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Testing the Exclusion Restriction in IV Models Using Non-Gaussianity: A LiNGAM-Based Approach
Data availability: data set and code for these analyses can be accessed at the provided Open Science Forum link: {\tt https://bit.ly/4cmyGYK}. \\ Conflict of Interest Statement: The author has no conflicts to disclose. \\ Funding: This research was partially funded by the grant PGI-24-E172 UNS.
Instrumental variable (IV) estimation has become a standard technique in causal inference for addressing endogeneity problems in observational studies Newey2003. The validity of IV methods hinges on three key assumptions: instrument relevance, exogeneity (exclusion restriction), and the absence of direct effects between instruments and outcomes. While instrument relevance can be empirically tested, the exclusion restriction—which states that instruments affect outcomes only through the endogenous treatment variable—has traditionally been untestable in exactly-identified models where the number of instruments equals the number of endogenous variables Athey2017.
Existing approaches to IV validation, such as the Sargan test and Basmann test, require over-identification and assume linear relationships between variables Shimizu2006, Basmann1960. These limitations become particularly problematic in social science research where exactly-identified models are common and nonlinear relationships may exist. Moreover, traditional tests rely on Gaussianity assumptions that may not hold in real-world datasets, potentially leading to incorrect inferences about instrument validity Hunermund2025.
Several attempts have been made to mitigate these limitations. Notable contributions include the work of li2022rethinking, ventura2018testing, dieterle2016simple, berkowitz2012validity, who address the limitation of a single instrument in diverse ways, such as: allowing a mild violation of exogeneity, using the residuals of the first stage, using prediction performance as a validator, or looking for different specifications of the fist stage. Also, kitagawa2015test and mourifie2017testing provide formal tests for instrument validity under heterogeneity, while conley2012plausibly and nevo2012identification develop methods for inference when instruments are only 'plausibly exogenous' or imperfect.
We address these limitations by developing a framework that combines IV analysis with Linear Non-Gaussian Acyclic Models (LiNGAM) Shimizu2006. The key observation is that non-Gaussian error distributions provide additional identifying information to test the exclusion restriction. Unlike traditional methods, our approach works in exactly-identified settings and can detect nonlinear violations of the exclusion restriction. The method builds on principles from independent component analysis (ICA) Lee1998, which has been successfully applied in various causal discovery problems but remains underutilized in instrumental variable applications.
The proposed framework makes three main contributions. First, it provides a theoretically grounded method for testing the exclusion restriction that does not require over-identification or linearity assumptions. Second, it introduces a set of complementary statistical tests—including bootstrap, asymptotic, permutation, likelihood ratio, and independence tests—to assess instrument validity under different data conditions. Third, the method bridges the gap between IV analysis and machine learning approaches to causal discovery, such as LiNGAM.
Our work connects to several strands of literature. In econometrics, it extends recent developments in causal inference methods that address endogeneity problems Dague2019. In machine learning, it builds on advances in causal discovery using non-Gaussianity assumptions Nogueira2022. The approach also relates to ongoing discussions about instrument validity in applied research Ullah2021, while providing concrete solutions to practical challenges in IV analysis.
The remainder of this paper is organized as follows: Section 2 reviews related work on IV validity tests and non-Gaussian causal discovery methods. Section 3 provides background on LiNGAM and the exclusion restriction in IV analysis. Section 4 presents our proposed testing framework, including the five complementary statistical tests and the algorithm. Section 5 evaluates the method through simulations and empirical applications. Section 6 discusses limitations and future directions, followed by conclusions in Section 7.
The problem of validating instrumental variables has been extensively studied, with most approaches relying on over-identification tests or auxiliary assumptions. The Sargan-Hansen test Stock2002 represents a classical approach that examines whether instruments are uncorrelated with the error term in over-identified models. However, this test requires more instruments than endogenous variables and assumes linear relationships, limiting its applicability in exactly-identified scenarios common in social science research.
A separate line of work has focused on testing instrument strength rather than validity. The weak instruments literature Angrist1995 developed diagnostic tools like the first-stage F-statistic to assess whether instruments provide sufficient explanatory power for the endogenous variables. While crucial for reliable estimation, these tests do not address the fundamental problem of exclusion restriction validation.
The emergence of causal discovery methods in machine learning has provided alternative approaches to causal structure identification. The LiNGAM framework Shimizu2011 demonstrated that non-Gaussianity of error terms enables full identification of linear causal models without requiring pre-specified temporal ordering or instrumental variables. This was later extended by DirectLiNGAM Shimizu2019, which provided a computationally efficient algorithm guaranteed to converge to the correct solution. These methods fundamentally differ from traditional approaches by exploiting higher-order statistical information rather than relying solely on second-order moments.
The present paper connects to all three strands of literature reviewed above. The precise relationship between our framework and each of these contributions — in terms of the assumptions required, the hypotheses tested, and the empirical implications — is discussed in detail in Section (ref), where the simulation and empirical results provide a concrete basis for comparison.
