Thanks to the suggestion from @Ariel in this discussion, I visit this paper and face a problem.

The DID equation is $$Y_{i,g,t} = \alpha_g + \alpha_t + \beta D_{g,t}+e_{igt}$$

$Y_{i,g,t}$ is the outcome (dependent variable) of unit $i$ in group $g$ at period $t$ on group fixed effects, period fixed effects, and $D_{g,t}$ is the treatment in group $g$ at period $t$

And, from the page 2965, they describe $\beta$ as

$$\beta = \mathbb{E}{\Large{[}}\sum_{(g,t);D_{g,t}=1}w_{g,t}\Delta_{g,t}\Large{]}$$

where $\Delta_{g,t}$ is the average treatment effect (ATE) in group $g$ and period $t$ and the weight

$w_{g,t}$ sum to 1 but maybe negative.

I have two questions as below:

1> From this discussion, ATE should be the average treatment over time rather than the average treatment of a group $g$ at a specific time $t$.

2> I did not see where in this paper the author mentioned the way to calculate $w_{g,t}$ so I did not understand why its sum is 1 but maybe negative


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