2.1 ICM Metrics

For a random variable $U\in \mathbb {R}$ and a random vector $Z\in \mathbb {R}^{p_z},$ we say that U is mean-independent of Z, if

(1) $$ \begin{align} {\mathbb{E}}[U \mid Z]={\mathbb{E}}[U] \ almost \ surely\ (a.s.), \end{align} $$

otherwise, U is mean-dependent on Z. To characterize the relationship (1), the existing literature puts much effort into studying ICM mean dependence metrics of the form:

$$ \begin{align*} T(U \mid Z;\nu)=\int_{\Pi} \Big|{\mathbb{E}}[(U-{\mathbb{E}} U)w(s,Z)]\Big|^2\,\nu(ds), \end{align*} $$

where $w(s,Z)$ is a weight function indexed by $s \in \Pi $ , and $\nu $ is a measure on $\Pi $ . A notable feature of $T(U \mid Z;\nu )$ is its omnibus property, namely, $T(U \mid Z;\nu )=0$ if and only if (1) holds (see, e.g., Shao and Zhang, Reference Shao and Zhang2014, Thm. 1.2). The omnibus property guarantees the consistency of ICM tests. Therefore, a larger value of ${T(U \mid Z;\nu )}$ indicates a stronger mean dependence of U on Z.

Under suitable conditions, the ICM metric has the general form:

$$ \begin{align*} T(U \mid Z;\nu) &= \int_{\Pi} \Big({\mathbb{E}}[(U-{\mathbb{E}} U)w(s,Z)]\Big)\overline{\Big({\mathbb{E}}[(U-{\mathbb{E}} U)w(s,Z)]\Big)}\nu(ds)\\ &= {\mathbb{E}}\Big[(U-{\mathbb{E}} U)(U^\dagger-{\mathbb{E}} U)\int_{\Pi} \big(w(s,Z)\overline{w(s,Z^\dagger)}\big)\nu(ds) \Big]\\ &:= {\mathbb{E}}\Big[(U-{\mathbb{E}} U)(U^\dagger-{\mathbb{E}} U)K(Z,Z^\dagger)\Big]\\ &:= \mathrm{ICM}(U \mid Z), \end{align*} $$

where $\overline {w(\cdot ,\cdot )}$ denotes the complex conjugate of $w(\cdot ,\cdot )$ and $K(\cdot ,\cdot )$ denotes the kernel. Table 1 provides a few examples of commonly used ICM kernels (see Li et al., Reference Li, Ke, Yin and Yu2023 for a recent review).

Table 1 Examples of ICM kernels

Note: $\phi (x) = \frac {1}{\sqrt {2\pi }} e^{-\frac {1}{2}x^2} $ is the standard normal probability density function, and $ c_p = \frac {\pi ^{(1+p)/2}}{\Gamma ((1+p)/2)}, \ p \geq 1, $ where $ \Gamma (\cdot ) $ is the complete gamma function.

Examples of weight functions from the literature include the step function $w(s,Z) = \mathrm {I}(Z\leq s) $ (e.g., Stute, Reference Stute1997; Domínguez and Lobato, Reference Domínguez and Lobato2004; Delgado et al., Reference Delgado, Domínguez and Lavergne2006; Escanciano, Reference Escanciano2006b; Zhu et al., Reference Zhu, Li, Li and Zhu2011); a one-dimensional projection in the step function $w(s,Z) =\mathrm {I}(Z^{\top }s_{-1}\leq s_1)$ (e.g., Escanciano, Reference Escanciano2006a; Kim, Balakrishnan, and Wasserman, Reference Kim, Balakrishnan and Wasserman2020); the real exponential $w(s,Z) = \exp (Z^{\top }s)$ (e.g., Bierens, Reference Bierens1990); and the complex exponential $w(s,Z)=\exp (\mathrm {i}Z^{\top }s)$ (e.g., Bierens (Reference Bierens1982), Shao and Zhang (Reference Shao and Zhang2014), and Antoine and Lavergne (Reference Antoine and Lavergne2023)). The space $\Pi $ in Escanciano (Reference Escanciano2006a) and Kim et al. (Reference Kim, Balakrishnan and Wasserman2020) is given by $\Pi = \mathbb {R}\times \mathbb {S}_{p_z}$ , where $\mathbb {S}_{p_z}$ denotes the space of $p_z\times 1$ vectors with unit Euclidean norm while $\Pi =\mathbb {R}^{p_z}$ for the other works mentioned above. See Escanciano (Reference Escanciano2006b, Lemma 1) for a general characterization of ICM weight functions.

The omnibus property of ICM metrics thus translates as

(2) $$ \begin{align} \mathrm{ICM}(U \mid Z)=0 \quad \text{if and only if }~~ {\mathbb{E}}[U \mid Z]={\mathbb{E}}[U] \ almost \ surely \ (a.s). \end{align} $$

When $ U $ and $ Z $ are observed, a natural empirical estimator of $ \mathrm {ICM}(U \mid Z) $ is given by the (modified) U-statistic

$$ \begin{align*} \widehat{\mathrm{ICM}}_n(U \mid Z) = \frac{1}{n(n-1)} \sum_{i \neq j} (U_i - {\mathbb{E}}_n[U])(U_j - {\mathbb{E}}_n[U]) K(Z_i, Z_j). \end{align*} $$

As with most existing ICM-based tests, the asymptotic null distribution of the corresponding test statistic (under (1)) is non-pivotal (Bierens and Ploberger, Reference Bierens and Ploberger1997, Thm. 3):

(3) $$ \begin{align} n\widehat{\mathrm{ICM}}_n(U \mid Z) \overset{d}{\rightarrow} \sum_{k=1}^{\infty}\lambda_k G_k^2, \end{align} $$

granted ${\mathbb {E}} [U^2U^{\dagger 2}K^2(Z,Z^{\dagger })]<\infty $ . Here, $\{G_k\}_{k=1}^{\infty }$ is a sequence of $iid$ standard Gaussian random variables, and $\{\lambda _k\}_{k=1}^{\infty }, \ \lambda _k\geq 0$ , is a sequence of non-increasing coefficients that depend on the distribution of $[U,Z^\top ]^\top $ and the kernel $K(\cdot ,\cdot )$ . Thus, the limiting distribution of $ n\widehat {\mathrm {ICM}}_n(U \mid Z) $ under (1) is non-pivotal, and re-sampling/bootstrap techniques are usually required to obtain valid $ p $ -values for inference.

