# How to get the weights of coefficients of treatment effect from De Chaisemartin,2020?

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