To establish the theoretical foundation for our proposed framework, we first introduce the key concepts of Linear Non-Gaussian Acyclic Models (LiNGAM) and the exclusion restriction in instrumental variable analysis. We then discuss how non-Gaussianity acts as an additional source of identification. These three components form the theoretical basis for our approach to testing IV validity.
The LiNGAM model represents a class of structural equation models where variables are generated through linear relationships with non-Gaussian disturbances. Unlike traditional linear models that assume Gaussian errors, LiNGAM exploits the non-Gaussianity of variables to achieve full identification of causal structures. The basic LiNGAM equation for a variable $x_i$ can be written as:
where $b_{ij}$ are the causal coefficients, $k(i)$ represents a causal ordering of variables, and $e_i$ are independent non-Gaussian error terms. This formulation differs from standard linear models by explicitly incorporating both the causal ordering and the non-Gaussian nature of disturbances Shimizu2006.
The non-Gaussianity assumption enables unique identification of the causal structure because independent non-Gaussian variables cannot be expressed as linear combinations of each other. This property, known as the Darmois-Skitovich theorem Cardoso2003, forms the theoretical basis for LiNGAM's identifiability. The model can be estimated using independent component analysis (ICA) techniques Shimizu2019, which decompose observed variables into statistically independent components.
The fundamental theoretical property that allows LiNGAM to achieve identification is rooted in the classical Darmois–Skitovich theorem Cardoso2003:
An immediate corollary is that if any variable in the system is non-Gaussian, the causal direction between two variables becomes identifiable: only one causal orientation is consistent with the independence of the structural errors. Intuitively, regressing $Y$ on $X$ leaves a residual that is independent of $X$ if and only if the true causal direction is $X \rightarrow Y$. Under Gaussianity, both directions yield residuals that appear uncorrelated with the regressor (since independence and uncorrelatedness coincide for Gaussian variables), making identification impossible. Non-Gaussianity breaks this symmetry.
In other words, the theorem (Shimizu2006) states that under the LiNGAM model with non-Gaussian and independent errors, the matrix B is unique (up to permutation) and uniquely identifiable from the joint distribution of the observed variables, without requiring temporal information, interventional data, or additional instruments. This result is not achievable under Gaussian assumptions alone. This means that the causal structure is identifiable without additional experiments or assumptions. Lingam use Independent Component Analysis (ICA) to obtain the results.
In particular, the Direct LiNGAM Algorithm can be stated in a symplified fashion as follows Shimizu2011:
The DirectLiNGAM algorithm is guaranteed to converge to the correct causal ordering under the LiNGAM assumptions (Shimizu2011), and runs in $O(p^3)$ time, making it computationally feasible for moderate-dimensional applications. In our three-variable setting $(Z, X, Y)$, the algorithm is particularly tractable.
In instrumental variable analysis, the exclusion restriction represents a critical identifying assumption that requires the instrument $Z$ to affect the outcome $Y$ only through the treatment $X$. Formally, this can be expressed as:
where $\epsilon$ and $\eta$ are error terms, and $Z$ is independent of $\epsilon$. The exclusion restriction implies that there should be no direct path from $Z$ to $Y$ that bypasses $X$. While this assumption is fundamental to IV estimation, it has traditionally been untestable in exactly-identified models without additional restrictions Angrist1996.
In exactly-identified models — where there is exactly one instrument per endogenous variable — the exclusion restriction is not testable using conventional methods. This is because the overidentifying restriction tests require more instruments than endogenous variables to produce testable predictions. When the model is exactly identified, the 2SLS estimator imposes the exclusion restriction by construction, and any residual direct effect of Z on Y is absorbed into the estimation without generating a detectable specification error. Violations of the exclusion restriction can arise for many reasons in applied research. For example, the instrument may affect the outcome through channels other than the treatment (e.g., proximity to college might affect wages through local labor market networks, not just through education). Even small violations can lead to substantial biases in IV estimates, particularly when the instrument is weak. Detecting such violations is therefore of first-order importance for applied researchers.
The connection between LiNGAM and IV analysis emerges when we consider the joint distribution of $(Z, X, Y)$. Under the exclusion restriction, the system forms a specific causal structure where $Z$ influences $Y$ only through $X$. However, if the exclusion restriction is violated, an additional direct path $Z \to Y$ would exist. The key insight of our approach is that non-Gaussianity allows us to distinguish between these two scenarios by examining the independence structure of residuals Angrist1995.
The power of non-Gaussian distributions in causal identification stems from their richer information content compared to Gaussian distributions. While Gaussian variables are fully characterized by their first and second moments, non-Gaussian variables contain additional information in higher-order moments. This additional information enables the identification of causal directions even in cases where traditional methods would fail Wang2020.
To see this more concretely, consider a bivariate system in which $X$ causes $Y$:
$$Y = \alpha * X + e_Y$$, where $e_Y$ is independent of $X$. If we attempt to 'reverse' this relationship and write $X = (1/\alpha) * Y + e_X$, the implied residual $e_X = X - (1/\alpha)*Y$ will generally be correlated with $Y$, unless both $X$ and $e_Y$ are Gaussian. Under non-Gaussianity, only the true causal direction yields statistically independent residuals. This asymmetry is the precise mechanism through which LiNGAM identifies causal structures.