2.2 A New Characterization of Mean Independence

Consider the following test hypotheses of mean independence:

(4) $$ \begin{align} \begin{aligned} \mathbb{H}_o&: {\mathbb{E}}[U \mid Z] - {\mathbb{E}}[U] = 0 \ a.s.; \\ \mathbb{H}_a&: \mathbb{P}\big({\mathbb{E}}[U \mid Z]={\mathbb{E}}[U]\big) < 1. \end{aligned} \end{align} $$

The above hypotheses of interest, in view of the omnibus property (2), can be restated as testing $ \mathrm {ICM}(U \mid Z)=0.$ Indeed, by the law of iterated expectations (LIE),

(5) $$ \begin{align} \begin{aligned} \mathrm{ICM}(U \mid Z)=&\ {\mathbb{E}}\left\{K(Z,Z^\dagger) (U^\dagger-{\mathbb{E}} U) {\mathbb{E}}[(U-{\mathbb{E}} U) \mid Z,Z^\dagger,U^\dagger]\right\} \\ =&\ {\mathbb{E}}\left\{K(Z,Z^\dagger)(\underbrace{{\mathbb{E}}[U \mid Z]-{\mathbb{E}} U}_{=\,0 \, a.s. \text{ under } \mathbb{H}_o})(U^\dagger-{\mathbb{E}} U)\right\}. \end{aligned} \end{align} $$

Under $\mathbb {H}_o$ , ${\mathbb {E}}[U \mid Z]$ is degenerate, that is, ${\mathbb {E}}[U \mid Z]-{\mathbb {E}} U=0\ a.s.$ This is the first-order degeneracy problem in U/V-statistic-based estimator for (5) that accounts for the non-pivotal limiting distribution in (3). In this regard, we propose to replace the role of U in the ICM metric with a random vector $V\in \mathbb {R}^{p_v}$ constructed such that ${\mathbb {E}}[V \mid Z]$ is non-degenerate under both the null and alternative hypotheses while preserving the omnibus property (2). In other words, we propose to measure the mean independence of U on Z with the assistance of a random vector V by

$$ \begin{align*}\delta(V):= {\mathbb{E}}\left[K(Z,Z^\dagger) (U^\dagger-{\mathbb{E}} U) (V - {\mathbb{E}} V) \right]. \end{align*} $$

The resulting test relies on the empirical estimator of $\delta (V)$ .

The two key properties—first-order non-degeneracy and omnibusness—ensure a pivotal limiting distribution under the null and the consistency of the test, respectively. Specifically, we require (1) V to be mean dependent on Z to rule out first-order degeneracy and (2) V to be chosen such that $\delta (V)\neq 0$ under $\mathbb {H}_a$ . The first condition is easy to satisfy by selecting V to be a measurable function of Z. In contrast, the second condition is non-trivial: randomly selecting V may lead to the same limitation as the classical CM test if it happens to be orthogonal to the direction of departure under $\mathbb {H}_a$ . This trade-off mirrors the strengths and weaknesses of existing methods: while the CM test is first-order non-degenerate, it may lack consistency under $\mathbb {H}_a$ ; conversely, the ICM test is consistent under $\mathbb {H}_a$ , but suffers from first-order degeneracy, resulting in a non-pivotal limiting distribution under $\mathbb {H}_o$ .

To this end, the proposed testing procedure integrates the respective strengths of two existing approaches in the literature: the pivotality of the CM test and the omnibus property of the ICM test. In particular, we focus on the quantity

(6) $$ \begin{align} V_h = \big[ h(Z), \ U - h(Z) \big]^\top, \end{align} $$

such that

(7) $$ \begin{align} \delta_h:=\delta(V_h)= \begin{bmatrix} {\mathbb{E}}[K(Z,Z^\dagger) (h(Z)-{\mathbb{E}}[h(Z)]) ({U}^\dagger-{\mathbb{E}} U)]\\ \mathrm{ICM}(U \mid Z) - {\mathbb{E}}[K(Z,Z^\dagger) (h(Z)-{\mathbb{E}}[h(Z)]) ({U}^\dagger-{\mathbb{E}} U)] \end{bmatrix}. \end{align} $$

The two components in $V_h$ , namely, $h(Z)$ and $U - h(Z) $ , serve different purposes. First, one can view $h(Z)$ as the assistant function where prior information about the alternative can be incorporated, as done in the classical CM test. Second, the inclusion of $ U $ in the second element draws inspiration from ICM tests, which gives the omnibus property of our test, that is, it has non-trivial power in all possible directions of the alternative. This structure acts as a safeguard in scenarios where the chosen assistant $h(Z)$ is (nearly) orthogonal to the alternative, a situation in which traditional CM tests suffer from trivial power. Moreover, the $-h(Z)$ in the second element ensures that $\delta _h$ is first-order non-degenerate and that the proposed test statistic is pivotal ( $\chi ^2$ -distributed) under the null hypothesis. We remark that the choice of V can be readily extended beyond two dimensions, although in this article, we focus on the specific two-dimensional case for clarity and analytical tractability.

