I am looking for an empirical paper that perform a difference-in-difference regression where the whole did regression (treated+post+post*treated) is interacted with a continuous variable (or a categorical one)... Do you have an idea of such a paper ?
My own not-yet-published paper!
Sorry, I can't immediately think of a more authoritative paper, though mine does fit the bill (assuming that it is non-wrong).
(I formulate the DID slightly differently -- rather than including a dummy for treated (implicitly assuming that any difference between T and C is a level effect) I weight the control group by propensity scores for being in the treatment group.)
Price and Wolfers (2010) measured NBA referees on how many fouls they called based on their race and the race of the players they called fouls on. See Table 3:
Joseph Price & Justin Wolfers, 2010. "Racial Discrimination Among NBA Referees," The Quarterly Journal of Economics, MIT Press, vol. 125(4), pages 1859-1887, November. http://www.nber.org/papers/w13206
The continuous variable is the foul rate, and the "treatment" is the change in fouls called over the change in % white referees.