pi_{z,j}$ to have the same sign across all the strong IV clusters.
We notice that this condition is not needed for the bootstrap Wald test studentized by CCE described in Section <ref>.

Second, we also need a sufficient number of strong IV clusters.
For instance, the condition $\lceil |\textbf{G}|(1-\alpha) \rceil \leq |\textbf{G}|-2^{q-q_s+1}$ requires that
$q_s \geq 5$ and $q_s \geq 6$ for $\alpha=10\%$ and $5\%$, respectively. Theorem <ref> suggests that although the size of the wild bootstrap test is well controlled even with only one strong IV cluster, its power depends on the number of strong IV clusters.

Third, the wild bootstrap test has resemblance to the group-based $t$-test in IM and the randomization test with sign changes in CRS. IM and CRS approaches separately estimate the parameters using the samples in each cluster (say, $\hat{\beta}_{1}, ..., \hat{\beta}_{q}$), and therefore requires $\beta_n$ to be strongly identified in all clusters. In contrast, our method allows for weak IV clusters in the sense of [69], where $\Pi_{z,j,n}$ has the same order of magnitude as $n_j^{-1/2}$.[The cluster-level IV estimators of such weak clusters would become inconsistent and have highly nonstandard limiting distributions. Also, if there exist both strong and “semi-strong" clusters, in which the (unknown) convergence rates of IV estimators can vary among clusters and be slower than $\sqrt{n}_j$ [5],
then the estimators with the slowest convergence rate will dominate in the test statistics that are based on the cluster-level estimators.] Specifically, the size result in Theorem <ref> holds even with only one strong IV cluster, and the power result in Theorem <ref> holds with 5 and 6 strong IV clusters for 10% and 5% levels, respectively. On the other hand, if $\beta_n$ is strongly identified in all clusters and the cluster-level IV estimators have minimal finite sample bias, IM and CRS have an advantage over the wild bootstrap when there are multiple endogenous variables as they do not require Assumption <ref>(ii).
Furthermore, as remarked after Assumption <ref>, if we use a full set of interactions between baseline covariates and cluster dummies as $W_{i,j}$, then Assumption <ref>(i)(1) holds automatically. In this case, like IM and CRS, we do not require the homogeneity of the joint distribution of $(Z_{i,j}^\top,W_{i,j}^\top)_{i \in I_{n,j}}$ across clusters. Specifically, the IV regression in Example <ref> satisfies both Assumptions <ref> and <ref>, and the joint distribution of $(Z_{i,j}^\top,W_{i,j}^\top)_{i \in I_{n,j}}$ can be heterogeneous across clusters. In Section <ref>, we confirm the good finite sample performance of our wild bootstrap inference procedure while allowing the
heterogeneity in the distribution of $(Z_{i,j}^\top,W_{i,j}^\top)_{i \in I_{n,j}}$ across clusters.  Therefore, the two types of approaches could be considered as complements, and practitioners may choose between them according to the characteristics of their data and models.

§ ASYMPTOTIC RESULTS FOR THE WALD TESTS WITH CCE

Now we consider a wild bootstrap test for the Wald statistic with CCE in which the weighting matrix $\hat{A}_{r,CR}$  is defined as the inverse of the CCE:

\begin{align}\label{eq: Wald-hat-A}
%W_{n} &\equiv & \left(\sqrt{n} \left( \hat{\beta} - \beta_0 \right) \right)' \widehat{V} \left( \sqrt{n} \left(\hat{\beta} - \beta_0\right) \right), \notag \\
%W_n(\lambda) & \equiv &  \frac{|\sqrt{n}(c'\hat{\beta} - \lambda)|}{\sqrt{ \lambda^{\top}_{\beta} \widehat{V} c}},
%T_n(\lambda_0) & = & n (\lambda^{\top}_{\beta} \hat{\beta} - \lambda_0)^{\top} \left( \lambda^{\top}_{\beta} \widehat{V} \lambda_{\beta} \right)^{-1} (\lambda^{\top}_{\beta} \hat{\beta} - \lambda_0),
\hat{A}_{r,CR} & = \left( \lambda^{\top}_{\beta} \widehat{V} \lambda_{\beta} \right)^{-1} ,\quad  \widehat{V}
\widehat{Q}^{-1}
\widehat{Q}^{\top}_{\widetilde{Z}X} \widehat{Q}^{-1}_{\widetilde{Z}\widetilde{Z}}
\widehat{\Omega}_{CR}
\widehat{Q}^{-1}_{\widetilde{Z}\widetilde{Z}}\widehat{Q}_{\widetilde{Z}X}
\widehat{Q}^{-1}, \quad \widehat{Q} = \widehat{Q}^{\top}_{\widetilde{Z}X} \widehat{Q}^{-1}_{\widetilde{Z}\widetilde{Z}}\widehat{Q}_{\widetilde{Z}X},
\end{align}

$\widehat{\Omega}_{CR} = n^{-1} \sum_{j \in J} \sum_{i \in I_{n,j}} \sum_{k \in I_{n,j} } \widetilde{Z}_{i,j} \widetilde{Z}_{k,j}^{\top} \hat{\eps}_{i,j} \hat{\eps}_{k,j}$,
and $\hat{\eps}_{i,j}$ is the unrestricted residual defined in (<ref>).

The corresponding Wald statistic with CCE is denoted as

\begin{align*}
T_{CR,n} = ||\sqrt{n}(\lambda_\beta^\top \hat{\beta} - \lambda_0)||_{\hat{A}_{r,CR}}.
\end{align*}

To obtain the wild bootstrap critical value for the Wald statistic with CCE, the bootstrap weighting matrix $\hat{A}^*_{r,CR,g}$ in (<ref>) is defined as

\begin{eqnarray}
\label{eq: boot-Wald-hat-A}
\hat{A}^*_{r,CR,g} &=& \left( \lambda^{\top}_{\beta} \widehat{V}^*_g \lambda_{\beta} \right)^{-1}, \quad
\widehat{V}^*_g
=  \widehat{Q}_g^{*-1}
\widehat{Q}^{*\top}_{\widetilde{Z}X}(g) \widehat{Q}^{-1}_{\widetilde{Z}\widetilde{Z}}
\widehat{\Omega}^*_{CR,g}
\widehat{Q}^{-1}_{\widetilde{Z}\widetilde{Z}}\widehat{Q}^*_{\widetilde{Z}X}(g)
\widehat{Q}_g^{*-1},
\end{eqnarray}

where for $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$,

\begin{align}
& \widehat{\Omega}^*_{CR,g} = \frac{1}{n} \sum_{j \in J} \sum_{i \in I_{n,j}} \sum_{k \in I_{n,j} } \widetilde{Z}_{i,j} \widetilde{Z}_{k,j}^{\top} \hat{\eps}^*_{i,j}(g) \hat{\eps}^*_{k,j}(g), \quad  \widehat{Q}^*_{\widetilde{Z}X}(g) = \frac{1}{n} \sum_{j \in J} \sum_{i \in I_{n,j}}  \widetilde{Z}_{i,j} X_{i,j}^{*\top}(g),
\notag \\
&   \hat{\eps}^*_{i,j}(g) = y^*_{i,j}(g) - X_{i,j}^{*\top}(g) \hat{\beta}^*_{g,L} - W_{i,j}^{\top} \hat{\gamma}^*_{g,L}, \quad \text{and} \quad \widehat{Q}^*_g = \widehat{Q}_{\widetilde{Z}X}^{*\top}(g) \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1} \widehat{Q}^*_{\widetilde{Z}X}(g). \label{eq:Qstar}
\end{align}

Then, we denote the corresponding bootstrap Wald statistic with CCE as

\begin{align*}
T^*_{CR,n}(g) = || \sqrt{n} (\lambda_{\beta}^{\top} \hat{\beta}^*_g - \lambda_0) ||_{\hat{A}^*_{r,CR,g}},
\end{align*}

and define the bootstrap critical value $\hat{c}_{CR,n}(1-\alpha)$ as the $1-\alpha$ quantile of $\{T_{CR,n}^*(g): g \in \textbf{G}\}$.
Unlike the Wald statistic without CCE considered in the previous section, here we need to bootstrap the weighting matrix because under our asymptotic framework with a small number of clusters $\hat{A}_{r,CR}$ has a random limit, which depends on the limits of the scores and the IV estimators $(\hat{\beta}_L,\hat{\gamma}_L)$.

Suppose that Assumptions <ref>-<ref> hold, and $q > d_r$.
Then under $\mathcal{H}_0$, for all four estimation methods (namely, TSLS, LIML, FULL, and BA),

\begin{align*}
\alpha - \frac{1}{2^{q-1}} & \leq \liminf_{n \rightarrow \infty} \mathbb{P} \{ T_{CR,n} > \hat{c}_{CR,n}(1-\alpha) \} \\
& \leq \limsup_{n \rightarrow \infty} \mathbb{P} \{ T_{CR,n} > \hat{c}_{CR,n}(1-\alpha) \} \leq \alpha + \frac{1}{2^{q-1}}.
\end{align*}

We require $q>d_r$ because otherwise CCE and its bootstrap counterpart are not invertible. Theorem <ref> states that with at least one strong IV cluster, the $T_{CR,n}$-based wild bootstrap test controls size asymptotically up to a small error. Next, we turn to the local power.

(i) Suppose that Assumptions <ref>-<ref> hold, $q > d_r$, and that there exists a subset $J_s$ of $J$ such that $\min_{j \in J_s} |a_j| > 0$,
$a_j=0$ for each $j \in J \backslash J_s$,
and $\lceil |\textbf{G}|(1-\alpha) \rceil \leq |\textbf{G}|-2^{q-q_s+1}$, where $|\textbf{G}| = 2^q$, $q_s = |J_s|$, and $a_j$ is defined in Assumption <ref>.[When $d_x= 1$, we define $a_j = Q^{-1}Q_{\widetilde{Z}X}^\top Q_{\widetilde{Z}\widetilde{Z}}^{-1} Q_{\widetilde{Z}X,j}$.]
Then under $\mathcal{H}_{1,n}$, for all four estimation methods (namely, TSLS, LIML, FULL, and BA),

\begin{eqnarray*}
\lim_{|| \mu ||_2 \rightarrow \infty} \liminf_{n \rightarrow \infty} \mathbb{P} \{ T_{CR,n} > \hat{c}_{CR,n}(1-\alpha) \} =1.
\end{eqnarray*}

(ii) Further suppose that $d_r=1$. Then under $\mathcal{H}_{1,n}$,
for any $e>0$, there exists a constant $c_{\mu}>0$ such that when $||\mu||_2 >c_\mu$,

\begin{align*}
\liminf_{n \rightarrow \infty}\mathbb{P}(\phi^{cr}_n \geq \phi_n) \geq 1-e,
\end{align*}

where $\phi^{cr}_n = 1 \{ T_{CR,n} > \hat{c}_{CR,n}(1-\alpha)\}$ and $\phi_n = 1 \{ T_{n} > \hat{c}_{n}(1-\alpha) \}$.

Two remarks are in order.
First, different from Theorem <ref>, the power result in Theorem <ref> does not require the homogeneity condition on the sign of first-stage coefficients for the strong IV clusters (i.e., it only requires $\min_{j \in J_s} |a_j|>0$).

Therefore, the bootstrap Wald test studentized by CCE has an advantage over its unstudentized counterpart, given that the homogeneous sign condition may not hold in some empirical studies (e.g., Figure <ref>).
To establish the power without the sign condition, we rely on arguments very different from those in [21].

