Abstract: In this paper, we investigate optimal estimator of regression coefficient in a general Gauss-Markov model under balanced loss function. Firstly, necessary and sufficient conditions for linear estimators to be best linear unbiased estimator (BLUE) are provided. Secondly, we prove the best linear unbiased estimator is unique in the sense of almost everywhere, and also a balance between least squares estimator and optimal estimator under quadratic loss. Thirdly, loss robustness of the optimal estimator is discussed in terms of relative losses and relative saving losses. Finally, we give some conditions about the robust BLUE on the mis-specification of covariance matrix.