Computing maximum likelihood estimates in recursive linear models with correlated errors

Details Computing maximum likelihood estimates in recursive linear models with correlated errors Computing maximum likelihood estimates in recursive linear models with correlated errors

2006-01-26

arxiv.org/abs/math/0601631v4

Journal of Machine Learning Research 2009, Vol. 10(Oct), 2329-2348

Mathematics

Statistics Theory

22 pages; removed an incorrect identifiability claim

Scientific paper

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.

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