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Suppose that in an observational study with $N$ units, we have that $X_i$ are our covariates. I have read in several places that in order to calculate an unbiased treatment effect, the $X_i$ are assumed to be iid. I am wondering why this is necessarily the case, and specifically why the independence part is needed. What happens if we do not have:

$$ X_i \overset{iid}{\sim} F $$

for some distribution $F$? What if it is the case that there is a dependency structure for $X_i$ and $X_j$?

The unbiased estimation procedure I am referring to is from Rosenbaum 1983. Thank you.

user321627
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  • What kind of statistical inference is the causal inference? Is link or ref for mentioned paper available? – user158565 Oct 16 '18 at 19:37
  • The article that you refer to is 16 pages. Could you refer more precisely to the point where it is stated that the covariates need to be independent or assumed to be independent. Or do you mean the part in section 1.2 with the equation below? $$x \perp !!! \perp z \vert b (x)$$ The covariates can be dependent (dependent on each other), but they should not be different, dependent, on the different treatments (dependent on the treatment). – Sextus Empiricus Jan 27 '19 at 10:17
  • The kind of statistical inference is mentioned in this landmark paper. It contrasts it from say Pearl's framework. – user321627 Jan 27 '19 at 20:25
  • @MartijnWeterings I do not believe the paper mentions it explicitly. It is something I've gained through talks. I will try to see if I can find an explicit source. – user321627 Jan 27 '19 at 20:27
  • So you do get the necessity of the independence between the treatment and the covariates? And your question is not about that but more specifically about whether is necessary to also have an independence among the covariates? – Sextus Empiricus Jan 27 '19 at 21:53

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