Robustness Checks #SmartPLS4 - Endogeneity using Gaussian Copula in #SmartPLS4?

Endogeneity checks in SmartPLS4 can be made more reliable by testing not just single paths, but every combination of paths using the Gaussian copula approach. The workflow starts by selecting “Gaussian copula,” then running an endogeneity test for each individual relationship (path) in the structural model. After that, the process scales up: it tests pairs of relationships together, then triples, and continues through to the maximum number of relationships present in the model. This matters because endogeneity can hide in specific paths or emerge only when multiple paths are considered jointly, leading to misleading path significance if it’s not detected.

In practice, the procedure is iterative. A user first saves the output after testing one relationship and runs bootstrapping to obtain path coefficients and p-values. If the resulting path coefficient is insignificant, that path is treated as not showing endogeneity concerns. The transcript also notes a practical reporting step: copy the relevant output row into Excel for tracking, including the p-value and the Gaussian copula results. The output is interpreted in terms of which variable is exogenous versus endogenous (for example, “TL” as the endogenous variable and the Gaussian copula output as the test statistic/indicator).

Next, the user removes that tested relationship and adds a different path, then reruns bootstrapping. In the example, the second relationship produces a significant result, which flags endogeneity issues for that specific path—meaning the model may violate assumptions that underpin standard inference. The transcript emphasizes that resolving endogeneity is a separate topic, to be handled later, but the detection step is treated as complete once the problematic paths are identified.

After single-path testing, the workflow moves to combinations. The user runs the Gaussian copula endogeneity test for two relationships at a time, then checks the resulting path coefficients again. In the example, one relationship in the pair shows no endogeneity problems while the other still requires attention, illustrating why combination testing is useful: it helps map where endogeneity is concentrated and how it interacts across paths. The same logic is repeated for all possible pairs, then for all possible triples, and then for all possible quadruples (and so on), until every relevant combination has been assessed.

The final deliverable is a table summarizing endogeneity test results across single paths and multi-path combinations. The decision rule is straightforward: p-values above 0.05 indicate no endogeneity issue, while p-values at or below 0.05 suggest endogeneity concerns. With those results compiled, the transcript frames this as the second step in robustness checking—after which the next robustness dimension to examine is heterogeneity.