For a theoretical treatment see Sarndal and Lundstrom’s 2006 textbook, which shows that the approximate bias under calibration is zero only if the outcome or inverse response probabilities are a linear function of the auxiliary vector. In raking the aux vec is the marginal dist; in PS the joint.
Theoretically either is sufficient to drive the approximate bias to zero, but given that this is impossible in practice, it’s definitely better to do both. Also, doing it for the outcome has the additional advantage of reducing variance.
Comments
Shouldn't it be if both the outcome and inverse response probabilities are a linear function of the auxiliary vector though?