In the context of IV analysis, non-Gaussianity allows us to test whether the residuals from the structural equations satisfy the independence conditions implied by the exclusion restriction. Specifically, if the instrument is valid, certain independence relationships must hold among the estimated components. Violations of these relationships indicate potential exclusion restriction violations, providing a basis for statistical testing Shimizu2019.
It is worth emphasizing that the non-Gaussianity assumption, while necessary for identification, is not particularly restrictive in practice. Many economic variables of interest — including wages, income, test scores, firm sizes, and financial returns — exhibit significant departures from normality due to heavy tails, skewness, or multimodality. Our framework includes formal tests to verify this prerequisite before proceeding with the structural analysis. Wang2020 have shown that the LiNGAM identification results extend to high-dimensional settings, and that even mild departures from Gaussianity provide substantial identification power. This suggests that our framework will be applicable in a wide range of empirical settings, even when non-Gaussianity is not extreme.
The proposed framework for testing the exclusion restriction combines the structural identification power of LiNGAM with instrumental variable analysis. This approach enables direct estimation of potential exclusion restriction violations through the parameter $\alpha_{ZY}$, which represents any direct effect of the instrument $Z$ on the outcome $Y$ that bypasses the treatment $X$. The methodology consists of five complementary statistical tests designed to assess different aspects of the exclusion restriction under varying data conditions. As shown in Figure 1, the framework replaces traditional validation modules with a LiNGAM-based approach that explicitly models non-Gaussian distributions.
The LiNGAM framework provides a natural mechanism for testing the exclusion restriction through its explicit modeling of direct causal effects. We consider the structural equation system:
where $\alpha_{ZX}$ represents the instrument's effect on the treatment, $\alpha_{XY}$ captures the treatment's effect on the outcome, and $\alpha_{ZY}$ quantifies any direct effect violating the exclusion restriction. The error terms $\epsilon_Z, \epsilon_X, \epsilon_Y$ are mutually independent and non-Gaussian. The key insight is that non-zero $\alpha_{ZY}$ directly indicates a violation of the exclusion restriction, as it implies $Z$ affects $Y$ through pathways other than $X$.
Under the null hypothesis $H_0 : a_ZY = 0$, the system reduces to the standard IV model and $Z$ is a valid instrument. Under the alternative $H_1 : a_ZY \neq 0$, the exclusion restriction is violated and the instrument is invalid. Our testing framework is designed to provide inference on this single parameter.
The validity of our approach fundamentally depends on the non-Gaussianity of the error terms. The Jarque-Bera test serves as our primary diagnostic tool thadewald2007jarque:
where $n$ is the sample size, $S$ represents standardized skewness, and $K$ denotes kurtosis. The Jarque-Bera test is particularly sensitive to departures in the tails of the distribution and is well-suited for the heavy-tailed distributions common in economic data. For more robust assessment, we supplement this with the Shapiro-Wilk test shapiro1965analysis, which is based on the correlation between the data and the scores of a normal distribution:
If the null hypothesis of normality cannot be rejected for one or more variables, the LiNGAM identification may be compromised. In such cases, we recommend either (i) investigating whether the apparent normality is genuine or a finite-sample artifact, (ii) exploring transformations or alternative distributional assumptions, or (iii) reporting the results with explicit acknowledgment that the identification assumption may not hold. In practice, economic variables rarely satisfy strict normality over large samples, and the tests typically confirm non-Gaussianity in empirical applications.
The sequential nature of the protocol — applying LiNGAM-based tests only after rejecting normality — introduces a form of pre-test bias. The nominal size of the subsequent tests is calibrated under the assumption that the non-Gaussianity condition holds unconditionally, but the actual inference is conducted conditional on the event {normality rejected}. When non-Gaussianity is pronounced, this event occurs with probability approaching one and the distortion is negligible. However, in small samples or when the true distribution is only mildly non-Gaussian, the conditional size of the exclusion restriction tests may differ non-trivially from the nominal level. We therefore recommend treating the reported p-values with caution in settings where the normality tests yield borderline rejections (e.g., p between 0.01 and 0.10), and suggest reporting the non-Gaussianity test statistics alongside the exclusion restriction results so that readers can assess the robustness of the identification assumption.
Strictly speaking, the identification result requires non-Gaussianity of the structural errors $(e_Z, e_X, e_Y)$, which are unobservable. In practice, however, the non-Gaussianity tests are applied to the observed variables $(Z, X, Y)$. This is justified by the linear structure of the model: since each observed variable is a linear combination of the independent structural errors, the distributional properties of the errors are transmitted to the observables. Specifically, a linear combination of independent non-Gaussian variables is itself non-Gaussian, provided that no Gaussian component dominates in the sense of the Darmois-Skitovich theorem. Conversely, if the observed variables were Gaussian, the structural errors would also have to be Gaussian by the same theorem, since a linear combination of independent variables is Gaussian if and only if all non-Gaussian components have zero weight. Therefore, rejecting normality for the observed variables is sufficient evidence that at least one structural error is non-Gaussian, which is the condition required for identification. This does not rule out the possibility that individual error terms are closer to Gaussian than the observed variables, so we recommend reporting the test statistics for all three observed variables and interpreting them jointly rather than requiring rejection for each variable separately.
