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December 2018 Causal inference in partially linear structural equation models
Dominik Rothenhäusler, Jan Ernest, Peter Bühlmann
Ann. Statist. 46(6A): 2904-2938 (December 2018). DOI: 10.1214/17-AOS1643


We consider identifiability of partially linear additive structural equation models with Gaussian noise (PLSEMs) and estimation of distributionally equivalent models to a given PLSEM. Thereby, we also include robustness results for errors in the neighborhood of Gaussian distributions. Existing identifiability results in the framework of additive SEMs with Gaussian noise are limited to linear and nonlinear SEMs, which can be considered as special cases of PLSEMs with vanishing nonparametric or parametric part, respectively. We close the wide gap between these two special cases by providing a comprehensive theory of the identifiability of PLSEMs by means of (A) a graphical, (B) a transformational, (C) a functional and (D) a causal ordering characterization of PLSEMs that generate a given distribution $\mathbb{P}$. In particular, the characterizations (C) and (D) answer the fundamental question to which extent nonlinear functions in additive SEMs with Gaussian noise restrict the set of potential causal models, and hence influence the identifiability.

On the basis of the transformational characterization (B) we provide a score-based estimation procedure that outputs the graphical representation (A) of the distribution equivalence class of a given PLSEM. We derive its (high-dimensional) consistency and demonstrate its performance on simulated datasets.


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Dominik Rothenhäusler. Jan Ernest. Peter Bühlmann. "Causal inference in partially linear structural equation models." Ann. Statist. 46 (6A) 2904 - 2938, December 2018.


Received: 1 July 2016; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06968603
MathSciNet: MR3851759
Digital Object Identifier: 10.1214/17-AOS1643

Primary: 62G99, 62H99
Secondary: 68T99

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 6A • December 2018
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