Can i combine related treatment covariates in one staggered difference-in-difference regression?

[Santy Beans]

I am carrying out staggered diff in diff regressions, looking at how legalization of cannabis has affected different variables, one of which being state expenditure. I also have another treated group, which are states which sell cannabis from retail shops. At the moment, I am running separate regressions for each treatment, however I am wondering if it would be possible to add both treatments into one regression. Since all states which retail sell cannabis have to have legalised in the past, these two treatments are very closely related; I know that in normal regression having controls which are linked to the independent variable are not good. However, i was wondering if this was different for staggered diff in diff.

At the moment, I am wondering which regression would be the correct one to use:

1) xtreg TotExpend i.legalizedtreated i.activelegalized i.Year, fe cluster(States)

2) xtreg TotExpend i.legalizedtreated i.activelegalized i.salestreated i.activesales i.Year, fe cluster(States)

, where I have two treatments (sales and legalized), and am looking to estimate their relationship with my dependent variable (TotExpend). 'legalizedtreated' is the dummy which switches to 1 for every treated state at any period, and 0 for all control states. 'activelegalized' is the dummy which only switches to 1 for treated states in their post-treatment years, and 0 for all else.

In 2), I have used both treatments in the same regression.

Please let me know if running something like 2) would be feasible, and don't hesitate to tell me if i need to provide any more info. Many thanks, Santy!

[Clyde Schechter]

It's not that you can't do it, but the interpretation of the results is complicated by the fact that salestreated is necessarily 0 whenever legalizedtreated is 0 (and also for the corresponding active* variables.

I would do it differently. I would create a three level variable, called treatment, coded 0 for no legalization at all, 1 for legalization without retail shops, and 2 for legalization with retail shops. Then I would have an active variable which is 0 whenever a state is untreated (either because it never well be, or just isn't yet) and 1 when it is treated. Then I would run it as:

xtreg TotExpend i.treatment#i.active i.year, fe cluster(state)

The coefficients of the 2.treatment#1.active and 3.treatment#1.active variables in the regression table will be the generalized DID estimates of the effects of legalization without retail stores, and legalization with retail stores, respectively. And if you want the difference between them (which would be the incremental effect of retail stores given legalization) you can get that as

lincom 3.treatment#1.active - 2.treatment#1.active

I know that in normal regression having controls which are linked to the independent variable are not good.

Nonsense! In fact, if a covariate is unassociated with the independent variable, there is no point to including it in the model, as its omission cannot lead to omitted variable bias. OK, there might be some benefit to adding it to the model if it reduces the residual variance of the outcome variable. But it is precisely those variables that are associated with both the independent and dependent variable that need to be included in the model to reduce confounding bias.

[Santy Beans]

Wow thanks so much Clyde, I've run the code and it's definitely a lot more efficient than what I was trying to do! Your reply is much appreciated, cheers

[Jared Greathouse]

Another way of doing this that's on the menu would be to use the interrupted time series command itsa by Ariel Linden. It allows you to model the impact of multiple interventions, where the simple law would be a phase in period and retail would be the "real" policy of interest, perhaps.

[Santy Beans]

Thank you for the suggestion Jared, I will look into this