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From de Chaisemartin and D'Haultfoeuille 2020, p.2969

I saw an equation

$D_{g,t}$ $=$ $\alpha$ + $\gamma_g$ + $\delta_t$ + $\epsilon_{g,t}$

$D_{g,t}$ is the treatment in group $g$ at period $t$

They said that

$\epsilon_{g,t}$ arises from a unit-level regression, where the dependent and independent variables only vary at the ($g,t$) level. Therefore, all the units in the same ($g,t$) cell have the same value of $\epsilon_{g,t}$

What does it mean "where the dependent and independent variables only vary at the ($g,t$) level" and why "all the units in the same ($g,t$) cell have the same value of $\epsilon_{g,t}$"

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Consider a regression: $$ y_{i,g,t} = \alpha + \beta x_{i,g,t} + \varepsilon_{i,g,t}. $$ where $i$ is observation, $g$ is group and $t$ is time.

What does it mean "where the dependent and independent variables only vary at the (g,t) level"

This means that if I take two observations $i$ and $j$ that belong to the same group and the same time (i.e. same $(g,t)$ cell), then: $$ y_{i,g,t} = y_{j,g,t} \tag{1} $$ and $$ x_{i,g,t} = x_{j,g,t} \tag{2} $$

why "all the units in the same (g,t) cell have the same value of $\varepsilon_{g,t}"

By definition: $$ \varepsilon_{i,g,t} = y_{i,g,t} - \alpha - \beta x_{i,g,t}. $$ Then if we take two observations $i$ and $j$ and use $(1)$ and $(2$), we get: $$ \varepsilon_{i,g,t} = \varepsilon_{j,g,t}. $$ So the error is the same within each $(g,t)$ cell.

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  • $\begingroup$ So, do you mean "where the dependent and independent variables only vary at the (g,t) level" means that dependent and independent variables only varies outside a (g,t) box rather than inside, intuitively speaking? $\endgroup$ Jun 8, 2021 at 7:08
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    $\begingroup$ @dif-in-diff To be clear, it means that for all observations that are part of the same (group and time), the values of the dependent and independent variables are the same. For example, if I have an observation $i$ at time 1 for group 1 and an observation $j$ also at time 1 for group 1, then the values of the dependent and independent covariates for these two observations are identical. $\endgroup$
    – tdm
    Jun 8, 2021 at 7:59

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