Endogeneity in PLS-SEM: What Gaussian Copulas Can Address without Instrumental Variables

PLS-SEM estimation assumes independent variables are uncorrelated with structural equation residuals. When this assumption fails and no valid instrumental variable is available, Gaussian copulas allow testing and correction of endogeneity using existing data, provided the suspected endogenous regressor is nonnormally distributed.

PLS-SEM & SMARTPLSPrepared by Lê Minh7 min

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Endogeneity: define the problem before the remedy

Mechanism
Identify plausible sources
Design
Assess data and identification
Method
Check applicability conditions
Sensitivity
Compare assumptions and results

Conceptual diagram. A Gaussian copula does not resolve every source of endogeneity.

Source: Lê Minh · Conceptual illustration, not software output.

1. What Is Endogeneity, and Why Does It Distort Path Coefficients?

Like OLS regression, PLS-SEM rests on an assumption often left unstated in theses: the independent variables in each structural equation are uncorrelated with that equation's error term. Hair, Risher, Sarstedt, and Ringle (2019) identify this as a condition to check before interpreting path coefficients causally. When it fails, the independent variable is endogenous, and its estimated coefficient no longer accurately reflects its true effect on the dependent variable.

The consequences do not have a fixed direction. Depending on the correlation between the independent variable and error term, endogeneity can inflate or attenuate a path coefficient, or even reverse the observed relationship's sign. A statistically significant coefficient with the theoretically expected sign can therefore still be biased if endogeneity remains unchecked.

Brief definition

An independent variable is endogenous in a structural equation when it correlates with that equation's error term. This can arise from omitted variables, measurement error, reverse causality, or shared features of data collection.

2. Four Common Sources of Endogeneity in PLS-SEM

Endogeneity rarely has a single cause. SmartPLS's Gaussian copula documentation lists four main source groups, and most social science survey models face at least one.

Table 1
Source	Mechanism	Survey model example
Omitted variables	A variable affecting the dependent variable is excluded while correlating with the independent variable under examination.	A model estimates Service Quality → Repurchase Intention but omits Brand Reputation, which affects Repurchase Intention and correlates with Service Quality.
Measurement error	Observed variables do not accurately measure the latent concept, allowing measurement error into the structural equation residual.	Self-reported work engagement scales contain systematic bias because respondents interpret questions differently.
Simultaneity	Independent and dependent variables affect each other at the same measurement occasion, rather than in the single direction assumed.	Job satisfaction and performance may influence each other, but the model includes only Satisfaction → Performance.
Unobserved heterogeneity	Groups differ in unmeasured behavior or characteristics, mixing distinct mechanisms in the observed correlations.	Two customer groups with different purchase motivations are pooled, producing an average path that represents neither accurately.

These four groups follow SmartPLS's Gaussian copula documentation. Common method variance, arising when independent and dependent variables share respondents, timing, and scale formats, can also appear as a form of simultaneity or unobserved heterogeneity.

Control variables do not replace endogeneity testing

Adding controls may reduce some omitted-variable risk, but this is risk mitigation rather than a statistical endogeneity test. A model with many controls still requires separate examination when endogeneity is suspected.

3. Why Valid Instrumental Variables Are Not Always Available

The classical econometric remedy uses an instrumental variable (IV), which correlates with the suspected endogenous predictor but not directly with the equation's error term. Good instruments are difficult to find, particularly in cross-sectional social science surveys, where measured variables often relate to multiple concepts within the same theoretical model.

Many studies select weak or theoretically tenuous instruments merely to satisfy procedural requirements. A weak instrument may create more serious estimation problems than leaving endogeneity untreated. Gaussian copulas address this gap by detecting and correcting endogeneity without an external instrument.

4. Gaussian Copula Principles and Three Variants

Park and Gupta (2012) propose using the suspected endogenous regressor's own distribution rather than finding an external instrument. If it is nonnormal, a copula term can represent its correlation with the error and enter the model as an additional predictor. The copula coefficient reflects the estimated regressor-error correlation scaled by error variance. With the copula included, the main regressor's path coefficient is treated as endogeneity-corrected.

Table 2
Variant	Characteristics	Scope
Original maximum likelihood (Park & Gupta, 2012)	Joint maximum likelihood estimation.	Handles only one endogenous regressor per estimation.
Control function (Becker, Proksch & Ringle, 2022)	Adds a separate copula term for each suspected endogenous regressor, allowing simultaneous treatment of several variables. This is the SmartPLS implementation used in most practical applications.	PLS-SEM and regression models with multiple predictors.
Extended framework (Liengaard et al., 2025)	Provides an adjusted estimator addressing limitations of the original control-function variant.	Integrated into SmartPLS and applicable beyond conventional regression, including PLS-SEM and path analysis.

The SmartPLS Gaussian Copula button implements the control-function variant. Researchers can therefore examine several suspected endogenous paths simultaneously, with a separate copula term for each selected path.

5. A Prerequisite before Use

Gaussian copulas do not apply to every regressor. The suspected endogenous regressor must be nonnormally distributed. If it is close to normal, the copula term cannot separate correlation with the error from random variation, making estimation unreliable.

Check nonnormality first

Before constructing copula terms, test each suspected endogenous regressor's distribution, for example with the Cramér-von Mises test. Becker, Proksch, and Ringle (2022) and Eckert and Hohberger (2023) emphasize that omitting this step commonly causes misapplication.

Through simulation, Eckert and Hohberger (2023) identify further limitations: correction effectiveness depends on the degree of nonnormality, sample size, and actual endogeneity. The method works better with clearly nonnormal regressors and samples large enough for stable estimation. Results may be less reliable when departures from normality are slight.

6. Implementation in SmartPLS

Build the PLS-SEM model as usual

Construct the measurement and structural models according to theory without adding copula terms yet.

