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

Question

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 $\epsilon_{g,t}$"

Answer 1

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.