Good morning, I am replicating the identification strategy of Lavy, Victor, and Analia Schlosser. 2011. “Mechanisms and Impacts of Gender Peer Effects at School.” American Economic Journal: Applied Economics 3 (2): 1–33.
I have a dataset on test scores taken in grade 8 by 10 cohorts (2010-2019) of around 500,000 students each, from around 5,000 schools. Therefore I have only one observation for each individual, but I observe the whole cohort of grade 8 of each school every year. So the dataset is a repeated cross-section of the population of grade 8 students in all Italian schools. The dataset looks something like this:
student_id school_id test_score year proportion_females
1 1 100 2010 0.49
2 1 103 2010 0.49
1001 2 98 2010 0.52
1002 2 100 2010 0.52
...
500,001 1 102 2011 0.50
500,002 1 101 2011 0.50
501,001 2 97 2011 0.51
501,002 2 99 2011 0.51
...
1,000,001 1 98 2012 0.48
1,000,002 1 100 2012 0.48
1,001,001 2 101 2012 0.49
1,001,002 2 97 2012 0.49
I run the following regression to exploit the supposedly exogenous variation of the proportion of female peers in a school in each year compared to the school-specific time trend to capture the effect on test scores of having more female peers. My regression needs to include (1) school FE, (2) year FE, and (3) a school-specific time trend

I thought of running the following regression:

Code:
reghdfe test_score proportion_females, absorb (school_id year)
But I saw that some papers using this strategy used something like
Code:
xi: reg test_score proportion_females i.school_id i.year
and some others used (1) just reg or (2) areg

My questions are:
(0) Any general comments on this approach?
(1) Which regression command is suitable for this strategy? I know xtreg and reghdfe are used for panel regressions, and I am wondering whether my dataset can be considered a "school-panel" and therefore those commands would be ok.
(2) how to include school-specific time-trend? By adding c.year#school_id?

Thanks in advance,
Pietro