Event study examining selection bias(?) in DD specification

[Emily Morton]

I have a school district-year panel and am running a diff-in-diff (DD) looking at a policy change (a switch to four-day school weeks) that happens at the school district-level ("fourday" is my treatment dummy). Districts switch schedules at different years and always stay on that schedule after switching. I use year and district fixed effects in a DD specification (with standard errors clustered at the district level) to estimate the effect of four-day weeks on achievement outcomes (avg).

The code I've used for that specification is the following:

reghdfe avg fourday ${covariates}, absorb(year district_id) vce(cluster district_id)

Now, I am trying to perform robustness checks and have been struggling with what to do. I first perform a parallel trends event study specification for achievement outcomes using the following code:

reghdfe avg fourday_lead_3 fourday_lead_2 fourday_lead_1 fourday_time_0 fourday_lag_1 ///
              fourday_lag_2plus ${covariates}, absorb(year district_id) vce(cluster district_id)
test fourday_lead_3=fourday_lead_2=fourday_lead_1
test fourday_time_0=fourday_lag_1=fourday_lag_2plus

I interpret the point estimates as the "effect" of being x years pre ("lead")- or post ("lag")- fourday treatment (fourday_time_0 = adoption year) relative to never having been treated or being four or more years pre-receiving treatment.
Is it correct to use a F-test that equates the pre-treatment years to each other but not to zero because I want to allow for different levels in treatment without allowing for different trends in treatment?

Here is where I am most stuck: I want to check for selection into treatment (or other general threats to the original DD?) based on demographic variables such as the percent of students receiving free lunch (perfrl). I think it would invalidate the original DD if there were differential trends in perfrl leading up to treatment between the treatment and control districts. I think it would also be problematic if perfrl were changing after treatment, as that could be an indication that the districts exposed to four-day week are demographically different than the control districts (which I do not expect to be the case)? Could I use a similar event study specification as above to examine this question? Would a single significant point estimate be problematic or only a significant F-test? Would the F-test in this case be that all point estimates are equal to each other? Or should I test the equivalence of pre-treatment and post-treatment point estimates separately to allow for different trends pre- and post-treatment? Or should I do something else entirely?

reghdfe perfrl fourday_lead_3 fourday_lead_2 fourday_lead_1 fourday_time_0 fourday_lag_1 ///
           fourday_lag_2plus ${covariates}, absorb(year district_id) vce(cluster district_id)
test fourday_lead_3=fourday_lead_2=fourday_lead_1=fourday_time_0=fourday_lag_1=fourday_lag_2plus

Any advice would be greatly appreciated -- thanks!