2.1 Model and Test Statistics

We consider the following linear IV model:

(2.1) $$ \begin{align} y = X\theta + u, \quad X= Z \pi + v, \end{align} $$

where $y\in \mathbb {R}^n$ and $X\in \mathbb {R}^n$ are vectors of dependent and endogenous variables, respectively, $Z \in \mathbb {R}^{n \times k}$ is a matrix of instruments ( $k \geq 1$ ), $(\theta , \pi ')' \in \mathbb {R}^{k+1}$ are unknown parameters, and n is the sample size. Denote by $u_i$ , $v_i$ , $y_i$ , $X_i$ , and $Z_i$ the ith rows of u, v, y, X, and $Z,$ respectively, written as column vectors or scalars. For notational simplicity, we assume that the other exogenous variables have already been partialled out from the model.

The object of inferential interest is the structural parameter $\theta $ , and we consider the problem of testing the null hypothesis $H_0: \theta =\theta _0$ . We study the two-stage testing procedure for assessing $H_0$ , where an exogeneity test is conducted in the first stage to determine whether the OLS or IV-based method is appropriate for testing $H_0$ in the second stage. Assume that the instruments Z are exogenous, i.e., $E_F[u_iZ_i]=0$ , where $E_F$ denotes expectation under the distribution $F.$ Under this orthogonality condition of the instruments, X is endogenous in (2.1) if and only if v and u are correlated. Consider the following linear projection of u on v:

(2.2) $$ \begin{align} u = v a + e, \; a = (E_F[v^2_i])^{-1} E_F[v_i u_i], \end{align} $$

where e is uncorrelated with v. Notice that the exogeneity of X in (2.1) can be assessed by testing the null hypothesis $H_a:\;a=0$ in (2.2). Substituting (2.2) into (2.1), we obtain

(2.3) $$ \begin{align} y = X \theta + v a + e, \end{align} $$

where X and v are uncorrelated with e. Therefore, the null hypothesis of exogeneity $H_{a}:\; a = 0$ can be assessed using a standard Wald statistic in the extended regression (2.3) (see, e.g., Doko Tchatoka and Dufour, Reference Doko Tchatoka and Dufour2014). To account for possible conditional heteroskedasticity, we consider the following control function-based Wald statisticFootnote ¹² :

(2.4) $$ \begin{align} H_n = \hat{a}^2/\hat{V}_{a}, \end{align} $$

where $\hat {a} = (\tilde {v}' \tilde {v})^{-1} \tilde {v}' y$ , $\hat {V}_{a} = (n^{-1} \tilde {v}' \tilde {v})^{-1} \left ( n^{-2} \sum _{i=1}^n \tilde {v}^2_i \hat {e}^2_i \right ) (n^{-1} \tilde {v}'\tilde {v})^{-1}$ is the heteroskedasticity-robust estimator of the variance of $\hat {a}$ , $\tilde {v} = M_X \hat {v}$ , $\hat {v}=M_ZX,$ and $\hat {e}=M_{[X : \hat {v}]}y.$ Note that $\hat {e}$ is the residual vector from the OLS regression of y on X and $\hat {v}$ . If $\theta $ is strongly identified (Z being strong instruments) and X is exogenous, $H_n$ follows a $\chi ^2_{1}$ distribution asymptotically. The pretest rejects the null hypothesis that X is exogenous in (2.1) if $H_n>\chi ^2_{1, 1-\beta }$ , where $\chi ^2_{1, 1-\beta }$ is the $(1-\beta )$ th quantile of $\chi ^2_{1}$ -distributed random variable for some $\beta \in (0, 1)$ .

Let $\hat {\theta }_{2sls} = (X^{'}P_{Z}X)^{-1} X^{'}P_{Z}y,$ and $\hat {\theta }_{ols} = (X^{'}X)^{-1} X^{'} y$ be the 2SLS and OLS estimators of $\theta $ in (2.1), respectively. Also, define their corresponding variance estimators as

(2.5) $$ \begin{align} \hat{V}_{2sls} &= \left( n^{-1} X^{'}P_{Z}X \right)^{-1} \hat{\pi}' \left( n^{-2} \sum_{i=1}^n Z_i Z_i' \hat{u}^2_i(\hat{\theta}_{2sls}) \right) \hat{\pi} \left( n^{-1} X^{'}P_{Z}X \right)^{-1}, \notag \\ \hat{V}_{ols} &= \left( n^{-1}X^{'}X \right)^{-1} \left( n^{-2} \sum_{i=1}^n X_iX_i' \hat{u}^2_i(\hat{\theta}_{ols}) \right) \left( n^{-1}X^{'}X \right)^{-1}, \end{align} $$

where $\hat {u}_i(\hat {\theta }_{2sls}) = y_i - X_i \hat {\theta }_{2sls}$ , $\hat {u}_i(\hat {\theta }_{ols}) = y_i - X_i \hat {\theta }_{ols}$ , and $\hat {\pi } = (Z'Z)^{-1}Z'X$ . Then, the two-stage test statistic associated with the $H_n$ -based pretest of exogeneity in the first stage is given by

(2.6)

where $T_{ols}(\theta )$ and $T_{2sls}(\theta )$ are the Wald statistics with OLS and 2SLS estimates, respectively, i.e.,

(2.7) $$ \begin{align} T_{ols}(\theta) = ( \hat{\theta}_{ols} - \theta )^2 / \hat{V}_{ols}, \quad \text{and} \quad T_{2sls}(\theta) = ( \hat{\theta}_{2sls} - \theta )^2 / \hat{V}_{2sls}. \end{align} $$

Related to the two-stage procedure, Hansen (Reference Hansen2017) proposed a Stein-like shrinkage approach in the context of IV regressions. His estimator follows Maasoumi (Reference Maasoumi1978) in taking a weighted average of the 2SLS and OLS estimators, with the weight depending inversely on the test statistic for exogeneity, and the proposed shrinkage estimator is found to have substantially reduced finite-sample median squared error relative to the 2SLS estimator. Following Hansen’s (Reference Hansen2017) approach, we define the Stein-like shrinkage test statistic as follows:

(2.8) $$ \begin{align} T^S_{\tau,n}(\theta_0) = T_{ols}(\theta_0) w(H_n) + T_{2sls}(\theta_0) (1-w(H_n)), \end{align} $$

where the weight function takes the form $w(H_n) = \begin {cases} \tau / H_n \; \; \; \text {if} \; \; \; H_n \geq \tau \\ 1 \; \; \; \text {if} \; \; \; H_n < \tau \end {cases}$ , and $\tau $ is a shrinkage parameter chosen by the researcher. The shrinkage statistic has a relatively smooth transition between the OLS and 2SLS test statistics. In Section 4, we evaluate the performance of the shrinkage procedure with different choices of $\tau $ .

In addition, we define the weak-identification-robust version of the two-stage/shrinkage test statistics $T^S_{\beta ,n}(\theta _0)$ and $T^S_{\tau ,n}(\theta _0)$ . Specifically, we first propose the following weak-identification-robust version of $H_n$ by imposing $H_0: \theta = \theta _0$ :

(2.9) $$ \begin{align} H_n(\theta_0) = \hat{a}^2(\theta_0)/\hat{V}_{a}(\theta_0), \end{align} $$

where $\hat {a}(\theta _0) = (\hat {v}' \hat {v})^{-1} \hat {v}' \tilde {y}(\theta _0)$ , $\hat {V}_{a}(\theta _0) = (n^{-1} \hat {v}' \hat {v})^{-1} \left ( n^{-2} \sum _{i=1}^n \hat {v}^2_i \tilde {e}^2_i(\theta _0) \right ) (n^{-1} \hat {v}'\hat {v})^{-1}$ is the (null-imposed) heteroskedasticity-robust estimator of the variance of $\hat {a}(\theta _0)$ , $\tilde {y}(\theta _0) = y - X \theta _0$ , $\hat {v}=M_ZX,$ and $\tilde {e}(\theta _0)=M_{\hat {v}}\tilde {y}(\theta _0)$ . Note that $\tilde {e}(\theta _0)$ is the residual vector from the OLS regression of $\tilde {y}(\theta _0)$ on $\hat {v}$ . When $H_0$ is true, and X is exogenous (i.e., $a=0$ ), $H_n(\theta _0)$ follows a $\chi ^2_{1}$ distribution asymptotically, no matter whether the IVs are strong or weak.Footnote ¹³

Now, let us define the heteroskedasticity-robust AR statistic as

(2.10) $$ \begin{align} T_{ar}(\theta_0) = \left( n^{-1/2} \tilde{y}(\theta_0)'Z \right) \left( \hat{V}_{ar}(\theta_0) \right)^{-1} \left( n^{-1/2} Z'\tilde{y}(\theta_0) \right), \end{align} $$

where $\hat {V}_{ar}(\theta _0) = n^{-1} \sum _{i=1}^n Z_iZ_i' \tilde {y}^2_i(\theta _0)$ . Then, the weak-IV-robust two-stage/shrinkage test statistics associated with the $H_n(\theta _0)$ pretest statistic are given by

(2.11)

