2.1 Setup

Let $\nu $ be a probability measure on $\left( \mathbb {R}, \mathscr {B}(\mathbb {R}) \right)$ . We first introduce the following assumptions.

Assumption 2.4. For every $\theta _0\in \Theta $ and every $\varepsilon>0$ , there exists $\delta>0$ such that

$$ \begin{align*} \sup_{x\in\mathbb{R}} P\left[ (\psi_{x,\theta}-\psi_{x,\theta_0})^2 \right] <\varepsilon \end{align*} $$

for all $\theta \in \Theta $ with $\left\Vert \theta -\theta _0 \right\Vert _2 <\delta $ .

Assumption 2.1 shows that we focus on moment restrictions that are continuous in x for every $\theta \in \Theta $ . Assumption 2.2 requires the absolute continuity of the Lebesgue measure $\mu $ with respect to the probability measure $\nu $ . For example, $\nu $ could be set as the probability measure corresponding to a normal distribution with a large variance.Footnote ² Assumption 2.3 is a common condition on the compactness of $\Theta $ . Assumption 2.4 can be understood as the continuity of $\psi _{x,\theta }$ with respect to $\theta $ under a certain metric.

Define a function space

$$ \begin{align*} \mathbb{D}_{\mathcal{L}0} = \left\{ \varphi\in\ell^\infty(\mathbb{R}\times\Theta): \theta\mapsto \varphi(\cdot,\theta), \text{ as a map from } \Theta \text{ to } {L^2(\nu)}, \text{ is continuous} \right\}. \end{align*} $$

In the definition of $\mathbb {D}_{\mathcal {L}0}$ , the continuity of the map $\theta \mapsto \varphi (\cdot ,\theta )$ is understood in the sense that for every $\theta _0\in \Theta $ and every $\varepsilon>0$ , there exists $\delta>0$ such that

$$ \begin{align*} \int_{\mathbb{R}} \left[ \varphi(x,\theta)-\varphi \left( x, \theta_0\right) \right]^2 \;\mathrm{d} \nu(x)<\varepsilon \end{align*} $$

for all $\theta \in \Theta $ with $\left\Vert \theta -\theta _0 \right\Vert _2 <\delta $ . Note that for every $x\in \mathbb {R}$ and all $\theta ,\theta _0\in \Theta $ , by Jensen’s inequality,

$$ \begin{align*} [\phi_P(x,\theta)-\phi_P(x,\theta_{0})]^2 \le P\left[ (\psi_{x,\theta}-\psi_{x,\theta_0})^2 \right]. \end{align*} $$

Since $\nu $ is a probability measure, Assumption 2.4 implies that $\phi _P\in \mathbb {D}_{\mathcal {L} 0}$ .

The proposition below provides an equivalent characterization of the null hypothesis in (2). We construct the test based on this equivalent characterization to avoid estimating the nuisance parameter $\theta $ under the null.

Proposition 2.1. If Assumptions 2.1–2.4 hold, then the null hypothesis in (2) is equivalent to

(5) $$ \begin{align} \mathrm{H}_0: \inf_{\theta\in\Theta}\int_{\mathbb{R}} \left[\phi_P(x,\theta) \right]^2 \;\mathrm{d} \nu(x)=0. \end{align} $$

It is worth noting that different measures $\nu $ may deliver different power properties of the test. However, searching for the optimal $\nu $ to maximize power is challenging, as it may depend in a complicated manner on the DGP.

The measure $\nu (\mathbb {R})$ is assumed to be finite (Assumption 2.2) to obtain the theoretical results in the article. In practice, we suggest setting $\nu $ to a normal probability measure with a large variance so that it does not heavily concentrate on some region of the real line, given no prior information about the DGP. Other probability measures satisfying Assumption 2.2 also work asymptotically for the proposed method. For finite samples, the simulation results in Section 4 show that normal probability measures with different variances ( $\mathcal {N}(0,1)$ , $\mathcal {N}(0,5^2)$ , $\mathcal {N}(0,10^2)$ ) perform well.

2.2 Test Statistic

We first restrict our attention to independent and identically distributed (i.i.d.) samples and will extend the results to dependent data in Section 3. Let $\widehat {P}_n$ be the empirical probability measure of the sample $\mathbf {Z}_n$ , which assigns weight $1/n$ to each observation $Z_i$ with $i\in \{1,\ldots ,n\}$ . Then the sample analog of $\phi _P$ is defined as

$$ \begin{align*} \widehat{\phi}_n(x,\theta)=\widehat{P}_n(\psi_{x,\theta})=\frac{1}{n}\sum_{i=1}^n \psi_{x,\theta}(Z_i) \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ . We present the exact function form of $\widehat {\phi }_n$ in every example.