Second, we further establish in Theorem <ref>(ii) that in the case of a $t$-test (i.e., when the null hypothesis involves one restriction), the rejection of the $T_{CR,n}$-based bootstrap test dominates that based on $T_n$ with large probability when the local parameter $\mu$ is sufficiently different from zero. To see this, we note that, when there is just one restriction, $
1\{T_n > \hat{c}_{n}(1-\alpha)\} = 1\{T_{CR,n} >  \tilde{c}_{CR,n}(1-\alpha)\},$ where $\tilde{c}_{CR,n}(1-\alpha)$ denotes the $(1-\alpha)$ quantile of $\left\{|\sqrt{n}(\lambda_{\beta}^{\top}\hat{\beta}^*_g - \lambda_0)|/\sqrt{\lambda_{\beta}^{\top}\widehat{V}\lambda_{\beta}}: g \in \textbf{G}\right\}.$ Then, Theorem <ref>(ii) follows because $\tilde{c}_{CR,n}(1-\alpha) > \hat{c}_{CR,n}(1-\alpha)$ with large probability as $||\mu||_2$ becomes sufficiently large.

§ ASYMPTOTIC RESULTS FOR WEAK-INSTRUMENT-ROBUST TESTS

The size control of the bootstrap Wald tests with or without CCE relies on Assumption <ref>(i), which rules out overall weak identification in which all clusters are weak. In the case that the parameter of interest may be weakly identified in all clusters, we may consider the bootstrap AR tests for the full vector of $\beta_n$ as defined in Section <ref>.

$|| \hat{A}_z - A_z ||_{op} = o_p(1)$, where
$A_z$ is a $d_z \times d_z$ symmetric deterministic weighting matrix such that $0 < c \leq \lambda_{min}(A_z) \leq \lambda_{max}(A_z) \leq C < \infty$ for some constants $c$ and $C$.

Theorem <ref> below shows that, in the general case with multiple IVs, the limiting null rejection probability of the $AR_{n}$-based bootstrap test does not exceed the nominal level $\alpha$, and that of the $AR_{CR,n}$ test does not exceed $\alpha$ by more than $1/2^{q-1}$ when $q>d_z$, irrespective of IV strength.

Suppose Assumption <ref> holds and $\beta_n=\beta_0$.
For $AR_n$, further suppose Assumption <ref> holds.
For $AR_{CR,n}$, further suppose $q > d_z$.

\begin{align*}
%\alpha - \frac{1}{2^{q-1}} \leq & \liminf_{n \rightarrow \infty} \mathbb{P} \{ AR_{CR,n} > \hat{c}_{AR,CR,n}(1-\alpha) \}
%\leq  \limsup_{n \rightarrow \infty} \mathbb{P} \{ AR_{CR,n} > \hat{c}_{AR,CR,n}(1-\alpha) \} \leq \alpha, \\
\alpha - \frac{1}{2^{q-1}} & \leq \liminf_{n \rightarrow \infty} \mathbb{P} \{ AR_{n} > \hat{c}_{AR,n}(1-\alpha) \} \notag \\
& \leq \limsup_{n \rightarrow \infty} \mathbb{P} \{ AR_{n} > \hat{c}_{AR,n}(1-\alpha) \} \leq \alpha, \; \text{and}
\notag \\
\alpha - \frac{1}{2^{q-1}} & \leq  \liminf_{n \rightarrow \infty} \mathbb{P} \{ AR_{CR,n} > \hat{c}_{AR,CR,n}(1-\alpha) \} \\
& \leq  \limsup_{n \rightarrow \infty} \mathbb{P} \{ AR_{CR,n} > \hat{c}_{AR,CR,n}(1-\alpha) \} \leq \alpha + \frac{1}{2^{q-1}}.
\end{align*}

Several remarks are in order.
First, for the bootstrap AR test studentized by the CCE, we require the number of IVs to be smaller than the number of clusters because otherwise, the CCE is not invertible.
Second, the behavior of wild bootstrap for other weak-IV-robust statistics proposed in the literature is more complicated as they depend on an adjusted sample Jacobian matrix
(e.g., see [49], [7], [9], and [6], among others). We study the asymptotic properties of wild bootstrap for these statistics in Section  <ref> in the Online Supplement and show that their validity would require at least one strong IV cluster.
This provides theoretical support to the
findings in the literature that the bootstrap AR test has better finite sample size properties than other bootstrap weak-IV-robust tests for clustered data
(e.g., see [31, 32]).

Third, for the weak-IV-robust subvector inference, one may use a projection approach
[29]
after implementing the wild bootstrap AR tests for $\beta_n$, but the result may be conservative.[
Alternative subvector inference methods (e.g., see Section 5.3 in [10] and the references therein) provide a power improvement over the projection approach with a large number of observations/clusters.
However, it is unclear whether they can be applied to the current setting.
It is unknown whether the asymptotic critical values given by these approaches will still be valid with a small number of clusters.
Also, [72] show that bootstrap tests based on the subvector statistics therein may not be robust to weak IVs
even under conditional homoskedasticity.]

Next, to study the power of the $AR_n$-based bootstrap test against the local alternative,
we let $\lambda_{\beta} = I_{d_x}$ for $\mathcal{H}_{1,n}$ in (<ref>)
so that $\mu = \mu_{\beta}$ and impose the following condition.

(i) $Q_{\widetilde{Z}X} \neq 0$.
(ii) There exists a scalar $a_j$ for each $j \in J$ such that $Q_{\widetilde{Z}X,j} = a_jQ_{\widetilde{Z}X}$.

If $d_x = d_z = 1$, Assumption <ref>(i) implies strong identification.
When $d_z>1$ or $d_x >1$,
Assumption <ref>(i) only rules out the case that $Q_{\widetilde{Z}X}$ is a zero matrix,
while allowing it to be nonzero but not of full column rank.
As noted below, this means the AR test can still have local power in some direction even without strong identification.

Assumption <ref>(ii) is similar to Assumption <ref>(ii). In particular, it holds automatically if Assumption <ref>(i) holds and
$d_x= d_z=1$ (i.e., single endogenous regressor and single IV).

Suppose Assumptions <ref>, <ref>, and <ref> hold.

Further suppose that there exists a subset $J_s$ of $J$ such that $\min_{j \in J_s} a_j > 0$,
$a_j=0$ for each $j \in J \backslash J_s$,
and $\lceil |\textbf{G}|(1-\alpha) \rceil \leq |\textbf{G}|-2^{q-q_s+1}$, where $q_s = |J_s|$ and $a_j$ is defined in Assumption <ref>.
Then, under $\mathcal{H}_{1,n}$ with $\lambda_{\beta} = I_{d_x}$,

\begin{eqnarray*}
\lim_{|| Q_{\widetilde{Z}X}\mu ||_2 \rightarrow \infty} \liminf_{n \rightarrow \infty} \mathbb{P} \{ AR_{n} > \hat{c}_{AR,n}(1-\alpha) \} =1.
\end{eqnarray*}

Two remarks are in order. First, notice that the bootstrap AR test not studentized by CCE has power against local alternatives as long as $||Q_{\widetilde{Z}X}\mu||_2 \rightarrow \infty$, which may hold even when $\beta_n$ is not strongly identified and $Q_{\widetilde{Z}X}$ is not of full column rank. Second, when $d_z = 1$, we have
$1\{AR_n > \hat{c}_{AR,n}(1-\alpha) \} = 1\{AR_{CR,n} > \hat{c}_{AR,CR,n}(1-\alpha) \}$,
which implies the $AR_n$ and $AR_{CR,n}$-based bootstrap tests have the same power against local alternatives. However, such a power equivalence does not hold when $d_z>1$.
Unlike the Wald test with CCE in Section <ref>,

we cannot establish the power of the AR statistics with CCE for the general case due to the fact that the CCE for the Wald test is computed using the estimated residual $\hat{\eps}_{i,j}$ whereas the CCE for the AR test is computed by imposing the null (see Section <ref>). Consequently, unlike the Wald test, the AR test with CCE is not theoretically guaranteed to have better power than the AR test without CCE.
Indeed, we observe in Section <ref> that the $AR_{CR,n}$-based bootstrap test has inferior finite sample power properties compared with its $AR_n$-based counterpart.

Finally, we show that in the specific case with one endogenous regressor and one IV, if the wild bootstrap procedure for the $AR_n$ statistic is applied to the unstudentized Wald statistic $T_n$, then the resulting test will be asymptotically equivalent to the $AR_n$-based bootstrap test, both under the null and the alternative.

More precisely, for $T_n = | \sqrt{n}(\hat{\beta} - \beta_0) |$, where $\hat{\beta}$ is the TSLS estimator, the wild bootstrap generates

\begin{align}\label{eq: score-boot}
T^{s*}_{n}(g) = \left|\widehat{Q}_{\widetilde{Z}X}^{-1} \frac{1}{\sqrt{n}} \sum_{j \in J} \sum_{i \in I_{n,j}} g_j \widetilde{Z}_{i,j} \hat{\eps}^r_{i,j}  \right|.
\end{align}

Notice that in this case, the restricted TSLS estimator $\hat{\gamma}^r$ and the restricted OLS estimator $\bar{\gamma}^r$ are the same, which implies the residuals $\hat{\eps}^r_{i,j}$ and $\Bar{\eps}^r_{i,j}$ defined in Sections <ref> and <ref>, respectively, are the same.
Let $\hat{c}^s_n(1-\alpha)$ denote the $(1-\alpha)$-th quantile of
$\{ T^{s*}_n(g) \}_{g \in \textbf{G}}$.
We show this equivalence below.

Suppose that $d_x=d_z=1$, $\liminf_{n \rightarrow \infty}\mathbb{P}(\widehat{Q}_{\widetilde{Z}X} \neq 0) = 1$, $\hat{\beta}$ in the Wald test is computed via TSLS, and Assumption <ref>(iv) holds. Then,

\begin{eqnarray*}
\liminf_{n \rightarrow \infty}
\mathbb{P} \{ \phi_n^s = \phi_n^{ar} \} =1,
\end{eqnarray*}

where $\phi_n^s = 1 \{ T_n > \hat{c}^s_n(1-\alpha) \}$ and $\phi_n^{ar} = 1 \{ AR_n > \hat{c}^{ar}_n(1-\alpha) \}$.

Several remarks are in order.
First, with one endogenous variable and one IV, the Jacobian matrix $\widehat{Q}_{\widetilde{Z}X}$ is just a scalar, which shows up in both the Wald statistic $T_n$ and the bootstrap critical value. After the cancellation of $\widehat{Q}_{\widetilde{Z}X}$,
$T_n$ and its critical value are numerically the same as their AR counterparts, which leads to Theorem <ref>.
Second, for $\widehat{Q}_{\widetilde{Z}X}$ to be cancelled, we only need
$\liminf_{n \rightarrow \infty}\mathbb{P} \{ \widehat{Q}_{\widetilde{Z}X} \neq 0 \} = 1$, which is very mild.

It holds when $\widehat{Q}_{\widetilde ZX}$ is continuously distributed.
Even when both $\widetilde{Z}_{i,j}$ and $X_{i,j}$ are discrete, it still holds if there exists at least one strong IV cluster, i.e., $Q_{\widetilde{Z}X} \neq 0$, where $Q_{\widetilde{Z}X}$ is the probability limit of $\widehat{Q}_{\widetilde{Z}X}$.
When both $\widetilde{Z}_{i,j}$ and $X_{i,j}$ are discrete and $Q_{\widetilde{Z}X} = 0$, this condition still holds if some type of CLT holds such that $\sqrt{n} \widehat{Q}_{\tilde{Z}X} \convD N(c,\sigma^2)$, as $N(c,\sigma^2)$ is continuous. Third, the robustness of the $T_n$-based bootstrap test in (<ref>) does not carry over to the general case with multiple IVs as $T_n$ and its bootstrap statistic can no longer be reduced to their AR counterparts.

Fourth, the robustness to weak IV cannot be extended to the $T_{CR,n}$-based bootstrap test, for which we have to further bootstrap the CCE.