2.2 Parameter Space and Asymptotic Size

Assume that $\{(u_i, v_i, Z_i): i\leq n\}$ in (2.1) are i.i.d. with distribution F. To characterize the asymptotic size of the two-stage and shrinkage tests, we define the parameter space $\Gamma $ of the nuisance parameter vector $\gamma $ following the seminal studies by Andrews and Guggenberger (Reference Andrews and Guggenberger2009a, Reference Andrews and Guggenberger2009b, Reference Andrews and Guggenberger2010a, Reference Andrews and Guggenberger2010b), Guggenberger (Reference Guggenberger2012), and Guggenberger and Kumar (Reference Guggenberger and Kumar2012). For the current testing problem, define the vector of nuisance parameters $\gamma = (\gamma _1, \gamma _2, \gamma _3)$ by

(2.12) $$ \begin{align} \gamma_1 = a, \;\; \gamma_2 = (\gamma_{21},\gamma_{22},\gamma_{23},\gamma_{24},\gamma_{25}),\;\; \gamma_3 = F, \end{align} $$

where a is defined in (2.2), $\gamma _{21}=\pi , \gamma _{22}=E_Fe_i^2Z_iZ_i', \gamma _{23}=E_Fe^2_iv_i^2, \gamma _{24}= E_F Z_i Z^{\prime }_i,$ and $\gamma _{25}=E_F v_i^2$ . Here, $\gamma _1$ measures the degree of endogeneity of X and is the key parameter in the current testing problem, as it determines the point of discontinuity of the null limiting distributions of the two-stage and shrinkage test statistics. For the parameter space, let

(2.13) $$ \begin{align} \Gamma_1= \mathbb{R}, \;\; \Gamma_2 & = \Big\{(\gamma_{21},\gamma_{22},\gamma_{23},\gamma_{24},\gamma_{25}): \gamma_{21}=\pi_n \in \mathbb{R}^k, \gamma_{22}=E_Fe_i^2Z_iZ_i' \in \mathbb{R}^{k \times k}, \Big. \notag \\ &\quad \gamma_{23}=E_Fe^2_iv_i^2 \in \mathbb{R}, \gamma_{24}= E_F Z_i Z^{\prime}_i \in \mathbb{R}^{k \times k}, \gamma_{25}= E_F v_i^2 \in \mathbb{R}, \Big. \notag \\ &\quad\textit{s.t.}\;\; \lambda_{min}(\gamma_{22}) \geq \underline{\kappa}, \gamma_{23}> 0, \lambda_{min}(\gamma_{24}) \geq \underline{\kappa}, \; \text{and} \; \gamma_{25} > 0 \Big\}, \end{align} $$

for some $\underline {\kappa }>0 $ that does not depend on n. We note that the current framework allows for weak identification. In addition, $\Gamma _3(\gamma _1, \gamma _2)$ is defined as follows:

(2.14) $$ \begin{align} \Gamma_3(\gamma_1, \gamma_2) &= \Big\{ F: \; E_F e_i v_i = E_F e_i Z_i = E_F v_i Z_i = 0,\; \notag \\ & \quad E_F e_i^2 v_i Z_i = E_F e_i v_i^2 Z_i = E_F e_iv_i Z_i Z_i' =0,\; \Big. \notag \\ & \quad E_Fv_i^2Z_iZ_i' \in \mathbb{R}^{k \times k} \;\; \text{with} \;\; \lambda_{min}(E_Fv_i^2Z_iZ_i') \geq M^{-1}, \; \Big. \\ & \quad \left\lVert E_F \left( ||Z_i e_i||^{2+\xi}, ||Z_iv_i||^{2+\xi}, |v_ie_i|^{2+\xi}, ||Z_iZ_i'||^{2+\xi}, |X_i|^{2(2+\xi)} \right)'\right\rVert \leq M \Big\},\notag \end{align} $$

for some constant $\xi> 0$ and $M<\infty $ . We then define the whole nuisance parameter space $\Gamma $ of $\gamma $ as

(2.15) $$ \begin{align} \Gamma = \{ \gamma = (\gamma_1, \gamma_2, \gamma_3): \gamma_1 \in \Gamma_1, \gamma_2 \in \Gamma_2, \gamma_3 \in \Gamma_3(\gamma_1, \gamma_2)\}, \end{align} $$

where $\Gamma _j,\;j=1,2,3$ are given in (2.13) and (2.14). This nuisance parameter space extends the one defined in Guggenberger (Reference Guggenberger2010a) to allow for conditional heteroskedasticity and is similar to those defined in Guggenberger (Reference Guggenberger2012) and Guggenberger and Kumar (Reference Guggenberger and Kumar2012), which also allow for heteroskedastic errors. In Section 3.3, we further extend the analysis to the case with clustered data and show that our size-adjusted wild bootstrap is also uniformly valid in that case. The condition that $E_F e_i^2 v_i Z_i = E_F e_i v_i^2 Z_i = E_F e_iv_i Z_i Z_i' =0$ in (2.14) is similar to that imposed for $\Gamma _3(\gamma _1, \gamma _2)$ in Guggenberger (Reference Guggenberger2010a).Footnote ¹⁴

Now, we define the asymptotic size. Let $T_n(\theta _0)$ denote a generic test statistic. Let $c_n$ denote a (possibly data-dependent) CV being used for the two-stage testing or shrinkage procedure. Then, the finite sample null rejection probability (NRP) of $T_n(\theta _0)$ evaluated at $\gamma \in \Gamma $ is given by $P_{\theta _0, \gamma }\left [ T_{n} (\theta _0)> c_n\right ]$ , where $P_{\theta _0, \gamma } \left [ E_n \right ]$ denotes the probability of event $E_n$ when $\theta _0$ and $\gamma $ are the true values of the parameters. Then, the asymptotic NRP of the test evaluated at $\gamma \in \Gamma $ is given by $\underset {n \rightarrow \infty }{lim sup} P_{\theta _0, \gamma } \left [T_{n} (\theta _0)> c_n\right ]$ , while the asymptotic size is given by

(2.16) $$ \begin{align} AsySz[c_n] = \underset{n \rightarrow \infty}{lim sup} \underset{\gamma \in \Gamma}{sup} P_{\theta_0, \gamma} \left[ T_{n} (\theta_0)> c_n \right]. \end{align} $$

In general, asymptotic NRP evaluated at a given $\gamma \in \Gamma $ is not equal to the asymptotic size of the test. To control the asymptotic size, one needs to control the null limiting behavior of $T_{n}(\theta _0)$ under drifting parameter sequences $\{ \gamma _n: n \geq 1\}$ indexed by the sample size (see, e.g., Andrews and Guggenberger, Reference Andrews and Guggenberger2009a, Reference Andrews and Guggenberger2009b, Reference Andrews and Guggenberger2010a, Reference Andrews and Guggenberger2010b, Guggenberger, Reference Guggenberger2012, and Guggenberger and Kumar, Reference Guggenberger and Kumar2012).

Following the arguments used in these papers, to derive $AsySz[c_n],$ we can study the asymptotic NRP along certain parameter sequences of the type $\{\gamma _{n,h}\}$ (defined below) for some $h\in \mathscr {H}$ , as the highest asymptotic NRP is materialized under such a sequence, where

(2.17) $$ \begin{align} \mathscr{H} &= \Bigg\{ h = (h_1, h^{\prime}_{21},vec(h_{22})', h_{23},vec(h_{24})', h_{25})' \in \mathbb{R}^{2k^2+k+3}_{\infty}: \notag \\ & \quad \exists \{\gamma_n = (\gamma_{n,1}, \gamma_{n,2}, \gamma_{n,3}) \in \Gamma: n \geq 1\} \quad s.t. \Big.\notag\\ &\quad n^{1/2}\gamma_{n,1} \rightarrow h_1 \in \mathbb{R}_{\infty},\;\;\gamma_{n,2} \rightarrow h_2=(h_{21},h_{22}, h_{23},h_{24},h_{25}), \; \lambda_{min}(A)\geq \underline{\kappa} \Big.\notag\\ & \;\;\;\; \text{for}\;\;A\in \{ h_{22}, h_{24}\}, \; h_{23}>0, \; h_{25}>0 \Bigg\} \end{align} $$

for some $\underline {\kappa }>0$ and $\mathbb {R}_{\infty } = \mathbb {R} \cup \{ \pm \infty \}.$ Then, for $h \in \mathscr {H}$ , the relevant sequence of parameters $\{\gamma _{n,h}\} \subset \Gamma $ is defined following Guggenberger (Reference Guggenberger2010a) as $\gamma _{n,h} = ( \gamma _{n,h,1}, \gamma _{n,h,2}, \gamma _{n,h,3} ),$ where

(2.18) $$ \begin{align} \gamma_{n,h,1} = (E_{F_n}[v^2_i])^{-1} E_{F_n}[v_i u_i], \;\; \gamma_{n,h,2} = (\gamma_{n,h,21} ,\gamma_{n,h,22} , \gamma_{n,h,23} ,\gamma_{n,h,24}, \gamma_{n,h,25}), \end{align} $$

with $\gamma _{n,h,21}=\pi _n$ , $\gamma _{n,h,22}=E_{F_n}e_i^2Z_iZ_i'$ , $\gamma _{n,h,23}=E_{F_n}e^2_iv_i^2, \gamma _{n,h,24}= E_{F_n} Z_i Z^{\prime }_i$ , $\gamma _{n,h,25}=E_{F_n} v_i^2$ , s.t.

