I'm not sure of your question.  The idea is to use least squares (better, weighted least squares) regression to minimize the sum of squared estimated residuals (better, the sum of the squares of the random factors of estimated residuals), while also having the expected sum of estimated residuals be zero.  For some cases, the sum of estimated residuals will always be exactly zero (see Särndal, CE, Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling, Springer-Verlang, Result 6.5.1, pages 231-232), such as the classical ratio estimator, but for unbiasedness, it is enough that the 'expected' total is zero.  (Note: You may be more interested in BLUP, the best linear unbiased predictor.  "Estimators" indicate regression coefficients, while a "predictor" is for the dependent variable, y.)

Perhaps parts of the following will help:

 https://www.researchgate.net/publication/313925533_Comparing_Model_Performances_Graphically

Thank you so much sharing and information, James R Knaub ,regards.