  In recursive linear models, the multivariate normal joint distribution of all
variables exhibits a dependence structure induced by a recursive (or acyclic)
system of linear structural equations. These linear models have a long
tradition and appear in seemingly unrelated regressions, structural equation
modelling, and approaches to causal inference. They are also related to
Gaussian graphical models via a classical representation known as a path
diagram. Despite the models' long history, a number of problems remain open. In
this paper, we address the problem of computing maximum likelihood estimates in
the subclass of `bow-free' recursive linear models. The term `bow-free' refers
to the condition that the errors for variables $i$ and $j$ be uncorrelated if
variable $i$ occurs in the structural equation for variable $j$. We introduce a
new algorithm, termed Residual Iterative Conditional Fitting (RICF), that can
be implemented using only least squares computations. In contrast to existing
algorithms, RICF has clear convergence properties and finds parameter estimates
in closed form whenever possible.
