Let's say we are considering the following model: $$(\beta^{\star},f^{\star}) := \arg\min_{\beta,f \in \mathcal{F}} \mathbb{E}[\left(Z_i - f(X_i, E_i) - \beta^\top \boldsymbol{\tau}_{i,E_i}\right)^2|S_i = 1],$$$$ (\beta^{\star},f^{\star}) := \arg\min_{\beta,f \in \mathcal{F}} \mathbb{E}[\left(Z_i - f(X_i, E_i) - \beta^\top \boldsymbol{\tau}_{i,E_i}\right)^2|S_i = 1], $$ where $S_i$ is some certain sample selector. Specfically, I'm consider the case in the paper https://arxiv.org/pdf/2403.19563.
- Dmitry Arkhangelsky, Kazuharu Yanagimoto, Tom Zohar, Flexible Analysis of Individual Heterogeneity in Event Studies: Application to the Child Penalty, https://arxiv.org/abs/2403.19563
The author saysauthors say by the FWL theorem, we can write $$\beta^{\star} = \left(\mathbb{E}[\mathbb{V}[\boldsymbol{\tau}_{i,E_i}|X_i,E_i]|S_i^{tr} = 1]\right)^{-1}\mathbb{E}[\left(\boldsymbol{\tau}_{i,E_i} - \mathbb{E}[\boldsymbol{\tau}_{i,E_i}|X_i,E_i]\right)Z_i|S_i = 1]$$ but$$ \beta^{\star} = \left(\mathbb{E}[\mathbb{V}[\boldsymbol{\tau}_{i,E_i}|X_i,E_i]|S_i^{tr} = 1]\right)^{-1}\mathbb{E}[\left(\boldsymbol{\tau}_{i,E_i} - \mathbb{E}[\boldsymbol{\tau}_{i,E_i}|X_i,E_i]\right)Z_i|S_i = 1] $$ but I do not understand why and how exactly this can solve the problem. My guess is to isolate the effect of a set of variables on both sides of a regression equation. Yet, I want some mathematical insights to this way of solving this problem. Any help or explanation is appreciated.
