  We develop estimation for potentially high-dimensional additive structural
equation models. A key component of our approach is to decouple order search
among the variables from feature or edge selection in a directed acyclic graph
encoding the causal structure. We show that the former can be done with
nonregularized (restricted) maximum likelihood estimation while the latter can
be efficiently addressed using sparse regression techniques. Thus, we
substantially simplify the problem of structure search and estimation for an
important class of causal models. We establish consistency of the (restricted)
maximum likelihood estimator for low- and high-dimensional scenarios, and we
also allow for misspecification of the error distribution. Furthermore, we
develop an efficient computational algorithm which can deal with many
variables, and the new method's accuracy and performance is illustrated on
simulated and real data.
