DiD with heterogenous treatmens where the treatment switch on and off with leads and lags

[Ludvig Olsson]

Hello,

Before I explain my issues I want to informyou that I cannot share my data due to confidentiality reasons.

For my master's thesis I'm studying if the physicians as operation managers affect the turnover rate of physicians at primary care centers (PCCs) with the usage of a balanced panel dataset with the years 2010-2018 (yearly format) and 316 PCCs. PCCs are treated when they have a physician manager and is untreated otherwise. 30 PCCs will be treated, 15 out of those will switch out of treatment and 2 of those will get a second treatment. No PCC starts treated and we have no time gaps.

Our idea is to perform a DiD with heterogenous treatmens where the treatment is allowed to switch on and off with leads and lags so we create an event study graph. I have tried to acquire as much information as possible but I have som questions regarding this:

1. In this thread Clyde Schechter states that the code I'm using below should be able to handle heterogenous DiD with reversal of treatment: https://www.statalist.org/forums/for...hed-on-and-off

 xtreg physician_turnover i.year##i.treatment $covariates, fe VCE(cluster PCC_ID)

However, I have a hard time to find a paper to cite that confirms which leads to my first question. Every paper I read seems to assume that treated subjects should remain treated (for example "Differences in Differences with variation in treatment timing" Goodman Bacon (2021)). Are there any papers you know on top of your head that I could cite for this?

2. My second question is if I can add leads and lags by using the code below?:

 xtreg physician_turnover $leads $lags i.year##i.treatment $covariates, fe VCE(cluster PCC_ID)

3. In the thread below Jeff Wooldridge have been working on a heterogenous DiD where reversal of treatment is possible. I have checked the coding from the dropbox and paper linked in the thread "Simple Approaches to Nonlinear Difference-in-Differences with Panel Data." Wooldridge (2023). My question is if this is applicable with leads and lags?

https://www.statalist.org/forums/for...ing-on-and-off

[Clyde Schechter]

Unfortunately, the link to a paper from Northwestern University that I posted in the https://www.statalist.org/forums/for...hed-on-and-off thread no longer works. And, even more unfortunately, I have not preserved my download of that paper. I can assure you that that paper does explicitly refer to the validity of this approach to DID with intermittent treatment. I realize, however, that this is not as good as having a citation, and I'm afraid I don't know of another paper that specifically deals with this off the top of my head. And I don't have time to do a literature search to find one any time soon.

Let me clarify a few things, though, about intermittent treatment. This approach is usable with intermittent treatment if certain assumptions are met:

1. Treatment effects turn on and off immediately. That is, treatment effect occurs in the same time period that treatment begins, and after subsequent discontinuation, the effect disappears immediately upon discontinuation.
2. The effect of a second or later course of treatment is the same as the effect of an initial course of treatment. Previous treatment neither enhances nor interferes with the effect of later treatments.

If these assumptions are not met, more complicated models are required.

[Jeff Wooldridge]

Ludvig: Very perceptive of you. I wish in that paper I had commented on combing the method I proposed for exit with the leads-lags approach. I'm currently working on a survey where I discuss it explicitly. I've also incorporated it into my teaching.

So, yes. You can included the leads -- what I call the pre-treatment indicators -- in the usual ES way, with the cohort dummies indexed by the first period treated and the last period treated -- so D(g,h) where g is the first period of treatment, h is the last. You can include D(g,h)*fs_t for all s except g = 1, which sets g - 1 as the comparison period. I don't know if anyone who's applied it yet. The non-leads estimator works well under parallel trends and no anticipation.

As Clyde mentions, the other methods assume the effect of the treatment is zero if the treatment is not in place. My approach allows you to estimate persistent effects after treatment, and so you can test whether the effect disappears.

If you implement it, let me know how it works!