I'm doing a difference-in-differences analysis with one pre-treatment time (0), and two post-treatment time points (1,2).
My basic regression model is:
= $β_0+β_1T_1 +β_2T_2 +β_3S+ β_4(S∗T_1)+β_5(S∗T_2)+ε$
where $T_1$ is a dummy (equals 1 for time 1, 0 otherwise) $T_2$ is a dummy (equals 2 for time 2, 0 otherwise) $S$ is a dummy (equals 1 for treatment group, 0 otherwise)
The DiD coefficient is $β_4$ for the first post period, and $β_5$ for the second period.
However, in $T_2$ a second policy affected a subset of the treatment group ($S$). I'd like to isolate the effect of this separate policy from the main policy, by comparing the affected individuals in the treatment group to the unaffected individuals from the treatment group (so basically, a DiD analysis within the original treatment group). The second policy did not affect the original control group.
Would this be the correct regression model for analysis:
$= β_0+β_1T_1 +β_2T_2 +β_3S_1+ β_4(S_1∗T_1)+β_5(S_1∗T_2)+β_6(S_2∗T_2)+ε$
where $β_6$ is the DiD coefficient comparing the affected individuals and unaffected individuals (from the original treatment group) between the second 2 time points, and:
$S_1$ is a dummy variable that equals 1 if the individual is in the treatment group for the first policy, and
$S_2$ is a dummy variable which equals 1 only if the individual is in the subset of the treated group that is affected by the second policy.