Check suspected endogenous regressors' distributions

Run a distributional test, such as Cramér-von Mises, for each suspected regressor. Proceed only for variables meeting the nonnormality requirement.

Select paths and create copula terms

Click Gaussian Copula on the SmartPLS toolbar, then left-click each structural path to examine. Each selected path gains a circle labeled GC representing its copula term.

Run the PLS-SEM algorithm

Estimate the model with GC terms using PLS-SEM as usual, obtaining corrected path coefficients and each copula coefficient.

Bootstrap for significance testing

Use bootstrapping to obtain standard errors and significance tests for copula coefficients as well as main path coefficients.

Assess endogeneity for each relationship

Use each GC term's significance to assess endogeneity in its corresponding structural relationship and decide whether to report the corrected or original path coefficient.

7. Interpreting the Copula Term

The copula coefficient reflects the estimated regressor-error correlation scaled by error variance. A significant coefficient supports endogeneity in the relationship and use of the corrected path coefficient. A non-significant coefficient indicates insufficient evidence of endogeneity in the available data, rather than conclusive proof of its absence.

Example wording for a report or thesis

To assess endogeneity in [independent variable] → [dependent variable], the study used the Gaussian copula control-function approach (Becker, Proksch & Ringle, 2022), after confirming nonnormality of [independent variable] with the Cramér-von Mises test (p [</≥] 0,05). The copula coefficient was [value], [statistically significant/not statistically significant] at [significance level] using [number of repetitions] bootstrap subsamples. This indicates [endogeneity/no evidence of endogeneity] in the available data for this relationship; the study reports the [corrected/original] path coefficient.

8. Common Interpretation Mistakes

Table 3
Weak wording	Why it is inappropriate	Recommended wording
"Add copula terms to every path to be safe."	The method is meaningful where theoretical or empirical grounds suggest endogeneity and the regressor meets the nonnormality requirement.	Justify selected paths and apply the method to relationships with specific grounds for concern.
"Skip distribution checks; run Gaussian copulas and inspect the results."	Regressor nonnormality is mandatory rather than optional.	Report nonnormality test results before copula results.
"The copula term is not significant, so the model definitely has no endogeneity."	Non-significance indicates no evidence in the available data, rather than proof of complete absence.	State that no evidence of endogeneity was found for this relationship and explain the conclusion's limits.
"The model has controls, so endogeneity testing is unnecessary."	Controls reduce some omitted-variable risk but do not constitute an endogeneity test.	Conduct Gaussian copula testing where endogeneity is suspected, even with controls.
"Report only the copula coefficient without discussing changes in the main path."	The objective is to assess bias in the focal path coefficient, beyond identifying a copula term.	Compare and report path coefficients before and after adding copula terms.

9. Checklist before Reporting

Have you stated theoretical or empirical grounds for suspecting endogeneity in the tested relationship?
Have you tested regressor nonnormality with a specified test, such as Cramér-von Mises?
Have you identified the Gaussian copula variant used: the control-function approach of Becker, Proksch & Ringle (2022)?
Have you bootstrapped copula significance rather than reporting point estimates alone?
Have you compared path coefficients before and after adding copula terms?
Have you interpreted non-significant copula terms as insufficient evidence rather than proof of no endogeneity?
Have you stated limitations related to sample size and the actual degree of nonnormality?

Conclusion

Gaussian copulas provide a feasible endogeneity test for PLS-SEM models lacking valid instruments, a common situation in cross-sectional social science surveys. Their value lies in enabling direct testing with existing data rather than assuming away or overlooking endogeneity.

The nonnormality prerequisite and limited ability to identify the specific source of endogeneity require attention when using this method in a thesis or article. A significant copula term signals a need for correction, but does not fully explain the mechanism causing endogeneity.

References

Original Methodological Sources

Park, S., & Gupta, S. (2012). Handling endogenous regressors by joint estimation using copulas. Marketing Science, 31(4), 567-586. https://doi.org/10.1287/mksc.1120.0718
Becker, J.-M., Proksch, D., & Ringle, C. M. (2022). Revisiting Gaussian copulas to handle endogenous regressors. Journal of the Academy of Marketing Science, 50, 46-66. https://doi.org/10.1007/s11747-021-00805-y
Liengaard, B. D., Becker, J.-M., Bennedsen, M., Heiler, P., Taylor, L. N., & Ringle, C. M. (2025). Dealing with regression models' endogeneity by means of an adjusted estimator for the Gaussian copula approach. Journal of the Academy of Marketing Science, 53, 279-299. https://doi.org/10.1007/s11747-024-01055-4

Applications and Evaluations of Limitations

Hair, J. F., Risher, J. J., Sarstedt, M., & Ringle, C. M. (2019). When to use and how to report the results of PLS-SEM. European Business Review, 31(1), 2-24. https://doi.org/10.1108/EBR-11-2018-0203
Hult, G. T. M., Hair, J. F., Proksch, D., Sarstedt, M., Pinkwart, A., & Ringle, C. M. (2018). Addressing endogeneity in international marketing applications of partial least squares structural equation modeling. Journal of International Marketing, 26(3), 1-21. https://doi.org/10.1509/jim.17.0151
Eckert, C., & Hohberger, J. (2023). Addressing endogeneity without instrumental variables: An evaluation of the Gaussian copula approach for management research. Journal of Management, 49(4), 1460-1495. https://doi.org/10.1177/01492063221085913

Academic note: This article presents general Gaussian copula principles and procedures based on Park and Gupta (2012), Becker, Proksch, and Ringle (2022), and related evaluations. Application to a specific model requires checking actual data nonnormality and considering limitations in the cited sources rather than applying the method mechanically to every relationship.