Specifying a Diff-In-Diff with multiple treated groups over multiple years

Question

Suppose I have a group of 100 counties. Further suppose that the state decides to allocate a lump sum amount of money to some counties each year to boost spending on education in that county (until, say, 50 of the 100 counties have received money). Suppose my goal is to use a dif-in-dif approach to analyze the effect of the spending increase.

How would I specify this DID regression? The basic idea is that, for each year, I want to compare educational outcomes in counties that receive increased funding with those that do not. I want to do this so that I eventually get to evaluate the effect for all 50 counties that receive this educational funding boost.

My guess is something like this: I need to specify a typical DID regression with year and county FE s.t. for any given year, a county that receives funding is in the treated group and any county without the funding boost is in the control group. However, the effect of that funding boost means that a county treated in year $t$ ought not be included in the control group in year $t+1$, despite not receiving a funding boost in $t+1$.

Am I over complicating this? It doesn't seem to me like specifying a normal DID with year and county FE will quite work.

Answer 1

Firstly, the diff-in-diff already is a fixed effects estimation. If you run a regression with FE and your time variant terms ($S_i*T_t$ and $T_t$) as in the plain and simple diff-in-diff setup, as such:

$$ Y_{it}=B_0+B_1T_t+B_2S_i+B_3S_i*T_t+e_{it} (1)$$

You should obtain similar results. Secondly, your specification -- since it will be on a multiple treatment setup -- should consider if your treated groups in different periods of time are disjoint, that is, if a unit of observation might receive the treatment multiple times. Considering that adding a fixed effects in time might be convenient for your estimation, as you were considering, I would work with the following specification if the the treated groups are disjoint:

$$ Y_{it}=\alpha_0+\alpha_1S_i*T_t+\alpha_2S2_i*T2_t+v_t+u_i+e_{it} (2)$$

And if they are not disjoint, I would consider a interaction between the treatment effects:

$$ Y_{it}=\beta_0+\beta_1S_i*T_t+\beta_2S2_i*T2_t+\beta_3(S_i*T_t)*(S2_i*T2_t)+v_t+u_i+e_{it} (3)$$

You should also consider testing the common trend hypothesis, since you still need an adequate control group.