Directed Cyclic Graphical Representations of Feedback Models

TL;DR

This paper extends the graphical modeling framework to include directed cyclic graphs, enabling representation of non-recursive and dependent-error systems, with implications for economic and non-linear models.

Contribution

It introduces a characterization of conditional independence in directed cyclic graphs, generalizing previous acyclic models to systems with dependent errors and non-linearities.

Findings

Characterization of conditional independence in cyclic graphs

Extension to systems with dependent error variables

Sufficient conditions for independence in non-linear models

Abstract

The use of directed acyclic graphs (DAGs) to represent conditional independence relations among random variables has proved fruitful in a variety of ways. Recursive structural equation models are one kind of DAG model. However, non-recursive structural equation models of the kinds used to model economic processes are naturally represented by directed cyclic graphs with independent errors, a characterization of conditional independence errors, a characterization of conditional independence constraints is obtained, and it is shown that the result generalizes in a natural way to systems in which the error variables or noises are statistically dependent. For non-linear systems with independent errors a sufficient condition for conditional independence of variables in associated distributions is obtained.