(2.19) $$ \begin{align} n^{1/2}\gamma_{n,h,1} \rightarrow h_1, \; \gamma_{n,h,2} \rightarrow h_2, \; \text{and} \; \gamma_{n,h,3} = F_n\in \Gamma_3(\gamma_{n,h,1}, \gamma_{n,h,2}). \end{align} $$

More specifically, under $\{ \gamma _{n,h} \}$ satisfying (2.19) with $|h_1| = \infty $ (i.e., strong endogeneity), $H_n \rightarrow ^P \infty $ , and the two-stage and shrinkage test statistics $T_{l,n}^S(\theta _0)$ , $l \in \{\beta , \tau \}$ , are asymptotically equivalent to the 2SLS-based t-statistic. On the other hand, under $\{\gamma _{n,h}\}$ satisfying (2.19) with $|h_1| < \infty $ (i.e., local endogeneity) and $||h_{21}|| \geq \kappa $ , where $\kappa> 0$ (i.e., strong IVs), the following joint convergence results hold for the two-stage and shrinkage statistics $T^S_{l,n}(\theta _0)$ for $l \in \{\beta ,\tau \}$ ,

(2.20) $$ \begin{align} \begin{pmatrix} T^S_{\beta,n}(\theta_0) \\ T^S_{\tau,n}(\theta_0) \end{pmatrix} & \rightarrow^{d} \begin{pmatrix} \tilde{T}^S_{\beta,h} \\ \tilde{T}^S_{\tau,h} \end{pmatrix}, \end{align} $$

where

$\eta ^S_{1,h}\sim \chi ^2_{1}$ , $\eta ^S_{2,h}\sim \chi ^2_1 \Big ( (h^{\prime }_{21} h_{22}h_{21} + h_{23} )^{-1} h^2_{25} h^2_1 \Big )$ , and $\eta ^S_{3,h}\sim \chi ^2_1 \Big (\big (\frac { h^{\prime }_{21} h_{22} h_{21}}{(h^{\prime }_{21} h_{24}h_{21})^{2}} + h_{23} h_{25}^{-2} \big )^{-1}h^2_1\Big )$ .Footnote ¹⁵ Furthermore, the following joint convergence results hold for $T^W_{l,n}(\theta _0)$ for $l \in \{\beta ,\tau \}$ under local endogeneity, no matter whether IVs are strong or weak (as both $H_n(\theta _0)$ and $T_{ar}(\theta _0)$ are weak-identification-robust):

(2.21) $$ \begin{align} \begin{pmatrix} T^W_{\beta,n}(\theta_0) \\ T^W_{\tau,n}(\theta_0) \end{pmatrix} & \rightarrow^{d} \begin{pmatrix} \tilde{T}^W_{\beta,h} \\ \tilde{T}^W_{\tau,h} \end{pmatrix}, \end{align} $$

where

$\eta ^W_{1,h} \sim \chi ^2_{k}$ , $\eta ^W_{2,h} = \eta ^S_{2,h}$ , and $\eta ^W_{3,h}\sim \chi ^2_1 \Big ( h_{23}^{-1} h_{25}^2 h_1^2 \Big )$ .

In the next section, we will study the asymptotic behaviors of the two-stage and shrinkage tests under bootstrap-based CVs.

3.1 Standard Wild Bootstrap

In this section, we study the asymptotic behavior of the standard wild bootstrap for the two-stage testing and shrinkage procedures.

Wild Bootstrap Algorithm:

1. 1. Compute the (null-restricted) residuals from the first-stage and structural equations: $\hat {v} = X -Z \hat {\pi }$ , $\hat {u}(\theta _0) = y - X \theta _0$ , where $\hat {\pi }=(Z^{\prime }Z)^{-1}Z^{\prime }X$ denotes the least squares estimator of $\pi $ .

2. 2. Generate the bootstrap pseudo-data following $X^{\ast } = Z \hat {\pi } + v^{\ast }, y^{\ast } = X^{\ast }\theta _0 + u^{\ast },$ where there are two options to generate the bootstrap disturbances:

   1. (a) $v^{\ast }$ and $u^{\ast }$ are generated independently from each other. Specifically, in the current case with heteroskedastic data, we set for each observation i: $v^*_i = \hat {v}_i \omega ^*_{1i},$ and $u^*_i = \hat {u}_i(\theta _0) \omega ^*_{2i}$ , where $\omega ^*_{1i}$ and $\omega ^*_{2i}$ are two random variables with mean 0 and variance $1$ , i.e., $E^*\left [\omega ^*_{1i}\right ]=E^*\left [\omega ^*_{2i}\right ]=0$ and $Var^*\left [\omega ^*_{1i}\right ]=Var^*\left [\omega ^*_{2i}\right ]=1$ , and they are independent from the data and independent from each other.

   2. (b) $v^{\ast }$ and $u^{\ast }$ are drawn dependently from each other. We set for each observation i: $v^*_i = \hat {v}_i \omega ^*_{1i} $ , and $u^*_i = \hat {u}_i(\theta _0) \omega ^*_{1i}$ .

   Following Young (Reference Young2022), we refer to (a) as independent transformation of disturbances and (b) as dependent transformation of disturbances.Footnote ¹⁶

3. 3. Compute the bootstrap analogs of the two-stage and shrinkage test statistics:

   (3.1)

   
   where $T^*_{ols}(\theta _0)$ , $T^*_{2sls}(\theta _0)$ , $T^*_{ar}(\theta _0)$ , $H^*_n$ , and $H^*_n(\theta _0)$ are bootstrap analogs of $T_{ols}(\theta _0)$ , $T_{2sls}(\theta _0)$ , $T_{ar}(\theta _0)$ , $H_n$ , and $H_n(\theta _0)$ , respectively, which are obtained from the bootstrap samples generated in Step 2.

4. 4. For $l \in \{\beta ,\tau \}$ and $s \in \{S, W \}$ , repeat Steps 2 and 3 B times and obtain $\{ T^{s*(b)}_{l,n}(\theta _0), b=1, \ldots , B \}$ . The bootstrap test with the test statistic $T^s_{l,n}(\theta _0)$ rejects $H_0$ if the corresponding bootstrap p-value is less than the nominal level $\alpha .$

Following the standard arguments for bootstrap validity, to check whether (conditional on the data) the bootstrap can consistently estimate the distribution of the two-stage or shrinkage test statistic, one needs to check whether under $H_0$ and both cases of strong endogeneity ( $|h_1|=\infty $ ) and local endogeneity ( $|h_1| < \infty $ ), $ \sup _{x \in \mathbb {R}} \left | P^*\left (T^{s*}_{l,n}(\theta _0) \leq x \right ) - P\left (T^s_{l,n}(\theta _0) \leq x\right ) \right | \rightarrow ^P 0, $ for $l \in \{\beta ,\tau \}$ and $s \in \{S, W\}$ . However, we note below that neither bootstrap procedure consistently estimates the distribution of interest under local endogeneity, even with strong IVs (i.e., even when $|| \gamma _{21} ||> 0$ ).

More specifically, it holds for the bootstrap statistics with dependent or independent transformation (for the dependent transformation, we further require $E^*\left [\omega _{1i}^{*3} \right ]=0$ and $E^*\left [\omega _{1i}^{*4}\right ] =1$ ; see Lemma S.5 in the Supplementary Material for details) that

(3.2) $$ \begin{align} n^{-1/2}\begin{pmatrix} Z^{'}u^* \\ \left(u^{*'}v^* - E^*[u^{*'}v^*]\right) \end{pmatrix} \rightarrow^{d^*} \begin{pmatrix} \psi^*_{Ze} \\ \psi^*_{ve} \end{pmatrix}, \end{align} $$

in probability P (i.e., with probability approaching one according to P), where the bootstrap (conditional) weak limit $(\psi ^{*'}_{Ze}, \psi ^*_{ve})'$ is the same as $(\psi ^{\prime }_{Ze}, \psi _{ve})'$ , i.e., the weak limit of $n^{-1/2} \left ( (Z'u)', (u'v - E_F[u'v]) \right )'$ . Therefore, the bootstrap procedures do replicate well the randomness in the original sample.

By contrast, under local endogeneity, the standard wild bootstrap does not well mimic the key localization parameter $h_1$ , leading to a discrepancy between the original and bootstrap samples. In particular, let $h^b_1$ denote the localization parameter of endogeneity in the bootstrap world, then we have:

1. 1. $h^b_1=0$ for the bootstrap with independent transformation. This is because the random weights $\omega ^*_{1i}$ and $\omega ^*_{2i}$ under this transformation are independent from each other. As a consequence, the bootstrap disturbances $u^*_i$ and $v^*_i$ are independent from each other (conditional on the data), which implies that $E^*[u_i^* v_i^*] = E^*[\hat {u}_i(\theta _0) \omega _{1i}^* \hat {v}_i \omega _{2i}^*] = \hat {u}_i(\theta _0) \hat {v}_i E^*[\omega _{1i}^*] E^*[\omega _{2i}^*] = 0$ .

2. 2. $h^b_1 = h_1 + h_{25}^{-1} \psi _{ve}$ for the one with dependent transformation, where $\psi _{ve} \sim N(0, h_{23})$ . This is due to the fact that the same random weight $\omega ^*_{1i}$ is used to generate the bootstrap disturbances $u^*_i$ and $v^*_i$ , so that $E^*[u_i^* v_i^*] = E^*[\hat {u}_i(\theta _0) \omega _{1i}^* \hat {v}_i \omega _{1i}^*] = \hat {u}_i(\theta _0) \hat {v}_i E^*[\omega _{1i}^{*2}] = \hat {u}_i(\theta _0) \hat {v}_i$ . Intuitively, while the bootstrap with dependent transformation can mimic the situation of local endogeneity in the original sample ( $h^b_1$ is finite with probability approaching one when $h_1$ is finite), the approximation is imprecise and results in an extra error term $h_{25}^{-1}\psi _{ve}$ , whose value depends on the actual realization of the sample (e.g., see Theorems S.1 and S.2 in the Supplementary Material for details).