Example 1.1 (Cont)

With the known function g, it follows by definition that

$$ \begin{align*} \widehat{\phi}_n(x,\theta)=\widehat{P}_n(\psi_{x,\theta})=\frac{1}{n}\sum_{i=1}^n \psi_{x,\theta}(Z_i)=\frac{1}{n}\sum_{i=1}^n g(Y_i,\theta)\mathbb{1}\{X_i\le x\} \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

Example 1.2 (Cont)

The cumulative distribution function G can be estimated by the empirical distribution function $\widehat {G}_{n}$ such that for every $x\in \mathbb {R}$ ,

$$ \begin{align*} \widehat{G}_{n}(x)=\widehat{P}_n(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( Z_i \right). \end{align*} $$

Then

$$ \begin{align*} \widehat{\phi}_n(x,\theta)=\widehat{P}_n(\psi_{x,\theta})=\widehat{P}_n(\mathbb{1}_{(-\infty,x]})+\widehat{P}_n(\mathbb{1}_{(-\infty,2\theta-x]})-1= \widehat{G}_{n}(x)+\widehat{G}_{n}(2\theta-x)-1 \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

Example 1.3 (Cont)

The cumulative distribution function G can be estimated by the empirical distribution function $\widehat {G}_{n}$ such that for every $x\in \mathbb {R}$ ,

$$ \begin{align*} \widehat{G}_{n}(x)=\widehat{P}_n(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( Z_i \right). \end{align*} $$

Then $\widehat {\phi }_n(x,\theta )=\widehat {P}_n(\psi _{x,\theta })=\widehat {P}_n(\mathbb {1}_{(-\infty ,x]})-G_0(x,\theta )=\widehat {G}_{n}(x)-G_0(x,\theta )$ for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

Example 1.4 (Cont)

For every $i\in \{1,\ldots , n\}$ , the observation $Z_i=(X_i,Y_i)$ . Let $\widehat {P}_n$ be the empirical distribution of $\{Z_i\}_{i=1}^n$ , and $\widehat {H}_{n}$ be its empirical distribution function so that

$$ \begin{align*} \widehat{H}_{n}(x,y)=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]\times(-\infty,y]}\left( X_i,Y_i \right) \end{align*} $$

for all $(x,y)\in \mathbb {R}^2$ . Let $\widehat {P}_{X,n}$ and $\widehat {P}_{Y,n}$ be the marginal distributions of $\widehat {P}_n$ , that is, the (marginal) empirical distributions of $\left\{ X_i \right\}_{i=1}^{n}$ and $\left\{ Y_i \right\}_{i=1}^{n}$ , respectively. It follows that

$$ \begin{align*} \widehat{\phi}_n(x,\theta)=\widehat{P}_n(\psi_{x,\theta})=\widehat{P}_{X,n}(\mathbb{1}_{(-\infty,x]})-\widehat{P}_{Y,n}(\mathbb{1}_{(-\infty,(x-\theta_1)/\theta_2]}) \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ . The marginal distribution functions F and G can be estimated by the empirical distribution functions $\widehat {F}_n$ and $\widehat {G}_n$ , respectively, where for every $x\in \mathbb {R}$ ,

$$ \begin{align*} &\widehat{F}_{n}(x)=\widehat{P}_{X,n}(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( X_i \right) \text{ and }\\ &\widehat{G}_{n}(x)=\widehat{P}_{Y,n}(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( Y_i \right). \end{align*} $$

This implies that $\widehat {\phi }_n(x,\theta )=\widehat {F}_{n} (x)-\widehat {G}_{n} [ (x-\theta _1)/\theta _2 ]$ for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

We may also test this null hypothesis with two independent samples of different sizes. This case will be discussed in Section C of the Supplementary Material, where we present the results for comparing multiple samples.

To obtain the asymptotic law of the stochastic process $\widehat {\phi }_n$ , we need the following assumption on the function class $\Psi $ .

Assumption 2.5. The function class $\Psi =\{\psi _{x,\theta }:(x,\theta )\in \mathbb {R}\times \Theta \}$ satisfies that

(6) $$ \begin{align} \sup_{f\in\Psi} |P(f)|<\infty \text{ and } \sup_{f\in\Psi} \left\vert f(z)-P(f) \right\vert <\infty \end{align} $$

for all $z\in \mathbb {R}^{d_z}$ and is P-Donsker in the sense that

(7) $$ \begin{align} \sqrt{n}(\widehat{P}_n-P)\rightsquigarrow \mathbb{W} \text{ in } \ell^\infty(\Psi) \end{align} $$

as $n\to \infty $ , where $\mathbb {W}$ is a P-Brownian bridge in $\ell ^\infty (\Psi )$ .

