Question on RDD

In the ‘Estimation using local and global least squares regression’ there’s a sentence:
“As Lee and Lemieux [2010] note, allowing different functions on both sides of the discontinuity should be the main results in an RDD paper.”

Why are different functions on either side the main results of an RDD paper?

different functional forms give different answers so an author needs to take a side in the argument and advance a specific causal theory consistent with the data at hand. Assuming the author is correct, it should replicate on future studies.

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