identification with ordinal independent variable


New Member

Can you help me find the relevant literature for this problem?
My regression model is: Yi=Xi*b+Ei, where Yi is income and Xi is a score, Ei the error.
I don't observe Xi but only Ri=rank(Xi), that is instead of (13.4,13.5,14,17.5,18,..) I only observe (1st,2nd,3rd,..). [I lose the distance].
Are there methods to say something about b? Suppose that I would like to argue that "b" is actually different from the first three observations, is there a way to test this without observing Xi?

Thanks very much for your help!