However, even if the bootstrap is inconsistent conditional on the data, it may still be valid in the unconditional sense (see, e.g., Cavaliere and Georgiev, Reference Cavaliere and Georgiev2020 and the references therein). More precisely, the bootstrap might still be able to provide a valid test in the current context if its asymptotic NRP does not exceed the nominal level $\alpha $ under any parameter sequence $\{ \gamma _{n,h} \}$ in (2.19). To further shed light on the behavior of bootstrap statistics with the dependent transformation, we apply the results in (2.20) and (2.21) and Theorems S.1 and S.2 in the Supplementary Material to plot the quantiles of the null limiting distributions of the original and bootstrap test statistics for the case of strong identification and conditional homoskedasticity studied in Guggenberger (Reference Guggenberger2010a). The limiting distributions of both the two-stage and shrinkage test statistics are substantially simplified in this case and depend on only two scalar parameters, say, $h_{1,ho}$ and $h_{2,ho}$ . $h_{1,ho}$ captures the degree of local endogeneity and $h_{2,ho}$ captures the IV strength, respectively.Footnote ¹⁷

Figure 1 reports the 95% quantiles of $\tilde {T}^S_{l,h}$ defined in (2.20) and its bootstrap counterpart $\tilde {T}^{S*}_{l,h}$ for $l \in \{\beta , \tau \}$ , as a function of $h_{1,ho}$ with $h_{2,ho} \in \{0.5, 1, 2\}$ , $\beta =0.05$ for the two-stage test statistic, and $\tau = 0.5$ for the shrinkage test statistic. The results are based on 1,000,000 simulation replications. We highlight some findings below. First, we observe that the quantiles of $\tilde {T}^{S*}_{l,h}$ for the dependent bootstrap turn out to be rather close to those of $\tilde {T}^S_{l,h}$ across various values of $h_{1,ho}$ and $h_{2,ho}$ . However, the figure suggests that this bootstrap procedure can have over-rejection when the quantiles of $\tilde {T}^S_{l,h}$ are relatively high (e.g., when $h_{2,ho}=0.5$ and $h_{1,ho}$ is between 5 and 6). In addition, we note that the quantiles of $\tilde {T}^{S*}_{l,h}$ for the dependent bootstrap converge in each sub-figure to the standard normal CV when the value of $h_{1,ho}$ increases: when $|h_{1,ho}|$ is large, the Hausman pretest rejects with high probability and the weight $w(H_n)$ shrinks toward zero, so that both two-stage and shrinkage tests become the 2SLS-based test, and the dependent bootstrap does mimic well such behavior.

Figure 1

95% quantiles of $\tilde {T}^S_{l,h}$ and $\tilde {T}^{S*}_{l,h}$ under strong identification and homoskedasticity.

Note: “Hausman” and “Hansen” denote the Hausman-type two-stage statistic $T^S_{\beta ,n}(\theta _0)$ with $\beta =0.05$ and Hansen’s (Reference Hansen2017) shrinkage statistic $T^S_{\tau ,n}(\theta _0)$ with $\tau =0.5$ , respectively. “BS-Hausman” and “BS-Hansen” denote their wild bootstrap analogs $T^{S*}_{\beta ,n}(\theta _0)$ and $T^{S*}_{\tau ,n}(\theta _0)$ with the dependent transformation. The results are based on 1,000,000 simulation replications.

Furthermore, the quantiles corresponding to the shrinkage test statistics and their bootstrap analogs (i.e., $\tilde {T}^S_{\tau ,h}$ and $\tilde {T}^{S*}_{\tau ,h}$ ) have smoother shapes than their two-stage counterparts (i.e., $\tilde {T}^S_{\beta ,h}$ and $\tilde {T}^{S*}_{\beta ,h}$ ), and this may be due to the fact that the two-stage test statistic uses an abrupt transition between the OLS and 2SLS-based statistics (especially when the IV strength is relatively low, e.g., when $h_{2,ho}=0.5$ ). In addition, Figure 2 shows the 95% quantiles of $\tilde {T}^W_{l,h}$ and $\tilde {T}^{W*}_{l,h}$ for $l \in \{\beta ,\tau \}$ as a function of $h_{1,ho}$ (from 0 to 20) with $h_{2,ho} \in \{0.5, 1, 2\}$ . We also observe smoother quantile shapes for the shrinkage statistics than for the two-stage statistics. This suggests that the shrinkage statistics may yield greater power than the Hausman-type two-stage statistics when appropriate CVs are used. Moreover, we note that similar to Figure 1, the two quantiles intersect at certain values of $h_{1,ho}$ , suggesting that the bootstrap tests with $T^W_{\beta ,n}(\theta _0)$ or $T^W_{\tau ,n}(\theta _0)$ do not have valid size control in general. In sum, Figures 1 and 2 suggest that the standard wild bootstrap-based two-stage or shrinkage tests are invalid in the unconditional sense, even under strong identification and conditional homoskedasticity.

Figure 2

95% quantiles of $\tilde {T}^W_{l,h}$ and $\tilde {T}^{W*}_{l,h}$ under strong identification and homoskedasticity.

Note: “Hausman” and “Hansen” denote the weak-identification-robust Hausman-type two-stage statistic $T^W_{\beta ,n}(\theta _0)$ with $\beta =0.05$ and Hansen’s (Reference Hansen2017) shrinkage statistic $T^W_{\tau ,n}(\theta _0)$ with $\tau =0.5$ , respectively. “BS-Hausman” and “BS-Hansen” denote their wild bootstrap analogs $T^{W*}_{\beta ,n}(\theta _0)$ and $T^{W*}_{\tau ,n}(\theta _0)$ with the dependent transformation. The results are based on 1,000,000 simulation replications.

In addition, we report in Table 1 below the asymptotic null rejection probabilities of the two-stage and shrinkage tests with the independent and dependent bootstrap CVs for the homoskedastic and strong-IV case, with $\alpha =0.05$ for $\underline {\kappa }_{ho} \in \{0.001, 0.1, 0.5, 1, 2, 10\}$ , where $\underline {\kappa }_{ho}$ denotes the lower bound of $h_{2,ho}$ , $\beta \in \{0.05, 0.1, 0.2\}$ for the two-stage test, and $\tau \in \{0.1, 0.25,0 .5\}$ for the shrinkage test. First, we note that both $"T^S_{\beta }"$ and $"T^S_{\tau }"$ with independent bootstrap CVs have asymptotic rejection probabilities much larger than $0.05$ ; e.g., when $\underline {\kappa }_{ho} =0.001$ , the probabilities are $100\%, 95.1\%, 85.1\%$ , and $100\%, 100\%, 89\%$ , respectively. The corresponding probabilities of $"T^W_{\beta }"$ and $"T^W_{\tau }"$ for $\underline {\kappa }_{ho} =0.001$ are $13.5\%, 8.9\%, 8.7\%$ , and $12.3\%, 10.4\%, 8.4\%$ , respectively. Second, we note that although the dependent bootstrap CVs yield asymptotic size distortions much smaller than those of the independent bootstrap CVs, they, in general, fail to achieve uniform size control. Third, as the heteroskedastic case nests the homoskedastic case, these results could be regarded as the lower bounds of the asymptotic size distortions under heteroskedasticity and strong identification.

Table 1

Asymptotic null rejection probabilities (in %) for $\alpha =0.05$ under homoskedasticity

Note: “ $T^S_{\beta }$ ” and “ $T^S_{\tau }$ ” denote the Hausman-type two-stage test statistic and Hansen’s (Reference Hansen2017) shrinkage test statistic, respectively, and “ $T^W_{\beta }$ ” and “ $T^W_{\tau }$ ” denote their weak-identification-robust counterparts. “BS-ind” and “BS-dep” denote the standard wild bootstrap with independent and dependent transformation, respectively. The results are based on 1,000,000 simulation replications.

3.2 Size-Adjusted Wild Bootstrap

As illustrated in the previous section, standard wild bootstrap procedures do not provide uniform size control, even under strong identification and conditional homoskedasticity. To achieve uniform validity, including the scenarios with weak identification and heteroskedasticity, we propose, in this section, Bonferroni-based size-correction methods for the bootstrap tests based on $T^W_{\beta ,n}(\theta _0)$ and $T^W_{\tau ,n}(\theta _0)$ , the identification-robust two-stage and shrinkage statistics. As explained in McCloskey (Reference McCloskey2017), the idea behind such size-correction is to construct CVs that use the data to determine how far the key nuisance parameter (i.e., the endogeneity parameter in the current testing problem) is from the point that causes the discontinuity in the limiting distributions of the test statistics. Although the key nuisance parameter cannot be consistently estimated under the drifting sequences in (2.19), it is still possible to construct an asymptotically valid confidence set for it and then construct adaptive CVs that control the asymptotic size.

