I have a SEM where various latent variables are correlated with each other. I want to use a subset of these correlations to run a regression. This can be done easily enough to find point estimates, but standard regressions from correlation matrices assume that the only source of error in the correlations is from sampling error (i.e., not also error in estimating the latent variable), so the standard errors of the regression coefficients are biased.
Is there a way to adjust standard errors for such a regression to account for the additional source of error due to using latent variable correlations?
Note: In this case, I cannot simply run the regression within the SEM as it is somewhat susceptible to 'Interpretational Confounding' (Burt, 1976), whereby the measurement parameters change to optimize the fit when run as a regression, where the problem is not present when run as correlations. I can clarify this in the comments if anyone's interested.
Burt, R. S. (1976). Interpretational Confounding of Unobserved Variables in Structural Equation Models. Sociological Methods & Research, 5(1), 3–52. https://doi.org/10.1177/004912417600500101