We introduce five distinct tests for $H_0 : a_{ZY} = 0$, each exploiting different aspects of the sampling distribution of the LiNGAM estimator. Together, they provide a multi-faceted assessment of instrument validity. The motivation for using multiple tests is that each has different power properties: parametric tests are more powerful when distributional assumptions hold, while nonparametric tests are more robust to model misspecification.
The bootstrap percentile test constructs confidence intervals for the direct effect parameter $\alpha_{ZY}$ without requiring knowledge of its exact sampling distribution:
The procedure is as follows:
The bootstrap percentile test is attractive because it makes no parametric assumptions about the distribution of $\hat{a}_{ZY}$ and directly accounts for finite-sample variability in the LiNGAM estimates. However, it requires re-estimating the ICA model $B$ times (typically $B = 500$ or $B = 1000$), which may be computationally intensive for large datasets. In our implementation, we set $B = 1000$ by default.
An important consideration is the sign indeterminacy of ICA-based estimators: the columns of the estimated mixing matrix $A$ can be negated without affecting model fit. To ensure comparability across bootstrap iterations, we apply a sign normalization — specifically, we require that the diagonal elements of A are positive.
For larger samples, as the LiNGAM estimator is consistent and asymptotically normal Shimizu2006,Wang2020, we implement an asymptotic Wald test:
The asymptotic normal test is computationally efficient and provides exact p-values without resampling. However, its finite-sample performance depends on the rate of convergence of the LiNGAM estimator to its asymptotic distribution, which in turn depends on the degree of non-Gaussianity and the sample size \footnote{In our simulations, the standard error is estimated via bootstrap rather than the analytical formula of Shimizu2006, since the latter requires knowledge of the full score function of the error distributions, which is unavailable in practice without strong distributional assumptions.}. We recommend supplementing this test with the bootstrap test when sample sizes are moderate ($n < 500$) or when non-Gaussianity is mild.
The permutation test provides a distribution-free assessment of whether the estimated $a_{ZY}$ differs significantly from zero. The key insight is that, under the null hypothesis of exclusion restriction validity, the instrument Z should be independent of the LiNGAM residuals of $Y$ after accounting for $X$. Permuting the values of $Z$ therefore breaks any spurious correlation while preserving the marginal distributions of all variables.
The procedure is:
where $I\{\}$ denotes an indicator function. The permutation test has exact Type I error control under the null hypothesis (given that $Z$ is exchangeable), regardless of the underlying distributions. It is particularly valuable as a robustness check when the parametric and bootstrap tests yield inconclusive results. We use $R = 1000$ permutations by default.
The likelihood ratio test compares the log-likelihood of the unrestricted LiNGAM model (which freely estimates $a_{ZY}$) with that of the restricted model (which imposes $a_{ZY} = 0$). Under regularity conditions, the test statistic:
where $\theta$ are the structural parameters of the model. The log-likelihood for the LiNGAM model can be constructed from the densities of the estimated independent components. For non-Gaussian error distributions, parametric density estimates (e.g., using kernel methods or a flexible parametric family) are used to evaluate the log-likelihood.
The likelihood ratio test exploits the full distributional information in the data, making it the most powerful test when the distributional model is correctly specified. Its main limitation is sensitivity to misspecification of the error distribution. We recommend using the LR test in conjunction with the permutation test, which provides a nonparametric reference distribution.
The Hilbert-Schmidt Independence Criterion (HSIC) provides a non-parametric test for statistical dependence between the instrument Z and the residuals from the LiNGAM regression of Y on X. The HSIC is defined as:
where $H = I - \frac{1}{n} \mathbf{1}\mathbf{1}^\top$. $K$ y $L$ are the kernel or Gram matrices for $X$ e $Y$, respectively. Each element, $K_{ij}=k(x_i,x_j)$ and $L_{ij}=l(y_i,y_j)$ where $k$ and $l$ are kernel functions (e.g., gaussian). Under the null hypothesis of independence, $HSIC = 0$; under the alternative, $HSIC > 0$. Statistical significance is assessed via a permutation test on the HSIC statistic.
The HSIC test has several important properties. First, it is consistent against all types of dependence — not just linear correlations — making it sensitive to non-linear violations of the exclusion restriction. Second, the use of kernel methods allows it to detect dependencies in higher-order moments and tail behavior. Third, its permutation-based calibration provides finite-sample Type I error control.
However, the HSIC test's sensitivity comes at a cost: in large samples, it may detect substantively minor dependencies that are statistically significant but economically negligible. This is analogous to the well-known problem of tests becoming 'too powerful' in large samples. We therefore recommend interpreting HSIC rejections alongside the effect size estimate $\hat{a}_{ZY}$: a rejection with a near-zero point estimate suggests mild, potentially non-linear, violations.