First, we will construct a size-adjusted wild bootstrap CV by using the wild bootstrap CVs with the independent transformation in Section 3.1 and Bonferroni bounds. Note that although the localization parameter $h_1$ cannot be consistently estimated, we may still construct an asymptotically valid confidence set for $h_1$ by defining $\hat {h}_{n,1}(\theta _0) = n^{1/2} \hat {a}(\theta _0)$ , where $\hat {a}(\theta _0) = \left ( \hat {v}'\hat {v} \right )^{-1}\hat {v}'\tilde {y}(\theta _0)$ . A confidence set of $h_1$ can be constructed by using the fact that under the drifting parameter sequences and $H_0: \theta = \theta _0$ ,

(3.3) $$ \begin{align} \hat{h}_{n,1}(\theta_0) \rightarrow^d \tilde{h}_1\sim N \Big( h_1, h^{-2}_{25}h_{23} \Big). \end{align} $$

Then, uniformly valid size-adjusted bootstrap CVs for testing $H_0: \theta = \theta _0$ under $T^W_{\beta ,n}(\theta _0)$ or $T^W_{\tau ,n}(\theta _0)$ can be constructed by using Bonferroni bounds: we may construct a $1-(\alpha - \delta )$ level first-stage confidence set for $h_1$ , and then take the maximal $(1-\delta )$ th quantile of appropriately generated bootstrap statistics over the first-stage confidence set. Specifically, let $\hat {h}_{n,2} = \left (\hat {h}^{\prime }_{n,21}, vec(\hat {h}_{n,22})', \hat {h}_{n,23}, vec(\hat {h}_{n,24})',\hat {h}_{n,25} \right )'$ be the estimators of $h_{2}$ , and define the $1-(\alpha -\delta )$ level confidence set of $h_1$ for some $0<\delta \leq \alpha <1$ as

$$ \begin{align*} CI_{_{\alpha-\delta}}(\hat{h}_{_{n,1}}(\theta_0) )& = \left[ \hat{h}_{n,1}(\theta_0) - z_{1-(\alpha-\delta)/2} \cdot \left( n \hat{V}_{a}(\theta_0) \right)^{1/2}, \; \hat{h}_{n,1}(\theta_0) \right.\\&\quad\left.+\ z_{1-(\alpha-\delta)/2} \cdot \left( n \hat{V}_{a}(\theta_0) \right)^{1/2}\right], \end{align*} $$

where $\hat {V}_{a}(\theta _0)$ is defined in (2.9). The wild bootstrap-based simple Bonferroni CV (SBCV) is defined as

(3.4) $$ \begin{align} c_l^{B-S} (\alpha,\alpha-\delta,\hat{h}_{n,1}(\theta_0), \hat{h}_{n,2}) = \underset{h_1 \in CI_{_{\alpha-\delta}}(\hat{h}_{n,1} (\theta_0))} \sup c^*_{l, (h_1, \hat{h}_{n,2})}(1-\delta),\; \end{align} $$

for $l \in \{\beta ,\tau \}$ , where $c^*_{l, (h_1, \hat {h}_{n,2})}(1-\delta )$ is the $(1-\delta )$ th quantile of the distribution of $T^{W*}_{l,n,(h_1, \hat {h}_{n,2})} (\theta _0)$ , i.e., the distribution of the bootstrap analog of $T^W_{l,n}(\theta _0)$ generated under the specific value of the localization parameter equal to $h_1$ .

As we saw in the previous section, standard wild bootstrap procedures do not accurately mimic the localization parameter $h_1$ , whether it is an independent or a dependent transformation. Therefore, for a given value of the localization parameter $h_1$ , we propose to generate the bootstrap two-stage test statistic $T^{W*}_{\beta ,n,(h_1, \hat {h}_{n,2})} (\theta _0)$ as follows:

(3.5)

where $T^*_{ols, (h_1, \hat {h}_{n,2})}(\theta _0)$ and $H^*_{n,(h_1, \hat {h}_{n,2})}(\theta _0)$ are the bootstrap analogs of $T_{ols}(\theta _0)$ and $H_n(\theta _0)$ , respectively, evaluated at the specific value of the localization parameter equal to $h_1$ . More precisely, to obtain these bootstrap analogs, we first generate the bootstrap counterparts of the OLS and regression endogeneity parameter estimators under $h_1$ :

(3.6) $$ \begin{align} \hat{\theta}^*_{ols, (h_1, \hat{h}_{n,2})} &= \hat{\theta}^*_{ols} +\big( \hat{h}^{\prime}_{n,21}\hat{h}_{n,24}\hat{h}_{n,21} + \hat{h}_{n,25}\big)^{-1}\hat{h}_{n,25} \left( n^{-1/2}h_1 \right), \quad \notag \\ \hat{a}^*_{(h_1, \hat{h}_{n,2})}(\theta_0) &= \hat{a}^*(\theta_0) +n^{-1/2}h_1,\; \end{align} $$

where $\hat {\theta }^*_{ols}$ and $\hat {a}^*(\theta _0)$ are generated by the standard wild bootstrap procedure in Section 3.1 with independent transformation of disturbances, so that $\hat {\theta }^*_{ols}$ and $\hat {a}^*(\theta _0)$ have localization parameter equal to zero in the bootstrap world. By doing so, $\sqrt {n} \left ( \hat {\theta }^*_{ols,(h_1, \hat {h}_{n,2})} - \theta _0 \right )$ and $\sqrt {n} \hat {a}^*_{(h_1, \hat {h}_{n,2})} (\theta _0)$ have appropriate null limiting distributions conditional on the data. Then, we obtain

(3.7) $$ \begin{align} T^*_{ols,(h_1, \hat{h}_{n,2})}(\theta_0) = (\hat{\theta}^{*}_{ols,(h_1, \hat{h}_{n,2})} - \theta_0)^2 / \hat{V}^{*}_{ols}, \quad H^*_{n,(h_1, \hat{h}_{n,2})}(\theta_0) = \hat{a}^{\ast^2}_{(h_1, \hat{h}_{n,2})}(\theta_0) / \hat{V}^{*}_{a}(\theta_0). \end{align} $$

$T^{W*}_{\tau ,n,(h_1, \hat {h}_{n,2})} (\theta _0)$ , the bootstrap analog of the shrinkage statistic, is generated similarly.

Furthermore, we can show that the following (conditional) convergence in distribution holds:

$$ \begin{align*} \begin{pmatrix} T^*_{ols,(h_1, \hat{h}_{n,2})}(\theta_0) \\ H^*_{n,(h_1, \hat{h}_{n,2})}(\theta_0) \end{pmatrix} \rightarrow^{d^*} \begin{pmatrix} \left(h^{\prime}_{21} h_{22}h_{21} + h_{23} \right)^{-1} \left( h^{\prime}_{21} \psi_{Ze}^{\ast} + \psi_{ve}^{\ast} + h_{25}h_1 \right)^2 \\ h_{23}^{-1} \left(\psi_{ve}^{\ast} + h_{25} h_1 \right)^{2} \end{pmatrix}, \; \notag \end{align*} $$

in probability P, where $\psi _{Ze}^*$ and $\psi _{ve}^*$ are bootstrap analogs of $\psi _{Ze}$ and $\psi _{ve}$ , respectively. This implies that conditional on the data, $T^{W*}_{\beta , n,(h_1, \hat {h}_{n,2})}(\theta _0)$ and $T^{W*}_{\tau , n,(h_1, \hat {h}_{n,2})}(\theta _0)$ , the bootstrap counterparts of the two-stage and shrinkage test statistics, have the desired null limiting distributions evaluated at the value of localization parameter equal to $h_1$ .

As seen in (3.4), the bootstrap SBCV equals the maximal quantile $c^*_{l, (h_1, \hat {h}_{n,2})} (1-\delta )$ over the values of the localization parameter $h_1$ in the set $CI_{_{\alpha -\delta }}(\hat {h}_{n,1}(\theta _0) )$ . We can now state the following asymptotic size result for $c_l^{B-S}(\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0), \hat {h}_{n,2})$ , where $l \in \{\beta ,\tau \}$ .

Theorem 3.1 states that tests based on $c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0),\hat {h}_{n,2})$ control the asymptotic size. In practice, $c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0),\hat {h}_{n,2})$ can be obtained by using the following algorithm.

Wild Bootstrap Algorithm for $\boldsymbol {c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{_{n,1}}(\theta _0),\hat {h}_{_{n,2}})}$ :

1. 1. Generate the bootstrap statistics $\left \{\hat {\theta }^{*(b)}_{ols}, \hat {a}^{*(b)}(\theta _0), \hat {V}^{*(b)}_{ols}, \hat {V}^{*(b)}_{a}(\theta _0), T^{*(b)}_{ar}(\theta _0) \right \}, b=1, \ldots , B,$ using the standard wild bootstrap procedure with independent transformation of disturbances.

2. 2. Choose $\alpha $ , $\delta $ , and compute $CI_{_{\alpha -\delta }}(\hat {h}_{n,1} (\theta _0))$ . Create a fine grid for $CI_{_{\alpha -\delta }}(\hat {h}_{n,1} (\theta _0))$ and call it $\mathscr {C}^{grid}_{_{\alpha -\delta }}.$

3. 3. For $l \in \{\beta , \tau \}$ and for $h_1 \in \mathscr {C}^{grid}_{_{\alpha -\delta }}$ , generate $T^{W*(b)}_{l, n, (h_1, \hat {h}_{n,2})}(\theta _0)$ , $b=1, \ldots , B$ , using the bootstrap statistics generated in Step 1. The same set of bootstrap statistics can be used repeatedly for each $h_1$ .

4. 4. Compute $c^*_{l, (h_1, \hat {h}_{n,2})}(1-\delta )$ , the $(1-\delta )$ th quantile of the distribution of $T^{W*}_{l, n, (h_1, \hat {h}_2)}(\theta _0)$ from these B draws of bootstrap samples.

