  In this paper we study linear non-Gaussian graphical models from the
perspective of algebraic statistics. These are acyclic causal models in which
each variable is a linear combination of its direct causes and independent
noise. The underlying directed causal graph can be identified uniquely via the
set of second and third order moments of all random vectors that lie in the
corresponding model. Our focus is on finding the algebraic relations among
these moments for a given graph. We show that when the graph is a polytree
these relations form a toric ideal. We construct explicit trek-matrices
associated to 2-treks and 3-treks in the graph. Their entries are covariances
and third order moments and their 2-minors define our model set-theoretically.
Furthermore, we prove that their 2-minors also generate the vanishing ideal of
the model. Finally, we describe the polytopes of third order moments and the
ideals for models with hidden variables.
