Regression discontinuity questions I am considering a regression discontinuity design (RD) where the "treatment" has a definite sorting rule (below the threshold, you are not fined - above the threshold, you are fined). The outcome I am considering is how much you are fined the FOLLOWING year. There are 4 options: 1) remain without a fine, 2) go from no fine to fined, 3) fined less than you were the previous year, or 4) fined more than you were the previous. While I have listed these as categorical, I have the continuous amount of the fine as well.

Here are my questions:
1.) I have never seen a "longitudinal" RD design where the treatment is at a time period before the outcome. Are there any methodological issues with this approach?
2.) Assuming this approach is valid, how best should I model the outcome given the multiple possibilities?

Thanks in advance for any advice!
The fact that your outcome is measured at a later point in time than the treatment is not a problem, and not unusual either - this is true to some extent in all RD studies (if this were not the case, there couldn't be causation). 

As for how to analyze your outcome, I suggest a couple of different specifications. First, you should use the continuous fine measure as an outcome. Second, you could use an indicator variable equal to 1 if they got a fine, and equal to zero if they didn't. You might be able to do similar dummy regressions for the more/less cases too, but that's a little more complicated. If you want to be a little more advanced, it would certainly be interesting to take the RD framework to a tobit model. Shouldn't be too complicated, I expect, but I haven't thought that much about it.