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If we had a model:

$y=x\beta +\eta$

and assumed exogeneity, so $E[\eta|x]$=0, is the fact that x or treatment intensity is now uncorrelated with $\eta$ equivalent to saying that x is 'independent of potential outcomes?'

Steve
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1 Answers1

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Yes, if we call the model

$$Y = X\beta + \eta$$

"structural" or "causal", we can use it to define potential outcomes as

$$Y(x) = x\beta + \eta.$$

(I'm using upper-case letters $Y$ and $X$ for random variables, and lower-case $x$ for a realization or fixed constant).

Then assuming $E[\eta|X] = 0$, it follows that

$$E[Y(x)|X] = x\beta + E[\eta|X] = x\beta,$$

which does not depend on $X$, so mean ignorability $E[Y(x)|X] = E[Y(x)]$ holds.

This "structural definition of counterfactuals" was proposed by Judea Pearl, see for example his book "Causality", or his book with Jewell and Glymour, "Causality: A Primer".

Julian Schuessler
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    Thanks! just one quick follow up, when people use the terminology 'structural' , that is synonymous with saying it just some true theoretical population relationship, correct? – Steve Aug 21 '20 at 02:14
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    Not quite, the conditional mean function E[Y|X] is also a population concept, but not necessarily structural. One definition of a structural model is that the model describes how $Y$ changes when $X$ is controlled physically. This entails that the value of the structural $\beta$ is "invariant" to changes in the distribution of $X$, while a regression coefficient would not be (if the regression coefficient is equal to $cov(X, Y)/var(X)$, this will change when the distribution of $X$ changes). More generally, "structural" is almost synonymous with "theoretical" (especially in economics). – Julian Schuessler Aug 21 '20 at 06:54
  • @Steve Infact the exogeneity condition must be related to the structural equation and not to the regression one. If population or only a sample are available do not matters. Frequently this point is misunderstood. Read here can help: https://stats.stackexchange.com/questions/262609/what-is-the-actual-definition-of-endogeneity/402015#402015 – markowitz Aug 26 '20 at 05:52