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I have a question about DDD (triple difference) with multiple time periods. There are several useful answers regarding DD with multiple time periods and this question (3 related questions about DDD (TD, triple-diff) estimators) regarding DDD with multiple time periods, but I am still confused about how to model this. Suppose I want to study the differential impact of a policy (implemented at different times across different states and not implemented at all in some states) on old vs young people:

  1. treatment vs control

  2. pre and post policy (with the policy being implemented at different times in different states)

  3. old vs young

If the policy dates were the same, I could just run a normal DDD:

$y_{ist}=\beta_0+\beta_1 post + \beta_2 trt_s + \beta_3 old_i +\beta_4 post \times trt + \beta_5 post \times old_i +\beta_6 trt_s \times old_i + \beta_7 post \times trt \times old_i$

where $post$, $trt$ and $old$ are dummies for post, treatment and old, respectively and $\beta_7$ is my parameter of interest. However, if the policy is implemented at different times, the variable $post$ is not well-defined. I tried running something like this:

$y_{ist}=\alpha_0+ \alpha_1 old_i +\alpha_2 policy_{st} + \alpha_3 policy_{st} \times old_i$

where $policy$ takes the value of 1 if state s is treated AND t is after the implementation of the policy in the state s. However, this is not really satisfactory, as I am not controlling for potential level differences in treatment vs control states. From my understanding, Wooldridge is suggesting "In a DDD analysis, a full set of dummies is included for each of the two kinds of groups and all time periods, as well as all pairwise interactions. Then, a policy dummy (or sometimes a continuous policy variable) measures the effect of the policy.", which I find very confusing. I take this to mean:

$y_{ist}=\gamma_0+ \sum_t \eta_t year_t + \gamma_2 old_i +\gamma_3 trt_s + \gamma_4 old_i \times trt_s +\sum_t \alpha_t (year_t \times old_i) +\sum_t \beta_t (year_t \times trt_s)+ \gamma_5 policy_{st} \times old_i $

The issue I have with this model is that by interacting the treatment and the year dummies, I am basically "killing off" most of the variation I hope to capture with the $\gamma_5$. I am not sure I understand the model correctly. How would you model a DDD in this context? (Apologies for the long question, I just wanted to be as clear as possible)

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  • $\begingroup$ I found an article called "Medicaid’s effect on single women’s labor supply: Evidence from the Introduction of Medicaid", where the author uses a triple difference design and is explained more clearly than in Wooldridge. Basically, the last specification I mentioned in my question seems to be the correct one. I have found several papers using this, but I could not find any in which, after adding time x "town or state" interactions, the triple difference term is significant. There does not seem to be a lot of variation left after adding all the fixed effects. $\endgroup$
    – Ndr2501
    Commented Feb 7, 2020 at 18:43

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Actually I am also confusing by this problem. I read some papers with triple difference in differences (especially the cycling program in India) and think maybe it can not be used to answer the heterogeneity effects of policy on different group but a better solution when the parrel trend are rejected in regular did. I think a grouped regression divided by the group can partly identify whether there is heterogeneity between groups.

some literature for your references, Econometrics Journal (2022), volume 25, pp. 531–553. https://doi.org/10.1093/ectj/utac010 https://brittarude.github.io/blog/2020/07/20/britta-rude-Triple-difference

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