Complicated Difference-in-Differences design

[Nikolaos Kanellopoulos]

Dear all,

I have some people who I observe five times (t0, t1, t2, t3, t4) and can participate in some financial aid programs. These programs are up to 5. Some of them do not participate in any program or may participate in all 5. The first program starts in period t1, the second in period t2, while the rest in period t3, and the entrance to the programs can be made at any period. For example, one may decide to participate in program 1 in period t3 and one may decide to participate in all five programs in period t4. To clarify, participation in each program is not affected by participation or not in another program. For example, one can participate in programs 1,3,5 and one in programs 2,5.

Can I use Difference-in-Differences to assess the impact of each program?
Is there an alternative approach that I might consider?

Many thanks,
Nikos

[Maxence Morlet]

I think you'll find a solution in this paper: https://papers.ssrn.com/sol3/papers....act_id=3906345.

[Fei Wang]

Jeff's paper cited by Maxence flexibly handles complicated designs of DiD. I think the design in #1 may be more complicated than the major cases in Jeff's paper. Following the paper's idea, I'm trying to figure out how to implement DiD for the design in #1, as below.

For Program 1, all individuals can be divided into five groups.
- Group 1-1: participating in Program 1 in t1
- Group 1-2: participating in Program 1 in t2
- Group 1-3: participating in Program 1 in t3
- Group 1-4: participating in Program 1 in t4
- Group 1-5: not participating in Program 1 at all

You may define four binary indicators for the five groups, say d11, d12, d13, and d14. For example, d13 = 1 for Group 1-3 and d13 = 0 for other groups; the control group has zero values in all the indicators. You may further define four indicators for periods, say p1, p2, p3, and p4. For example, p2 = 1 for t2 and after, and p2 = 0 for t0 and t1.

Finally, you may define the treatment indicator for each group: w11 = d11*p1, w12 = d12*p2, w13 = d13*p3, and w14 = d14*p4. For example, w14 = 1 when individuals in Group 1-4 are exposed to Program 1, and w14 = 0 if not.

Assume there is only Program 1, you may implement the DiD as below (without covariates).

xtset id
xtreg y i.w11 i.w12 i.w13 i.w14 i.t, fe vce(cluster id)

where "i.t" are the time fixed effects. Coefficients of w11-w14 record the effects of Program 1 on y for the individuals in Groups 1-1 to 1-4, respectively. You may extend the model to compute the program effects for a particular group in a particular period, as below.

xtreg y w11#t1 w11#t2 w11#t3 w11#t4 w12#t2 w12#t3 w12#t4 w13#t3 w13#t4 w14#t4 i.t, fe vce(cluster id)

t1, t2, t3 and t4 are binary indicators for being at corresponding periods (different from p1-p4.). For example, the coefficient of w12#t3 is the effect of Program 1 for the individuals of Group 1-2 in period t3. Actually you may generate the average effect of Program 1 by computing the weighted average of all such group-period specific effects.

Above displays the main idea if there is only Program 1. For other programs, similar indicators should be defined and added to the model. Below I added Program 2.

- Group 2-2: participating in Program 2 in t2 --> d22 = 1
- Group 2-3: participating in Program 2 in t3 --> d23 = 1
- Group 2-4: participating in Program 2 in t4 --> d24 = 1
- Group 2-5: not participating in Program 2 at all

Define w22 = d22*p2, w23 = d23*p3, w24 = d24*p4.

DiD estimation combining Programs 1 and 2:

xtreg y i.w11 i.w12 i.w13 i.w14 i.w22 i.w23 i.w24 i.t, fe vce(cluster id)

[Nikolaos Kanellopoulos]

Dear Maxence and Fei,

Thank you very much for the paper and the proposed solution.

Regards,
Nikos