Linear Recursive Equations, Covariance Selection, and Path Analysis

Abstract

Abstract By defining a reducible zero pattern and by using the concept of multiplicative models, we relate linear recursive equations that have been introduced by econometrician Herman Wold (1954) and path analysis as it was proposed by geneticist Sewall Wright (1923) to the statistical theory of covariance selection formulated by Arthur Dempster (1972). We show that a reducible zero pattern is the condition under which parameters as well as least squares estimates in recursive equations are one-to-one transformations of parameters and of maximum likelihood estimates, respectively, in a decomposable covariance selection model. As a consequence, (a) we can give a closed-form expression for the maximum likelihood estimate of a decomposable covariance matrix, (b) we can derive Wright's rule for computing implied correlations in path analysis, and (c) we can describe a search procedure for fitting recursive equations.