§ MONTE CARLO SIMULATION

In this section, we investigate the finite sample performance of the wild bootstrap tests and alternative methods.
We consider a simulation design similar to that in Section IV of [21] and extend theirs to the IV model.
The data are generated as

\begin{eqnarray*}
%\label{eq: simu-model}
X_{i,j} = \gamma + Z^{\top}_{i,j} \Pi_{z,j} + \sigma(Z_{i,j}) \left( a_{v,j} + v_{i,j} \right), \quad
y_{i,j} = \gamma + X_{i,j} \beta + \sigma(Z_{i,j}) \left( a_{\eps,j} + \eps_{i,j} \right),
\end{eqnarray*}

for $i=1, ..., n$ and $j=1, ..., q$.
The number of clusters $q$ equals 10, and the cluster sizes $n_j$
are set to be 100, 40, 40, 30, 30, 30, 20, 20, 10, and 10,

respectively, to allow for unbalanced clusters.
The total sample size $n$ therefore equals 310.
The disturbances $(\eps_{i,j}, v_{i,j})$, cluster fixed effects $(a_{\eps,j}, a_{v,j})$,
IVs $Z_{i,j}$, and $\sigma(Z_{i,j})$ are specified as follows:

\begin{align*}
& (\eps_{i,j}, u_{i,j})^{\top} \sim N(0, I_2), \quad
v_{i,j} = \rho \eps_{i,j} + (1-\rho^2)^{1/2} u_{i,j}, \quad
(a_{\eps,j}, a_{u,j})^{\top} \sim N(0, I_2), \quad \notag \\
& a_{v,j} = \rho a_{\eps, j} + (1-\rho^2)^{1/2} a_{u,j}, \quad
Z_{i,j} \sim N(0, \Sigma_{d_z,j}), \quad \text{and} \quad \sigma(Z_{i,j}) = \left(\sum_{k=1}^{d_z} Z_{i,j,k}\right)^2,
\end{align*}

$\Sigma_{d_z,j} = c_j \cdot diag(1,...,d_z)$ for $j \in \{1,2,3,4,5,6\}$,
$diag(1,...,d_z)$ is a diagonal matrix with its diagonal elements being $1, ..., d_z$,
$c_j=2.5$ for $j=1$, $c_j=2$ for $j \in \{2,3\}$, and $c_j=1.5$ for $j \in \{4,5,6\}$, respectively,
$\Sigma_{d_z,j} = c_j \cdot I_{d_z}$ for $j \in \{7,8,9,10\}$,
$I_{d_z}$ is a $d_z \times d_z$ identity matrix,

$c_j=1$ for $j \in \{7,8\}$, and $c_j=0.5$ for $j \in \{9,10\}$, respectively.
$Z_{i,j,k}$ denotes the $k$-th element of $Z_{i,j}$,

and  $\rho \in \{0, 0.1, 0.2, \cdots, 0.9, 0.99\}$ corresponds to the degree of endogeneity.

We let $W_{i,j}$ be just all the cluster dummies.
In addition, for the first cluster ($j=1$), we let the $d_z \times 1$ first-stage coefficients
$\Pi_{z,1} = (\Pi_0, ..., \Pi_0)^{\top}$ for $\Pi_0 \in \{2, 4, 6\}$.
Then, for the rest clusters, we let

\begin{align}\label{eq: IV-strength}
& \Pi_{z,2} = \Pi_{z,3} = 0.4 \cdot \Pi_{z,1}, \;\; \Pi_{z,4} = \Pi_{z,5} = \Pi_{z,6} = 0.3 \cdot \Pi_{z,1}, \notag \\
& \Pi_{z,7} = \Pi_{z,8} = -0.2 \cdot \Pi_{z,1}, \;\;
\Pi_{z,9} = \Pi_{z,10} = -0.1 \cdot \Pi_{z,1},
\end{align}

Therefore, we treat the first cluster as the strong IV cluster and the rest clusters as relatively weak IV clusters.

Such a DGP satisfies our assumptions[Specifically, $\widetilde{Z}_{i,j} = Z_{i,j} - \frac{1}{n_j}\sum_{i \in I_{n,j}}Z_{i,j}$ and $\frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}\widetilde{Z}_{i,j}\sigma(Z_{i,j})(a_{\eps,j}+\eps_{i,j})$ is asymptotically normal conditional on $a_{\eps,j}$. This is because $\left(\frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}Z_{i,j}\left(\sum_{k=1}^{d_z}Z_{i,j,k}^2\right), \left[\frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}Z_{i,j}\right] \left[\frac{1}{n_j} \sum_{i \in I_{n,j}} \left(\sum_{k=1}^{d_z}Z_{i,j,k}^2\right)\right],  \frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}Z_{i,j}\left(\sum_{k=1}^{d_z}Z_{i,j,k}^2\right)\eps_{i,j}\right)$ are jointly asymptotically normal given $(a_{\eps,j})_{j \in J}$. Then, we have

\begin{align*}
\frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}\widetilde{Z}_{i,j}\sigma(Z_{i,j})(a_{\eps,j}+\eps_{i,j}) & = \frac{a_{\eps,j}}{\sqrt{n_j}}\sum_{i \in I_{n,j}}Z_{i,j}\left(\sum_{k=1}^{d_z}Z_{i,j,k}^2\right) + \frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}Z_{i,j}\left(\sum_{k=1}^{d_z}Z_{i,j,k}^2\right)\eps_{i,j} \\
& - \left[\frac{1}{\sqrt{n_j}}\sum_{i \in I_{n,j}}Z_{i,j}\right] \left[\frac{a_{\eps,j}}{n_j}\sum_{i \in I_{n,j}}  \left(\sum_{k=1}^{d_z}Z_{i,j,k}^2\right)\right]+ o_p(1),
\end{align*}

which is asymptotically normal given $(a_{\eps,j})_{j \in J}$.] and allows for heterogeneous cluster sizes, IV strengths, and joint distributions of $(Z_{i,j},W_{i,j})_{i \in I_{n,j}}$ across clusters.

The number of Monte Carlo and bootstrap replications equal 5,000 and 499, respectively. The nominal level $\alpha$ is set at 10%. The values of $\beta$ and $\gamma$ are set at $0$ and $1$, respectively. For $T_n$ and $AR_n$,
we set the weighting matrices as $\hat{A}_r =1$
and $\hat{A}_{z} = I_{d_z}$, respectively.

Null Rejection Probabilities of the Tests based on IV Estimators with 1 Strong IV Cluster

Note: “WB-US" (solid line with circle), “WB-S" (dashed line with star), “IM" (dotted line with upward-pointing triangle), and “CRS" (dash-dotted line with downward-pointing triangle) denote the $T_n$ and $T_{CR,n}$-based wild bootstrap tests, IM tests, and CRS tests, respectively.

Null Rejection Probabilities of the AR Tests with 1 Strong IV Cluster

Note: “WB-AR-US" (solid line with circle) and “WB-AR-S" (dashed line with star) denote the $AR_n$ and $AR_{CR,n}$-based wild bootstrap tests, while “ASY-AR-S" (dash-dotted line with downward-pointing triangle) denote the $AR_{CR,n}$-based asymptotic tests, respectively.

Figure <ref> reports the null empirical rejection frequencies of the tests that are based on the IV estimators, including the $T_n$ and $T_{CR,n}$-based wild bootstrap procedures,  the group-based $t$-tests of IM, and the randomization tests of CRS.

The results of the tests are based on TSLS when $d_z=1$,
and we further report the results based on TSLS, LIML, and FULL when $d_z=3$.
Following the recommendation in the literature, we set the tuning parameter of FULL to be 1.[In this case FULL is best unbiased to a second order among $k$-class estimators under normal errors [68].]

Several observations are in order.

First, in general size distortions increase when the IVs become
weak, the degree of endogeneity becomes high, or the number of IVs becomes large. Second, in
the empirically leading case with one IV, both $T_n$ and $T_{CR,n}$-based bootstrap Wald tests always have null rejection frequencies very close to the nominal level, while IM and CRS tests can have considerable over-rejections when $\Pi_0$ is small and $\rho$ is large. Third, in the case with three IVs, IM and CRS tests with LIML or FULL show a substantial size improvement over their counterparts with TSLS, as these tests are based on cluster-level estimates (i.e., we run the cluster-by-cluster IV regressions), which could produce serious finite sample bias if TSLS is employed for the over-identified cases. Similarly, the LIML or FULL-based bootstrap tests, no matter studentized or
unstudentized, show a size improvement over their TSLS-based counterparts. Fourth, overall both studentized and unstudentized wild bootstrap procedures compare favorably with the alternatives, and the unstudentized bootstrap Wald tests ($T_n$) have the smallest size distortions across different settings of IV strength, degree of endogeneity, and number of IVs.
We report size results for the cases with more strong IV clusters and/or larger $q$ in Section <ref> in the Online Supplement, and the overall pattern remains quite similar to that found here.

Figure <ref> reports the null rejection frequencies of AR tests,
including the $AR_{CR,n}$-based asymptotic tests,

which reject the null when the square of the corresponding test statistic exceeds $\chi^2_{d_z,1-\alpha}$, the $1-\alpha$ quantile of the chi-squared distribution with $d_z$ degrees of freedom.
Additionally, it reports the rejection frequencies of the wild bootstrap AR tests in Section <ref> that are based on $AR_n$ and $AR_{CR,n}$,
We notice from Figure <ref> that $AR_{CR,n}$-based asymptotic tests have rejection probabilities very close to zero in the over-identified cases ($d_z=3$).[The null rejection probabilities of the $AR_{CR,n}$-based asymptotic test decrease toward zero when $d_z$ approaches $q$. When $d_z$ is equal to $q$, the value of $AR_{CR,n}$ will be exactly equal to $d_z$ (or $q$), and thus has no variation (for $\bar{f} = \left(\bar{f}_{1}, ..., \bar{f}_{q} \right)^{\top}$
and $\bar{f}_{j} = n^{-1} \sum_{i \in I_{n,j}} f_{i,j},$
AR_{CR,n} =
\iota_q^{\top} \bar{f} \left( \bar{f}^{\top} \bar{f} \right)^{-1}
\bar{f}^{\top} \iota_q = \iota_q^{\top} \iota_q = d_z
as long as $\bar{f}$ is invertible, where $\iota_q$ denotes a $q$-dimensional vector of ones). By contrast, the $AR_{n}$-based bootstrap test works well even when $d_z$ is  larger than $q$.]

By contrast, both bootstrap AR tests always have rejection frequencies very close to 10%.

Power of Wild Bootstrap Wald and AR Tests for $d_z=2$ with 1 Strong IV Cluster

“WB-FULL-US" (dashed line with upward-pointing triangle), and “WB-FULL-S" (dashed line with downward-pointing triangle) denote the $T_n$ and $T_{CR,n}$-based wild bootstrap tests with FULL, respectively.
“ASY-AR-S" (dotted line with cross) denotes the $AR_{CR,n}$-based AR tests with asymptotic CVs,
while “WB-AR-US" (dash-dotted line with plus) and “WB-AR-S" (dash-dotted line with square) denote the $AR_n$ and $AR_{CR,n}$-based wild bootstrap AR tests, respectively.