Lemma 2.1 establishes the consistency of $\widehat {\phi }_n$ and the weak convergence of $\sqrt {n}( \widehat {\phi }_n-\phi _P )$ in $\ell ^\infty (\mathbb {R}\times \Theta )$ as $n\to \infty $ .

Lemma 2.1. If Assumptions 2.4 and 2.5 hold, then $(\widehat {\phi }_n-\phi _P)\in \ell ^\infty (\mathbb {R}\times \Theta )$ for all $n\in \mathbb {Z}_+$ . In addition,

$$ \begin{align*} \sup_{(x,\theta)\in\mathbb{R}\times\Theta} \left\vert \widehat{\phi}_n(x,\theta)-\phi_P(x,\theta) \right\vert \xrightarrow{\mathbb{P}} 0 \text{ and } \sqrt{n}( \widehat{\phi}_n-\phi_P )\rightsquigarrow \mathbb{G}_0 \text{ in } \ell^\infty(\mathbb{R}\times\Theta) \end{align*} $$

as $n\to \infty $ , where $\mathbb {G}_{0}$ is some tight random element which almost surely takes values in $\mathbb {D}_{\mathcal {L}0}$ .

Define a function space

$$ \begin{align*} \mathbb{D}_{\mathcal{L}}=\left\{ \varphi\in\ell^{\infty}(\mathbb{R}\times\Theta): \int_{\mathbb{R}} \left[ \varphi (x,\theta) \right]^2 \;\mathrm{d} \nu(x)<\infty \text{ for all } \theta\in\Theta \right\}. \end{align*} $$

Define a map $\mathcal {L}$ on $\mathbb {D}_{\mathcal {L}}$ such that $\mathcal {L}(\varphi )= \inf _{\theta \in \Theta } \int _{\mathbb {R}} \left[ \varphi (x,\theta ) \right]^2 \;\mathrm {d} \nu (x)$ for every $\varphi \in \mathbb {D}_{\mathcal {L}}$ . Then under Assumptions 2.1–2.5, the null and the alternative hypotheses can be expressed as

(8) $$ \begin{align} \mathrm{H}_0: \mathcal{L} (\phi_P)=0 \text{ and } \mathrm{H}_1: \mathcal{L} (\phi_P)>0. \end{align} $$

To test the null hypothesis in (8), we set the test statistic to $n \mathcal {L} ( \widehat {\phi }_n )$ .

Next, we show that the map $\mathcal {L}$ is Hadamard directionally differentiable, but its Hadamard directional derivative is degenerate under $\mathrm {H}_0$ .Footnote ³ Define

$$ \begin{align*}\mathbb{D}_0=\left\{ \varphi\in \mathbb{D}_{\mathcal{L}0}:\mathcal{L}(\varphi)=0 \right\}.\end{align*} $$

The following lemma provides the Hadamard directional derivative of $\mathcal {L}$ and its first-order degeneracy under $\mathrm {H}_0$ .

Lemma 2.2. If Assumptions 2.3 and 2.4 hold, then $\mathcal {L}$ is Hadamard directionally differentiable at $\phi _P\in \mathbb {D}_{\mathcal {L}}$ tangentially to $\mathbb {D}_{\mathcal {L}0}$ with the Hadamard directional derivative

$$ \begin{align*} \mathcal{L}^{\prime}_{\phi_P}(h)=2\inf_{\theta\in \Theta_0(\phi_P)} \int_{\mathbb{R}} \phi_P(x,\theta)h(x,\theta) \;\mathrm{d} \nu(x) \text{ for all } h\in \mathbb{D}_{\mathcal{L}0}, \end{align*} $$

where $\Theta _0(\phi _P)= {\mathrm{arg\,min}}_{\theta \in \Theta }\int _{\mathbb {R}} \left[ \phi _P (x,\theta ) \right]^2 \;\mathrm {d} \nu (x)$ . Moreover, if $\phi _P\in \mathbb {D}_0$ , then the derivative $\mathcal {L}^{\prime }_{\phi _P}$ is well defined on the whole of $\ell ^\infty (\mathbb {R}\times \Theta )$ with $\mathcal {L}^{\prime }_{\phi _P}(h)=0$ for every $h\in \ell ^\infty (\mathbb {R}\times \Theta )$ .

The first-order degeneracy of $\mathcal {L}$ under $\mathrm {H}_0$ implies that we may need to find the second-order Hadamard directional derivative of $\mathcal {L}$ .Footnote ⁴ We assume the following conditions to guarantee the existence of the second-order Hadamard directional derivative of $\mathcal {L}$ .

