Computing maximum likelihood estimates in recursive linear models with correlated errors

TL;DR

This paper introduces RICF, a new algorithm for efficiently computing maximum likelihood estimates in bow-free recursive linear models, with guaranteed convergence and closed-form solutions when feasible.

Contribution

The paper presents RICF, a novel least squares-based algorithm for maximum likelihood estimation in bow-free recursive linear models, improving convergence and solution clarity.

Findings

RICF guarantees convergence in bow-free models.

RICF computes estimates in closed form when possible.

The algorithm outperforms existing methods in efficiency.

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 finds parameter estimates in closed form whenever possible.