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

2026-06-25

AI Summary

This study addresses the absence of semiparametric inference for the half-trek criterion (HTC) estimator in existing linear structural equation models, which has precluded derivation of its asymptotic distribution, standard errors, and confidence intervals. Focusing on directed mixed graph models with latent confounding, this work establishes the first semiparametric inference framework for HTC-identifiable structural coefficients. By deriving a semiparametric influence function applicable to both cyclic and acyclic graphs and integrating structural residuals with instrumental variables, the approach enables recursive correction for estimation uncertainty. The resulting HTC estimator is shown to be asymptotically normal and admits a closed-form variance expression. The method is successfully applied to the Fulton Fish Market dataset, yielding a fully identified causal effect of supply on demand.

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 all HTC-identified 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.

Problem

• linear structural equation models
• half-trek criterion
• asymptotic distribution
• latent confounding
• causal inference

Innovation

• half-trek criterion
• semiparametric inference
• influence function
• asymptotic normality
• structural equation models