  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 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.