The following fundamental result establishes the omnibus and first-order non-degeneracy properties of $\delta _h $ .

Lemma 2.1 implies that, with our construction of $V_h$ in (6), $\delta _h$ is nonzero under the alternative. Therefore, $\|\delta _h\|$ can be interpreted as a metric for conditional mean independence, that is, $\|\delta _h\|=0$ if and only if $\mathbb {E}[U\mid Z]=\mathbb {E}[U] \ a.s. $ While the performance of $\delta _h$ depends on the choice of $h(\cdot )$ , we note that similar concerns arise in ICM tests, particularly regarding the selection of kernels. A practitioner who is agnostic about the alternative can use , where denotes a $p\times 1$ vector of ones.Footnote ¹ The Maclaurin’s series expansion, namely, $\exp (z) = \sum _{l=0}^{\infty } z^l/{l!}$ shows the potential of to capture different directions under the alternative.Footnote ²

A key message of Lemma 2.1 is that symmetrizing any non-degenerate measurable function $h(\cdot )$ of Z about U guarantees the omnibus and first-order non-degeneracy properties of $\delta _h$ . The requirement on $h(Z)$ in Lemma 2.1 can hardly be termed a “condition,” as it imposes only measurability and non-degeneracy. The proof of Lemma 2.1 provides an insight into the inclusion of U linearly in the construction of V. This brings in the term $\mathrm {ICM}(U \mid Z)$ in $\delta _h$ that is strictly positive under $\mathbb {H}_a$ and contributes power under the alternative. The proposed test thus draws its consistency from the ICM metric.

Since $h(Z)$ can be chosen arbitrarily, the practitioner does not bear the burden of “carefully” selecting functions such as polynomials that provide power by approximating ${\mathbb {E}}[U \mid Z]$ under $\mathbb {H}_a$ , as required in nonparametric specification tests (e.g., Wooldridge, Reference Wooldridge1992; Yatchew, Reference Yatchew1992; Zheng, Reference Zheng1996). In addition, the user-specified $h(Z)$ provides extra flexibility as the practitioner can use it to augment the power of the test in given directions, unlike existing ICM-based bootstrap tests.

2.3 Extension to Testing the Nullity of ${\mathbb {E}}[U \mid Z]$

In some applications, such as specification testing, one may be interested in the almost sure nullity of ${\mathbb {E}}[U \mid Z]$ directly, that is,

(8) $$ \begin{align} \mathbb{H}_o^*&: {\mathbb{E}}[U \mid Z]=0 \ a.s.; \quad \mathbb{H}_a^* : \mathbb{P}({\mathbb{E}}[U \mid Z]=0) <1. \end{align} $$

This is an augmented version of (4) which further imposes ${\mathbb {E}}[U]=0$ , that is, a joint hypothesis of conditional mean independence and nullity of the unconditional mean. To this end, we follow Su and Zheng (Reference Su and Zheng2017) by augmenting $\delta _h$ with an additional quantity that accounts for ${\mathbb {E}}[U] = 0$ under $\mathbb {H}_o^*$ . In particular, one may consider the metric

$$ \begin{align*}\delta_h^*=\delta_h + {\mathbb{E}}|K(Z,Z^{\dagger})|{\mathbb{E}} U {\mathbb{E}}\big[h(Z), \ U - h(Z)) \big]^\top. \end{align*} $$

The following result shows that ${\delta }_h^*$ has the omnibus property with respect to (8).

To conserve space, the discussion below focuses on $\delta _h$ . Theoretical properties of the $\chi ^2$ -test based on $\delta _h^*$ are presented in Section S.3 of the Supplementary Material.

3.1 A Unified Framework

The mean independence testing problem and the specification testing problem are two closely related problems in econometric analyses. Here, we present a unified framework for both tasks. Let $X\in \mathbb {R}^{p_x}$ and $Z\in \mathbb {R}^{p_z}$ be two random vectors. We consider an econometric model defined by the following CM restriction:

(9) $$ \begin{align} \mathbb{E}[U(X;\theta_o)|Z]=0\quad a.s., \end{align} $$

for a unique model parameter $\theta _o\in \Theta \subset \mathbb {R}^k$ . Here, $U:\mathbb {R}^{p_x}\times \Theta \mapsto \mathbb {R}^{p_u}$ is the econometric model that is assumed to be known, and $\Theta $ is the parameter space. Given $iid$ observations $\{X_i,Z_i\}_{i=1}^n$ , and an empirical estimator $\widehat {\theta }_n$ for $\theta _o$ , we are interested in assessing the model specification in (9).

The above framework encompasses a wide range of applications, including treatment effect analyses (e.g., Sant’Anna and Song, Reference Sant’Anna and Song2019; Callaway and Karami, Reference Callaway and Karami2023), Euler and Bellman equations (Hansen and Singleton, Reference Hansen and Singleton1982; Escanciano, Reference Escanciano2018), the hybrid New Keynesian Phillips curve (Choi, Escanciano, and Guo, Reference Choi, Escanciano and Guo2022), forecast rationality (Hansen and Hodrick, Reference Hansen and Hodrick1980), and tests of conditional equal and predictive ability (Giacomini and White, Reference Giacomini and White2006) (see Li et al., Reference Li, Liao and Zhou2026 for a comprehensive discussion).

