# What is the standard literature on Inverse Probability Weighting Estimators?

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?

• – StasK Sep 22 '17 at 22:29
• Is this a dupe? This question is looking for a reference, that one was looking for an explanation. I'm not sure. – Peter Flom Sep 23 '17 at 13:26

• 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? – user321627 Sep 24 '17 at 2:49
• @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. – Dimitriy V. Masterov Sep 25 '17 at 16:36