In this paper by Angrist a stratification estimator is used (page 16 formula (4)) to calculate the Average Treatment Effect on the Treated (ATOT). The formula is given by:
\begin{align} \widehat{ATOT}_{Stratification}=\sum_{k=1}^K \frac{\delta_{k} N_{1k}}{\sum_{k=1}^K\delta_{k} N_{1k}}(\overline{Y}_{1k}-\overline{Y}_{0k}) \end{align}
I do not understand this formula. My questions:
What does it do? What is the idea?
What is the difference to the general matching estimator, which uses a k-nearest neighbour matching or a kernel matching?
Is there any simple calculation example? I just found those papers, but with these huge datasets I cannot understand what this formula does, so I would be happy if I could calculate an example manually to understand what happens.