In the mean independence testing problem outlined in Section 2.2, the expressions for $ \mathrm {ICM}(U \mid Z) $ and $ \delta _h $ involve the nuisance parameter $ \mathbb {E} U $ . In this case, we can simply set $X=U$ , $\theta _o=\mathbb {E}U$ , and $U(X;\theta )=X-\theta $ , then testing for (9) reduces to the hypothesis (4). In the class of nonlinear models with additively separable errors $X^{(1)} = g(X^{(-1)};\beta _o) + U $ , where $X^{(1)}$ denotes the first element of X, $X^{(-1)}$ is the sub-vector of X that excludes $X^{(1)}$ , and U is the model error (up to a location shift), we let

$$ \begin{align*} U(X;\theta) &= X^{(1)} - g(X^{(-1)};\beta) - \theta_c, \end{align*} $$

where $\theta = [ \theta _c,\ \beta ^\top ]^\top $ , and $\theta _c$ is the location parameter such that $\mathbb {E}[X_1-g(X_{-1};\beta )-U]=\theta _c$ . Including the location parameter $\theta _c$ in $\theta $ becomes necessary as the ICM metric and $\delta _h$ can only identify mean independence up to an unknown nuisance mean shift. Nullity testing is discussed in Section 2.3.

In practice, a natural estimator for $\delta _h$ in (7) is given by

(10) $$ \begin{align} \widehat{\delta}_h& = \frac{1}{n(n-1)}\sum_{i=1}^n\sum_{j\neq i}K(Z_i,Z_j)U(X_i;\widehat{\theta}_n)\nonumber\\&\quad\big[(h(Z_j) - {\mathbb{E}}_n[h(Z)]) , \ U(X_j;\widehat{\theta}_n) - (h(Z_j) - {\mathbb{E}}_n[h(Z)]) \big]^\top. \end{align} $$

We remark that, for the mean independence testing problem in (4), $U(X_i;\widehat {\theta }_n)=U_i-\widehat {\theta }_n$ with $\widehat {\theta }_n={\mathbb {E}}_nU.$

3.2 Asymptotic Distribution

In this section, we analyze the asymptotic behavior of $ \widehat {\delta }_h $ under the null hypothesis, local alternatives, and fixed alternatives. Let $ D := [U, X^\top , Z^\top ]^\top $ , and define $ \widetilde {U} := U(X; \theta _o) $ such that ${\mathbb {E}}\widetilde {U}=0.$ We consider the following sequences of local alternatives: Pitman local alternatives $ \mathbb {H}_{an}: \mathbb {E}[\widetilde {U} \mid Z] = n^{-1/2} a(Z)$ and milder (non-Pitman) local alternatives $ \mathbb {H}_{an}': \mathbb {E}[\widetilde {U} \mid Z] = n^{-1/4} a(Z) $ , where $ a(Z) $ is a non-degenerate measurable function of $ Z $ satisfying $ \mathbb {E}[a(Z)] = 0 $ . For completeness, we impose the following regularity conditions.

Assumption 3.3. $ U(X; \theta ) $ is differentiable in $ \theta $ and admits the following first-order expansion:

$$\begin{align*}U(X; \theta) = \widetilde{U} + \frac{\partial U(X; \bar{\theta})}{\partial \theta'} (\theta - \theta_o) := \widetilde{U} - G(X; \bar{\theta})^\top (\theta - \theta_o), \end{align*}$$

where $ G(x; \theta ) $ is continuous in $ \theta $ for all $ x $ in the support of $ X $ , $ {\mathbb {E}}\big [\|G(X; \theta )\|^2\big ] <\infty $ for all $\theta $ , and $ \bar {\theta } $ lies on the line segment between $ \theta $ and $ \theta _o $ , that is, $ \| \bar {\theta } - \theta _o \| \leq \| \theta - \theta _o \| $ .

Assumption 3.4. $\widehat {\theta }_n$ is a $\sqrt {n}$ -consistent estimator of $\theta _o$ with the asymptotically linear representation

$$\begin{align*}\sqrt{n}(\widehat{\theta}_n - \theta_o) = \frac{1}{\sqrt{n}}\sum_{i=1}^n \xi_{\theta,i} + o_p(1), \end{align*}$$

where $ \xi _{\theta ,i}\in \mathbb {R}^k$ satisfies ${\mathbb {E}}[\xi _{\theta ,i}]=\textbf {0}$ and $ {\mathbb {E}}\big [\|\xi _{\theta ,i}\|^2\big ] < \infty $ .

The assumption of $iid$ observations in Assumption 3.1 simplifies the theoretical analyses.Footnote ³ Assumption 3.2 is standard (e.g., Serfling, Reference Serfling1980, Sect. 5.5.1 Thm. A); it is needed to establish the asymptotic normality of $\sqrt {n}(\widehat {\delta }_h-\delta _h)$ . Although the moment restriction on U cannot be relaxed, that of $K(\cdot )$ can be relaxed through the choice of bounded kernels or suitable transformations of Z. Assumption 3.3 regulates the smoothness and moment conditions of the possibly nonlinear parameterized error $U(X;\theta )$ . Assumption 3.4 is akin to Escanciano (Reference Escanciano2009a, Assump. A3); it assumes that the estimator $\widehat {\theta }_n$ is consistent with respect to its probability limit $\theta _o$ . The Bahadur linear representation holds for a broad class of estimators, including commonly used estimators, such as (nonlinear) least squares, maximum likelihood, and the (generalized) method of moments, under standard regularity conditions.

The result on the limiting behavior of the proposed test statistic relies on the following asymptotically linear representation of $\widehat {\delta }_h$ in the following lemma. Define $\widetilde {h}(Z):=h(Z)-\mathbb {E}h(Z)$ .

