Diff in Diff: Why do we control for unit and time fixed effects instead of just treatment status and pre/post indicator?

Question

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?

Answer 1

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.