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I am really having a hard time realizing the difference between OLS regression and SEM (structural equation modeling) from a causal inference point of view. I do understand some features of SEM such as it includes measurement model, etc.

But the thing I would really want to know is what happens if I have an omitted covariate in the SEM model. I have a firm grasp of what it would cause in OLS regression; we will get a biased estimator so that we cannot establish a causal effect in confidence. But what about in SEM? why is it so hard to find articles talking about the same endogeneity problem in SEM? Please help.

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  • $\begingroup$ Kaplan (1988) discusses this a bit. $\endgroup$
    – Noah
    Sep 17 '19 at 21:20
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As the other answer says, it's the same. But your confusion is understandable. In order not to get confused about such issues, it's important to keep distinct the following two concepts:

1) The distribution of observables, say $\Pr(Y,X,Z)$ and quantities like $E[Y|X=x, Z=x]$ and $E[Y|X=x]$ which we often approximate with OLS regression.

2) A causal model that specifies how, if at all, $Y,X,Z$ are related. We could formulate such a model in terms of structural equations, eg $$Y = f_x(X) + f_z(Z) + \epsilon_Y$$ $$X = h(Z)+ \epsilon_X$$ $$Z = \epsilon_Z$$ These 3 equations say that $Z$ and $X$ affect the value of $Y$ and that $Z$ also affects the value of $Y$.

We see data $Y, X, Z$ and can estimate features of $\Pr(Y,X,Z)$. But without some sort of assumptions about the causal model connecting them (such as the 3 structural equations) we have no idea about what causing what.

If we do OLS regression and there is talk of endogeneity, then the goal is to estimate a causal effect. Say that $Y$ is wages at age 30, $X$ is years of schooling and $Z$ is IQ. Then everyone would agree that $E[Y|X=x]$ does not estimate the causal effect of schooling on wages. There is an omitted variable bias: $Z$ positively affects both $X$ and $Y$. The association between $X$ and $Y$ will reflect both the effect of $X$ on $Y$ and the effect of $Z$ on both. The system of structural equations I've listed above fits this causal model of wages, schooling and IQ. Another way of expressing the omitted variable bias is that $E[Y|X=x]$ is not equal to $f_x(x)$, the part of the structural equation that specifies how $X$ affects $Y$

Had we mistakenly assumed that $X = \epsilon_X$ instead of $X = h(Z)+ \epsilon_X$ in our system of structural equations, ie that $X$ isn't affected by $Z$ and proceeded to estimate the effect of $X$ on $Y$ by SEM, the problem would be exactly the same.. assuming the true model is the one I've written down above.

A great resource for this sort of material is Pearl's The Causal Foundations of Structural Equation Modeling available here: https://ftp.cs.ucla.edu/pub/stat_ser/r370.pdf

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It's the same problem. Omitting variables means that your estimates are biased in SEM and in OLS.

It's worse in SEM, because you can also misspecify your model - you can have an item in there, but with the wrong arrows.

Endogeneity in SEM should be discussed under model misspecification.

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