Lemma 3.1. Under Assumptions 3.1–3.4, $\widehat {\delta }_h$ has the following asymptotically linear representation:

$$ \begin{align*} \sqrt{n}\big(\widehat{\delta}_h - \delta_h\big) = \frac{1}{\sqrt{n}}\sum_{i=1}^{n}\xi_h(D_i) + o_p(1), \end{align*} $$

where $ \xi _h(D_i):= 2\big [\xi _{h,1}(D_i), \ \xi _{h,2}(D_i) - \xi _{h,1}(D_i) \big ]^\top $ , $ \delta _h = \big [\delta _h^{(1)},\delta _h^{(2)}\big ]^{\top } $ ,

$$ \begin{align*} \xi_{h,1}(D_i): &= \psi_h^{(1)}(D_i) - \delta_h^{(1)} - H_1^\top[\xi_{\theta,i}^\top, \widetilde{h}(Z_i)]^\top , \\ \xi_{h,2}(D_i): &= \psi_U^{(1)}(D_i)-\mathrm{ICM}(U\mid Z) - H_2^\top\xi_{\theta,i},\\ \psi_h^{(1)}(D_i) : &={\mathbb{E}}[\psi_h(D_i,D_j)|D_i],\\ \psi_U^{(1)}(D_i): &={\mathbb{E}}[\psi_U(D_i,D_j)|D_i], \end{align*} $$

$\delta _h^{(1)}={\mathbb {E}}[K(Z,Z^{\dagger })\widetilde {h}(Z)\widetilde {U}^\dagger ], \ \delta _h^{(2)}=\mathrm {ICM}(U \mid Z)-\delta _h^{(1)}$ , $ \psi _h(D_i,D_j):= K(Z_i,Z_j)\big \{ \widetilde {h}(Z_j)\widetilde {U}_i + \widetilde {h}(Z_i)\widetilde {U}_j \big \}/2 $ , $\psi _U(D_i,D_j) := K(Z_i,Z_j)\widetilde {U}_i\widetilde U_j$ , $ H_1= (1/2)\Big [ {\mathbb {E}}\big [K(Z, Z^\dagger )\widetilde {h}(Z^\dagger ) G(X;\theta _o)\big ]^\top ,\ {\mathbb {E}}\big [K(Z,Z^\dagger )\widetilde {U}\big ] \Big ]^\top $ , and $H_2= {\mathbb {E}}\big [K(Z,Z^\dagger ){\widetilde {U}^\dagger }G(X;\theta _o)\big ] $ .

The proof of Lemma 3.1 exploits the classical Hoeffding decomposition (Serfling, Reference Serfling1980) for U-statistics and provides intuition for the two different rates of local alternatives. If the user-defined assistant $h(Z)$ is non-orthogonal to the alternative direction ${\mathbb {E}}\big [a(Z^{\dagger })K(Z,Z^{\dagger }) \mid Z\big ]$ under the Pitman local alternatives $ \mathbb {H}_{an}: \ \mathbb {E}[\widetilde {U}|Z]=n^{-1/2}a(Z)$ , then $\sqrt {n}\delta _h^{(1)}$ is non-diminishing, that is,

$$ \begin{align*}\sqrt{n}\delta_h^{(1)} = {\mathbb{E}}\big[K(Z,Z^{\dagger})\widetilde{h}(Z)\sqrt{n}\mathbb{E}(\widetilde{U}^\dagger\mid Z^{\dagger})\big] = {\mathbb{E}}[K(Z,Z^{\dagger})\widetilde{h}(Z)a(Z^{\dagger})] \not \to 0, \end{align*} $$

which drives the power in a manner similar to the classical CM test. If instead $h(Z)$ is (nearly) orthogonal to the alternative with $\sqrt {n}\delta _h^{(1)}\to 0$ , then for the milder local alternatives $ \mathbb {H}_{an}': \ \mathbb {E}[\widetilde {U}|Z] = n^{-1/4}a(Z)$ ,

$$ \begin{align*}\sqrt{n}\delta_h^{(2)}\to \sqrt{n}\mathrm{ICM}(U\mid Z)&= \sqrt{n}\mathbb{E}[K(Z,Z^{\dagger})\mathbb{E}(\widetilde{U}|Z)\mathbb{E}(\widetilde{U}^{\dagger}|Z^{\dagger})]\\&=\mathbb{E}[K(Z,Z^{\dagger})a(Z)a(Z^{\dagger})]>0.\end{align*} $$

In this case, the classical ICM test drives the power. As such, it provides robustness to the user’s choice of direction, though this comes at the cost of a slower detection rate under local alternatives.

To study the limiting behavior of $\sqrt {n}(\widehat {\delta }_h-\delta _h)$ , we define

$$ \begin{align*}\Omega_h:= \operatorname{Var}[\xi_h(D_i)].\end{align*} $$

We distinguish $\Omega _{h,o}$ and $\Omega _{h,a}$ , corresponding to specific expressions of $\Omega _h$ under $\mathbb {H}_o$ and $\mathbb {H}_a$ , respectively. Both are generically referred to as $\Omega _h$ whenever the distinction is not necessary. In particular, under $\mathbb {H}_o$ , $\psi _U^{(1)}(D_i)=\mathrm {ICM}(U\mid Z)=H_2^\top \xi _{\theta ,i} =0 \ a.s.$ by the LIE, and $\sqrt {n}\big (\widehat {\delta }_h - \delta _h\big )$ reduces to $ \sqrt {n}\widehat {\delta }_h = [1, -1]^\top \frac {2}{\sqrt {n}}\sum _{i=1}^{n} \xi _{h,1}(D_i) + o_p(1) $ , with

(11) $$ \begin{align} \Omega_{h,o}: = 4{\mathbb{E}}\big[\xi_{h,1}(D)^2\big]\times \begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix}. \end{align} $$

The following result builds upon Lemma 3.1.