Assumption 2.6. The function $\phi _P$ is twice differentiable with respect to $\theta $ , and the second partial derivative satisfies

(9) $$ \begin{align} \int_{\mathbb{R}} \sup_{\theta\in\Theta} \left\Vert \left. \frac{\partial^2 \phi_P (z,\vartheta)}{\partial \vartheta \partial \vartheta^{\texttt{T}}} \right\vert_{(z,\vartheta)=(x,\theta)} \right\Vert_{2}^2 \,\mathrm{d}\nu(x) <\infty, \end{align} $$

where $\left\Vert \cdot \right\Vert _{2}$ denotes the $\ell ^2$ operator norm of a matrix.

Assumption 2.7. The set $\Theta _0\equiv \{\theta \in \Theta :\int _{\mathbb {R}} \left[ \phi _P (x,\theta ) \right]^2 \;\mathrm {d} \nu (x)=0\}\subset \mathrm {int}(\Theta )$ , and there exist $\kappa \in (0,1]$ , $\overline {\varepsilon }>0$ , and $C>0$ such that for all $\varepsilon \in (0,\overline {\varepsilon })$ ,

(10) $$ \begin{align} \inf_{\theta\in\Theta\setminus\Theta_0^\varepsilon} \left\{\int_{\mathbb{R}} \left[ \phi_P(x,\theta) \right]^2 \;\mathrm{d} \nu(x)\right\}^{1/2} \ge C \varepsilon^\kappa. \end{align} $$

We provide Assumptions 2.6 and 2.7 following the basic idea of Chen and Fang (Reference Chen and Fang2019b). Assumption 2.6 requires the boundedness of the second partial derivative of $\phi _P$ in the sense of (9). Assumption 2.7 requires that the set $\Theta _0$ is in the interior of $\Theta $ and it is well separated. The condition in (10) is similar to the partial identification assumption used in Chernozhukov et al. (Reference Chernozhukov, Hong and Tamer2007, p. 1265). It is worth noting that these conditions are sufficient but not necessary for our results, as also mentioned by Chen and Fang (Reference Chen and Fang2019b). We impose such high-level conditions for theoretical completeness. In Section 4, we verify these assumptions for a conditional moment restriction model.

Lemma 2.3. If Assumptions 2.3, 2.4, 2.6, and 2.7 hold, and $\phi _P\in \mathbb {D}_0$ , then the function $\mathcal {L}$ is second-order Hadamard directionally differentiable at $\phi _P$ tangentially to $\mathbb {D}_{\mathcal {L}0}$ with the second-order Hadamard directional derivative

$$ \begin{align*} \mathcal{L}^{\prime\prime}_{\phi_P}(h)=\inf_{\theta\in\Theta_0(\phi_P)} \inf_{v\in \mathbb{R}^{d_{\theta}}} \left\Vert \left[ \Phi'(\theta) \right]^{\texttt{T}} v +\mathscr{H} (\theta) \right\Vert_{L^2(\nu)}^2 \mathrm{ for all } h\in \mathbb{D}_{\mathcal{L}0}, \end{align*} $$

where $\Phi '(\theta ): \mathbb {R}\to \mathbb {R}^{d_{\theta }}$ with

$$ \begin{align*} \Phi'(\theta)(x)=\frac{\partial \phi_P (z,\vartheta)}{\partial \vartheta}\Bigg\vert_{(z,\vartheta)=(x,\theta)}\quad \mathrm{for every } (x,\theta)\in\mathbb{R}\times\Theta, \end{align*} $$

and $\mathscr {H}:\Theta \to \ell ^\infty (\mathbb {R})$ with $\mathscr {H}(\theta )(x)=h(x,\theta )$ for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

With Lemma 2.3, the asymptotic null distribution of the test statistic $n\mathcal {L}( \widehat {\phi }_n )$ is obtained by applying the second-order delta method.

Proposition 2.2. If Assumptions 2.1–2.7 hold and $\mathrm {H}_0$ is true ( $\phi _P\in \mathbb {D}_0$ ), then

$$ \begin{align*} n \mathcal{L}(\widehat{\phi}_n ) \rightsquigarrow \mathcal{L}^{\prime\prime}_{\phi_P}\left( \mathbb{G}_0 \right) \text{ as } n\to \infty. \end{align*} $$