Figures <ref>–<ref> show the power properties of the wild bootstrap tests with one, three, and six strong IV clusters, respectively. Specifically, for the case with three strong IV clusters, we modify (<ref>) with $\Pi_{z,2} = \Pi_{z,3} = \Pi_{z,1}$, keeping other settings unchanged. For the case with six strong IV clusters, we further set
$\Pi_{z,4} = \Pi_{z,5} = \Pi_{z,6} = \Pi_{z,1}$.
For the Wald tests, we focus on the FULL estimator as it has better size control than the TSLS estimator in the over-identified case.[For the ease of graphical comparison, we omit the result for the Wald tests with LIML as they have patterns very similar to FULL, but those with FULL have slightly higher power.]
According to Figure <ref>, the $AR_{CR,n}$-based asymptotic test also controls size. We therefore include the test to compare its power with those of the bootstrap AR tests.
We let the number of IVs be 2, $\Pi_0 \in \{2, 4, 6\}$, and $\rho \in \{0.2, 0.4, 0.6\}$.
Some observations are in order.
First, all the tests become more powerful when the number of strong IV clusters increases.
Second, in line with our theory in Section <ref>, when the alternative is sufficiently distant, the $T_{CR,n}$-based bootstrap tests (the bootstrap Wald tests studentized by CCE) are more powerful than the other tests, including their $T_n$-based counterparts (the unstudentized bootstrap Wald tests), especially when the value of $\Pi_0$ and/or the number of strong clusters are small.[We notice that in some cases such as $\Pi_0 =2, \rho=0.6,$ and one strong IV cluster, the minimum of the power curve for the bootstrap Wald test with CCE occurs to the left of zero.
Similar pattern is observed in <cit.> and <cit.>.
This may be due to the combined effect of relatively weak IVs and a relatively high degree of endogeneity.]
Third, we notice that among the AR tests, the $AR_n$-based bootstrap tests (the bootstrap AR tests without CCE) have the highest power, followed by the
$AR_{CR,n}$-based bootstrap tests.
The $AR_{CR,n}$-based asymptotic tests have the lowest power among the AR tests, which is in line with the under-rejections found for the over-identified cases in Figure <ref>.

Fourth, the unstudentized bootstrap tests (the $T_n$ and $AR_n$-based tests) have quite similar behaviours. The $T_n$-based tests seem to have higher power when $\Pi_{0}$ is small and/or the number of strong clusters is large but the $AR_n$-based tests perform better in some other cases.
In Section <ref> in the Online Supplement, we report further power results, which have very similar pattern.

Power of Wild Bootstrap Wald and AR Tests for $d_z=2$ with 3 Strong IV Clusters

“WB-FULL-US" (dashed line with upward-pointing triangle), and “WB-FULL-S" (dashed line with downward-pointing triangle) denote the $T_n$ and $T_{CR,n}$-based wild bootstrap tests with FULL, respectively.
“ASY-AR-S" (dotted line with cross) denotes the $AR_{CR,n}$-based AR tests with asymptotic CVs,
while “WB-AR-US" (dash-dotted line with plus) and “WB-AR-S" (dash-dotted line with square) denote the $AR_n$ and $AR_{CR,n}$-based wild bootstrap AR tests, respectively.

Power of Wild Bootstrap Wald and AR Tests for $d_z=2$ with 6 Strong IV Clusters

“WB-FULL-US" (dashed line with upward-pointing triangle), and “WB-FULL-S" (dashed line with downward-pointing triangle) denote the $T_n$ and $T_{CR,n}$-based wild bootstrap tests with FULL, respectively.
“ASY-AR-S" (dotted line with cross) denotes the $AR_{CR,n}$-based AR tests with asymptotic CVs,
while “WB-AR-US" (dash-dotted line with plus) and “WB-AR-S" (dash-dotted line with square) denote the $AR_n$ and $AR_{CR,n}$-based wild bootstrap AR tests, respectively.

§ EMPIRICAL APPLICATION

In an influential study, [15] analyze the effect of rising Chinese import competition on US local labor markets between 1990 and 2007, when the share of total US spending on Chinese goods increased substantially from 0.6%  to 4.6%. The dataset of [15] includes 722 commuting zones (CZs) that cover the entire mainland US.
In this section, we further analyze the region-wise effects of such import exposure by applying IV regression with the proposed wild bootstrap procedures to three Census Bureau-designated regions: South, Midwest and West, with 16, 12, and 11 states, respectively, in each region.[The Northeast region is not included in the study because of the relatively small number of states ($9$) and small number of CZs in each state (e.g., Connecticut and Rhode Island have only 2 CZs).]

Following [15], we consider the linear IV model

\begin{align}
y_{i,j} = X_{i,j}\beta + W_{i,j}^{\top}\gamma + \eps_{i,j},
\end{align}

where the outcome variable $y_{i,j}$ denote the decadal change in average individual log weekly wage in a given CZ and the endogenous variable $X_{i,j}$ is the change in Chinese import exposure per worker in a CZ, where imports are apportioned to the CZ according to its share of national industry employment. The Bartik-type instrument $Z_{i,j}$
(e.g., see [35])
proposed by [15]
is Chinese import growth in other high-income countries, where imports are apportioned to the CZs in the same way as that for $X_{i.j}$.[See Sections I.B and III.A in [15] for a detailed definition of these variables.]

As can be seen from Figure <ref>, the predictive power of the IV varies considerably among states.
The exogenous variables $W_{i,j}$ include the characteristic variables of CZs and decade specified in [15] as well as state dummies.
Our regressions are based on the CZ samples in each region, and the samples are clustered at the state level, following [15].
Besides the results for the full sample, we also report those for female and male samples separately. Although we do not use the set of full interactions between the baseline covariates and cluster dummies as $W_i$, Assumption <ref>(i)(2) is still plausible in this application. Specifically, Tables <ref>–<ref> in the Online Supplement show the values of $\widehat{Q}_{\tilde{Z}W,j}$ for each cluster in the three regions, and all of them are close to zero.

The main result of the IV regression for the three regions is given in Table <ref>,
with the number of observations ($n$) and clusters ($q$) for each region.
As we have only one endogenous variable and one IV, the TSLS estimator is used throughout this section.

We further construct the 90% bootstrap confidence sets (CSs) by inverting the corresponding $T_n$,
$T_{CR,n}$, and $AR_n$-based wild bootstrap tests with a 10% nominal level.
The computation of the CSs is conducted over the parameter space $[-10,10]$ with a step size of 0.01, and the number of bootstrap draws is set at 2,000 for each step.

We highlight the main findings below. First, the results in Table <ref> suggest that there exists regional heterogeneity in terms of the average effect of Chinese imports on wages in local labor markets. For instance, the TSLS estimates for the South and West regions equal $-0.97$ and $-1.05$, respectively, while that for the Midwest region equals $-0.025$. That is, a $$1,000$ per worker increase in a CZ's exposure to Chinese imports is estimated to reduce average weekly earnings by $0.97$, $1.05$, and $0.025$ log points, respectively, for the three regions (the corresponding TSLS estimate in [15] for the entire mainland US is $-0.76$). Second, only the effect on CZs in the South is significantly different from zero at the 10% level
under all three bootstrap CSs,
while the effects on CZs in the other two regions are not.[Figure <ref> also suggests that the identification is strongest for the South region.]
Third, we notice that the effect on West is only significant under the studentized bootstrap Wald CS (the CS is $[-1.22, -0.47]$).

Fourth, compared with that for the South, the wider CSs for the Midwest and West may be due to relatively weak identification, which our bootstrap AR procedure is able to guard against.
Table <ref> also reports the results for female and male samples.
We find that across all the regions, the effects are more substantial for the male samples.
Furthermore, the effects for both female and male
samples in the South are significantly different from zero.
Finally, the studentized Wald CSs have the shortest length among the three types of CSs in most cases in Table <ref>, which is in line with our power results in Section <ref>.

max width=

Gender     Region      $n$       $q$      Estimate     Unstud Wald CS              Stud Wald CS                Unstud AR CS

All     South       578     16     -0.97        [-1.71, -0.58]     [-1.61, -0.43]     [-1.70, -0.58]

Midwest     504     12     -0.025       [-0.69, 0.83]      [-0.60, 0.75]      [-0.69, 0.83]

West        276     11     -1.05        [-1.50, 0.25]      [-1.22, -0.47]     [-1.50, 0.24]

Female     South       578     16     -0.81        [-1.48, -0.41]     [-1.40, -0.26]     [-1.48, -0.41]

Midwest     504     12     0.024        [-0.64, 0.74]      [-0.56, 0.67]      [-0.64, 0.74]

West        276     11     -0.61        [-1.46, 0.74]      [-0.91, 0.30]      [-1.46, 0.75]

Male       South       578     16     -1.08        [-1.90, -0.65]     [-1.78, -0.53]     [-1.91, -0.65]

Midwest     504     12     -0.17        [-1.02, 0.80]      [-0.86, 0.77]      [-1.01, 0.82]

West        276     11     -1.26        [-1.99, 0.68]      [-1.58, 0.42]      [-2.00, 0.69]

IV regressions of [15] with all, female, and male samples for three US regions

§ CONCLUSION AND PRACTICAL RECOMMENDATIONS

In many empirical applications of IV regressions with clustered data, the IVs may be weak for some clusters and the number of clusters may be small.
In this paper, we provide valid inference methods in this context.

For the Wald tests with and without CCE,
we extend the WREC procedure to allow for the setting of few clusters and cluster-level heterogeneity in IV strength.
For the full-vector inference, we develop wild bootstrap AR tests that control size asymptotically irrespective of IV strength, and also show that the bootstrap validity of other weak-IV-robust tests requires at least one strong IV cluster.
Furthermore, we establish the power properties of the Wald and AR tests.

Our results have several important implications for applied works.
First, for inference based on Wald statistics,

we recommend the bootstrap Wald test studentized by CCE because of its superior power properties.
Second, for the over-identified case, instead of using the bootstrap Wald tests with TSLS,
we recommend to use those based on estimators with reduced finite sample bias such as Fuller's modified LIML estimator. Third, for the full-vector inference, if researchers are concerned that all clusters are weak and thus would like to implement a weak-identification-robust procedure, then we recommend the bootstrap AR test without CCE because of its power advantage over alternative asymptotic and bootstrap AR tests that are studentized by CCE.

Finally, recent studies by [41] and [56] suggest that asymptotic or bootstrap inference based on jackknife variance matrix estimators can be more reliable than that based on the conventional CCE for OLS models. Therefore, it may be interesting to consider extending the jackknife inference to the setting with a fixed number of clusters. We leave this line of investigation for future research.

§ PROOF OF THEOREM <REF>

For the power analysis, we focus on the TSLS estimator. The results for other $k$-class estimators can be derived in the same manner given Lemma <ref>. Recall $a_j$ defined in Assumption <ref>. We further define

\begin{align}\label{eq: thm-2-2-T_CR_g}
T_{CR,\infty}(g) = \left\Vert \lambda_\beta^\top \tilde{Q} \left[\sum_{j \in J} g_j \sqrt{\xi}_j \mathcal{Z}_j \right]+ c_{0,g} \mu \right\Vert_{A_{r,CR,g}},
\end{align}

where $\tilde{Q} = Q^{-1} Q_{\widetilde{Z}X}^{\top} Q_{\widetilde{Z}\widetilde{Z}}^{-1}$, $Q = Q_{\widetilde{Z}X}^\top Q_{\widetilde{Z}\widetilde{Z}}Q_{\widetilde{Z}X}$,
$c_{0,g} =  \sum_{j \in J} \xi_j g_j a_j$, and

\begin{align*}
A_{r,CR,g}  = \begin{pmatrix}
\sum_{j \in J} \xi_j\left\{\lambda_\beta^\top \tilde{Q} \left[g_j\mathcal{Z}_j - a_j\sqrt{\xi}_j \sum_{\tilde{j} \in J}\sqrt{\xi}_{\tilde{j}} g_{\tilde{j}}\mathcal{Z}_{\tilde{j}}\right]+\sqrt{\xi}_j (g_j - c_{0,g} )a_j \mu \right\} \\
\times \left\{\lambda_\beta^\top \tilde{Q} \left[g_j\mathcal{Z}_j - a_j \sqrt{\xi}_j \sum_{\tilde{j} \in J}\sqrt{\xi}_{\tilde{j}} g_{\tilde{j}}\mathcal{Z}_{\tilde{j}} \right]+\sqrt{\xi}_j (g_j - c_{0,g})a_j \mu \right\}^\top
\end{pmatrix}^{-1}.
\end{align*}

