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I understand that Imbens has several papers on IPW, but was wondering if there was a default text one would recommend to understand IPW. For example, where does the estimator:

$$ \tau = \frac{1}{N^T}\sum_{i=1}^N \frac{W_iY_i}{e_i(X_i)} - \frac{1}{N^C}\sum_{i=1}^N \frac{(1-W_i)Y_i}{1-e_i(X_i)} $$

come from?

Specifically, it appears that normally an estimator will $\frac{1}{N}$ on the outside of both sums instead of what I have above as $\frac{1}{N^T}$ and $\frac{1}{N^C}$. If I replaced $\frac{1}{N^T}$ and $\frac{1}{N^C}$ with $\frac{1}{N}$, is it estimating something else?

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There's a derivation of the average treatment effect formula on p. 240-241 of Guo and Fraser's Propensity Score Analysis, which has a chapter devoted to IPW. This is a useful book aimed at the applied researcher, but it also contains references to the primary literature.

I don't know of any other textbook that covers this material very well.

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  • $\begingroup$ It appears that in pg. 240 they have $\frac{1}{N}$ on the outside of both sums instead of what I have above as $\frac{1}{N^T}$ and $\frac{1}{N^C}$. Are these the same? $\endgroup$ – user321627 Sep 24 '17 at 2:49
  • $\begingroup$ @user321627 They are not. All my sources have $\frac{1}{N}$. Typically, the other two pop up in formulas for ATT. It's possible that your source has non-standard notation, and there is some normalization/stabilization that makes the formulas the same, but hard to know without a citation. $\endgroup$ – Dimitriy V. Masterov Sep 25 '17 at 16:36
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Imbens and Rubin give it on page 274 of Causal Infernece without any real attribution. It looks like the estimator is there in Hirano, Imbens and Ridder (2003). Whether it could be traced any earlier is left as an exercise to the interested reader.

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    $\begingroup$ I think the usual proto-reference is the Horvitz and Thompson 1952 JASA paper, where the application is re-weighting to deal with sample attrition rather treatment-control imbalance. $\endgroup$ – Dimitriy V. Masterov Sep 22 '17 at 23:28
  • $\begingroup$ I need to rethink my life, @DimitriyV.Masterov. Or at least read the freaking paper. $\endgroup$ – StasK Sep 25 '17 at 4:05

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