Embedding common factors in a path model

In models containing unobservable variables with multiple indicators--common factors, true scores--it seems to be common practice not to allow, except in special cases, nonzero covariances of residuals of indicators of the common factors and other variables in the model. Indeed, those models, such as LISREL and LISCOMP, that separate a “measurement” model from a “structural” model, exclude these by their defining matrix structure. Yet it is not at all obvious that such covariances should not be allowed, and it seems desirable to look for some explicit principles to apply to such cases, rather than settle the matter by default. This question will be examined in the context of the reticular action model. The following section sets out some necessary preliminaries from the case of path models without common factors, then section 3 treats the problem of embedding a block of common factors in a path model. Section 4 gives a numerical example.