Interaction between two Difference-in-differences treatment variables

[Michelle Chen]

I am interested in estimating the effects of adoption of different medical technologies (MTs) on clinical outcomes. I have data on patient-level clinical outcomes and hospital-level technology adoptions. I am using Stata 14 for windows.

I first use command xtreg to run a difference-in-differences (DID) specification based on my panel data (with year fixed effects and hospital fixed effects), to estimate the changes in clinical outcomes of hospitals after they adopt technologies compared to changes in outcomes of those hospitals that do not adopt any technology. If there is only one technology, then I am done.

However, there are multiple technologies and to simplify, we can classify them to be basic technology and advanced technology. A hospital may adopt none, either or both during our study period and may also change their adoption status from none to either or both. I suspect that there may be some interaction effect between the basic technology and advanced technology. In other words, basic technology makes a difference in clinical outcomes when advanced technology is adopted (or vice versa). I added interaction terms between BasicMT(ht) (a dummy variable which equals to 1 if a basic technology is adopted at hospital h at time t) and AdvancedMT(ht) (a dummy variable which equals to 1 if an advanced technology is adopted at hospital h at time t) to my xtreg regression and Stata14 did produce estimation results.

My question is: in a difference-in-difference setting, are we violating the underlying identifying assumption by interacting two treatment variables? Some studies that use triple-difference estimation examine how the DID effect differ based on exogenous factors such as age or gender, but I have never seen any DID study that estimates the interaction effect of two treatment variables. I would also greatly appreciate any econometric reference that either justify or disprove such practice.

Thank you very much for your time and help in advance!

[Clyde Schechter]

There is no mathematical reason why you cannot do this, provided you are careful to write the -xtreg- command correctly.

But for what you describe, I would take a different approach. I would have a four level treatment variable: none, basic only, advanced only, and both. And I would just do the DID model with that, plus time, plus the interaction of this treatment with time. It will produce the same results as a correctly specified model with interaction terms, but it will be easier to write the code correctly, and the output will be much easier to interpret.

[Michelle Chen]

Thanks so much for your advice, Clyde. My baseline specification is as follows: xtreg p_dep HT_adopt p_char* i.id_t1, fe i(hospital_id) cl(hospital_id), where p_dep measures certain patient outcome, HT_adopt is a dummy variable that equals to 1 if hospital h adopts any technology at time t; and 0 otherwise. p_char* are patient level controls and i.id_t1 are time fixed effects. My understanding is that in a panel data setting, the estimates of HT_adopt would directly capture the effect of hospital technology adoption and is essentially the same as including both time, treatment and the interaction of the two, except that in the latter I need to sum up the estimates of the treatment variable and the interaction term to get the effect of hospital technology adoption.

If I understand your suggestion correctly, the specification should be enriched to: xtreg p_dep HT_adopt_none HT_adopt_basic HT_adopt_advance HT_adopt_both p_char* i.id_t1, fe i(hospital_id) cl(hospital_id). Is this right? And how should I interpret the estimates and identify the interaction effects between basic technology and more advanced technology? In other words, how do I tell if basic technology makes a difference in clinical outcomes when advanced technology is adopted (or vice versa)? Any help is greatly appreciated!

[Clyde Schechter]

I agree with your proposed model. Each of the HT_adopt_* variables is a treatment#time interaction, and you have both hospital and time fixed effects in the model. So that looks right. To tell if basic technology makes a difference in clinical outcomes when advanced technology is adopted, you would look at the difference between the coefficients of HT_adopt_both and HT_adopt_advance. Similarly, to tell if advance technology makes a difference when basic is already adopted, you would look at the difference between HT_adopt_both and HT_adopt_basic. You can calculate those differences using -lincom- after -xtreg- runs. I think this is simpler than having three-way- interactions, where the interpretation gets more complex.

[Michelle Chen]

Thank you very much for the helpful suggestions, Clyde. It is nice to know that the -lincom- command can be used to calculate the differences after -xtreg- runs.