Equivalence of Specifications in Triple Difference Estimation

[Maxence Morlet]

Dear all,

Suppose I run the following saturated triple difference equation:

reg Y post treated_industry treated_state treated_industry#treated_state treated_industry#post treated_state#post treated_industry#treated_state#post

the triple difference coefficient we want will be that of treated_industry#treated_state#post.

Suppose the outcome thus varies on a state, year and industry basis. We therefore run the following specification, also saturated:

reg Y treated_industry#treated_state#post i.year#i.state i.year#i.industry i.industry#i.state

Is it normal to obtain the identical coefficient on the triple difference coefficient treated_industry#treated_state#post across both estimations?

Might this be the case because both models are saturated, and therefore the variation of the triple difference variable must be identical within clusters defined in the abovementioned estimations?

[Jeff Wooldridge]

This doesn't surprise me. In my work on the usual DiD, with covariates, I showed that the extended TWFE is the same as just controlling for whether units are treated or not -- when fully flexible effects are assumed. By choosing the covariate X to be binary gives the triple DD estimator. I think that's what you're finding here.

[Maxence Morlet]

Dear Professor Wooldridge,

Your explanation makes a lot of sense. Thank you!

[Maxence Morlet]

This doesn't surprise me. In my work on the usual DiD, with covariates, I showed that the extended TWFE is the same as just controlling for whether units are treated or not -- when fully flexible effects are assumed. By choosing the covariate X to be binary gives the triple DD estimator. I think that's what you're finding here.

Dear Jeff Wooldridge, might you please have a link to your paper you mentioned?

Many thanks in advance!

[Andrew Musau]

https://papers.ssrn.com/sol3/papers....act_id=3906345