Linear structural equation model

In a linear structural equation model, each variable is a linear combination of its graphical parents and an exogenous error term. Edge coefficients quantify direct effects.

Wright path tracing rule computes covariances in a linear structural equation model by adding contributions along the admissible paths of its graph. For an acyclic system $X=BX+\epsilon$ with centered disturbances and disturbance covariance matrix $\Omega$,Acyclicity makes $B$ nilpotent in a topological ordering, so $(I-B{)}^{-1}=I+B+\cdots +B^{p-1}$. Every matrix entry sums products of coefficients along directed paths. Substituting this finite expansion into the covariance formula gives the path-tracing construction. With uncorrelated disturbances, contributions join two directed paths at a common disturbance; correlated disturbances add the corresponding cross-disturbance terms.

For a worked example, let $X={\epsilon }_X$, $M=aX+{\epsilon }_M$ and $Y=bM+cX+{\epsilon }_Y$, with mutually uncorrelated centered disturbances. Substitution gives $Y=(ab+c)X+b{\epsilon }_M+{\epsilon }_Y$, henceThe direct path contributes $c$ and the mediated path contributes $ab$, each times the source variance. For standardized variables this becomes a sum of coefficient products for a correlation coefficient. Reading those coefficients as causal effects requires the structural intervention interpretation; an arbitrary covariance-fitting graph does not supply it by itself.