A simple expression is developed for the difference between the least squares and minimum variance linear unbiased estimators obtained in linear models in which the covariance operator of the observation vector is nonsingular. Bounds and series expansion for this difference are obtained, and bounds for the efficiency of least squares estimates are also obtained.
"How Much Do Gauss-Markov and Least Square Estimates Differ? A Coordinate-Free Approach." Ann. Statist. 3 (4) 982 - 990, July, 1975. https://doi.org/10.1214/aos/1176343201