Diff in Diff with a continous treatment variable

[Nel Del]

Hi everyone!
I am studying the relation between qualitity of school infrastructions and student performance in rural primary schools in Angola. I have observation at school level in two time perionds(2015 and 2018). I want to use a Difference-in Difference model with a continous indipendent variable, which is an index for the quality of infrastructure. However, I am not really confident with the diff-in-diff with a non-binary treatment effect and I am not sure f I am interpreting the model correctly. The model I would estimate is the following:

test_scores = time + index + time*index

where time is a dummy equal one if year=2018, index is a continous variable for school infrastructure and the interaction term would be the variable of interest.

Q1. How do I run this model in Stata?
Can I use the -diff- command or should I use a fe model? Do DiD and FE reach the same results?

Q2. Can I use school level controls in this model? Do they need to be time variant controls?

Q3. How would I test the common trend assumption?

Q4. Alternatively, could I generate a dummy variable for schools characterized by a higher infratructure index (higher than the mean for example) and such dummy would be my tratment variabe for diff in diff. How do I do this in Stata?

[Matt Mulligan]

Hi Nel, I'm curious how you solved this (if you were able to). I have a similar problem.