2.3 Bootstrap Procedure

The distribution of $\mathcal {L}^{\prime \prime }_{\phi _P}\left( \mathbb {G}_0 \right)$ in Proposition 2.2 is unknown because both the function $\mathcal {L}^{\prime \prime }_{\phi _P}$ and the stochastic process $\mathbb {G}_0$ depend on the unknown underlying distribution P. Motivated by Hong and Li (Reference Hong and Li2018) and Chen and Fang (Reference Chen and Fang2019b), we propose to approximate $\mathcal {L}^{\prime \prime }_{\phi _P}$ by a consistent estimator and approximate the distribution of $\mathbb {G}_0$ by bootstrap.Footnote ⁵ We use the numerical second-order Hadamard directional derivative $\widehat {\mathcal {L}}^{\prime \prime }_n$ to approximate $\mathcal {L}^{\prime \prime }_{\phi _P}$ , which is defined as

$$ \begin{align*} \widehat{\mathcal{L}}^{\prime\prime}_n(h)=\frac{\mathcal{L}( \widehat{\phi}_n+\tau_n h )-\mathcal{L}( \widehat{\phi}_n )}{\tau_n^2} \end{align*} $$

for all $h\in \ell ^\infty (\mathbb {R}\times \Theta )$ , where $\left\{ \tau _n \right\}$ is a sequence of tuning parameters satisfying the assumption below.Footnote ⁶

Assumption 2.8 provides the rate at which $\tau _n\downarrow 0$ . Under this condition, we show that $\widehat {\mathcal {L}}^{\prime \prime }_n$ approximates $\mathcal {L}^{\prime \prime }_{\phi _P}$ well in the following lemma.

Lemma 2.4. If Assumptions 2.1–2.8 hold and $\mathrm {H}_0$ is true ( $\phi _P\in \mathbb {D}_0$ ), then for every sequence $\left\{ h_n \right\}\subset \ell ^\infty (\mathbb {R}\times \Theta )$ and every $h\in \mathbb {D}_{\mathcal {L}0}$ such that $h_n\to h$ in $\ell ^\infty (\mathbb {R}\times \Theta )$ as $n\to \infty $ , we have

$$ \begin{align*} \widehat{\mathcal{L}}^{\prime\prime}_n\left( h_n \right) \xrightarrow{\mathbb{P}} \mathcal{L}^{\prime\prime}_{\phi_P} (h)\text{ as } n\to\infty. \end{align*} $$

We next approximate the distribution of $\mathbb {G}_0$ via bootstrap. The bootstrap sample $\mathbf {Z}_n^*=\{Z_i^*\}_{i=1}^n$ is i.i.d. drawn from the empirical distribution $\widehat {P}_n$ of the original sample $\mathbf {Z}_n$ . Equivalently, $\mathbf {Z}_n^*$ is a random sample of size n, drawn from the set $\mathbf {Z}_n$ with replacement. Let $\widehat {P}_n^*$ be the empirical distribution of $\mathbf {Z}_n^*$ . Then the bootstrap version of $\widehat {\phi }_n$ is $\widehat {\phi }_n^*$ such that

$$ \begin{align*} \widehat{\phi}_n^*(x,\theta)=\widehat{P}_n^*(\psi_{x,\theta})=\frac{1}{n}\sum_{i=1}^n \psi_{x,\theta}(Z_i^*) \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

Example 1.1 (Cont)

It follows by definition that

$$ \begin{align*} \widehat{\phi}_n^*(x,\theta)=\widehat{P}_n^*(\psi_{x,\theta})=\frac{1}{n}\sum_{i=1}^n \psi_{x,\theta}(Z_i^*)=\frac{1}{n}\sum_{i=1}^n g(Y_i^*,\theta)\mathbb{1}\{X_i^*\le x\} \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ , where $Z_i^*=(X_i^*,Y_i^*)$ .

Example 1.2 (Cont)

Define

$$ \begin{align*} \widehat{G}_n^*(x)=\widehat{P}_n^*(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( Z^*_i \right) \end{align*} $$

for every $x\in \mathbb {R}$ . Then

$$ \begin{align*} \widehat{\phi}_n^*(x,\theta)=\widehat{P}_n^*(\psi_{x,\theta})=\widehat{P}_n^*(\mathbb{1}_{(-\infty,x]})+\widehat{P}_n^*(\mathbb{1}_{(-\infty,2\theta-x]})-1= \widehat{G}_n^*(x)+\widehat{G}_n^*(2\theta-x)-1 \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

Example 1.3 (Cont)

Define

$$ \begin{align*} \widehat{G}_n^*(x)=\widehat{P}_n^*(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( Z^*_i \right) \end{align*} $$

for every $x\in \mathbb {R}$ . Then

$$ \begin{align*} \widehat{\phi}_n^*(x,\theta)=\widehat{P}_n^*(\psi_{x,\theta})=\widehat{P}_n^*(\mathbb{1}_{(-\infty,x]})-G_0(x,\theta)=\widehat{G}_n^*(x)-G_0(x,\theta) \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

Example 1.4 (Cont)