5. 5. Find $c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{_{n,1}}(\theta _0), \hat {h}_{n,2}) = \underset { h_1 \in \mathscr {C}^{grid}_{_{\alpha -\delta }}}\sup c^*_{l, (h_1, \hat {h}_{n,2})}(1-\delta ).$

6. 6. Reject $H_0: \theta = \theta _0$ if $T^W_{l,n}(\theta _0)> c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{_{n,1}}(\theta _0), \hat {h}_{n,2})$ .

Note that, as shown in Theorem 3.1, although controlling the asymptotic size, the bootstrap SBCV defined above may yield a conservative test whose asymptotic size does not reach its nominal level. For further refinement on the Bonferroni bounds, we propose a size-adjustment method to adjust the bootstrap SBCV so that the resulting test is not conservative with an asymptotic size exactly equal to $\alpha $ . Specifically, for $l \in \{\beta , \tau \}$ , the size-adjustment factor for the bootstrap SBCV is defined as

(3.8) $$ \begin{align} \hat{\eta}_{l,n} = \inf \left\{\! \eta: \underset{h_1 \in \mathscr{H}_1} \sup P^* \left[\! T^{W*}_{l, n, (h_1, \hat{h}_{n,2})} (\theta_0)\!>\! c_l^{B-S} (\alpha, \alpha-\delta, \hat{h}^*_{n,1} (\theta_0, h_1), \hat{h}_{n,2}) + \eta \!\right] \!\leq \alpha \!\right\}, \end{align} $$

where $\hat {h}^*_{n,1}(\theta _0, h_1)$ denotes the bootstrap analog of $\hat {h}_{n,1}(\theta _0)$ with localization parameter equal to $h_1$ and is generated by the same bootstrap samples as those of $T^{W*}_{n, (h_1, \hat {h}_{n,2})} (\theta _0)$ . More precisely, we define

(3.9) $$ \begin{align} \hat{h}^*_{n,1} (\theta_0, h_1) = \hat{h}^*_{n,1}(\theta_0) + h_1, \end{align} $$

where $\hat {h}^*_{n,1}(\theta _0) = n^{1/2} \hat {a}^{\ast }(\theta _0) = \big (\hat {v}^{*\prime } \hat {v}^*\big )^{-1} \hat {v}^{*\prime }u^*$ , $\hat {v}^*=M_{Z}X^*,$ is generated by the standard wild bootstrap procedure with independent transformation so that the localization parameter equals zero in the bootstrap world. Notice that we have the following convergence in distribution (jointly with the other bootstrap statistics): $ \hat {h}^*_{n,1} (\theta _0, h_1) \rightarrow ^{d^*} N \Big ( h_1, h^{-2}_{25}h_{23} \Big ), $ in probability P, i.e., the same limiting distribution as that of $\hat {h}_{n,1}(\theta _0)$ in (3.3), under the specific value of $h_1$ .

The goal of the size-adjustment method is to decrease the bootstrap SBCV as much as possible by using the factor $\eta $ while not violating the inequality in (3.8), so that the asymptotic size of the resulting tests can be controlled. Then, the bootstrap size-adjusted CV (BACV) can be defined as

(3.10) $$ \begin{align} & c_l^{B-A} (\alpha,\alpha-\delta,\hat{h}_{n,1}(\theta_0), \hat{h}_{n,2}) \notag \\ &\quad= c_l^{B-S} (\alpha,\alpha-\delta,\hat{h}_{n,1}(\theta_0), \hat{h}_{n,2}) + \hat{\eta}_{l,n} \;\; \text{for} \;\; l \in \{\beta, \tau\}, \end{align} $$

and one can expect that relatively small $\hat {\eta }_{l,n}$ results in a relatively less conservative (and more powerful) test. Furthermore, we notice that the bootstrap-based size-adjustment method in (3.10), which employs a size-adjustment factor, is in the same spirit as the adjusted Bonferroni CV proposed in McCloskey (Reference McCloskey2017, Sect. 3.2), which is based on adjusting the quantile level of the underlying localized quantile in the SBCV.

Below, we state the theorem on the uniform size control of wild bootstrap CVs using the size-adjustment method. We assume a continuity condition on the NRP function, following similar continuity assumptions in Andrews and Cheng (Reference Andrews and Cheng2012, p. 2195, Assump. Rob2(i)) and Han and McCloskey (Reference Han and McCloskey2019, p. 1052, Assump. DF2(ii)). Define $c^{B-S}_l(\alpha , \alpha -\delta , \tilde {h}_1, h_2) = \sup _{h_1 \in CI_{\alpha -\delta } (\tilde {h}_1)} c_{l,h} (1-\delta )$ , where $c_{l,h}(1-\delta )$ is the $(1-\delta )$ th quantile of $\tilde {T}^W_{l,h}$ , and $\tilde {T}^W_{l,h}$ is the weak limit of $T^W_{l,n}(\theta _0)$ under the sequence $\{\gamma _{n,h}\} \subset \Gamma $ satisfying (2.19) for $l \in \{\beta , \tau \}$ .

Furthermore, let $CS_{l,n}(1-\alpha )$ denote the confidence set of the nominal level $1-\alpha $ for $\theta $ constructed by collecting all values of $\theta $ that cannot be rejected by the corresponding two-stage or shrinkage size-adjusted test at the nominal level $\alpha $ .

Theorem 3.2 shows that $c_l^{B-A} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0), \hat {h}_{n,2})$ yield two-stage and shrinkage bootstrap tests with the correct asymptotic size, irrespective of identification strength, and Corollary 3.1 states that the confidence sets constructed from inverting these tests have correct asymptotic coverage probability.Footnote ¹⁸ To implement such size-adjusted tests in practice, we must compute $c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{_{n,1}}(\theta _0),\hat {h}_{n,2})$ and $\hat {\eta }_{l,n}$ . These values can be computed sequentially starting with $c_{l}^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0),\hat {h}_{n,2})$ . Then the size-adjustment factor $\hat {\eta }_{l,n}$ can be calculated by evaluating (3.8) on a fine grid of $\mathscr {H}_1$ as follows.

Wild Bootstrap Algorithm for $\boldsymbol {c_l^{B-A} (\alpha ,\alpha -\delta ,\hat {h}_{_{n,1}}(\theta _0),\hat {h}_{_{n,2}})}$ :

1. 1. Generate bootstrap statistics $\left \{\hat {\theta }^{*(b)}_{ols}, \hat {a}^{*(b)}(\theta _0), \hat {V}^{*(b)}_{ols}, \hat {V}^{*(b)}_{a}(\theta _0), T^{*(b)}_{ar}(\theta _0) \right \}, b=1, \ldots , B,$ using the standard wild bootstrap procedure with independent transformation.

2. 2. For $l \in \{\beta , \tau \}$ , let $c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{n,1} (\theta _0),\hat {h}_{n,2})$ be the obtained SBCV.

3. 3. Create a fine grid of the set $\mathscr {H}_1$ in (3.8) and call it $\mathscr {H}^{grid}_1$ . For $l \in \{\beta , \tau \}$ and for each $h_1 \in \mathscr {H}^{grid}_1$ , obtain $T^{W*(b)}_{l, n,(h_1, \hat {h}_{n,2})} (\theta _0)$ and $c_l^{B-S} (\alpha , \alpha -\delta , \hat {h}^{*(b)}_{n,1} (\theta _0, h_1), \hat {h}_{n,2})$ , $b=1, \ldots , B$ , using the bootstrap statistics generated in Step 1. Note that the same set of bootstrap statistics can be used for each $h_1$ .

4. 4. Create a fine grid of $[-c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0), \hat {h}_{n,2}),\,0]$ and call it .

5. 5. Find all s.t.

   
   and set $\hat {\eta }_{l,n}$ equal to the smallest $\eta $ .

6. 6. The BACV is given by $c_l^{B-A} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0), \hat {h}_{n,2}) = c_l^{B-S} (\alpha ,\alpha -\delta ,\hat {h}_{n,1}(\theta _0), \hat {h}_{n,2}) + \hat {\eta }_{l,n}.$

7. 7. Reject $H_0: \theta = \theta _0$ if $T^W_{l,n}(\theta _0)> c_l^{B-A} (\alpha ,\alpha -\delta ,\hat {h}_{_{n,1}}(\theta _0), \hat {h}_{n,2})$ .

Several remarks are in order. First, we emphasize that $\hat {h}^*_{n,1}(\theta _0, h_1)$ needs to be generated simultaneously with $T^{W*}_{l, n, (h_1, \hat {h}_{n,2})} (\theta _0)$ using the same bootstrap samples, so that the dependence structure between the statistics $T^W_{l, n}(\theta _0)$ and $\hat {h}_{n,1}(\theta _0)$ is well mimicked by the bootstrap statistics. This is important for the size-adjustment procedure to correct the conservativeness of the Bonferroni bound. Similarly, for the implementation of the size adjustment, one cannot replace $c_l^{B-S} (\alpha , \alpha -\delta , \hat {h}^*_{n,1} (\theta _0, h_1), \hat {h}_{n,2}) $ in (3.8) with $c_l^{B-S} (\alpha , \alpha -\delta , \hat {h}_{n,1}(\theta _0), \hat {h}_{n,2})$ , as it also breaks down the dependence structure.