A distinctive advantage of the HSIC and distance covariance tests is their ability to detect non-linear violations of the exclusion restriction — cases where $a_{ZY} = 0$ in the linear structural model, but the instrument still affects the outcome through non-linear channels. Such violations would be invisible to parametric tests that focus exclusively on the linear direct effect parameter. Non-linear violations can arise in several ways. For example, the instrument might affect the variance of the outcome even when its effect on the mean is zero (a form of heteroskedastic violation), or it might interact non-linearly with the treatment variable. In our framework, such violations manifest as non-zero HSIC between $Z$ and $e_Y$ even when $\hat{a}_{ZY} \sim 0$ (which is precisely the pattern we observe in the Card dataset application in the illustration section).
When both the linear parameter $a_{ZY}$ is estimated to be zero and HSIC suggests non-linear dependence, researchers face a nuanced inference problem. Our recommendation is to report both findings and interpret them in light of economic theory: if there is no credible channel through which $Z$ could non-linearly affect $Y$, the HSIC rejection may reflect finite-sample noise or distributional artifacts. If such a channel exists, the non-linear violation warrants further investigation.
The finite-sample performance of the proposed tests depends on three key factors: the sample size $n$, the magnitude of the exclusion restriction violation $|a_{ZY}|$, and the degree of non-Gaussianity in the error terms.
Let $\mathbf{W} = [Z, X, Y]$ denote the vector of observed variables:
The power of this approach derives from the non-Gaussian structure of errors. Consider the reduced form:
The finite-sample performance is quantified using the negentropy measure:
where $H(.)$ denotes differential entropy and $e_i^{Gauss}$ is a Gaussian variable with the same variance as $e_i$. Negentropy is always non-negative and equals zero if and only if $e_i$ is Gaussian. High negentropy implies greater identification power for the LiNGAM-based tests.
For the asymptotic regime, the LiNGAM estimator satisfies:
where $sigma^2_{ZY}$ depends on the causal structure and the distribution of the errors. This asymptotic result confirms the consistency of the estimator and provides the theoretical basis for the asymptotic Wald test described previously.
In finite samples, the bootstrap and permutation tests typically provide better Type I error control than asymptotic tests, at the cost of greater computation. Our simulation results, show that all five tests maintain Type I error rates close to the nominal $5\%$ level when the exclusion restriction holds, and achieve good power for violations of size $|a_{ZY}| \geq 0.2$ with $n = 500$ and non-Gaussian errors from a $t(5)$ distribution.
The complete testing protocol is summarized as follows:
The proposed protocol is designed to be sequential and informative at each step. A practitioner who finds evidence of non-Gaussianity (Step 1), a strong instrument (Step 2), and consistently fails to reject $H_0$ across the five tests (Step 5) has consistent evidence that the exclusion restriction holds. A practitioner who observes mixed evidence — such as rejection by the permutation and LR tests but non-rejection by the bootstrap and HSIC tests — should proceed with caution and explore potential channels of instrument invalidity. The codes used to implement the framework are available at the OSF platform: https://bit.ly/4cmyGYK
To evaluate the finite-sample performance of our proposed framework, we conduct Monte Carlo simulations. We consider a variety of scenarios, including exactly-identified models with and without exclusion restriction violations. The data generating process (DGP) follows the structural equations defined in Section 4, with error terms drawn from several a non-Gaussian distribution such as t-student.
The causal structural model is defined in the following form:
$$
$$
Where the parameters where defined as: $n=500$, $\alpha_{zx} = 0.7$ if the simulation assumes a strong instrument, else is set to $0.2$, $\alpha_{xy} = 0.5$. Also, the parameter of interest when testing, $\alpha_{zy}^{\text{*}}$, is set to $0.3$ if the exclusion is not valid, else is $0$. Finally, the DGP:
$$
$$
Prior to the structural analysis, the distributional properties of the variables were examined. For both scenarios, significant non-Gaussianity was detected via Jarque-Bera and Anderson-Darling tests, satisfying the fundamental identification requirement for LiNGAM. Furthermore, the instrument $Z$ demonstrated high relevance in the first-stage regression, with $F$-statistics of $336.28$ and $272.75$ for Example 1 and Example 2, respectively, indicating the absence of weak instrument bias.
Following the steps enumerated by the proposed testing protocol, we have the results shown in (ref).
The protocol yielded contrasting results for the two case studies.
To further characterize the performance of the tests, we conduct a power analysis varying the magnitude of the exclusion violation $a_{ZY}$ and the sample size. We consider $a_{ZY}$ in ${0, 0.1, 0.2, 0.3, 0.5}$ and $n$ in ${100, 250, 500, 1000}$, using the strong instrument specification $(\alpha_{ZX} = 0.7$). The number of simulations were set to $1000$ times each scenario. Table (ref) reports the rejection rates of the five tests.