We order $\{T_{CR,\infty}(g)\}_{g \in \textbf{G}}$ in ascending order:
$(T_{CR,\infty})^{(1)} \leq \cdots \leq (T_{CR,\infty})^{|\textbf{G}|}$. In the proof of Theorem <ref>, we have already shown that, under $\mathcal{H}_{1,n}$,

\begin{align*}
\sqrt{n}(\lambda_\beta^\top \hat{\beta}_{tsls} - \lambda_0) \xrightarrow{\enskip d \enskip} \sum_{j \in J} \left[\sqrt{\xi_j} \lambda_\beta^\top \tilde{Q} \mathcal{Z}_j \right] + \mu, \quad
\sqrt{n}(\lambda_\beta^\top \hat{\beta}^*_{tsls,g} - \lambda_0) \xrightarrow{\enskip d \enskip} \sum_{j \in J} g_j \left[\sqrt{\xi_j} \lambda_\beta^\top \tilde{Q} \mathcal{Z}_j \right] + c_{0,g}\mu.
\end{align*}

Next, we derive the limit of $\hat{A}_{r,CR}$ and $\hat{A}^*_{r,CR,g}$. We first note that

\begin{align*}
\frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\hat{\eps}_{i,j} & = \frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\left[\eps_{i,j} -X_{i,j}^\top(\hat{\beta}_{tsls} - \beta_n) - W_{i,j}^\top(\hat{\gamma}_{tsls} -\gamma)\right]\\
& = \frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} - \xi_j\widehat{Q}_{\widetilde{Z}X,j} \sqrt{n}(\hat{\beta}_{tsls} - \beta_n) + o_p(1) \\
& = \frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} - \xi_j\widehat{Q}_{\widetilde{Z}X,j} \widehat{Q}^{-1}\widehat{Q}_{\widetilde{Z}X}\widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}\sum_{\tilde{j} \in J} \frac{1}{\sqrt{n}} \sum_{i \in I_{n,\tilde{j}}} \widetilde{Z}_{i,\tilde{j}}\eps_{i,\tilde{j}} + o_p(1) \\
& \xrightarrow{\enskip d \enskip} \sqrt{\xi_j}\mathcal{Z}_j -  \xi_jQ_{\widetilde{Z}X,j}Q^{-1}Q_{\widetilde{Z}X}Q_{\widetilde{Z}\widetilde{Z}}^{-1} \sum_{\tilde{j} \in J} \sqrt{\xi_{\tilde{j}}}\mathcal{Z}_{\tilde{j}} + o_p(1),
\end{align*}

where the first equality holds by Lemma <ref>. This implies

\begin{align*}
& \hat{\Omega}_{CR} \convD \sum_{j \in J} \xi_j \left(\mathcal{Z}_j -  \sqrt{\xi_j}Q_{\widetilde{Z}X,j}\tilde{Q} \sum_{\tilde{j} \in J} \sqrt{\xi_{\tilde{j}}}\mathcal{Z}_{\tilde{j}}\right)\left(\mathcal{Z}_j -  \sqrt{\xi_j}Q_{\widetilde{Z}X,j}\tilde{Q} \sum_{\tilde{j} \in J} \sqrt{\xi_{\tilde{j}}}\mathcal{Z}_{\tilde{j}}\right)^\top, \\
& \hat{A}_{r,CR}     \convD A_{r,CR,\iota}, \quad \text{and} \quad T_{CR,n} \convD T_{CR,\infty}(\iota),
\end{align*}

where $\iota$ is a $q \times 1$ vector of ones. Similarly, we have

\begin{align}
& \frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\hat{\eps}^*_{i,j}(g) \notag \\
& = \frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\left[g_j\hat{\eps}_{i,j}^r - X_{i,j}^{*\top}(g)(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls}) - W_{i,j}^\top(\hat{\gamma}_{tsls,g}^* - \hat{\gamma}^r_{tsls})\right] \notag \\
& = \frac{g_j}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\hat{\eps}_{i,j}^r - \xi_j\widehat{Q}_{\widetilde{Z}X,j}^*(g) \sqrt{n}(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls}) + o_p(1) \notag \\
& = \frac{g_j}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} - \xi_j g_j\widehat{Q}_{\widetilde{Z}X,j}\sqrt{n}(\hat{\beta}^r_{tsls} - \beta_n)-
\xi_j\widehat{Q}_{\widetilde{Z}X,j} \sqrt{n}(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls}) + o_p(1) \notag \\
& = \frac{g_j}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} - \xi_j g_j Q_{\widetilde{Z}X,j}\sqrt{n}(\hat{\beta}^r_{tsls} - \beta_n)-
\xi_j Q_{\widetilde{Z}X,j} \sqrt{n}(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls}) + o_p(1),
\label{eq:Zeps^star}
\end{align}

where $\widehat{Q}_{\widetilde{Z}X,j}^*(g) = \frac{1}{n_j}\sum_{i \in I_{n,j}}\widetilde{Z}_{i,j}X_{i,j}^{*\top}(g)$, the second equality is by Lemma <ref>, and the last equality holds because $\widehat{Q}_{\widetilde{Z}X,j} = Q_{\widetilde{Z}X,j} + o_p(1)$,  $\widehat{Q}_{\widetilde{Z}X,j}^*(g) = Q_{\widetilde{Z}X,j} + o_p(1)$ proved in Step 1 of the proof of Theorem <ref>, $\sqrt{n}(\hat{\beta}^r_{tsls}-\beta_n) = O_p(1)$, and $\sqrt{n}(\hat{\beta}^*_{tsls,g} - \beta_n) = O_p(1)$. In addition, following the same arguments that lead to (<ref>), we have

\begin{align}
\sqrt{n}(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls}) & = \tilde{Q} \sum_{\tilde{j} \in J} \sum_{i \in I_{n,\tilde{j}}}  \frac{g_{\tilde{j}} \widetilde{Z}_{i,\tilde{j}}\eps_{i,\tilde{j}}}{\sqrt{n}} + \tilde{Q}\sum_{\tilde{j} \in J} \xi_{\tilde{j}} g_{\tilde{j}} Q_{\widetilde{Z}X,\tilde{j}}\sqrt{n}(\beta_n - \hat{\beta}^r_{tsls})  + o_p(1) \notag \\
& = \tilde{Q} \sum_{\tilde{j} \in J} \sum_{i \in I_{n,\tilde{j}}}  \frac{g_{\tilde{j}} \widetilde{Z}_{i,\tilde{j}} \eps_{i,\tilde{j}}}{\sqrt{n}} +\sum_{\tilde{j} \in J} \xi_{\tilde{j}}g_{\tilde{j}}a_{\tilde{j}}\sqrt{n}(\beta_n - \hat{\beta}^r_{tsls})  + o_p(1).
\label{eq:b^*-b^r}
\end{align}

Note $\widehat{Q}^{*-1}_g\widehat{Q}_{\widetilde{Z}X}^{*\top}(g)\widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}  \convP     \tilde{Q}$. Therefore, combining (<ref>) and (<ref>), we have

\begin{align*}
& \lambda_\beta^\top \widehat{Q}_g^{*-1}\widehat{Q}_{\widetilde{Z}X}^{*\top}(g)\widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1} \left[\frac{1}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\hat{\eps}^*_{i,j}(g)\right] \\
& = \lambda_\beta^\top \tilde{Q}\left[\frac{g_j}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} - \xi_j g_j Q_{\widetilde{Z}X,j}\sqrt{n}(\hat{\beta}^r_{tsls} - \beta_n)-
\xi_j Q_{\widetilde{Z}X,j} \sqrt{n}(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls}) \right] + o_p(1) \\
& = \lambda_\beta^\top \tilde{Q}\left[\frac{g_j}{\sqrt{n}}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j}\right] - \xi_j g_j a_j \lambda_\beta^\top \sqrt{n}(\hat{\beta}^r_{tsls} - \beta_n)-
\xi_j a_j \lambda_\beta^\top \sqrt{n}(\hat{\beta}_{tsls,g}^* - \hat{\beta}^r_{tsls})  + o_p(1)\\
& \convD \lambda_\beta^\top \tilde{Q} \sqrt{\xi_j} g_j \mathcal{Z}_j - \xi_j a_j \lambda_\beta^\top \tilde{Q} \sum_{\tilde{j} \in J} \sqrt{\xi_{\tilde{j}}} g_{\tilde{j}} \mathcal{Z}_{\tilde{j}} + \xi_j (g_j - c_{0,g})a_j \mu,
\end{align*}

where the second equality is by the fact that $\tilde{Q}Q_{\widetilde{Z}X,j} = a_j I_{d_x}$ and the last convergence is by the fact that $\lambda_\beta^\top\sqrt{n}(\hat{\beta}^r_{tsls} - \beta_n) = -\mu$. This implies

\begin{align*}
& \hat{A}_{r,CR,g}^* \convD A_{r,CR,g},\quad \text{and thus,} \quad  (T_{CR,n},\{T_{CR,n}^*(g)\}_{g \in \textbf{G}}) \convD (T_{CR,\infty}(\iota),\{T_{CR,\infty}(g)\}_{g \in \textbf{G}}).
\end{align*}

By the Portmanteau theorem, we have

\begin{align*}
\liminf_{n \rightarrow \infty}\mathbb{P} \{ T_{CR,n} > \hat{c}_{CR,n}(1-\alpha) \} & \geq  \mathbb{P}\left\{ T_{CR,\infty} (\iota) > (T_{CR,\infty})^{(k^*)} \right\}.
\end{align*}

We aim to show that, as $||\mu ||_2\rightarrow \infty$, we have

\begin{align}
\mathbb{P}\left\{ T_{CR,\infty} (\iota) > \max_{g \in \textbf{G}_s}T_{CR,\infty} (g)\right\} \rightarrow 1,
\label{eq:GsCR-liniv}
\end{align}

where $\textbf{G}_s = \textbf{G} \backslash \textbf{G}_w$, and $\textbf{G}_w = \{g \in \textbf{G}: g_{j} = g_{j'}, \forall j,j' \in J_s\}$.
Then, given $|\textbf{G}_s| = |\textbf{G}|-2^{q-q_s+1}$ and $k^* = \lceil |\textbf{G}|(1-\alpha) \rceil \leq |\textbf{G}|-2^{q-q_s+1}$, (<ref>) implies that   as $ ||\mu||_2 \rightarrow \infty$,

\begin{align*}
\mathbb{P}\left\{ T_{CR,\infty}(\iota) > (T_{CR,\infty})^{(k^*)}  \right\} \geq \mathbb{P}\left\{ T_{CR,\infty}(\iota) > \max_{g \in \textbf{G}_s}T_{CR,\infty}(g) \right\} \rightarrow 1.
\end{align*}

Therefore, it suffices to establish (<ref>).

By (<ref>), we see that

$$T_{CR,\infty}(\iota) = \left\Vert \lambda_\beta^\top \tilde{Q} \left[\sum_{j \in J} g_j \sqrt{\xi}_j \mathcal{Z}_j \right]+ \mu \right\Vert_{A_{r,CR,\iota}},$$

and $A_{r,CR,\iota }$ is independent of $\mu$ as $c_{0, \iota}=1$. In addition, we have
$\lambda_{\min}(\tilde{Q}^\top \lambda_\beta^\top A_{r,CR,\iota} \lambda_\beta \tilde{Q})>0$ with probability one.
Therefore, for any $e>0$, we can find a sufficiently small constant $c>0$ and a sufficiently large constant $M>0$ such that with probability greater than $1-e$, for any $\mu$,

\begin{align}
T_{CR,\infty}(\iota) \geq  c ||\mu||^2_2 - M.
\label{eq:IV-Tiota}
\end{align}

On the other hand, for $g \in \textbf{G}_s$,

we can write  $T_{CR,\infty}(g)$ as

\begin{align*}
%\label{eq: power-wald-stud-1}
T_{CR,\infty}(g)& = \biggl\{(N_{0,g} + c_{0,g} \mu )^\top \left[\sum_{j \in J}\xi_j (N_{j,g} + c_{j,g} \mu )(N_{j,g} + c_{j,g} \mu )^\top\right]^{-1} (N_{0,g} + c_{0,g} \mu )\biggr\},
\end{align*}

where for $j \in J$,
$N_{0,g} = \lambda_\beta^\top \tilde{Q} \left[\sum_{j \in J} g_j \sqrt{\xi}_j \mathcal{Z}_j \right]$,
$N_{j,g} =  \lambda_\beta^\top \tilde{Q} \left[g_j\mathcal{Z}_j - a_j \sqrt{\xi}_j \sum_{\tilde{j} \in J}\sqrt{\xi}_{\tilde{j}} g_{\tilde{j}}\mathcal{Z}_{\tilde{j}} \right]$,
and $c_{j,g} =  \sqrt{\xi}_j (g_j - c_{0,g})a_j$.