Define $\widehat {P}_n^*$ as the empirical distribution of $\{Z_i^*\}_{i=1}^n$ with $Z_i^*=(X_i^*,Y_i^*)$ . Let $\widehat {P}^*_{X,n}$ and $\widehat {P}^*_{Y,n}$ be the marginal distributions of $\widehat {P}_n^*$ , that is, the (marginal) empirical distributions of $\left\{ X_i^* \right\}_{i=1}^{n}$ and $\left\{ Y_i^* \right\}_{i=1}^{n}$ , respectively. It follows that

$$ \begin{align*} \widehat{\phi}_n^*(x,\theta)=\widehat{P}_n^*(\psi_{x,\theta})=\widehat{P}_{X,n}^*(\mathbb{1}_{(-\infty,x]})-\widehat{P}_{Y,n}^*(\mathbb{1}_{(-\infty,(x-\theta_1)/\theta_2]}) \end{align*} $$

for every $(x,\theta )\in \mathbb {R}\times \Theta $ . Define $\widehat {F}_{n}^*$ and $\widehat {G}_{n}^*$ to be the (marginal) empirical distribution functions of $\left\{ X_i^* \right\}_{i=1}^{n}$ and $\left\{ Y_i^* \right\}_{i=1}^{n}$ , respectively, such that for every $x\in \mathbb {R}$ ,

$$ \begin{align*} &\widehat{F}_{n}^*(x)=\widehat{P}_{X,n}^*(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( X_i^* \right)\text{ and } \\ &\widehat{G}_{n}^*(x)=\widehat{P}_{Y,n}^*(\mathbb{1}_{(-\infty,x]})=\frac{1}{n}\sum_{i=1}^{n} \mathbb{1}_{(-\infty,x]}\left( Y_i^* \right). \end{align*} $$

This implies that $\widehat {\phi }_n^*(x,\theta )=\widehat {F}_{n}^* (x)-\widehat {G}_{n}^*\left( (x-\theta _1)/\theta _2 \right)$ for every $(x,\theta )\in \mathbb {R}\times \Theta $ .

The following lemma establishes the conditional weak convergence of $\sqrt {n}(\widehat {\phi }_n^*-\widehat {\phi }_n)$ in probability as $n\to \infty $ .

Lemma 2.5. If Assumption 2.5 holds, then as $n\to \infty $ ,

$$ \begin{align*} \sup_{\Gamma\in\mathrm{BL}_1\left( \ell^\infty(\mathbb{R}\times\Theta) \right)}\left\vert \mathbb{E} \left[ \left. \Gamma \left( \sqrt{n} \left( \widehat{\phi}_n^*-\widehat{\phi}_n \right) \right) \right\vert \mathbf{Z}_n \right]-\mathbb{E} \left[ \Gamma \left( \mathbb{G}_0 \right) \right] \right\vert \xrightarrow{\mathbb{P}} 0, \end{align*} $$

and $\sqrt {n}( \widehat {\phi }_n^*-\widehat {\phi }_n )$ is asymptotically measurable, where $\mathbb {G}_0$ is defined as in Lemma 2.1.

With the numerical estimator $\widehat {\mathcal {L}}^{\prime \prime }_n$ for $\mathcal {L}^{\prime \prime }_{\phi _P}$ and a suitable bootstrap approximation $\sqrt {n}( \widehat {\phi }_n^*-\widehat {\phi }_n )$ for $\mathbb {G}_0$ at hand, we can naturally approximate the distribution of $\mathcal {L}^{\prime \prime }_{\phi _P}\left( \mathbb {G}_0 \right)$ by the conditional distribution of the bootstrap test statistic $\widehat {\mathcal {L}}^{\prime \prime }_n \{ \sqrt {n}( \widehat {\phi }_n^*-\widehat {\phi }_n ) \}$ given the original sample. This is justified by the following proposition.

Proposition 2.3. If Assumptions 2.1–2.8 hold and $\mathrm {H}_0$ is true ( $\phi _P\in \mathbb {D}_0$ ), then

$$ \begin{align*} \sup_{\Gamma\in\mathrm{BL}_1\left( \mathbb{R} \right)} \left\vert \mathbb{E} \left[ \left. \Gamma \left( \widehat{\mathcal{L}}^{\prime\prime}_n \left[ \sqrt{n} \left( \widehat{\phi}_n^* -\widehat{\phi}_n \right) \right] \right) \right\vert \mathbf{Z}_n \right]-\mathbb{E} \left[ \Gamma \left( \mathcal{L}^{\prime\prime}_{\phi_P} \left( \mathbb{G}_0 \right) \right) \right] \right\vert \xrightarrow{\mathbb{P}} 0 \end{align*} $$

as $n\to \infty $ .

