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In the following equation, the outer expectation is over the distribution $X_i|T_i = 1$

$\tau|_{T = 1} = E(E(Y_i|X_i, T_i = 1) - E(Y_i|X_i, T_i = 0)|T_i =1)$

Are we taking the expected value of $X_i$ given $T_i = 1$ and then taking the expected values of the $Y$s or the other way around? Essentially, do we work out the inner expectations or the outer expectations first, or are they the same thing?

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The conditioning is made explicit in the inner expectation, so that inner expectation is with respect to whatever is not being conditioned on. The outer expectation is then over the distribution of the conditioning random variable.

It's hard to say much more without knowing what these random variables are.

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  • $\begingroup$ The variables come from a paper on propensity score matching where $Y_i$ is an outcome variable, $T_i$ is a treatment indicator, and $X_i$ is a pretreatment variable. The parameter of interest, $\tau|_{T = 1}$, is the average treatment effect on the treated $\endgroup$ – EconStats Jul 2 '15 at 19:23
  • $\begingroup$ Intuitively, I would have thought that the equation meant we take the expected value of $X_i$ for the treated observations, and then, take the expected values of the outcomes for the observations with respect to this specific value of $X$ and their treatment status? $\endgroup$ – EconStats Jul 2 '15 at 19:26

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