Second, when $|h_1| = \infty $ , $H_n(\theta _0)$ diverges to infinity, so that $T^W_{\beta ,n}(\theta _0)$ and $T^W_{\tau ,n}(\theta _0)$ are equal to $T_{ar}(\theta _0)$ with probability approaching 1. Furthermore, when $|h_1| = \infty $ , $\hat {h}_{n,1}(\theta _0)$ and thus the confidence interval $CI_{\alpha - \delta }(\hat {h}_{n,1}(\theta _0))$ also diverge to infinity. We note that the bootstrap SBCV is equal to $\sup _{ h_1 \in CI_{\alpha - \delta } (\hat {h}_{n,1}(\theta _0))} c^*_{l, (h_1, \hat {h}_{n,2})} (1-\delta ), $ where $l \in \{\beta , \tau \}$ , and $c^*_{l, (h_1, \hat {h}_{n,2})} (1-\delta )$ denotes the $(1-\delta )$ th quantile of the bootstrap version of the two-stage/shrinkage statistic under the specific value of $h_1$ . Therefore, when $|h_1| = \infty $ , the SBCV will be equal to the corresponding quantile of $T^*_{ar}(\theta _0)$ (i.e., the bootstrap version of $T_{ar}(\theta _0)$ ) with probability approach 1. As a result, the limiting NRP of our proposed bootstrap tests is controlled in this case as well. In addition, our size-adjustment factor $\hat {\eta }_{l,n}$ in the BACV will decrease the SBCV as much as possible to achieve the correct asymptotic size.

Third, we note that in finite samples, as the transition between the OLS and IV-based methods can be relatively abrupt for the two-stage statistic compared to the shrinkage statistic, the finite sample behavior of the size-adjusted bootstrap tests with $T^W_{\beta ,n}(\theta _0)$ may be closer to that of $T_{ar}(\theta _0)$ (this is also observed in our simulations).

3.3 Extension to Clustered Data

Many applications in economics involve error terms that are correlated within clusters (see, e.g., Cameron and Miller, Reference Cameron and Miller2015 and the references therein), and various studies in the literature on cluster-robust inference recommend using wild cluster bootstrap as a way to obtain more accurate inference, including Cameron et al. (Reference Cameron, Gelbach and Miller2008), MacKinnon and Webb (Reference MacKinnon and Webb2017), Djogbenou et al. (Reference Djogbenou, MacKinnon and Nielsen2019), and MacKinnon et al. (Reference MacKinnon, Nielsen and Webb2023), among others. However, by using arguments similar to those for the IV model with heteroskedastic errors, we can show that the standard wild cluster bootstrap is invalid in the current context for the two-stage testing and shrinkage procedures. In this section, we extend the (weak-identification-robust) size-adjusted bootstrap procedure proposed in Section 3.2 to the case with clustered samples.

To proceed, consider the following linear IV model with clustered data:

(3.11) $$ \begin{align} y_g = X_g \theta + u_g, \quad X_g = Z_g \pi + v_g, \end{align} $$

where $y_g = (y_{g1}, \ldots , y_{g n_g})'$ , $X_g=(X_{g1}, \ldots , X_{gn_g})'$ , and $Z_g=(Z_{g1}, \ldots , Z_{gn_g})'$ denote an $n_g \times 1$ vector of dependent variables, an $n_g \times 1$ vector of endogenous regressors, and an $n_g \times k$ matrix of instruments for the gth cluster, respectively. Let G denote the number of clusters and n denote the total number of observations. Similar to the case with heteroskedastic data, we can define the extended regression

(3.12) $$ \begin{align} y_g = X_g \theta + v_g a^c + e_g, \end{align} $$

where $a^c = \left (n^{-1} \sum _{g=1}^G E_F[v_g'v_g] \right )^{-1} \left (n^{-1} \sum _{g=1}^G E_F[v_g' u_g] \right )$ , and the weak identification and cluster-robust test statistic for the null of exogeneity $H^c_a: a^c=0$ takes the form of

(3.13) $$ \begin{align} H^c_n(\theta_0) = (\hat{a}^{c}(\theta_0))^2/ \hat{V}^c_a(\theta_0), \end{align} $$

where $\hat {V}^c_a(\theta _0) = (n^{-1} \hat {v}'\hat {v})^{-1} \left ( n^{-2} \sum _{g=1}^G \hat {v}_g' \hat {e}_g (\theta _0) \hat {e}(\theta _0)' \hat {v}_g \right ) (n^{-1} \hat {v}' \hat {v})^{-1}, \hat {a}^c(\theta _0) = (\hat {v}'\hat {v})^{-1} \hat {v}'\tilde {y}(\theta _0)$ , $\hat {v}'\hat {v} = \sum _{g=1}^G \hat {v}_g' \hat {v}_g$ , and $\hat {v}'\tilde {y}(\theta _0) = \sum _{g=1}^G \hat {v}_g'\tilde {y}_g(\theta _0)$ . In the same fashion, we define $T^c_{ols}(\theta _0)$ and $T^c_{ar}(\theta _0)$ , the cluster-robust OLS-based Wald and AR statistics, following the definitions in the heteroskedastic case. Then, the weak identification and cluster-robust two-stage test statistic $T^{Wc}_{\beta ,n}(\theta _0)$ and shrinkage test statistic $T^{Wc}_{\tau ,n}(\theta _0)$ can be defined according to the definitions in Section 2.1, i.e.,

(3.14)

For clustered data, we define the vector of nuisance parameters $\gamma _n^c = (\gamma ^c_{n,1}, \gamma ^c_{n,2}, \gamma ^c_{n,3})$ by

(3.15) $$ \begin{align} \gamma^c_{n,1} = a_n^c, \;\; \gamma^c_{n,2} = (\gamma^c_{n,21},\gamma^c_{n,22},\gamma^c_{n,23},\gamma^c_{n,24},\gamma^c_{n,25}),\;\; \gamma^c_{n,3} = F_n, \end{align} $$

and the corresponding parameter space by $\Gamma ^c = \{ \gamma ^c_n = (\gamma ^c_{n,1}, \gamma ^c_{n,2}, \gamma ^c_{n,3}): \gamma ^c_{n,1} \in \Gamma ^c_1, \gamma ^c_{n,2} \in \Gamma ^c_2, \gamma ^c_{n,3} \in \Gamma ^c_3(\gamma ^c_{n,1}, \gamma ^c_{n,2}) \}$ , where

(3.16) $$ \begin{align} & \Gamma^c_1= \mathbb{R}, \;\; \Gamma^c_2 = \Big\{(\gamma^c_{n,21},\gamma^c_{n,22},\gamma^c_{n,23},\gamma^c_{n,24},\gamma^c_{n,25}): \gamma^c_{n,21} = \pi_n \in \mathbb{R}^k, \notag \\ & \gamma^c_{n,22} = \mu_{n} \left( n^{-2} \sum_{g=1}^G E_{F_n} Z_g' e_g e_g' Z_g \right) \in \mathbb{R}^{k \times k}, \Big. \notag \\ & \gamma^c_{n,23}= \mu_n \left( n^{-2} \sum_{g=1}^G E_{F_n} v_g' e_g e_g' v_g \right) \in \mathbb{R}, \notag \\ & \gamma^c_{n,24}= n^{-1} \sum_{g=1}^G E_{F_n} Z_g' Z_g \in \mathbb{R}^{k \times k}, \gamma^c_{n,25}= n^{-1} \sum_{g=1}^G E_{F_n} v_g'v_g \in \mathbb{R}, \Big. \notag \\ & \textit{s.t.}\;\; \lambda_{min}(\gamma^c_{n,22}) \geq \underline{\kappa}, \gamma^c_{n,23}> 0, \lambda_{min}(\gamma^c_{n,24}) \geq \underline{\kappa}, \; \text{and} \; \gamma^c_{n,25} > 0 \Big\}, \end{align} $$

for some $\underline {\kappa }>0 $ that does not depend on n, and $\{ \mu _n \}$ is a non-random sequence, which plays a similar role as that used in Djogbenou et al. (Reference Djogbenou, MacKinnon and Nielsen2019) and is needed because different from the model with heteroskedastic errors, the convergence rate of the estimators $\hat {\theta }^c_{ols}$ and $\hat {a}^c(\theta _0)$ under clustering depends on various factors, such as the regressor, the relative cluster size, and the intra-cluster correlation (also see Hansen and Lee, Reference Hansen and Lee2019, Sect. 4 for related discussions). As pointed out by Djogbenou et al. (Reference Djogbenou, MacKinnon and Nielsen2019), the sequence $\{ \mu _n \}$ can be interpreted as the rate at which information accumulates, and because of the studentization of the test statistics, $\{ \mu _n \}$ needs not be known in practice, but only needs to exist. Furthermore, we allow for the case with weak identification under clustering.

