# Computing Maximum Likelihood Estimates in Recursive Linear Models with Correlated Errors

Mathias Drton, Michael Eichler, Thomas S. Richardson.

Year: 2009, Volume: 10, Issue: 81, Pages: 2329−2348

#### Abstract

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 yields exact maximum likelihood
estimates after the first iteration whenever the MLE is available in
closed form.