Theorem 3.1 offers a comprehensive asymptotic analysis of $\widehat {\delta }_h$ under the null, local alternatives, and fixed alternatives. In particular, part (i) shows that $\sqrt {n}\widehat {\delta }_h$ is non-degenerate and converges in distribution, which forms the basis of our $\chi ^2$ -distributed test. Under the Pitman local alternatives $\mathbb {H}_{an}$ , the test is only non-trivial in directions where the user-defined assistant $h(Z)$ is non-orthogonal to ${\mathbb {E}}\big [a(Z^{\dagger })K(Z,Z^{\dagger }) \mid Z\big ]$ , that is, ${\mathbb {E}}[\widetilde {h}(Z)a(Z^{\dagger })K(Z,Z^{\dagger })] \neq 0$ . This is expected, as the use of the assistant $h(Z)$ draws inspiration from classical CM tests. Part (iii) of Theorem 3.1 partially resolves this limitation: the local power becomes non-trivial in all directions due to the presence of $\mathrm {ICM}\big (a(Z)\mid Z\big )$ in the term $a_o' := [0, \ \mathrm {ICM}\big (a(Z)\mid Z\big )]^\top $ under a sequence of alternatives converging to the null at the $n^{-1/4}$ rate. In other words, the test becomes omnibus for alternative directions that converge to the null at a rate slower than $n^{-1/4}$ , thereby enhancing its ability to detect a broader class of local alternatives. The two types of local alternatives stem from our novel construction of the test, which combines components that operate at two different convergence rates. Part (iv) suggests that the resulting test is consistent under the fixed alternative.

3.3 Test Statistic

Theorem 3.1 justifies the asymptotic normality of $\widehat {\delta }_h$ , even under $\mathbb {H}_o$ , which naturally motivates the Wald test statistic:

(12) $$ \begin{align} \widetilde{T}_{V,n}=n\widehat{\delta}_h^{\top}\widetilde{\Omega}_{h,n}^{-1}\widehat{\delta}_h, \end{align} $$

where $\widetilde {\Omega }_{h,n}$ is a consistent estimator of $\Omega _h$ under both $\mathbb {H}_o$ and $\mathbb {H}_a$ . This testing procedure is valid when $\Omega _h$ is positive definite. However, from (11), $\Omega _h$ is singular under $\mathbb {H}_o$ with $\mathrm {rank}(\Omega _{h,o})=1$ .

Although it is tempting to replace the inverse matrix $\widetilde {\Omega }_{h,n}^{-1}$ in (12) with a generalized inverse matrix $\widetilde {\Omega }_{h,n}^{-}$ , for example, the Moore–Penrose inverse, the resulting Wald statistic may still not have an asymptotic $\chi ^2$ distribution unless the rank condition, $\mathbb {P}\big (\mathrm {rank}(\widetilde {\Omega }_{h,n})=\mathrm {rank}(\Omega _h)\big ) \to 1 \quad \text {as }n\to \infty $ is satisfied (Andrews, Reference Andrews1987). To ensure that the rank condition is satisfied and thereby address this problem, we adopt the thresholding technique described in Lütkepohl and Burda (Reference Lütkepohl and Burda1997) (see also Duchesne and Francq, Reference Duchesne and Francq2015; Dufour and Valéry, Reference Dufour and Valéry2016). Let

$$ \begin{align*} \widetilde{\Omega}_{h,n}:=\frac{1}{n-1}\sum_{i=1}^n \widehat{\xi}_h(D_i)\widehat{\xi}_h(D_i)^{\top} \end{align*} $$

be a consistent estimator of $\Omega _h$ (Sen, Reference Sen1960), where $\widehat {\xi }_h(D_i)$ denotes the sample analog of $\xi _h(D_i)$ with parameters replaced by estimates.

By the singular value decomposition,

(13) $$ \begin{align} \widetilde{\Omega}_{h,n}=\widetilde{\Gamma}_n\widetilde{\Lambda}_n\widetilde{\Gamma}_n^{\top}, \end{align} $$

where $\widetilde {\Lambda }_n = \mathrm {diag}(\widetilde {\lambda }_1,\widetilde {\lambda }_2)$ is the diagonal matrix comprising the eigenvalues $\widetilde {\lambda }_1\geq \widetilde {\lambda }_2 \geq 0$ of $\widetilde {\Omega }_{h,n}$ , and the columns of $\widetilde {\Gamma }_n$ are the corresponding eigenvectors. Let $ c_n = C n^{-1/2 + \iota } $ , where $ \iota \in (0, 1/2) $ is a small positive constant and $ C \in (0, \infty ) $ is a fixed constant, as in Dufour and Valéry (Reference Dufour and Valéry2016). Since $ \mathrm {rank}(\Omega _h) \geq 1 $ regardless of whether the null hypothesis $ \mathbb {H}_o $ holds, we define the regularized estimator of $ \Omega _h $ as

(14) $$ \begin{align} \widehat{\Omega}_{h,n} := \widetilde{\Gamma}_n \widehat{\Lambda}_{n,c_n} \widetilde{\Gamma}_n^{\top}, \quad \text{where} \quad \widehat{\Lambda}_{n,c_n} = \mathrm{diag}\big( \widetilde{\lambda}_1, \widetilde{\lambda}_2 \mathbf{1}(\widetilde{\lambda}_2> c_n) \big). \end{align} $$

The corresponding Moore–Penrose inverse is defined as

$$\begin{align*}\widehat{\Omega}_{h,n}^{-} := \widetilde{\Gamma}_n \widehat{\Lambda}_{n,c_n}^{-} \widetilde{\Gamma}_n^{\top}, \quad \text{where} \quad \widehat{\Lambda}_{n,c_n}^{-} = \mathrm{diag}\big( \widetilde{\lambda}_1^{-1}, \widetilde{\lambda}_2^{-1} \mathbf{1}(\widetilde{\lambda}_2> c_n) \big). \end{align*}$$

Finally, we define the regularized Wald test statistic:

(15) $$ \begin{align} T_{h,n} := n\widehat{\delta}_h^{\top}\widehat{\Omega}_{h,n}^{-}\widehat{\delta}_h. \end{align} $$