Several patterns emerge from the rejection rates analysis. First, the tests maintain rejection rates close to 5% under the null ($a_{ZY} = 0$), confirming Type I error control. Second, power increases monotonically with both $|a_ZY|$ and $n$. Third, for moderate violations ($a_ZY = 0.2$) and $n = 500$, four of the five tests achieve power above 60%. Fourth, the HSIC test tends to have higher power than the Asymptotic Normal test for small violations, reflecting its sensitivity to higher-order distributional features. Fifth, even with $n = 100$, violations as large as $a_{ZY} = 0.3$ are detectable with reasonable power ($> 60\%$ for Permutation and LR tests).
The weak invalid case $(\alpha_{ZY} = 0.1, 0.2$) shows substantially lower power across all tests and violation sizes, consistent with the well-known difficulty of testing IV assumptions in the presence of weak instruments. This represents an important limitation of the approach, discussed further in Discussion section. This findings are further illustrated in Figure (ref). In Figure (ref) we can see that when the scenario is clearly one of rejection or non-rejection all five tests behave well, while in a middle-scenario (one of a small $\alpha_{ZX} = 0.2$ and a moderate sample, the results are not uniform among the different tests. The Figure (ref) shows the progression of the rejection rates of the Table (ref). Finally, Figure (ref) takes two representative tests (one parametric and other non-parametric, to display the patterns previously mentioned.
We apply our framework to the classic card1995using dataset, which studies the causal effect of years of education (X) on log wages (Y), using proximity to a four-year college (Z) as an instrument. We will apply this simplified scenario, and not the full model with co-variates, to illustrate the framework. The sample consists of $n = 4,739$ observations from the National Longitudinal Survey of Young Men \footnote{Extracted from the R package AER.}. The central research question is: what is the causal effect of an additional year of schooling on wages? The exclusion restriction requires that proximity to college affects wages only through its effect on educational attainment, and not through any direct channel such as local labor market conditions or social networks.
As a prerequisite, we verify the non-Gaussianity of all three variables. The Jarque-Bera test strongly rejects normality for log wages ($JB = 312.4, p < 0.001$), years of education ($JB = 189.7, p < 0.001$), and the college proximity indicator ($JB = 2,451.3, p < 0.001$). These rejections confirm that the identification conditions for LiNGAM are satisfied, and that the structural parameters are identifiable.
The first-stage regression confirms that the instrument is statistically strong, with an $F$-statistic of $41.49$ ($p < 0.001$), well above the conventional threshold of 10. This ensures that the proximity to college significantly influences educational attainment, fulfilling the relevance condition required for instrumental variable estimation.
The DirectLiNGAM estimation yielded a direct effect of proximity on wages ($\alpha_{ZY}$) of $0.0000$. This result is supported by four out of five formal hypothesis tests: The Bootstrap Percentile and Permutation tests both yielded $p$-values of $1.0000$, failing to reject the null hypothesis of no direct effect. The Likelihood Ratio test similarly supported the exclusion restriction ($p = 0.9565$). However, the Independence-based test (HSIC) strongly rejected the null ($p < 0.001$), suggesting some remaining dependence between the instrument and the error term of the wage equation. The "mixed evidence" indicated by the consensus (1/5 tests rejecting $H_0$) suggests that while the linear direct effect is negligible, there may be non-linear dependencies or subtle structural violations that the HSIC test is more sensitive to.
This pattern — zero linear direct effect, but significant HSIC — has a natural interpretation in this context. College proximity is a binary (or discrete) variable, while wages are continuous. Even when there is no linear relationship between college proximity and wage residuals, the HSIC may detect non-linearities associated with the distributional asymmetry between treated (close to college) and untreated (far from college) groups. Given the very large sample size ($n = 4,739$), the HSIC test has extremely high power and may be detecting negligible higher-order dependencies. We therefore interpret the overall evidence as supporting the validity of the exclusion restriction for the linear causal effect of interest.
The LiNGAM structural estimate of the causal effect of education on wages is $\hat{a}_{XY} = 0.0000$. The 2SLS estimate using the same instrument is $0.0012 (SE = 0.0105, p = 0.908)$. Both estimates are statistically indistinguishable from zero, and the difference between them ($0.0012$) is negligible in magnitude. This close alignment between the LiNGAM and 2SLS estimates is expected when the exclusion restriction holds, since both estimators should converge to the same causal parameter.
The non-significance of the returns to schooling in this specification contrasts with Card's original finding of a positive and significant IV estimate. This discrepancy likely reflects differences in sample selection, control variables, and model specification. Our results do not contradict Card's findings but rather highlight the sensitivity of the IV estimate to these modeling choices.
When there are more instruments, this approach can still be used, with the exception of using a correction for repeated testings (e.g. Bonferroni tests). The proposed framework naturally extends to settings with multiple instruments. When testing the exclusion restriction for each instrument separately, the probability of a spurious rejection increases with the number of tests. We therefore apply a Bonferroni correction, which adjusts the significance threshold to $\alpha_{adj} = \alpha / K$, where $K$ is the number of instruments. With $K = 2$ instruments at a nominal $5\%$ level, the adjusted threshold is $\alpha_{adj} = 0.025$.
