<p>The 2-period model you wrote and the fixed effects (FE) models are equivalent for 2-periods and one treatment. The advantage of fixed effects, is that it extends to multiple periods and treatments (e.g., can be adapted for multiple/staggered treatments and other variations).</p>
<p>Suppose we apply the FE model for 2 periods. One way to apply FE is to include dummy variables for all possible units, minus one due to perfect colinearity. So you would have time dummies and treatment (i) dummies.</p>
<p>For two periods and one treatment, i.e. two groups (one treatment and one control), you can only use one dummy each, to avoid perfect multi-colinearity. For example, stata would automatically drop one dummy if you included time dummies for both (all) periods in this case.</p>
<p>So your FE model in the 2-period case would have 1 dummy (i.e. indicator variable) for time and 1 for the unit type (treatment vs. control).</p>
<p>So, in the FE model with 2 periods and one treatment, your time FE <span class="math-container">$\theta_{i}$</span> would entirely consist of 1 indicator variable for the second time period. Let’s call this <span class="math-container">$\theta_{post}$</span>. The FE for groups <span class="math-container">$\alpha_i$</span>, would similarly also have only 1 indicator for treatment, call it <span class="math-container">$\alpha_{treatmemt}$</span>,becuse there are no other treatment groups</p>
<p>It’s easy to see that this is then exactly the same as the two-period model.</p>
<p>Of course, having the opposite dummy (e.g. for pre instead of post and control instead of treatment), doesn’t affect the beta estimate for 2-periods with one treatment, because the pre and post dummies are perfectly colinear, so you can use either one. Same for the treatment and control dummies.</p>
<p>So the FE model is just a more general and flexible version of the 2-period model you mention, using the same logic.</p>

CC BY-SA 4.0
2023-03-18T04:41:59.857Z
2023-03-18T21:55:24.073Z
2023-03-18T21:55:24.073Z
BB King
<p>In a multi-period DiD analysis, the usual specification is:</p>
<p><span class="math-container">$$y_{it} = \alpha_i + \theta_t + \beta D_{it} + \epsilon_{it} $$</span>
where <span class="math-container">$D_{it}$</span> is the interaction  between <code>post</code> and <code>treated</code> variables.</p>
<p>Why don't we use:</p>
<p><span class="math-container">$$y_{it} = \alpha_{treated} + \theta_{t} + \beta D_{it} + \epsilon_{it} $$</span></p>
<p>where we only control for treatment status instead of unit fixed effects? Or why not use the specification used with 2-period DiD:
<span class="math-container">$$y_{it} = \alpha_{treated} + \theta_{post} + \beta D_{it} + \epsilon_{it} $$</span></p>
<p>When will I get the same estimates for <span class="math-container">$\beta$</span> using these specifications as the first one?</p>

<difference-in-difference>
Diff in Diff: Why do we control for unit and time fixed effects instead of just treatment status and pre/post indicator?
Rainroad
# Diff in Diff: Why do we control for unit and time fixed effects instead of just treatment status and pre/post indicator?

(Asked by: username_0 on 2023-03-16)

In a multi-period DiD analysis, the usual specification is:

$$y\_{it} = \alpha\_i + \theta\_t + \beta D\_{it} + \epsilon\_{it} $$
where $D\_{it}$ is the interaction between `post` and `treated` variables.

Why don't we use:

$$y\_{it} = \alpha\_{treated} + \theta\_{t} + \beta D\_{it} + \epsilon\_{it} $$

where we only control for treatment status instead of unit fixed effects? Or why not use the specification used with 2-period DiD:
$$y\_{it} = \alpha\_{treated} + \theta\_{post} + \beta D\_{it} + \epsilon\_{it} $$

When will I get the same estimates for $\beta$ using these specifications as the first one?

**Tags:** difference-in-difference


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## 1 answers

### Answer by username_1 on 2023-03-18. Score: 2

The 2-period model you wrote and the fixed effects (FE) models are equivalent for 2-periods and one treatment. The advantage of fixed effects, is that it extends to multiple periods and treatments (e.g., can be adapted for multiple/staggered treatments and other variations).

Suppose we apply the FE model for 2 periods. One way to apply FE is to include dummy variables for all possible units, minus one due to perfect colinearity. So you would have time dummies and treatment (i) dummies.

For two periods and one treatment, i.e. two groups (one treatment and one control), you can only use one dummy each, to avoid perfect multi-colinearity. For example, stata would automatically drop one dummy if you included time dummies for both (all) periods in this case.

So your FE model in the 2-period case would have 1 dummy (i.e. indicator variable) for time and 1 for the unit type (treatment vs. control).

So, in the FE model with 2 periods and one treatment, your time FE $\theta\_{i}$ would entirely consist of 1 indicator variable for the second time period. Let’s call this $\theta\_{post}$. The FE for groups $\alpha\_i$, would similarly also have only 1 indicator for treatment, call it $\alpha\_{treatmemt}$,becuse there are no other treatment groups

It’s easy to see that this is then exactly the same as the two-period model.

Of course, having the opposite dummy (e.g. for pre instead of post and control instead of treatment), doesn’t affect the beta estimate for 2-periods with one treatment, because the pre and post dummies are perfectly colinear, so you can use either one. Same for the treatment and control dummies.

So the FE model is just a more general and flexible version of the 2-period model you mention, using the same logic.


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