I am using the Stata rdrobust command for RDD analysis, aiming to perform a two-stage analysis. The first stage is to model the probability of receiving the treatment at cut-off and the second stage is to use the predicted value of the treatment variable on my dependent variables. I am taking a log of the forcing variable to remove skewness in the distribution.

Assume cut-off value is "C", "X" is forcing variable, "T" is treatment, "Y" is my dependent variable. I do not explicitly specify bandwidth and the command uses MSE optimal data-driven bandwidth selection algorithm. Regression fit is local linear polynomial.

I have tried two kinds of argument specifications in the rdrobust command:

Specification 1: I calculate cut_off_margin as log(X) - log(C) (i.e. distance between forcing variable and cut-off in terms of log difference). Here, command uses zero cut-off.

command: rdrobust Y cut_off_margin, fuzzy(T) p(1) c(0)

Specification 2: In this specification, the log of forcing variable "X" and log of cut-off "C" is directly passed onto the command. So, cut-off is not relative to zero.

command: rdrobust Y log(X), fuzzy(T) p(1) c(log(C))

There is some difference in output between specification 1 and 2 (both are in log scale). Could someone provide any insights about how is specification 2 handled by rdrobust? I thought that specification 2 should be handled in same as specification 1.
Is there any suggestions about which specification is more precise while using this STATA command?