In practice, $\mathbb {H}_o$ is rejected when $T_{h,n}> \chi ^2_{1,1-\alpha }$ , where $\chi ^2_{1,1-\alpha }$ denotes the $1-\alpha $ quantile of the $\chi ^2_1$ distribution at a pre-specified significance level $\alpha \in (0,1)$ . We remark that the ranks of $\Omega _{h,o}$ and $\Omega _{h,a}$ can differ: $\mathrm {rank}(\Omega _{h,o}) = 1$ , while $\mathrm {rank}(\Omega _{h,a}) = 2$ if $\mathbb {P}(a(Z)= c \cdot \widetilde {h}(Z))<1$ for any $c \in \mathbb {R}$ , and $\mathrm {rank}(\Omega _{h,a}) = 1$ otherwise. We use $c_n=\widetilde {\lambda }_1 n^{-1/3}$ following Lütkepohl and Burda (Reference Lütkepohl and Burda1997) and Dufour and Valéry (Reference Dufour and Valéry2016).Footnote ⁵

Remark 3.2. $\mathrm {rank}(\Omega _{h,o}) = 1$ and $\mathrm {ICM}_n(U \mid Z) = O_p(n^{-1}) $ under $\mathbb {H}_o$ . From the proof of part (i) of Theorem 3.1, $ \sqrt {n}\widehat {\delta }_h = \sqrt {n} \widehat {\delta }_h^{(1)} [1,\ -1]^{\top } + O_p(n^{-1/2}) $ , and

$$\begin{align*}T_{h,n} = n\widehat{\delta}_h^{\top}\widehat{\Omega}_{h,n}^{-}\widehat{\delta}_h = \Bigg(\frac{\widehat{\delta}_h^{(1)}}{\sqrt{\operatorname{Var}(\widehat{\delta}_h^{(1)})}}\Bigg)^2 + o_p(1) \xrightarrow{d} \chi_1^2 \quad \text{as}\quad n\rightarrow \infty \end{align*}$$

under $\mathbb {H}_o$ . This implies that $\sqrt {T_{h,n}}$ converges in distribution to the half standard normal under $\mathbb {H}_o$ and shares the interpretability (without a formal hypothesis test) of a two-sided t-test.

The next theorem justifies using (15) under the null, local alternative, and fixed alternative hypotheses.

Theorem 3.2 establishes the consistency of the thresholded estimator of the Moore–Penrose inverse of the covariance matrix and characterizes the asymptotic behavior of the resulting test statistic, in parallel with the analysis in Theorem 3.1. In particular, under the local alternatives $\mathbb {H}_{an}$ of part (ii), when $a_0 = {\mathbb {E}}[\widetilde {h}(Z)a(Z^{\dagger })K(Z,Z^{\dagger })] = 0$ , the test exhibits trivial power because $b_o = 0$ . However, this limitation is partially remedied in part (iii), where the modified limiting direction $b_o'$ remains positive.

3.4 Power Comparison with the Bootstrap-Based ICM Test

In this section, we examine the efficiency of the proposed test statistic compared with the bootstrap-based ICM test by adopting the approach of Bahadur (Reference Bahadur1960). Under a fixed alternative, both tests demonstrate consistency as n tends to infinity. The Bahadur slope in Bahadur (Reference Bahadur1960) allows us to further compare the rate of convergence of $ p $ -values to zero as n increases.

Let $ \displaystyle S_G(t):=\mathbb {P}\Big ( \sum _{k=1}^{\infty }\lambda _k G_k^2>t \Big ) $ and $ \displaystyle S_T(t):=\mathbb {P}\big ( \chi ^2_{1}> t \big ) $ be the survival functions of the asymptotic null distributions of the ICM test statistic $n\mathrm {ICM}_n(U \mid Z)$ , and the pivotal test statistic, $T_{h,n}$ , respectively. Their Bahadur slopes are, respectively, given by

$$ \begin{align*}c_G= \lim_{n\to\infty}-\frac{2}{n} \log S_G(n\mathrm{ICM}_n(U \mid Z)) \ \text{ and } \ c_T= \lim_{n\to\infty}-\frac{2}{n} \log S_G(T_{h,n}). \end{align*} $$

Theorem 3.3. Suppose $ \widehat {\Omega }_{h,n}^-\overset {p}{\rightarrow } \Omega _{h,a}^-, $ and the conditions of Theorem 3.2 hold, then under $\mathbb {H}_a$ , the (approximate) Bahadur slopes of the ICM test statistic $n\mathrm {ICM}_n(U \mid Z)$ and the pivotal test statistic $T_{h,n}$ are, respectively, given by

$$ \begin{align*}c_G= \frac{\mathrm{ICM}(U \mid Z)}{\lambda_1} \quad \text{ and } c_T= \delta_a^{\top}\Omega_{h,a}^{-}\delta_a, \end{align*} $$

where $\lambda _1$ is the leading eigenvalue associated with the limiting null distribution in (3), and

$$ \begin{align*}\delta_a = \Big[{\mathbb{E}}\big[\widetilde{h}(Z)a(Z^{\dagger})K(Z,Z^{\dagger})\big], \ \mathrm{ICM}(U \mid Z) - {\mathbb{E}}\big[\widetilde{h}(Z)a(Z^{\dagger})K(Z,Z^{\dagger})\big] \Big]^\top. \end{align*} $$

The approximate Bahadur slopes presented in Theorem 3.3 are primarily of theoretical interest. Conducting a comprehensive comparison of these slopes is challenging as they depend on data-dependent quantities, such as $\lambda _1$ , $a(Z)$ , and user-specified variables, such as $h(Z)$ and the kernel $K(\cdot ,\cdot )$ .

