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Bollen (1989) introduces a general structural equation model of the (matrix) form:

$$\eta = \beta \eta + \gamma \xi + \zeta$$ $$y = \lambda \eta + \epsilon$$

Most textbooks (e.g. Depaili 2021) presenting the general model describe $\beta$ is a $m \times m$ matrix of coefficients "relating the $m$ endogenous latent factors." However, the Bollen text and others don't describe this matrix in much detail. Accordingly, I'm hoping someone can help explain the contents and dimensionality of the matrix.

Hopefully, an answer would hit on the following points:

  1. Why the matrix is square and not lower-triangle since presumably we don't "regress variables on themselves."
  2. Why the matrix is not symmetric. Is this because these are directed paths (regressions) rather than covariances?
  3. Are the elements of this matrix the marginal, bivariate correlations between each factor? Or are they the partial correlations? For example, suppose $m > 2$ (more than 2 endogenous latent factors). How would we describe $\beta[1,2]$ or $\beta[2,1]$?
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    $\begingroup$ in addition to Jeremy Miles answer, I would add to (1) that the diagonal must be zeros (consistent with your intuition that nothing should predict itself). $\endgroup$
    – Terrence
    Commented Sep 8, 2022 at 7:48
  • $\begingroup$ @Terrence Ah, this is why the diagonal is zero! This is strange to me insofar as I expect the diagonal to be filled with 1s not 0s since Corr(x,x) = 1. $\endgroup$ Commented Sep 8, 2022 at 12:13

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  1. I think it is lower triangular (if the model has no feedback loops). See, for example: https://web.pdx.edu/~newsomj/semclass/ho_lisrel%20notation.pdf
  2. Yes, they're regressions.
  3. They're regression parameters, the same as if we were doing multiple regression.

It's less common for people to worry about the distinction between exogenous factors and endogenous factors - this was an issue when computers had less memory and were slower, but most programs don't care, so there's no need to distinguish the beta and gamma matrices.

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