The average treatment effect (ATE) of binary treatment T on outcome Y can be estimated using inverse propensity weights:

\begin{equation}\nonumber \frac{\sum_{i=1}^{N}t_i\hat{\pi}_i^{-1}y_i}{\sum_{i=1}^{N}t_i\hat{\pi}_i^{-1}}-\frac{\sum_{i=1}^{N}(1-t_i)(1-\hat{\pi}_i)^{-1}y_i}{\sum_{i=1}^{N}(1-t_i)(1-\hat{\pi}_i)^{-1}} \xrightarrow{p} E[Y^1]-E[Y^0] \end{equation}

where $\hat{\pi}_i$ is the estimated propensity for individual $i$, $Y^1$ is the outcome under treatment and $Y^0$ the outcome under the control condition.

To avoid extreme weights, there is a literature that suggests replacing the numerator of the treated weights with the marginal probability of treatment, $p(t=1)$, and the numerator of the control weights with $1-p(t=1)$. I see how this makes the weights milder, but why those particular numerators? What is the intution of this stabilization?

Also, the average treatment effect on the treated (ATT) can be estimated by weighting control units with the odds:

\begin{equation}\nonumber \frac{\sum_{i=1}^{N}t_iy_i}{\sum_{i=1}^{N}t_i}-\frac{\sum_{i=1}^{N}(1-t_i)\hat{\pi}_i(1-\hat{\pi}_i)^{-1}y_i}{\sum_{i=1}^{N}(1-t_i)\hat{\pi}_i(1-\hat{\pi}_i)^{-1}} \xrightarrow{p} E[Y^1|T=1]-E[Y^0|T=1] \end{equation}

How should one stabilize these $\hat{\pi}_i(1-\hat{\pi}_i)^{-1}$ weights?


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