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?


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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