We claim that for $g \in \textbf{G}_s$, $c_{j,g} \neq 0$ for some $j \in J_s$. Suppose it does not hold, then it implies that $g_j = c_{0,g}$ for all $j \in J_s$, i.e.,
for all $j \in J_s$, $g_j$ shares the same sign, and thus, contradicts the definition of $\textbf{G}_s$. Therefore, combining the claim with the assumption that $\min_{j \in J_s}|a_j| > 0$, we have

$\min_{g \in \textbf{G}_s} \sum_{j \in J} \xi_j c_{j,g}^2 >0.$

In addition, we note that

\begin{eqnarray*}
&& \sum_{j \in J}\xi_j (N_{j,g} + c_{j,g} \mu)(N_{j,g} + c_{j,g} \mu )^\top  \\
& =&  \sum_{j \in J}\xi_j N_{j,g} N_{j,g}^\top + \sum_{j \in J} \xi_j c_{j,g} N_{j,g} \mu^\top
+ \sum_{j \in J} \xi_j c_{j,g} \mu N_{j,g}^\top + (\sum_{j \in J} \xi_j c_{j,g}^2) \mu \mu^\top \\
&\equiv&  M_1 + M_2 \mu^\top +  \mu M_2^\top + \overline{c}^2 \mu \mu^\top,
\end{eqnarray*}

where we denote
$M_1 = \sum_{j \in J}\xi_j N_{j,g} N_{j,g}^\top$,
$M_2 = \sum_{j \in J} \xi_j c_{j,g} N_{j,g}$,  and $\overline{c}^2 = \sum_{j \in J} \xi_j c_{j,g}^2$.
For notation ease, we suppress the dependence of $(M_1,M_2,\overline{c})$ on $g$.
Then, we have

\begin{align*}
M_1 + M_2 \mu^\top +  \mu M_2^\top + \overline{c}^2 \mu\mu^\top = M_1 - \frac{M_2 M_2^\top}{\overline{c}^2} + \left(\frac{M_2}{\overline{c}} + \overline{c} \mu \right)\left(\frac{M_2}{\overline{c}} + \overline{c}\mu \right)^\top.
\end{align*}

Note for any $d_r \times 1$ vector $u$, by the Cauchy–Schwarz inequality,

\begin{align*}
u^\top \left( M_1 - \frac{M_2 M_2^\top}{\overline{c}^2}\right)   u = \sum_{j \in J} \xi_j (u^\top N_{j,g})^2 - \frac{\left(\sum_{j \in J}\xi_j u^\top N_{j,g} c_{j,g}\right)^2}{\sum_{j \in J} \xi_j c_{j,g}^2} \geq 0,
\end{align*}

where the equal sign holds if and only if there exist $(u,g) \in \Re^{d_r} \times \textbf{G}_s$ such that

\begin{align*}
\frac{u^\top N_{1,g}}{c_{1,g}} = \cdots = \frac{u^\top N_{q,g}}{c_{q,g}},
\end{align*}

which has probability zero if $q > d_r$ as $\{N_{j,g}\}_{j \in J}$ are independent and non-degenerate normal vectors.  Therefore, the matrix $\mathbb{M} \equiv M_1 - \frac{M_2 M_2^\top}{\overline{c}^2}$ is invertible with probability one. Specifically, denote $\mathbb{M}$ as $\mathbb{M}(g)$ to highlight its dependence on $g$.
We have $\max_{g \in \textbf{G}_s }(\lambda_{\min}(\mathbb{M}(g)))^{-1} = O_p(1)$.
In addition, denote $\frac{M_2}{\overline{c}} + \overline{c} \mu $ as $\mathbb{V}$, which is a $d_r \times 1$ vector. Then,  we have

\begin{align*}
& \left[\sum_{j \in J}\xi_j (N_{j,g} + c_{j,g} \mu )(N_{j,g} + c_{j,g} \mu )^\top\right]^{-1}
= [\mathbb{M} + \mathbb{V}\mathbb{V}^\top]^{-1}
= \mathbb{M}^{-1} - \mathbb{M}^{-1} \mathbb{V} (1+ \mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V})^{-1} \mathbb{V}^\top \mathbb{M}^{-1},
\end{align*}

where the second equality is due to the Sherman–Morrison–Woodbury formula.

Next, we note that

\begin{align*}
N_{0,g} + c_{0,g} \mu  = N_{0,g} + c_{0,g} \left(\frac{\mathbb{V}}{\overline{c}} - \frac{M_2}{\overline{c}^2}\right)
\equiv \mathbb{M}_0 + \frac{c_{0,g}}{\overline{c}}\mathbb{V},
\end{align*}

\mathbb{M}_0 = N_{0,g} - \frac{c_{0,g} M_2}{\overline{c}^2} = N_{0,g} - \frac{c_{0,g} (\sum_{j \in J}\xi_j c_{j,g} N_{j,g})}{\sum_{j \in J} \xi_j c_{j,g}^2}.$
With these notations, we have

\begin{align*}
& (N_{0,g} + c_{0,g} \mu )^\top \left[\sum_{j \in J}\xi_j (N_{j,g} + c_{j,g} \mu )(N_{j,g} + c_{j,g} \mu )^\top\right]^{-1}
(N_{0,g} + c_{0,g} \mu ) \\
&  = \left( \mathbb{M}_0 + \frac{c_{0,g}}{\overline{c}}\mathbb{V} \right)^\top(\mathbb{M}^{-1} - \mathbb{M}^{-1} \mathbb{V} (1+ \mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V})^{-1} \mathbb{V}^\top \mathbb{M}^{-1})\left( \mathbb{M}_0 + \frac{c_{0,g}}{\overline{c}}\mathbb{V} \right) \\
& \leq 2\mathbb{M}_0^\top (\mathbb{M}^{-1} - \mathbb{M}^{-1} \mathbb{V} (1+ \mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V})^{-1} \mathbb{V}^\top \mathbb{M}^{-1})\mathbb{M}_0 \\
& +  \frac{2c_{0,g}^2}{\overline{c}^2} \mathbb{V}^\top (\mathbb{M}^{-1} - \mathbb{M}^{-1} \mathbb{V} (1+ \mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V})^{-1} \mathbb{V}^\top \mathbb{M}^{-1})\mathbb{V}\\
& \leq  2\mathbb{M}_0^\top \mathbb{M}^{-1} \mathbb{M}_0 + \frac{2c_{0,g}^2}{\overline{c}^2} \frac{ \mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V}}{1+\mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V}}
\leq 2\mathbb{M}_0^\top \mathbb{M}^{-1} \mathbb{M}_0 + \frac{2c_{0,g}^2}{\overline{c}^2} \\
& \leq 2 (\lambda_{\min}(\mathbb{M}))^{-1}\left\Vert N_{0,g} - \frac{c_{0,g} (\sum_{j \in J}\xi_j c_{j,q}N_{j,g})}{\sum_{j \in J} \xi_j c_{j,g}^2}\right\Vert^2  + \frac{2c_{0,g}^2}{\sum_{j \in J} \xi_j c_{j,g}^2}
\equiv C(g),
\end{align*}

where the first inequality holds because $(u+v)^\top A (u+v) \leq 2 (u^\top A u + v^\top A v)$ for some $d_r\times d_r$ positive semidefinite matrix $A$ and $u,v \in \Re^{d_r}$, the second inequality holds due to the fact that $\mathbb{M}^{-1} \mathbb{V} (1+ \mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V})^{-1} \mathbb{V}^\top \mathbb{M}^{-1}$ is positive semidefinite, the third inequality holds because $\mathbb{V}^\top \mathbb{M}^{-1}\mathbb{V}$ is a  nonnegative scalar, and the last holds by substituting in the expressions for $\mathbb{M}_0$ and $\overline{c}$.

Then, we have

\begin{align}
\max_{g \in \textbf{G}_s} T_{CR,\infty}(g)  \leq   \max_{g \in \textbf{G}_s} C(g).
\label{eq:IV-Tiota*}
\end{align}

Combining (<ref>) and (<ref>), we have, as $||\mu||_2 \rightarrow \infty$,

\begin{align*}
& \liminf_{n \rightarrow \infty}\mathbb{P} \{ T_{CR,n} > \hat{c}_{CR,n}(1-\alpha) \}
\geq  \mathbb{P}\left\{ T_{CR,\infty}(\iota_q) > (T_{CR,\infty})^{(k^*)}  \right\}
\geq \mathbb{P}\left\{ T_{CR,\infty}(\iota_q) > \max_{g \in \textbf{G}_s} T_{CR,\infty}(g)  \right\} \\
&  = 1- \mathbb{P}\left\{ T_{CR,\infty}(\iota_q) \leq \max_{g \in \textbf{G}_s} T_{CR,\infty}(g)  \right\}
\geq 1- \mathbb{P}\left\{ c||\mu||^2_2 - M \leq \max_{g \in \textbf{G}_s} C(g) \right\}-e \rightarrow 1-e,
\end{align*}

where the second inequality is by the fact that $k^* \leq |\textbf{G}_s|$, and thus, $(T_{CR,\infty})^{(k^*)} \leq \max_{g \in \textbf{G}_s} T_{CR,\infty}(g)$ and the last convergence holds because $\max_{g \in \textbf{G}_s} C(g) = O_p(1)$ and does not depend on $\mu$. As $e$ is arbitrary, we can let $e \rightarrow 0$ and obtain the desired result.

For the proof of part (ii), let $\tilde{c}_{CR,n}(1-\alpha)$ denote the $(1-\alpha)$ quantile

\begin{align*}
\left\{|\sqrt{n}(\lambda_{\beta}^{\top}\hat{\beta}^*_{tsls,g} - \lambda_0)|/\sqrt{\lambda_{\beta}^{\top}\widehat{V}\lambda_{\beta}}: g \in \textbf{G}\right\},
\end{align*}

i.e., the bootstrap statistic $T_n^*(g)$ studentized by the original CCE instead of the bootstrap CCE.
Then, because $d_r = 1$, we have

\begin{align*}
1\{T_n > \hat{c}_{n}(1-\alpha)\} = 1\{T_{CR,n} >  \tilde{c}_{CR,n}(1-\alpha)\}.
\end{align*}

Therefore, it suffices to show that $\tilde{c}_{CR,n}(1-\alpha) > \hat{c}_{CR,n}(1-\alpha)$ with large probability as $||\mu||_2$ becomes sufficiently large. First, note that as $||\mu||_2 \rightarrow \infty$,
we have $\tilde{c}_{CR,n}(1-\alpha) \convP \infty$,
since $|\sqrt{n}(\lambda_{\beta}^{\top}\hat{\beta}^*_{tsls,g} - \lambda_0)| \convP \infty$ for all $g \in \textbf{G}$,  and
$\left( \lambda_{\beta}^{\top} \widehat{V} \lambda_{\beta} \right)^{-1} = O_p(1)$ as

\begin{align*}
\left( \lambda_{\beta}^{\top} \widehat{V} \lambda_{\beta} \right)^{-1}    \convD A_{r,CR,\iota},
\end{align*}

and $A_{r,CR,\iota}$ does not depend on $\mu$. On the other hand,
$\hat{c}_{CR,n}(1-\alpha)=O_p(1)$ as has been proved in Part (i).
Therefore, any $e>0$, there exists a constant $c_{\mu}>0$, such that when $||\mu||_2 > c_\mu$, $\tilde{c}_{CR,n}(1-\alpha) > \hat{c}_{CR,n}(1-\alpha)$
with probability greater than $1-e$. This concludes the proof. $\blacksquare$

§ $\WIDEHAT{Q}_{\TILDE{Z}W,J}$ IN THE EMPIRICAL APPLICATION

Tables <ref>–<ref> report the values of $\widehat{Q}_{\tilde{Z}W,j}$ for each cluster in the South, Midwest, and West regions, and all of them are close to zero.
For example, in Table <ref>, the column $``W_1"$ reports the values of $\widehat{Q}_{\tilde{Z}W_1,j}$, where $j=1, ..., 16$ for the South region.
Therefore, although we do not use the set of full interactions between the baseline covariates and cluster dummies as $W_i$, Assumption <ref>(i)(2) is still plausible in this application.

