  The field of causal discovery develops model selection methods to infer
cause-effect relations among a set of random variables. For this purpose,
different modelling assumptions have been proposed to render cause-effect
relations identifiable. One prominent assumption is that the joint distribution
of the observed variables follows a linear non-Gaussian structural equation
model. In this paper, we develop novel goodness-of-fit tests that assess the
validity of this assumption in the basic setting without latent confounders as
well as in extension to linear models that incorporate latent confounders. Our
approach involves testing algebraic relations among second and higher moments
that hold as a consequence of the linearity of the structural equations.
Specifically, we show that the linearity implies rank constraints on matrices
and tensors derived from moments. For a practical implementation of our tests,
we consider a multiplier bootstrap method that uses incomplete U-statistics to
estimate subdeterminants, as well as asymptotic approximations to the null
distribution of singular values. The methods are illustrated, in particular,
for the T\"ubingen collection of benchmark data sets on cause-effect pairs.