2.4 Asymptotic Properties

Now we construct the test for the null hypothesis $\mathrm {H}_0$ . For a given level of significance $\alpha \in (0,1)$ , define the bootstrap critical value

$$ \begin{align*} \widehat{c}_{1-\alpha,n}=\inf\left\{ c\in \mathbb{R}: \mathbb{P}\left(\left. \widehat{\mathcal{L}}^{\prime\prime}_n \left[ \sqrt{n} \left( \widehat{\phi}_n^*-\widehat{\phi}_n \right) \right] \le c \right\vert \mathbf{Z}_n\right) \ge 1-\alpha \right\}. \end{align*} $$

In practice, $\widehat {c}_{1-\alpha ,n}$ may be approximated by the $1-\alpha $ empirical quantile of the $n_B$ independently generated bootstrap test statistics, with $n_B$ set to be as large as computationally feasible. We reject $\mathrm {H}_0$ if and only if $n\mathcal {L} ( \widehat {\phi }_n )>\widehat {c}_{1-\alpha ,n}$ . The following theorem shows that the proposed test is asymptotically size controlled and consistent.

2.5 Local Power

In this section, we consider the local power of the test following the discussion in Chen and Fang (Reference Chen and Fang2019b). For each $n\in \mathbb {Z}_+$ , let the sample $\mathbf {Z}_n=\{Z_i\}_{i=1}^n$ be distributed according to the joint law $P_n^n=\prod _{i=1}^n P_n$ , where $P_n$ is a probability distribution on $(\mathbb {R}^{d_z}, \mathscr {B}(\mathbb {R}^{d_z}))$ with $P_n(B)=\mathbb {P}(Z_i\in B)$ for every Borel set B. That is, for each $n\in \mathbb {Z}_+$ , the observations $Z_1,\ldots ,Z_n$ are i.i.d. with distribution $P_n$ . We suppose that the null hypothesis $\mathrm {H}_0$ is false for each $P_n$ , that is, for all $\theta \in \Theta $ , $P_n(\psi _{x,\theta })\ne 0$ for some $x\in \mathbb {R}$ . Suppose that $P_n$ converges (in a way as described in the following assumption) to the probability measure P, and that P satisfies $\mathrm {H}_0$ , that is, for some $\theta \in \Theta $ , $P(\psi _{x,\theta })= 0$ for all $x\in \mathbb {R}$ .

Assumption 2.9. The probability distributions $P_n$ and P satisfy that

(11) $$ \begin{align} \lim_{n\to\infty}\int \left[ \sqrt{n}\left( \mathrm{d}P_n^{1/2} - \mathrm{d}P^{1/2} \right) -\frac{1}{2}v_0\,\mathrm{d}P^{1/2} \right]^2=0 \end{align} $$

for some measurable function $v_0:\mathbb {R}^{d_z}\to \mathbb {R}$ , where $\mathrm {d}P_n^{1/2}$ and $\mathrm {d}P^{1/2}$ denote the square roots of the densities of $P_n$ and P, respectively.

Our local power results rely on Assumption 2.9, which is similar to (3.10.10) of van der Vaart and Wellner (Reference van der Vaart and Wellner1996). The following proposition states formally the local power property of the test.

Proposition 2.4. Suppose that Assumptions 2.1–2.9 hold and $\sup _{f\in \Psi } |P_n(f^2)| =O(1)$ . Then $\sqrt {n}( \widehat {\phi }_n-\phi _P )\leadsto \mathbb {G}_0+\zeta _P$ , where $\mathbb {G}_0$ is some tight random element, and $\zeta _P(x,\theta )= P(\psi _{x,\theta }v_0)$ for every $(x,\theta )\in \mathbb {R}\times \Theta $ . Furthermore, if the CDF of $\mathcal {L}^{\prime \prime }_{\phi _P}(\mathbb {G}_0)$ is strictly increasing and continuous at its $1-\alpha $ quantile $c_{1-\alpha }$ , then it follows that

$$ \begin{align*} \liminf_{n\to\infty}\mathbb{P}\left( n \mathcal{L}( \widehat{\phi}_n )>\widehat{c}_{1-\alpha,n} \right) \ge \mathbb{P}(\mathcal{L}^{\prime\prime}_{\phi_P}(\mathbb{G}_0+\zeta_P)>c_{1-\alpha}). \end{align*} $$

Proposition 2.4 follows from Lemma C.1 of Chen and Fang (Reference Chen and Fang2019b) and provides lower bounds for the power of the test under local perturbations to the null.

