  Linear causal models are important tools for modeling causal dependencies and
yet in practice, only a subset of the variables can be observed. In this paper,
we examine the parameter identifiability of these models by investigating
whether the edge coefficients can be recovered given the causal structure and
partially observed data. Our setting is more general than that of prior
research - we allow all variables, including both observed and latent ones, to
be flexibly related, and we consider the coefficients of all edges, whereas
most existing works focus only on the edges between observed variables.
Theoretically, we identify three types of indeterminacy for the parameters in
partially observed linear causal models. We then provide graphical conditions
that are sufficient for all parameters to be identifiable and show that some of
them are provably necessary. Methodologically, we propose a novel
likelihood-based parameter estimation method that addresses the variance
indeterminacy of latent variables in a specific way and can asymptotically
recover the underlying parameters up to trivial indeterminacy. Empirical
studies on both synthetic and real-world datasets validate our identifiability
theory and the effectiveness of the proposed method in the finite-sample
regime.
