I am estimating a stacked difference in difference following slide 35 here:
I will call a subexperiment as a "cohort". Here, θsd are unit × cohort fixed effects and γtd are time × cohort fixed effects.
My empirical design is like this: I have a group of individuals (s) every year. In any given year (t), there could be a introduction of a new drug. I then rank each individual s based on their "exposure" (which is a score between 0 and 1) to the introduction of this new drug. The top two ranks (i.e., the two individuals with the highest and second highest score) are defined as the treated individuals. I set a window of (−2,+2) around the drug introduction year. Control individuals are all the individuals that are not in the top two ranks within this event window, i.e., they are "clean" controls in the context of a stacked difference in difference. My data essentially looks like this:
For example, a drug labeled 1001 was introduced in 2006, this is called cohort 1 (which is the same as the subexperiment d
in the equation specification above). Two individuals 5452 and 5723 are both treated because they are the top two exposed individuals to this drug. Control individuals are 6274 and 7882 because they are not in the top two ranks for any drug introduction in years 2004 to 2008. But another drug labeled 1006 was also introduced in 2006, this is called cohort 2. However, it turns out that individual 5452 is also one of the top two individuals exposed to this drug.
My question is: individual 5452 appears as a treated unit in multiple cohorts, that is, it is essentially duplicated across different cohorts, i.e., same values of outcome variable Ystd
are observed across different cohorts/subexperiments due to having the same treated unit s. Does repeated treated units in different cohorts invalidate this stacked difference in difference design? Or is it perfectly fine since I am including individual × cohort fixed effects, i.e., θsd, so that I am comparing treated units vs control units WITHIN a cohort, so the repeated treated units isn't a problem?
I will call a subexperiment as a "cohort". Here, θsd are unit × cohort fixed effects and γtd are time × cohort fixed effects.
My empirical design is like this: I have a group of individuals (s) every year. In any given year (t), there could be a introduction of a new drug. I then rank each individual s based on their "exposure" (which is a score between 0 and 1) to the introduction of this new drug. The top two ranks (i.e., the two individuals with the highest and second highest score) are defined as the treated individuals. I set a window of (−2,+2) around the drug introduction year. Control individuals are all the individuals that are not in the top two ranks within this event window, i.e., they are "clean" controls in the context of a stacked difference in difference. My data essentially looks like this:
For example, a drug labeled 1001 was introduced in 2006, this is called cohort 1 (which is the same as the subexperiment d
in the equation specification above). Two individuals 5452 and 5723 are both treated because they are the top two exposed individuals to this drug. Control individuals are 6274 and 7882 because they are not in the top two ranks for any drug introduction in years 2004 to 2008. But another drug labeled 1006 was also introduced in 2006, this is called cohort 2. However, it turns out that individual 5452 is also one of the top two individuals exposed to this drug.
My question is: individual 5452 appears as a treated unit in multiple cohorts, that is, it is essentially duplicated across different cohorts, i.e., same values of outcome variable Ystd
are observed across different cohorts/subexperiments due to having the same treated unit s. Does repeated treated units in different cohorts invalidate this stacked difference in difference design? Or is it perfectly fine since I am including individual × cohort fixed effects, i.e., θsd, so that I am comparing treated units vs control units WITHIN a cohort, so the repeated treated units isn't a problem?