4.1 Performance Improvement in Conditional Moment Restriction Models with Weak Instruments

Note that Cases 1 and 2 ( $n=200$ ) with $\delta =0$ (under the null hypothesis) are identical to the DGPs in Tables 1 and 2 of Jun and Pinkse (Reference Jun and Pinkse2009), respectively. Thus, the results in Table 1 with $n=100$ and Table E.12 in the Supplementary Material with $n=200$ can be compared with those in Tables 1 and 2 of Jun and Pinkse (Reference Jun and Pinkse2009), respectively. We present the comparisons in Tables 7 and 8, where $\widehat {T}_2(\widehat {\theta }_{*})$ and $\widehat {T}_{\mathrm {k}}(\widehat {\theta }_{*})$ with $\widehat {\theta }_{*}\in \{\widehat {\theta }_{\mathrm {2SLS}}, \widehat {\theta }_{\mathrm {SP}}\}$ are two-step plug-in test statistics computed by using either a 2SLS or a semi-parametric estimator of $\theta _0$ as described in Jun and Pinkse (Reference Jun and Pinkse2009), and $\widehat {T}_1(\widehat {\theta }_{\mathrm {CUE1}})$ , $\widehat {T}_2(\widehat {\theta }_{\mathrm {CUE2}})$ , and $\widehat {T}_{\mathrm {k}}(\widehat {\theta }_{\mathrm {CUEk}})$ are the test statistics proposed by Jun and Pinkse (Reference Jun and Pinkse2009).Footnote ⁹ Our test uses $\tau _n=n^{-1/4}$ for illustration. The plug-in method suffers from substantial size distortion. The tests of Jun and Pinkse (Reference Jun and Pinkse2009) improve upon the plug-in approach, but could be conservative as shown in their theoretical results. The proposed method achieves rejection rates closer to the nominal significance levels compared to the results of Jun and Pinkse (Reference Jun and Pinkse2009). These numerical observations provide supporting evidence for the theoretical results in the article.

Table 7

Comparison with Table 1 of Jun and Pinkse (Reference Jun and Pinkse2009)

Table 7 Long description

The table consists of seven columns. The first column is alpha. The next two columns are under the heading Plug-in, containing T-hat sub 2 (theta-hat sub 2 S L S) and T-hat sub k (theta-hat sub 2 S L S). The following three columns are under the heading Jun and Pinkse 2009, containing T-hat sub 1 (theta-hat sub C U E 1), T-hat sub 2 (theta-hat sub C U E 2), and T-hat sub k (theta-hat sub C U E k). The final column is the Proposed test where tau sub n equals n to the power of minus 1/4.

Data rows by alpha value:

* Alpha 0.01: Plug-in (0.511, 0.480); Jun and Pinkse (0.004, 0.012, 0.012); Proposed test (0.011).

* Alpha 0.025: Plug-in (0.533, 0.509); Jun and Pinkse (0.007, 0.016, 0.022); Proposed test (0.027).

* Alpha 0.05: Plug-in (0.551, 0.531); Jun and Pinkse (0.015, 0.025, 0.030); Proposed test (0.051).

* Alpha 0.1: Plug-in (0.584, 0.559); Jun and Pinkse (0.027, 0.049, 0.045); Proposed test (0.111).

* Alpha 0.2: Plug-in (0.626, 0.602); Jun and Pinkse (0.053, 0.078, 0.075); Proposed test (0.241).

Table 8

Comparison with Table 2 of Jun and Pinkse (Reference Jun and Pinkse2009)

Table 8 Long description

The table consists of seven columns. The first column is alpha. The next two columns are under the heading Plug-in, containing T-hat sub 2 (theta-hat sub S P) and T-hat sub k (theta-hat sub S P). The next three columns are under the heading Jun and Pinkse (2009), containing T-hat sub 1 (theta-hat sub C U E 1), T-hat sub 2 (theta-hat sub C U E 2), and T-hat sub k (theta-hat sub C U E k). The final column is the Proposed test with tau sub n = n to the power of minus 1/4.

Data rows by alpha value:

* Alpha 0.01: Plug-in (0.341, 0.362), Jun and Pinkse (0.018, 0.018, 0.024), Proposed (0.009).

* Alpha 0.025: Plug-in (0.360, 0.395), Jun and Pinkse (0.027, 0.027, 0.030), Proposed (0.023).

* Alpha 0.05: Plug-in (0.382, 0.419), Jun and Pinkse (0.036, 0.036, 0.046), Proposed (0.049).

* Alpha 0.1: Plug-in (0.422, 0.455), Jun and Pinkse (0.052, 0.056, 0.067), Proposed (0.109).

* Alpha 0.2: Plug-in (0.487, 0.519), Jun and Pinkse (0.077, 0.093, 0.106), Proposed (0.233).