Calibration weights have been studied extensively as adjusted survey weights which take auxiliary data into account.  A simple case of auxiliary data with linear regression through the origin is noted in Section 5.0 in

https://www.researchgate.net/publication/261508465_Use_of_Ratios_for_Estimation_of_Official_Statistics_at_a_Statistical_Agency

Many references for calibration weights one might find include

Deville, J-C. and Särndal, C-E. (1992). Calibration Estimators in Survey Sampling,

Journal of the American Statistical Association, 87, 376-382.

An Introduction to Calibration Weighting for Establishment Surveys

Phillip S. Kott, from ICES IV,

http://www.amstat.org/meetings/ices/2012/papers/302286.pdf

Kott from Pakistan Journal of Statistics,  2011 special issue in honor of Ken Brewer,

http://www.pakjs.com/journals//27(4)/27(4)5.pdf

and

Carl-Erik Särndal and Sixten Lundstrom. Estimation in Surveys with Nonresponse. Hoboken, NJ: Wiley, ISBN 0470011335

The regression weight, q, however, seems to often be set to 1, despite the references showing this is not necessary, but possibly for convenience.   q=1 implies homoscedasticity, which is not reasonable for establishment surveys with good data.  See

Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press.

So why set q to 1 for an establishment survey, when it is demonstrably untrue?  There are several ways to estimate the coefficient of heteroscedasticity, and for establishment surveys, both theory and empirical evidence show that heteroscedasticity should be at least as great as that of the classical ratio estimator, where we would have

q = 1/x

as a result. 

So why is q so often set to 1 when it can easily be shown that homoscedasticity for cross-sectional establishment surveys, in case after case, is not true? 

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