Triple Difference with Poisson PPML Interpretation of Interaction

[Maxence Morlet]

Hi all,

I would like to conduct a triple difference estimation. My dependent variable is a count variable. I am using the user-written command ppmlhdfe:

pmlhdfe count_appr 1.post#1.treated_occupation#1.treated_state, cluster(state1) abs(i.firsttwodig_occ_ind#i.yearmonth i.state1#i.firsttwodig_occ_ind i.state1#i.yearmonth)

count_appr | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------------------+----------------------------------------------------------------
post#treated_occupation#treated_state |
                  1 1 1  |  -.2446718   .3410114    -0.72   0.473    -.9130419    .4236983

I have read a lot of papers on the interpretation of coefficients on interaction terms, including this paper by Shang et al. (2017): https://onlinelibrary.wiley.com/doi/...?saml_referrer. It says that in the case of a normal interaction term between a continuous and a binary variable, the coefficient on the interaction term can be directly interpreted as a difference in semi-elasticity. They say this is not the case for the interaction between two binary variables.

The results shown here are the results from the triple interaction term; the interaction between three binary variables.

My main question is: how can I interpret this coefficient into something meaningful? Does this coefficient, as it is shown, mean anything meaningful? How could I calculate an interaction effect, and if I do, will it correspond to the average treatment effect on the treated? Would the margins command somehow be of any use? (I know the marginal effect of an interaction term does not exist, and interaction terms, in the sense of Ai and Norton, 2003, are difficult...)

[Jared Greathouse]

Perhaps this will be helpful?

[Maxence Morlet]

Thanks a lot Jared, that is a great reference indeed, I've given it a good read.

May I perhaps ask Joao Santos Silva about this? Unfortunately, Olden and Moen (2020) do not really focus on the interpretation of interaction terms between binary variables in nonlinear models.

Lee and Lee (2023) actually come close and discuss difference-in-difference for Poisson models (I would be curious to see corresponding Stata codes) but unfortunately I've not found a paper discussing triple difference Poisson models.

[Joao Santos Silva]

Dear Maxence Morlet,

I am afraid I do not have experience with triple differences, but the interpretation of PPML results should be similar to the case where you estimate a linear model in logs. Sorry not to be more helpful.

Best wishes,

Joao