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

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.