<p>Consider a regression:
<span class="math-container">$$
y_{i,g,t} = \alpha + \beta x_{i,g,t} + \varepsilon_{i,g,t}.
$$</span>
where <span class="math-container">$i$</span> is observation, <span class="math-container">$g$</span> is group and <span class="math-container">$t$</span> is time.</p>
<blockquote>
<p>What does it mean &quot;where the dependent and independent variables only vary at the (g,t) level&quot;</p>
</blockquote>
<p>This means that if I take two observations <span class="math-container">$i$</span> and <span class="math-container">$j$</span> that belong to the same group and the same time (i.e. same <span class="math-container">$(g,t)$</span> cell), then:
<span class="math-container">$$
y_{i,g,t} = y_{j,g,t} \tag{1}
$$</span>
and
<span class="math-container">$$
x_{i,g,t} = x_{j,g,t} \tag{2}
$$</span></p>
<blockquote>
<p>why &quot;all the units in the same (g,t) cell have the same value of $\varepsilon_{g,t}&quot;</p>
</blockquote>
<p>By definition:
<span class="math-container">$$
\varepsilon_{i,g,t} = y_{i,g,t} - \alpha - \beta x_{i,g,t}.
$$</span>
Then if we take two observations <span class="math-container">$i$</span> and <span class="math-container">$j$</span> and use <span class="math-container">$(1)$</span> and <span class="math-container">$(2$</span>), we get:
<span class="math-container">$$
\varepsilon_{i,g,t} = \varepsilon_{j,g,t}.
$$</span>
So the error is the same within each <span class="math-container">$(g,t)$</span> cell.</p>

CC BY-SA 4.0
2021-06-08T07:08:12.893Z
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?
Phil Nguyen
CC BY-SA 4.0
2021-06-08T07:59:21.510Z
@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.
tdm
CC BY-SA 4.0
2021-06-07T09:54:48.060Z
2021-06-07T09:54:48.060Z
tdm
<p>From de <a href="https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.20181169" rel="nofollow noreferrer">Chaisemartin and D'Haultfoeuille 2020, p.2969</a></p>
<p>I saw an equation</p>
<p><span class="math-container">$D_{g,t}$</span> <span class="math-container">$=$</span> <span class="math-container">$\alpha$</span> + <span class="math-container">$\gamma_g$</span> + <span class="math-container">$\delta_t$</span> + <span class="math-container">$\epsilon_{g,t}$</span></p>
<p><span class="math-container">$D_{g,t}$</span> is the treatment in group <span class="math-container">$g$</span> at period <span class="math-container">$t$</span></p>
<p>They said that</p>
<blockquote>
<p><span class="math-container">$\epsilon_{g,t}$</span> arises from a unit-level regression, where the
dependent and independent variables only vary at the (<span class="math-container">$g,t$</span>) level.
Therefore, all the units in the same (<span class="math-container">$g,t$</span>) cell have the same value
of <span class="math-container">$\epsilon_{g,t}$</span></p>
</blockquote>
<p>What does it mean &quot;<strong>where the dependent and independent variables only vary at the (<span class="math-container">$g,t$</span>) level</strong>&quot; and why &quot;<strong>all the units in the same (<span class="math-container">$g,t$</span>) cell have the same value of <span class="math-container">$\epsilon_{g,t}$</span></strong>&quot;</p>

<econometrics><mathematical-economics>
What does "dependent and independent variables only vary at the ($g,t$) level mean?
Phil Nguyen
# What does "dependent and independent variables only vary at the ($g,t$) level mean?

(Asked by: username_0 on 2021-06-07)

From de [Chaisemartin and D'Haultfoeuille 2020, p.2969](https://pubs.aeaweb.org/doi/pdfplus/10.1257/aer.20181169)

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 $\epsilon\_{g,t}$**"

**Tags:** econometrics, mathematical-economics


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## 1 answers

### Answer by username_1 on 2021-06-07. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: @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.  

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