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This crossed my mind when I was reading this stata forum post, at which it is written:

SEM does not allow any endogenous variable to directly covary with any other variable, only regression paths and covariances between their associated error variables are allowed.

I get that the reason must have something to do with the fact that the exogenous variables don't have their causes specified, whereas endogenous variables do. However, I am struggling to express the reasoning in an intuitive way.

Why isn't it allowed? Why are regression paths and covariances between the associated error variables OK?

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An endogenous variable is an outcome variable - it has some predictors, which explain some of its variance.

A correlation (or covariance) means that two variables share some of their variance.

If variables y1 and y2 are correlated, that might be because they have a common cause (x). So x is the cause of their correlation. In the model, that correlation is accounted for - that is, some of the variance of y1 and y2 can be thought of as belonging to x.

The variables can't share variance that comes from a predictor variable. They can only share variance that definitely belongs to them. The variance that definitely belongs to them is the residual (or error) variance - and you can correlate that variance.

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