This article describes a simple scenario where defining an instrumental variable is helpful for computing calibration weights (i.e., weights that satisfy the specified calibration equation yet are asymptotically identical to the inverse selection probabilities). The implicit model is simple regression with an intercept. The choice of instrumental variable can reduce the possibility that any calibration weight will be less than one.
A practical use for instrumental-variable calibration
Kott, P. (2003). A practical use for instrumental-variable calibration. Journal of Official Statistics, 19(3), 265-272.