4.1 Specifications

We consider the linear model

$$ \begin{align*}Y= \theta_c + \sum_{l=1}^5X_l\theta_l + U,\end{align*} $$

with and without excluded instruments.Footnote ⁷ The DGPs are varied through U:

1. LS1: $\displaystyle U = \frac {\mathcal {E}}{\sqrt {1+Z_1^2}} $ ;

2. LS2: $ \displaystyle U = \frac { \gamma }{5\sqrt {2}} \sum _{l=1}^5 Z_l^2 + \frac {\mathcal {E}}{\sqrt {1+Z_1^2}} $ ;

3. LS3: $ \displaystyle U = \gamma \sum _{l=1}^5 \frac {\cos (2Z_l)}{\sqrt {2(1-\exp (-8))}} + \frac {\mathcal {E}}{\sqrt {1+Z_1^2}} $ ; and

4. LS4: $ \displaystyle U = \gamma \sum _{l=1}^5 \big (\exp (-Z_l^2/3) - \sqrt {3/5}\big ) + \frac {\mathcal {E}}{\sqrt {1+Z_1^2}} $ ;

where $ \mathcal {E} \sim \mathcal {N}(0,1)$ independently of Z which follows the multivariate normal distribution with mean zero and covariance matrix $\Sigma _{ll'} = 0.25^{|l-l'|}$ , , and $\gamma \in [0,1]$ tunes the deviation away from $\mathbb {H}_o$ . $X=Z$ in DGP LS1 and $X_1 = (1.5Z_1 + \widetilde {\mathcal {E}}) / \sqrt {3.25} $ with $\widetilde {\mathcal {E}}\sim \mathcal {N}(0,1)$ and $\mathrm {Cov}(\widetilde {\mathcal {E}},{\mathcal {E}})=0.25$ in DGPs LS2 through LS4.Footnote ⁸ Thus, X is exogenous under DGP LS1 and endogenous under DGPs LS2 through LS4. LS1 is estimated via ordinary least squares (OLS) while the remaining DGPs are estimated using the IVs estimator with Z as the instrument. Heteroskedasticity of arbitrary form is imposed in all DGPs.

Different sample sizes $n \in \{200, 400, 600, 800\}$ serve to examine the empirical size and local power of the test using DGPs LS1 and LS2 (with $\gamma =0$ ). For the analyses of power in DGPs LS2 through LS4, the sample size is kept at $n=400$ , while $\gamma $ is varied in order to study the power of the proposed $\chi ^2$ -test in comparison to the bootstrap-based ICM procedures.

Table 2 Empirical size and local power

4.2 Empirical Size and Power

Table 2 presents the empirical sizes corresponding to DGPs LS1 (under strictly exogenous X) and LS2 (under endogenous X instrumented with Z) in addition to the local power under LS2 at three nominal levels: 10%, 5%, and 1%. One observes comparably good size control of the proposed $\chi ^2$ -test and the bootstrap-based procedures across all the sample sizes considered. This provides evidence in support of the proposed test’s validity under both exogenous X and endogenous X with valid instruments Z.

To ensure that the good size control of the $\chi ^2$ -test in Table 2 is not achieved at the expense of power under the alternative, we consider its performance under both local and fixed alternatives. The results, presented in the third panel of Table 2, indicate that the local power of the proposed $\chi ^2$ -test, along with the bootstrap-based procedures, is non-trivial. Figures 1–6 present power curves corresponding to DGPs LS2 through LS4 using the Gaussian and Negative Euclidean kernels. One observes that all tests demonstrate non-trivial power against the fixed alternative for variations of $\gamma $ away from zero. The proposed test exhibits highly competitive power performance across all three DGPs and both kernels considered. Notably, the relative power of the $\chi ^2$ -test compared to the bootstrap-based procedures demonstrates variability depending on the specific DGP and kernel, as highlighted in Theorem 3.3.

Figure 1 DGP LS2—Gaussian Kernel— $n=400$ .

Figure 2 DGP LS2—Negative Euclidean— $n=400$ .

Figure 3 DGP LS3—Gaussian Kernel— $n=400$ .

Figure 4 DGP LS3—Negative Euclidean— $n=400$ .

Figure 5 DGP LS4—Gaussian Kernel— $n=400$ .

Figure 6 DGP LS4—Negative Euclidean— $n=400$ .

Table 3 Running time—specification test—DGP LS1

Note: The second row (in parentheses) for each sample size and kernel includes standard deviations and inter-quartile ranges for the average running times and relative times (the $\chi ^2$ -test as the benchmark), respectively.

The $\chi ^2$ -test proves useful when researchers seek a measure of correct model specification or mean independence without resorting to a formal hypothesis test, owing to the pivotality of its test statistic. In summary, our simulations demonstrate that the proposed $\chi ^2$ -test, in addition to its enhanced interpretability compared to bootstrap-based ICM specification tests, exhibits good size control and comparable power performance.

4.3 Running Time

One key advantage of a pivotal test is its computational efficiency relative to bootstrap-based procedures, which becomes particularly critical in large samples. Accordingly, a comparison of the running times of competing tests is in order. Table 3 presents a comparison between the proposed pivotal $\chi ^2$ -test and the bootstrap-based ICM tests in terms of computational time for fixed ICM kernels.Footnote ⁹

Table 3 illustrates the substantial computational advantage of employing the $\chi ^2$ specification test. The average running times for the $\chi ^2$ -test are negligible compared to those of the bootstrap-based procedures across all considered sample sizes. A striking difference emerges in terms of median relative computational time (with the $\chi ^2$ -test serving as a benchmark). While the MB procedure is generally faster than the wild bootstrap procedure, our findings indicate that the median relative computational time of the MB tends to increase with the sample size, whereas that of the wild bootstrap appears to decrease. At particularly large sample sizes ( $n=600$ and $n=800$ ), the computational gain offered by the $\chi ^2$ -test becomes considerable.