Characterize the effects of misspecified outer transformations in PNL models

Characterize when misspecification of the outer transformation in a post-nonlinear causal model necessarily induces a departure from the additive-model null hypothesis used by the Sen et al. goodness-of-fit and independence test.

Background

The paper applies the goodness-of-fit and independence test of Sen et al., which was developed for additive models, to post-nonlinear models. When the outer transformation is correctly specified, the post-nonlinear model can be transformed into an additive-noise model, allowing the cited test theory to provide asymptotic Type I error control under additional regularity conditions.

The authors explain that this argument fails directly when the outer transformation is misspecified. Although an incorrectly specified inverse transformation may produce a misspecified regression function, predictor–error dependence, or both, the cited theory does not establish whether every form of outer-transformation misspecification necessarily produces such a departure. The paper therefore studies one particular misspecification empirically rather than providing a general characterization.

References

This argument does not directly apply when $f_2$ is misspecified. Applying an incorrectly specified inverse transformation may result in a misspecified regression function, dependence between the predictor and error, or both. Such departures correspond to alternatives considered by \citet{sen_testing_2014}, for which their test is asymptotically consistent under their regularity conditions. However, their theory does not characterize when misspecification of the PNL outer transformation necessarily induces one of these departures from the additive-model null.

— Statistical Inference for Bivariate Functional Causal Discovery  (2609.16562 - Prakash et al., 15 Sep 2026) in Appendix, Section “PNL Model Misspecification Simulation Results”