Semiparametric Inference for Half-Trek Estimators in Linear Structural Equation Models

Leopold Mareis, Nils Sturma, Mathias Drton
Proceedings of the 13th International Conference on Probabilistic Graphical Models, PMLR 343:575-598, 2026.

Abstract

Linear structural equation models on directed mixed graphs encode causal relationships among variables subject to latent confounding. The half-trek criterion (HTC) provides a graphical sufficient condition for the structural coefficients to be rationally identifiable from the observable covariance matrix, and yields a corresponding closed-form rational estimator. Despite this, the asymptotic distribution of the HTC estimator, and hence valid standard errors and confidence regions, have not been derived. We derive the semiparametric influence function of this estimator for every HTC-identified node, in arbitrary directed mixed graphs, including cyclic ones. The influence function combines the structural residual at the target node with the identification instruments, recursively corrected for uncertainty from earlier estimation stages. The HTC estimator is asymptotically normal with variance computable in closed form, yielding confidence regions, marginal intervals, and Wald tests for individual structural coefficients. Applied to the Fulton Fish Market dataset, our theory delivers a complete inferential summary for the causal effect of supply on demand.

Cite this Paper

BibTeX
@InProceedings{pmlr-v343-mareis26a, title = {Semiparametric Inference for Half-Trek Estimators in Linear Structural Equation Models}, author = {Mareis, Leopold and Sturma, Nils and Drton, Mathias}, booktitle = {Proceedings of the 13th International Conference on Probabilistic Graphical Models}, pages = {575--598}, year = {2026}, editor = {Camps-Valls, Gustau and Leonelli, Manuele and Varando, Gherardo}, volume = {343}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v343/main/assets/mareis26a/mareis26a.pdf}, url = {https://proceedings.mlr.press/v343/mareis26a.html}, abstract = {Linear structural equation models on directed mixed graphs encode causal relationships among variables subject to latent confounding. The half-trek criterion (HTC) provides a graphical sufficient condition for the structural coefficients to be rationally identifiable from the observable covariance matrix, and yields a corresponding closed-form rational estimator. Despite this, the asymptotic distribution of the HTC estimator, and hence valid standard errors and confidence regions, have not been derived. We derive the semiparametric influence function of this estimator for every HTC-identified node, in arbitrary directed mixed graphs, including cyclic ones. The influence function combines the structural residual at the target node with the identification instruments, recursively corrected for uncertainty from earlier estimation stages. The HTC estimator is asymptotically normal with variance computable in closed form, yielding confidence regions, marginal intervals, and Wald tests for individual structural coefficients. Applied to the Fulton Fish Market dataset, our theory delivers a complete inferential summary for the causal effect of supply on demand.} }
Endnote
%0 Conference Paper %T Semiparametric Inference for Half-Trek Estimators in Linear Structural Equation Models %A Leopold Mareis %A Nils Sturma %A Mathias Drton %B Proceedings of the 13th International Conference on Probabilistic Graphical Models %C Proceedings of Machine Learning Research %D 2026 %E Gustau Camps-Valls %E Manuele Leonelli %E Gherardo Varando %F pmlr-v343-mareis26a %I PMLR %P 575--598 %U https://proceedings.mlr.press/v343/mareis26a.html %V 343 %X Linear structural equation models on directed mixed graphs encode causal relationships among variables subject to latent confounding. The half-trek criterion (HTC) provides a graphical sufficient condition for the structural coefficients to be rationally identifiable from the observable covariance matrix, and yields a corresponding closed-form rational estimator. Despite this, the asymptotic distribution of the HTC estimator, and hence valid standard errors and confidence regions, have not been derived. We derive the semiparametric influence function of this estimator for every HTC-identified node, in arbitrary directed mixed graphs, including cyclic ones. The influence function combines the structural residual at the target node with the identification instruments, recursively corrected for uncertainty from earlier estimation stages. The HTC estimator is asymptotically normal with variance computable in closed form, yielding confidence regions, marginal intervals, and Wald tests for individual structural coefficients. Applied to the Fulton Fish Market dataset, our theory delivers a complete inferential summary for the causal effect of supply on demand.
APA
Mareis, L., Sturma, N. & Drton, M.. (2026). Semiparametric Inference for Half-Trek Estimators in Linear Structural Equation Models. Proceedings of the 13th International Conference on Probabilistic Graphical Models, in Proceedings of Machine Learning Research 343:575-598 Available from https://proceedings.mlr.press/v343/mareis26a.html.

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