I am trying to estimate the enrollment premium at high-fees schools i.e., impact of high-fee school enrollment on learning outcomes. I am instrumenting high-fee school enrollment by using a continuous treatment instrument. The treatment is a voucher of varying size randomly assigned to all applicants. So, the instrument takes a value of zero for all control applicants, and takes a continuous value for all treated applicants. Does this sound like a valid instrument? Or, would it be better if I made the instrument categorical, by say, dividing the continuous value to 3 or 4 categories?
In the first case, what would be the interpretation of the LATE?
Any thoughts would be greatly appreciated.