To benchmark the multiple-instrument protocol, we first apply it to a synthetic dataset in which both instruments violate the exclusion restriction, with direct effects $a_{Z1Y} = a_{Z2Y} = 0.50$. All tests — Bootstrap, Likelihood Ratio, and HSIC — return p-values of $0.000$ for both instruments, and all rejections survive the Bonferroni correction. This confirms the protocol's high power in detecting simultaneous exclusion violations across multiple instruments.
To enhance the reliability of the causal identification, the validation protocol was extended to a multiple-instrument setting using $Z_1$ (nearc4) and $Z_2$ (nearc2) as instruments for education ($X$). To account for the increased risk of Type I errors inherent in multiple hypothesis testing, a Bonferroni correction was applied, setting the adjusted significance threshold at $\alpha = 0.025$.
The application to the Card dataset reveals a more complex structure. For both "nearc4" and "nearc2", the LiNGAM structural estimation found direct effects of 0.0000. The parametric and resampling-based tests (Bootstrap and LR) consistently failed to reject the exclusion restriction for both instruments, with $p$-values ranging from 0.61 to 1.00. However, the non-parametric HSIC tests for both instruments remained highly significant ($p < 0.001$), suggesting that while there is no linear direct effect, some form of statistical dependency persists. As discussed in previous section, we attribute this result to non-linear distributional features of the binary instruments rather than genuine exclusion restriction violations. Despite the HSIC results, the consensus among the other tests—and the alignment between the LiNGAM causal estimate (0.0000) and the 2SLS estimate (0.0121)—provides evidence that the instruments are largely valid for identifying the returns to schooling in a linear framework.
The proposed framework makes a contribution to the toolkit available for IV validation by providing a procedure for testing the exclusion restriction in exactly-identified models. The key enabling insight — that non-Gaussianity provides additional identifying information beyond what is available under standard Gaussian assumptions — is well established in the causal discovery literature but has not previously been leveraged in the context of IV analysis. Our work bridges this gap and makes the LiNGAM identification strategy accessible to applied economists and social scientists.
An advantage of the framework is that it provides multiple, complementary tests that differ in their statistical properties. The bootstrap and asymptotic tests are most powerful when the parametric model is well-specified; the permutation test provides finite-sample validity regardless of distributional assumptions; the HSIC test extends sensitivity to non-linear violations. A practitioner who observes consensus rejection across all five tests can be highly confident that the exclusion restriction is violated; a practitioner who observes consensus non-rejection (with the possible exception of HSIC in large samples) can proceed with IV estimation with more support for instrument validity.
Our framework differs from existing approaches to IV validation in several important dimensions. The Sargan-Hansen test requires overidentification and is based solely on second-order moments (covariances), making it insensitive to non-linear violations and inapplicable when there is only one instrument. The approach of conley2012plausibly relaxes the exclusion restriction by assuming that the instrument is 'plausibly exogenous' within a specified range, which requires the researcher to specify priors about the violation. In contrast, our framework estimates the violation parameter directly from the data without requiring prior specification.
The methods of kitagawa2015test and mourifie2017testing test a related but different hypothesis: the validity of IV assumptions under treatment effect heterogeneity. Their approach does not require non-Gaussianity but applies to a different class of models (Local Average Treatment Effects). Our framework is more directly analogous to testing the structural exclusion restriction in a homogeneous-effects linear model.
The method of dieterle2016simple also addresses the exactly-identified case and uses the first-stage residuals as a diagnostic. While related in spirit, their approach focuses on detecting instrument heterogeneity rather than direct exclusion restriction violations, and does not exploit non-Gaussianity for identification.
li2022rethinking propose a method based on prediction validity, which tests whether the instrument improves out-of-sample prediction. This approach is particularly suitable for nonparametric models, but does not provide a direct test of the exclusion restriction. Our framework is more interpretable in the sense that it provides a point estimate of the violation parameter $a_{ZY}$ alongside the formal test, which directly quantifies the potential bias in the IV estimator.
Despite its advantages, the proposed framework has several important limitations that practitioners should be aware of. First and most fundamentally, the framework requires non-Gaussianity of the structural errors for identification. If the true error distributions are Gaussian, or if the sample is too small to reliably detect non-Gaussianity, the LiNGAM-based tests will have no power and may produce uninformative or misleading results. We therefore recommend always verifying the non-Gaussianity assumption as the first step of the protocol and reporting these diagnostic statistics.
Second, the framework assumes a linear structural model. If the true relationships among $Z$, $X$, and $Y$ are substantially non-linear, the linear LiNGAM model may misspecify the causal structure. In such cases, the estimated $a_{ZY}$ may not accurately capture the true exclusion violation. Extensions to nonparametric LiNGAM could address this limitation, but at the cost of additional assumptions and computational complexity.
Third, as noted previously and illustrated in the Card application, the HSIC test may reject $H_0$ in large samples even when the linear exclusion restriction holds, due to its sensitivity to non-linear and higher-order dependencies. This creates a risk of false positives in large datasets. We recommend interpreting HSIC results alongside the magnitude of $\hat{a}_{ZY}$: a rejection paired with a near-zero point estimate suggests that any violation, if present, is economically negligible.
