Identifying Direct Causal Effects in Linear Models

This paper deals with the problem of identifying direct causal effects in recursive linear structural equation models. Using techniques developed for graphical causal models, we show that a model can be decomposed into a set of submodels such that the identification problem can be solved independently in each submodel. We provide a new identification method that identifies causal effects by solving a set of algebraic equations. Introduction Structural equation models (SEMs) have dominated causal reasoning in the social sciences and economics, in which interactions among variables are usually assumed to be linear (Duncan 1975; Bollen 1989). This paper deals with one fundamental problem in SEMs, accessing the strength of linear cause-effect relationships from a combination of observational data and model structures. The problem has been under study for half a century, primarily by econometricians and social scientists, under the name “The Identification Problem”(Fisher 1966). Although many algebraic or graphical methods have been developed, the problem is still far from being solved. In other words, we do not have a necessary and sufficient criterion for deciding whether a causal effect can be computed from observed data. Most available methods are sufficient criteria which are applicable only when certain restricted conditions are met. The contribution of this paper consists of two parts. First, we show how a model can be decomposed into a set of submodels such that the identification problem can be solved separately in each submodel. The technique is orthogonal to the available identification methods, and it is useful in practice because it is possible for an identification method which can not be applied to the full model to become applicable in smaller submodels. Second, we show a reduction of the identification problem into a problem of solving a set of algebraic equations. These equations provide an alternative to the classic Wright’s rule (Wright 1934). We begin with an introduction to SEMs and the identification problem, and give a brief review to previous work before presenting our new results. Copyright c © 2005, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. X = x W = aX + w Z = bX + z Y = cW + dZ + y Cov( x, z) 6= 0 Cov( w, y) 6= 0