Diff-in-Diff vs Fixed Effects with dummy that is not time-varying

Question

In a simple diff-in-diff model, such as:

$$Y_{it}=\beta_0+\beta_1S_{it}+\beta T_{it}+\beta_3S_{it}T_{it}+\epsilon_{it}$$

In which $S_{it}$ represents a dummy for the units that received the treatment and $T_{it}$ is a dummy for the time in which ocurred. Consider the situation in which $S_{it}$ is time-invariant. If one was to run a fixed-effects version of this model on a common software package such as Stata, in which one would add a fixed effect term in the equation above, $S_{it}$ would be dropped out due to perfect multicollinearity, I believe. If that were to happen, does the model still 'works'?

The model makes sense to me, but I wonder if there is something that I should be worrying about.

Answer 1

In your difference in differences model some of the subscripts seem to be a little confused. For instance, group membership is something fixed over time, hence it should be $S_i$. Likewise the treatment (supposedly) starts at some time for everyone and therefore you can drop the $i$ subscript.

Whether you then use OLS to estimate $$Y_{it} = \beta_0 + \beta_1 S_i + \beta_2 T_t + \beta_3 (S_i\cdot T_t) + \epsilon_{it}$$ or if you estimate $$Y_{it} = \beta_2 T_t + \beta_3 (S_i\cdot T_t) + c_i + \epsilon_{it}$$ via fixed effects will give you the same result for the difference in differences coefficient. In the second model your $S_i$ is going to be absorbed by the individual fixed effects $c_i$ which would be differenced out anyway in the difference in differences setting.

Estimating DiD via a fixed effects model comes in handy when you have a non-binary treatment and/or the treatment comes into effect for some people at difference points in time.