Fourth, the framework currently handles the case of continuous variables most naturally. When instruments or treatments are binary or discrete, the LiNGAM model may not be the most appropriate structural model. Adapting our framework to discrete variable settings is an interesting direction for future research.
Fifth, the framework does not address the possibility of latent confounding — i.e., hidden common causes of $Z$ and $Y$. If such confounders exist, they would also constitute a violation of the exogeneity assumption, which is a distinct (and generally untestable) requirement from the exclusion restriction. Our framework focuses specifically on the exclusion restriction and assumes that the exogeneity condition holds.
As noted in Section 4, instrument weakness poses a challenge for the LiNGAM-based framework that is qualitatively distinct from its effect on 2SLS. In the 2SLS context, a weak instrument inflates standard errors and introduces finite-sample bias toward the OLS estimate, but the structural interpretation of the estimand is preserved. In the LiNGAM context, a weak instrument can cause the DirectLiNGAM algorithm to recover an incorrect causal ordering with non-trivial probability, which invalidates the structural interpretation of all estimated coefficients. Future work could characterize the finite-sample distribution of the LiNGAM causal order estimator as a function of instrument strength, analogously to the weak instruments literature.
The framework is developed and illustrated for the trivariate system $(Z, X, Y)$ without additional covariates. This is a deliberate simplification that reflects the paper's primary objective: to demonstrate that non-Gaussianity provides identifying information for the exclusion restriction when a single instrument is available, a setting where traditional tests are silent. The extension to models with covariates $W$ is conceptually straightforward — one can apply the LiNGAM-based protocol to the residuals obtained after projecting $Z$, $X$, and $Y$ onto $W$ — but a formal treatment of how the first-stage partialling affects the finite-sample properties of the proposed tests is left for future work. The multiple-instrument illustration provides a first step in this direction by showing that the protocol extends naturally beyond the baseline case, and the simulation design is internally consistent in the single-instrument setting that constitutes the paper's core contribution. Applied researchers working with richer specifications should treat the current framework as a diagnostic tool for the exclusion restriction conditional on their preferred first-stage specification, rather than as a complete IV estimation procedure.
Our framework connects IV analysis to the broader literature on causal discovery from observational data. The LiNGAM approach was originally developed as a machine learning tool for learning causal structure from non-experimental data, and has been applied successfully in areas including neuroscience, genomics, and social network analysis. Our work demonstrates that causal discovery methods can also provide valuable diagnostic information in settings where partial causal structure is already known (as in IV analysis).
This connection points to a broader research agenda: using causal discovery methods to validate, rather than replace, traditional identification strategies. Rather than treating causal discovery as an alternative to IV, our framework treats it as a complementary tool that can exploit distributional information to test the validity of IV assumptions.
Based on our theoretical analysis and simulation results, we offer the following practical recommendations for applied researchers.
This paper has introduced a framework for validating the exclusion restriction in instrumental variable analysis by leveraging the unique identification properties of non-Gaussian distributions. By integrating the LiNGAM method with traditional tools, we introduce a suite of five complementary statistical tests — bootstrap percentile, asymptotic normal, permutation, likelihood ratio, and HSIC — that can detect violations even in exactly-identified models where traditional overidentification tests are inapplicable.
The main theoretical result is that, under non-Gaussianity, the parameter $a_{ZY}$ capturing any direct effect of the instrument on the outcome is point-identified from the joint distribution of $(Z, X, Y)$, without requiring additional instruments or parametric assumptions beyond linearity and independence. This identification result, which follows directly from the Darmois-Skitovich theorem, allows us to formulate a testing problem for the exclusion restriction.
The Monte Carlo simulations demonstrate that the proposed tests maintain well-controlled Type I error rates under the null hypothesis and achieve reasonable power against exclusion restriction violations of economically relevant magnitudes. The empirical application illustrates the practical utility of the framework. The linear exclusion restriction is supported by four out of five tests, with the exception of the HSIC test — a pattern we attribute to the discreteness of the binary instrument and the large sample size rather than a genuine causal violation. The close alignment between the LiNGAM structural estimate and the 2SLS estimate provides additional evidence for the instrument's validity.
Several directions for future research emerge from this work. First, extending the framework to nonparametric structural models would allow it to handle genuine non-linear causal mechanisms. Second, incorporating latent confounding variables — hidden common causes that violate both exogeneity and the exclusion restriction simultaneously — would address an important limitation of the current approach. Third, developing a unified procedure that jointly tests instrument relevance, exogeneity, and the exclusion restriction within the LiNGAM framework would provide a comprehensive IV validation toolkit. Fourth, exploring the connections between our approach and the literature on sensitivity analysis and partial identification gangl2013partial may yield bounds on causal effects under mild exclusion restriction violations.
In conclusion, we believe that the integration of non-Gaussian causal discovery methods with classical tools offers a path toward more rigorous and transparent causal inference. The proposed framework demonstrates that non-Gaussianity, often treated as a nuisance in applied work, can be reinterpreted as a source of identifying information in IV settings where traditional tests are unavailable.