States     $W_1$     $W_2$     $W_3$     $W_4$     $W_5$     $W_6$     $W_7$

1    0.0027      0.0042      0.0009      0.0035      0.0012      0.0001        0.0001

2    -0.0008      0.0000        -0.0002     0.0003      0.0005       0.0000      0.0001

3    0.0019      -0.0076     -0.0017     -0.0026      -0.0018      -0.0001       -0.0005

4    0.0039      0.0035      0.0001      0.0099      0.0046      0.0002      -0.0005

5    -0.0007     -0.0061     -0.0017     -0.0066     -0.0033     -0.0001       -0.0001

6    -0.0025     -0.0028     -0.0001     -0.0063     -0.0039     0.0001        0.0001

7    0.0004      0.0006       -0.0001     0.0006     0.0005      0.0001        -0.0001

8    0.0068       -0.0614     -0.0384     -0.0421     -0.0180     -0.0015     -0.0025

9    0.0017      -0.0009     -0.0001       0.0003      0.0003       -0.0001      0.0001

10    -0.0047     0.0061      0.0026      0.0008      -0.0002     -0.0001      0.0003

11    0.0000      0.0041      0.0006       0.0044      0.0023      0.0001        0.0000

12    -0.0022     0.0002      0.0003      -0.0018     -0.0009     -0.0001       0.0002

13    0.0017      0.0022       0.0002      0.0024      0.0014      -0.0002     0.0003

14    0.0002      0.0042      0.0023      0.0039      0.0020      0.0001      0.0000

15    -0.0015     -0.0086      -0.0012     -0.0090     -0.0039      -0.0001       -0.0001

16    0.0015      0.0027       0.0001        0.0042      0.0021      -0.0001       0.0001

$\widehat{Q}_{\tilde{Z}W,j}$ for South in [15]

States     $W_1$     $W_2$     $W_3$     $W_4$     $W_5$     $W_6$     $W_7$

1    0.0041      0.0070      0.0038      0.0075      0.0048       0.0001      0.0001

2   -0.0015     0.0022      0.0005      -0.0016     -0.0004     -0.0001     0.0002

3   0.0005      0.0017      -0.0001     0.0022      0.0010        0.0001        0.0001

4   -0.0012     -0.0024     -0.0007     -0.0027     -0.0014      0.0000            -0.0001

5   -0.0056     -0.0170     -0.0025     -0.0187     -0.0096     -0.0001      -0.0003

6    0.0041       0.0090      0.0010      0.0114      0.0051      0.0001      0.0001

7    0.0020      0.0001        -0.0004     0.0022      0.0012      -0.0001       -0.0001

8   -0.0003      -0.0002     -0.0002     -0.0001       0.0002      0.0001        -0.0001

9   -0.0003     -0.0021     -0.0001     -0.0024     -0.0008     0.0000            -0.0001

10   -0.0015     0.0002      -0.0003     -0.0003     -0.0011     -0.0001       0.0003

11   0.0001      0.0007      0.0001        0.0011      0.0005      0.0000            -0.0001

12   -0.0015     -0.0002     -0.0002     -0.0012     -0.0007     -0.0001       0.0001

$\widehat{Q}_{\tilde{Z}W,j}$ for Midwest in [15]

States     $W_1$     $W_2$     $W_3$     $W_4$     $W_5$     $W_6$     $W_7$

1   0.0152      0.0535      0.0154      0.0632      0.0341      0.0005      0.0005

2   -0.0013     -0.0093     0.0016      -0.0151     -0.0092     0.0002       0.0001

3   0.0028      0.0061      -0.0001     0.0061      0.0033      -0.0001     0.0001

4   0.0015      0.0058      0.0006      0.0066      0.0031      0.0001        0.0001

5   0.0002      0.0016      0.0001      0.0021      0.0011       -0.0001       -0.0001

6   -0.0001       0.0041      -0.0022     0.0081      0.0030      0.0001        -0.0003

7   0.0016      0.0081      0.0013       0.0096      0.0051      0.0000            0.0001

8   0.0014      0.0048      0.0008      0.0056      0.0027      -0.0001       0.0001

9   -0.0014      0.0126      0.0015      0.0120      0.0060        0.0001        0.0001

10   -0.0124     -0.0510     -0.0122     -0.0522     -0.0247     -0.0004      -0.0006

11    0.0001        0.0004      0.0001        0.0006      0.0003      0.0001      -0.0001

$\widehat{Q}_{\tilde{Z}W,j}$ for West in [15]

§ EQUIVALENCE AMONG $K$-CLASS ESTIMATORS

We define $\hat{\beta}_L$ as the $k$-class estimator with $\hat{\kappa}_L$ for $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$.
Their null-restricted and bootstrap counterparts are denoted as $\hat{\beta}_L^r$ and $\hat{\beta}_{L,g}^*$, respectively, and $\hat{\gamma}_L$, $\hat{\gamma}_L^r$, and $\hat{\gamma}_{L,g}^*$ are similarly defined. In the following, we show that $\hat{\beta}_{tsls}$, $\hat{\beta}_{liml}$, $\hat{\beta}_{full}$, $\hat{\beta}_{ba}$ are asymptotically equivalent, and so be their null-restricted and bootstrap counterparts.

Suppose Assumptions <ref>, <ref>, and <ref>(i) hold.  Then, for $L \in \{\text{liml},\text{full},\text{ba}\}$,      we have

\begin{align*}
& \hat{\beta}_{L} = \hat{\beta}_{tsls} + o_p(n^{-1/2}), \quad  \hat{\beta}_{tsls} - \beta_n = O_p(n^{-1/2}),   \\
& \hat{\beta}_{L}^r = \hat{\beta}_{tsls}^r + o_p(n^{-1/2}),  \quad \text{and} \quad \hat{\beta}_{tsls}^r - \beta_n = O_p(n^{-1/2}).
\end{align*}

First, $L \in \{\text{liml},\text{full},\text{ba}\}$, we have $ \hat{\mu}_L = \hat{\kappa}_L-1$ and

\begin{align*}
\left( \hat{\beta}^{\top}_L, \hat{\gamma}^{\top}_L \right)^{\top} = & \left( \vec{X}^{\top} P_{\vec{Z}} \vec{X} - \hat{\mu}_L \vec{X}^{\top} M_{\vec{Z}}\vec{X} \right)^{-1} \left(\vec{X}^{\top} P_{\vec{Z}}Y - \hat{\mu}_L \vec{X}^{\top} M_{\vec{Z}}Y \right) \notag \\
= & \left( \Upsilon^{\top}\vec{X} \right)^{-1} \Upsilon^{\top}Y,
\;\; \text{where} \;\; \Upsilon =
\left[ P_{\vec{Z}} X - \hat{\mu}_L M_{\Vec{Z}}X : W \right].
\end{align*}

Then, by applying the Frisch–Waugh–Lovell Theorem we obtain that

\begin{align*}
\hat{\beta}_L = \left(X^{\top} P_{\widetilde{Z}} X - \hat{\mu}_L X^{\top} M_{\vec{Z}} X \right)^{-1}\left(X^{\top} P_{\widetilde{Z}} Y - \hat{\mu}_L X^{\top} M_{\vec{Z}} Y\right), \; \text{where} \; \tilde{Z} = M_W Z.
\end{align*}

By construction, $\hat{\beta}_{tsls}$ corresponds to $\hat{\mu}_{tsls} = 0$. For the LIML estimator, we have

\begin{align*}
\hat{\mu}_{liml}=\min_{r} r^{\top} \vec{Y}^{\top} M_W Z(Z^{\top} M_W Z)^{-1} Z^{\top} M_W \vec{Y} r / (r^{\top} \vec{Y}^{\top} M_{\vec{Z}}\vec{Y}r) \quad \text{and} \quad r=(1,-\beta^{\top} )^{\top},
\end{align*}

which implies that

\begin{align}
n \hat{\mu}_{liml} \leq \left(\frac{1}{\sqrt{n}} \eps^{\top} M_W Z \right) \left(\frac{1}{n} Z^{\top} M_W Z\right)^{-1} \left(\frac{1}{\sqrt{n}} Z^{\top} M_W \eps\right)/
\left(\frac{1}{n}\eps^{\top}M_{\vec{Z}}\eps\right).
\label{eq:muliml0}
\end{align}

We note that

\begin{align}
& \frac{1}{\sqrt{n}}Z^{\top}M_W\eps
= \frac{1}{\sqrt{n}}\sum_{j \in J}\sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\eps_{i,j} = O_p(1) \quad \text{and} \notag \\
& \left( \frac{1}{n} Z^{\top} M_W Z \right)^{-1} = \left( \frac{1}{n} \sum_{j \in J} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}\widetilde{Z}_{i,j}^{\top} \right)^{-1} = O_p(1).
\label{eq:muliml1}
\end{align}

In addition, let $\hat{\eps}_{i,j}$ be the residual from the full sample projection of $\eps_{i,j}$ on $W_{i,j}$. Then, we have

\begin{align}
\frac{1}{n} \eps^{\top} M_{\vec{Z}} \eps & = \frac{1}{n} \eps^{\top}\eps
- \frac{1}{n} \eps^{\top} \vec{Z} (\vec{Z}^{\top}\vec{Z})^{-1} \vec{Z}^{\top}\eps  \notag \\
& = \widehat{Q}_{\eps\eps} - \widehat{Q}_{\eps\widetilde{Z}} \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1} \widehat{Q}_{\eps\widetilde{Z}}^\top  - \widehat{Q}_{\eps W} \widehat{Q}_{WW}^{-1} \widehat{Q}_{\eps W}^\top \notag \\
& = \widehat{Q}_{\eps\eps}  - \widehat{Q}_{\eps W} \widehat{Q}_{WW}^{-1} \widehat{Q}_{\eps W}^\top + o_p(1) \notag \\
& = \widehat{Q}_{\tilde{\eps}\tilde{\eps}} + o_p(1)  \notag \\
& = \sum_{j \in J} \xi_j \widehat{Q}_{\tilde{\eps}\tilde{\eps},j} + o_p(1) \notag \\
& \geq \sum_{j \in J} \xi_j \widehat{Q}_{\dot{\eps}\dot{\eps},j} + o_p(1) \notag \\
& = \widehat{Q}_{\dot{\eps}\dot{\eps}} + o_p(1) \geq c~w.p.a.1,
\label{eq:muliml2}
\end{align}

where $c$ is a positive constant, the first inequality is by the definition that $\dot{\eps}_{i,j}$ is the residual from the cluster-level projection of $\eps_{i,j}$ on $W_{i,j}$,  and the second inequality is by Assumption <ref>(iii). Combining (<ref>)–(<ref>), we have $\hat{\mu}_{liml} = O_p(n^{-1})$. In addition, we have $\frac{1}{n} X^{\top} M_{\vec{Z}} X = O_p(1)$, $\frac{1}{n} X^{\top} M_{\vec{Z}} Y = O_p(1)$, and

\begin{align}
\left(\frac{1}{n}X^{\top} P_{\widetilde{Z}} X\right)^{-1} = \left(\widehat{Q}_{X\widetilde{Z}} \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}\widehat{Q}_{\widetilde{Z}X}    \right)^{-1} = O_p(1),
\end{align}

where the last equality holds by Assumptions <ref>(ii) and <ref>(i). This means

\begin{align*}
\hat{\beta}_{liml} =     \hat{\beta}_{tsls} + o_p(n^{-1/2}).
\end{align*}

In addition, we note $\hat{\mu}_{full} = \hat{\mu}_{liml} - \frac{C}{n-d_z-d_w} = O_p(n^{-1})$ and $\hat{\mu}_{ba} = O(n^{-1})$, respectively. Therefore, we have the same results for  $(\hat{\beta}_{ba},\hat{\gamma}_{ba})$.

