# endogeneity in SEM

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.

• Kaplan (1988) discusses this a bit.
– Noah
Sep 17 '19 at 21:20

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

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.