I had many questions about the mechanics of structural modeling, particularly because as a physicist I am always trying to figure out 1st what's 'given' and what do we need to obtain/get/calculate/estimate.
read recently in Les Hayduk's 1996 book (p. 15-16, e.g.) that Duncan
has proposed a very simple and elegant procedure for obtaining model predicted variances and covariances from the structural model
parameters, which is probably
even easier to follow/apply than Wright's tracing rules (see a great
modernized tutorial on Phillip Wood's web pages: standardized and unstandardized parts).
That simple rule allows you for instance to also "estimate the beta coefficient" in a simple regression by hand, like this:
= b*x + e , then multiply by x: x*y = x*b*x + x*e, take expectations (a
simple operation in fact, Duncan explains it, roughly speaking just sum
up across the entire sample and divide by sample size):
b*E(x,x) + E(x,e), which, if we assume that the covariance between
predictor and error is zero (common), takes us to the "quick estimation"
of b: b = Cov(x,y) /Var(x), a well known 'formula'.
Duncan, O. D. (1975). Introduction to structural equation models. New. York: Academic Press.
Hayduk, L. A. (1996). LISREL issues, debates, and strategies: Johns Hopkins University Press.