In addition, $\Gamma ^c_3(\gamma ^c_{n,1}, \gamma ^c_{n,2})$ is defined as follows:

(3.17) $$ \begin{align} & \Gamma^c_3(\gamma^c_{n,1}, \gamma^c_{n,2}) = \Big\{ F_n: \; E_{F_n} e_g' v_g = E_{F_n} Z_g'e_g = E_{F_n} Z_g'v_g = 0,\; \notag \\ & E_{F_n} Z_g'e_g e_g'v_g = E_{F_n} Z_g' v_g e_g'v_g = E_{F_n} Z_g' e_g v_g' Z_g =0,\; \Big. \notag \\ & \mu_n \left( n^{-2} \sum_{g=1}^G E_{F_n} Z_g'v_g v_g'Z_g \right) \in \mathbb{R}^{k \times k} \;\; \text{with} \;\; \lambda_{min} \left( \mu_n n^{-2} \sum_{g=1}^G E_{F_n} Z_g'v_g v_g'Z_g \right) \geq \underline{\kappa}, \; \Big. \notag \\ & \left\lVert \sup_{g,i} E_{F_n} \left( ||Z_{gi} e_{gi}||^{2+\xi}, ||Z_{gi} v_{gi}||^{2+\xi}, |v_{gi}e_{gi}|^{2+\xi}, || Z_{gi} Z_{gi}' ||^{2+\xi}, |X_{gi}|^{2(2+\xi)} \right)'\right\rVert \leq M \Big\}, \notag \\ \end{align} $$

for some constant $\underline {\kappa }>0, \xi >0$ , $M<\infty $ , and $\{ \mu _n \}$ is the non-random sequence defined above. We then define the whole nuisance parameter space $\Gamma ^c$ of $\gamma ^c_n$ as $ \Gamma ^c = \{ \gamma ^c_n = (\gamma ^c_{n,1}, \gamma ^c_{n,2}, \gamma ^c_{n,3}): \gamma ^c_{n,1} \in \Gamma ^c_1, \gamma ^c_{n,2} \in \Gamma ^c_2, \gamma ^c_{n,3} \in \Gamma ^c_3(\gamma ^c_{n,1}, \gamma ^c_{n,2})\}. $

Now, let us define

(3.18) $$ \begin{align} \mathscr{H}^c &= \Bigg\{ h^c = (h^c_1, h^{c'}_{21},vec(h^c_{22})', h^c_{23},vec(h^c_{24})', h^c_{25})' \in \mathbb{R}^{2k^2+k+3}_{\infty}: \notag \\ & \quad \exists \{\gamma^c_n = (\gamma^c_{n,1}, \gamma^c_{n,2}, \gamma^c_{n,3}) \in \Gamma^c: n \geq 1\} \quad s.t. \Big.\notag\\ &\quad \mu_n^{1/2}\gamma^c_{n,1} \rightarrow h^c_1 \in \mathbb{R}_{\infty},\;\;\gamma^c_{n,2} \rightarrow h^c_2=(h^c_{21},h^c_{22}, h^c_{23},h^c_{24},h^c_{25}), \; \lambda_{min}(A)\geq \underline{\kappa} \Big.\notag\\ & \;\;\;\; \text{for}\;\;A\in \{ h^c_{22}, h^c_{24}\}, \; h^c_{23}>0, \; h^c_{25}>0 \Bigg\} \; \end{align} $$

for some $\underline {\kappa }>0$ and $\mathbb {R}_{\infty } = \mathbb {R} \cup \{ \pm \infty \}.$ Similar to the heteroskedastic case, to derive the asymptotic size, it suffices to study the asymptotic NRP along a certain sequence $\{\gamma ^c_{n,h}\}$ for some $h^c \in \mathscr {H}^c$ , with $\gamma ^c_{n,h} = ( \gamma ^c_{n,h,1}, \gamma ^c_{n,h,2}, \gamma ^c_{n,h,3} )$ satisfying:

(3.19) $$ \begin{align} \mu_n^{1/2}\gamma^c_{n,h,1} \rightarrow h^c_1, \; \gamma^c_{n,h,2} \rightarrow h^c_2, \; \text{and} \; \gamma^c_{n,h,3} = F_n \in \Gamma^c_3(\gamma^c_{n,h,1}, \gamma^c_{n,h,2}). \end{align} $$

Now, we present the algorithm for the wild cluster bootstrap procedure with the independent transformation, which will be used to construct uniformly valid bootstrap CVs under clustering.

Wild Cluster Bootstrap Algorithm:

1. 1. Given $H_0: \theta = \theta _0$ , compute the residuals from the first-stage and structural equations: $\hat {v}_g = X_g -Z_g \hat {\pi }, \hat {u}_g(\theta _0) = y_g - X_g \theta _0$ , where $\hat {\pi }=(Z^{\prime }Z)^{-1}Z^{\prime }X=\left (\sum _{g=1}^G Z_g'Z_g \right )^{-1} \sum _{g=1}^G Z_g'X_g$ .

2. 2. Generate the cluster-level bootstrap pseudo-data following $X^{\ast }_g = Z_g \hat {\pi } + v^{\ast }_g, y^{\ast }_g = X_g^{\ast }\theta _0 + u^{\ast }_g$ , where $v^*_g = \hat {v}_g \omega ^*_{1g}$ , and $u^*_g = \hat {u}_g(\theta _0) \omega ^*_{2g}$ , for each $g=1, \ldots , G$ , where $\omega ^*_{1g}$ and $\omega ^*_{2g}$ are two random variables that have mean zero and variance $1$ , are independent from the data and independent from each other.

3. 3. Compute $\left \{\hat {\theta }^{c*}_{ols}(\theta _0), \hat {a}^{c*}(\theta _0), \hat {V}^{c*}_{ols}, \hat {V}^{c*}_{a}(\theta _0), T^{c*}_{ar}(\theta _0) \right \}$ by the bootstrap samples generated in Step 2.

Then, for a given value of $\dot {h}_1^c$ , we generate the bootstrap test statistics as follows:

(3.20) $$ \begin{align} & T^{c*}_{ols,(\dot{h}^c_1, \hat{h}^c_{n,2})}(\theta_0) \notag \\ &= \left( \left( \hat{\theta}^{c*}_{ols} - \theta_0 \right) / \hat{V}^{c*1/2}_{ols} + \left(\hat{h}_{n,21}^{c'} \hat{h}_{n,24}^c \hat{h}_{n,21}^c + \hat{h}_{n,25}^c\right)^{-1} \hat{h}_{n,25}^c \left(\hat{V}_a^{c*}(\theta_0) /\hat{V}_{ols}^{c*}\right)^{1/2} \dot{h}_1^c\right)^2, \quad \notag \\ & \text{and} \;\; H^{c*}_{n,(\dot{h}^c_1, \hat{h}^c_{n,2})} (\theta_0) = \left( \hat{a}^{c*}(\theta_0) / \hat{V}^{c*1/2}_a(\theta_0) + \dot{h}_{1}^c \right)^2, \end{align} $$

where $\hat {h}_{n,21}^c = \left (\sum _{g=1}^G Z_g'Z_g \right )^{-1} \sum _{g=1}^G Z_g'X_g$ , $\hat {h}_{n,24}^c = n^{-1} \sum _{g=1}^G Z_g'Z_g$ , and $\hat {h}_{n,25}^c = n^{-1} \sum _{g=1}^G \hat {v}_g'\hat {v}_g$ .

The bootstrap analogs of the two-stage and shrinkage test statistics evaluated at $\dot {h}_1^c$ , $T^{Wc*}_{l, n,(\dot {h}^c_1, \hat {h}^c_{n,2})} (\theta _0)$ , can be obtained subsequently. Notice that because of the appropriate studentization, $\{ \mu _n \}$ is also not needed in the procedure described by (3.20).

Now, let $CI_{_{\alpha -\delta }}(\hat {\dot {h}}^c_{n,1})$ denote the $1-(\alpha -\delta )$ level confidence set for $\dot {h}^c_1$ for some $0<\delta \leq \alpha <1$ , where $\hat {\dot {h}}^c_{n,1} = \hat {a}^c(\theta _0) / (\hat {V}^{c}_a(\theta _0))^{1/2}$ . The SBCV for clustered data is then defined as

(3.21) $$ \begin{align} c_l^{B-S} (\alpha,\alpha-\delta,\hat{\dot{h}}^c_{n,1}, \hat{h}^c_{n,2}) = \underset{\dot{h}^c_1 \in CI_{_{\alpha-\delta}}(\hat{\dot{h}}^c_{n,1})} \sup c^*_{l, (\dot{h}^c_1, \hat{h}^c_{n,2})}(1-\delta) \; \text{for} \; l \in \{\beta,\tau\}, \end{align} $$

where $c^*_{l, (\dot {h}^c_1, \hat {h}^c_{n,2})}(1-\delta )$ is the $(1-\delta )$ th quantile of the distribution of $T^{Wc*}_{l, n,(\dot {h}^c_1, \hat {h}^c_{n,2})} (\theta _0)$ , which is the bootstrap analog of $T^{Wc}_{l,n}(\theta _0)$ generated under the value of the localization parameter equal to $\dot {h}^c_1$ . The specific size-adjusted bootstrap algorithm for the SBCV closely follows that for heteroskedastic data in Section 3.2 and is thus omitted for conciseness.

For further refinement on the Bonferroni bound, we define the size-adjustment factor $\hat {\eta }^c_{l,n}$ following (3.8). Then, the BACV for the case with clustered data can be defined as

(3.22) $$ \begin{align} c_l^{B-A} (\alpha,\alpha-\delta,\hat{\dot{h}}^c_{n,1}(\theta_0), \hat{h}^c_{n,2}) = c_l^{B-S} (\alpha,\alpha-\delta,\hat{\dot{h}}^c_{n,1}(\theta_0), \hat{h}^c_{n,2}) + \hat{\eta}^c_{l,n}. \end{align} $$

Similarly, its algorithm follows closely that described in Section 3.2.

For the sake of conciseness, we present the asymptotic validity results for our size-adjusted bootstrap tests under clustering in Section S.4 of the Supplementary Material.