Graphical models, causal inference, and econometric models

Abstract

A graphical model is a graph that represents a set of conditional independence relations among the vertices (random variables). The graph is often given a causal interpretation as well. I describe how graphical causal models can be used in an algorithm for constructing partial information about causal graphs from observational data that is reliable in the large sample limit, even when some of the variables in the causal graph are unmeasured. I also describe an algorithm for estimating from observational data (in some cases) the total effect of a given variable on a second variable, and theoretical insights into fundamental limitations on the possibility of certain causal inferences by any algorithm whatsoever, and regardless of sample size.

Keywords:

• graphical models
• causal inference
• model search
• model testing

ACKNOWLEDGEMENT

I would like to thank Kevin Hoover for valuable comments and suggestions.

Notes

In the general case, DAG models actually entail conditional independence relations among sets of variables. In the multi‐variate normal case, all independence relations between sets of variables X and Y conditional on a set of variables Z are entailed by conditional independence relations among each individual variable X∈X and Y∈Y conditional on Z. This is not always the case for non‐normal distributions.

In counting degrees of freedom, it is assumed that no extra constraints (such as equality constraints among parameters) are imposed. The Bayesian Information Criterion for a DAG is defined as log

where D is the sample data, G is a DAG,

is the vector of maximum likelihood estimates of the parameters for DAG G, N is the sample size, and d is the dimensionality of the model, which in DAGs without latent variables is simply the number of free parameters in the model.

There are weaker versions of the Bayesian Causal Faithfulness Assumption (that assume a zero probability for zero partial correlations only between pairs of variables that are adjacent) that entail the existence of (weak) Bayes consistent estimators of the effects of manipulations, but at the cost of making the estimators more complicated and slower to compute. See Spirtes, Glymour and Scheines (2000), chapter 12.

X\Y is a set whose members are the members of X that are not also members of Y.