For the second statement in the lemma, we have

\begin{align*}
(\hat{\beta}_{tsls} - \beta_n)  &
= \left(X^{\top} P_{\widetilde{Z}} X \right)^{-1} X^{\top} P_{\widetilde{Z}} \eps = \left(\widehat{Q}_{X\widetilde{Z}} \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}\widehat{Q}_{\widetilde{Z}X}    \right)^{-1} \widehat{Q}_{X\widetilde{Z}} \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}\widehat{Q}_{\widetilde{Z}\eps}  = O_p(n^{-1/2}),
\end{align*}

where the last inequality holds because $\widehat{Q}_{\widetilde{Z}\eps} = \frac{1}{n}\sum_{j \in J} \sum_{i \in I_{n,j}}\widetilde{Z}_{i,j}\eps_{i,j} = O_p(n^{-1/2})$.

Next, we turn to the third statement in the lemma. We note that, for $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$ and
$\lambda_{\beta}^{\top}\hat{\beta}^r_L = \lambda_0$,

\begin{align*}
& \hat{\beta}^r_L = \hat{\beta}_L - \left(X^{\top}(P_{\widetilde{Z}} - \hat{\mu}_L M_{\Vec{Z}})X\right)^{-1} \lambda_{\beta}
\left(\lambda_{\beta}^{\top}(X^{\top}(P_{\widetilde{Z}} - \hat{\mu}_L M_{\vec{Z}})X)^{-1}
\lambda_{\beta}\right)^{-1} (\lambda_{\beta}^{\top}\hat{\beta}_L - \lambda_0) = O_p(1).
\end{align*}

As $\hat{\mu}_{L} = O_p(n^{-1})$ and $\hat{\beta}_L = \hat{\beta}_{tsls}+o_p(n^{1/2})$ for $L \in \{\text{liml},\text{full},\text{ba}\}$, we have

\begin{align*}
\hat{\beta}^r_L = \hat{\beta}_{tsls} - \left(X^{\top}P_{\widetilde{Z}}X\right)^{-1} \lambda_{\beta} \left(\lambda_{\beta}^{\top} (X^{\top}P_{\widetilde{Z}}X)^{-1} \lambda_{\beta}\right)^{-1}
(\lambda_{\beta}^{\top} \hat{\beta}_{tsls} - \lambda_0) + o_p(n^{-1/2}) = \hat{\beta}^r_{tsls} + o_p(n^{-1/2}).
\end{align*}

For the last statement in the lemma, we note that

\begin{align*}
\hat{\beta}^r_{tsls} - \beta_n & =   (\hat{\beta}_{tsls} - \beta_n)   - \left(X^{\top}P_{\widetilde{Z}}X\right)^{-1} \lambda_{\beta} \left(\lambda_{\beta}^{\top} (X^{\top}P_{\widetilde{Z}}X)^{-1} \lambda_{\beta}\right)^{-1}
\lambda_{\beta}^{\top} (\hat{\beta}_{tsls} - \beta_n) \\
& - \left(X^{\top}P_{\widetilde{Z}}X\right)^{-1} \lambda_{\beta} \left(\lambda_{\beta}^{\top} (X^{\top}P_{\widetilde{Z}}X)^{-1} \lambda_{\beta}\right)^{-1}
(\lambda_{\beta}^{\top}\beta_n - \lambda_0) = O_p(n^{-1/2}),
\end{align*}

where the last equality holds because $\lambda_{\beta}^{\top}\beta_n - \lambda_0= \mu n^{-1/2}$ by construction and $\hat{\beta}_{tsls} - \beta_n = O_p(n^{-1/2})$. $\blacksquare$

Suppose Assumptions <ref>, <ref>, and <ref>(i) hold and $\widehat{Q}_{\widetilde{Z}W,j}(\hat{\gamma}_L^r-\gamma) = o_p(n^{-1/2})$. Then, for $L \in \{\text{liml},\text{full},\text{ba}\}$ and $g \in \textbf{G}$, we have

\begin{align}
\hat{\beta}_{L,g}^* = \hat{\beta}_{tsls,g}^* + o_p(n^{-1/2}) \quad \text{and} \quad \hat{\beta}_{tsls,g}^*- \beta_n = O_p(n^{-1/2}). \label{eq:equiv_b}
\end{align}

Proof. By the same argument in the proof of Lemma <ref>, for $L \in \{\text{tsls},\text{liml},\text{full},\text{ba}\}$ and $g \in \textbf{G}$, we have

\begin{align}
\hat{\beta}^*_{L,g}= \left(X^{*\top}(g) P_{\widetilde{Z}} X^*(g) - \hat{\mu}^*_{L,g} X^{*\top}(g) M_{\vec{Z}} X^*(g) \right)^{-1}\left(X^{*\top}(g) P_{\widetilde{Z}} Y^*(g) - \hat{\mu}^*_{L,g} X^{*\top}(g) M_{\vec{Z}} Y^*(g)\right).
\label{eq:equiv_b1}
\end{align}

such that $\hat{\mu}_{tsls,g}^* = 0$, $\hat{\mu}_{full,g}^* = \hat{\mu}_{liml,g}^* - \frac{C}{n-d_z-d_x}$, $\hat{\mu}_{ba,g}^* = \hat{\mu}_{ba}$, and

$$\hat{\mu}^*_{liml,g}=\min_{r} r^{\top} \vec{Y}^{*\top}(g) M_W Z(Z^{\top} M_W Z)^{-1} Z^{\top} M_W \vec{Y}^*(g) r / (r^{\top} \vec{Y}^{*\top}(g) M_{\vec{Z}}\vec{Y}^*(g)r),$$

where $\vec{Y}^{*}(g) = [Y^*(g) : X^*(g)]$, $Y^*(g)$ is an $n \times 1$ vector formed by $Y_{i,j}^*(g)$, and $r=(1,-\beta^{\top} )^{\top}$.

Following the same argument previously, we have

\begin{align}\label{eq:muhatliml*}
n \hat{\mu}_{liml,g}^* \leq \left(\frac{1}{\sqrt{n}} \eps_g^{*r\top} M_W Z \right) \left(\frac{1}{n} Z^{\top} M_W Z\right)^{-1} \left(\frac{1}{\sqrt{n}} Z^{\top} M_W \eps_g^{*r}\right)/
\left(\frac{1}{n}\eps_g^{*r\top}M_{\vec{Z}}\eps_g^{*r}\right), %\label{muhatliml*}
\end{align}

where $\eps_g^{*r}$ is an $n\times 1$ vector formed by $g_j\hat{\eps}_{i,j}^r$. We first note that

\begin{align}
\frac{1}{n} \eps_g^{*r\top} M_W Z & = \frac{1}{n}\sum_{j \in J} \sum_{i \in I_{n,j}} g_j \widetilde{Z}_{i,j} \hat{\eps}^r_{i,j} \notag \\
& = \sum_{j \in J}\xi_j g_j\left[\frac{1}{n_j} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j} \eps_{i,j}
+ \frac{1}{n_j} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}X_{i,j}^{\top}(\beta_n - \hat{\beta}^r_L)
+ \frac{1}{n_j} \sum_{i \in I_{n,j}} \widetilde{Z}_{i,j}W_{i,j}^{\top}(\gamma - \hat{\gamma}^r_L) \right] \notag \\
& = O_p(n^{-1/2}), \label{eq:eps_g*r}
\end{align}

where the last line is by Assumptions <ref>(ii), Lemma <ref>, and Lemma <ref>. Second, we have

\begin{align}
\frac{1}{n}\eps_g^{*r\top}M_{\vec{Z}}\eps_g^{*r} & = \frac{1}{n} \eps_g^{*r\top} \eps_g^{*r}
- \frac{1}{n} \eps_g^{*r\top} \vec{Z} (\vec{Z}^{\top}\vec{Z})^{-1} \vec{Z}^{\top} \eps_g^{*r} \notag \\
& =     \frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r})^2  - \left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} \widetilde{Z}_{i,j}^\top)\right] \widehat{Q}_{\widetilde{Z}\widetilde{Z}}^{-1}\left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} \widetilde{Z}_{i,j})\right] \notag \\
& - \left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} W_{i,j}^\top)\right] \widehat{Q}_{WW}^{-1}\left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} W_{i,j})\right] \notag \\
& =        \frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(\hat{\eps}_{i,j}^{r})^2   - \left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} W_{i,j}^\top)\right] \widehat{Q}_{WW}^{-1}\left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} W_{i,j})\right] + o_p(1), \label{eq:eps_g*r2}
\end{align}

where the last equality is by $\frac{1}{n_j} \sum_{i \in I_{n,j}}\hat{\eps}^{r}_{i,j} \widetilde{Z}_{i,j} = O_p(n^{-1/2})$, as in (<ref>). We further note that

\begin{align*}
\hat{\eps}^{r}_{i,j} = \eps_{i,j} - X_{i,j}^\top(\hat{\beta}_L^r - \beta_n) - W_{i,j}^\top (\hat{\gamma}_L^r - \gamma),
\end{align*}

where $\hat{\beta}_L^r - \beta_n = O_p(n^{-1/2})$ and

\begin{align*}
\hat{\gamma}_L^r - \gamma = \widehat{Q}_{WW}^{-1} \left[ \frac{1}{n}\sum_{j \in J} \sum_{i \in I_{n,j}}W_{i,j}\left(Y_{i,j} - X_{i,j}^\top \hat{\beta}_L^r\right)  \right] - \gamma = \widehat{Q}_{WW}^{-1} \widehat{Q}_{W\eps} + O_p(n^{-1/2}).
\end{align*}

Therefore, we have

\begin{align*}
\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(\hat{\eps}_{i,j}^{r})^2 = \frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}\mathscr{E}_{i,j}^2 + o_p(1),
\end{align*}

where $\mathscr{E}_{i,j} = \eps_{i,j} - W_{i,j}^\top\widehat{Q}_{WW}^{-1} \widehat{Q}_{W\eps}.$ Similarly, we can show that

\begin{align*}
& \left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} W_{i,j}^\top)\right] \widehat{Q}_{WW}^{-1}\left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\hat{\eps}_{i,j}^{r} W_{i,j})\right] \\
& = \left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\mathscr{E}_{i,j} W_{i,j}^\top)\right] \widehat{Q}_{WW}^{-1}\left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\mathscr{E}_{i,j} W_{i,j})\right]  + o_p(1),
\end{align*}

which, combined with (<ref>), implies

\begin{align*}
\frac{1}{n}\eps_g^{*r\top}M_{\vec{Z}}\eps_g^{*r} & = \frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}\mathscr{E}_{i,j}^2 - \left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\mathscr{E}_{i,j} W_{i,j}^\top)\right] \widehat{Q}_{WW}^{-1}\left[\frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(g_j\mathscr{E}_{i,j} W_{i,j})\right]  + o_p(1) \notag \\
& = \frac{1}{n} \sum_{j \in J}\sum_{i \in I_{n,j}}(\mathscr{E}_{i,j} - g_j W_{i,j}^\top \hat{\theta}_g )^2 + o_p(1),
\end{align*}