They may exist, but they must be biased or non-normally distributed.
The ordinary least squares (OLS) estimator has the lowest sampling variance among the class of linear unbiased estimators if the errors in the linear regression model are uncorrelated, have equal variances, and have an expected value of zero, according to the Gauss-Markov theorem (also known as the Gauss theorem for some authors). The errors do not have to be independent and evenly distributed, nor do they have to be normal (only uncorrelated with mean zero and homoscedastic with finite variance). Since biased estimators exist and have lower variance, the requirement that the estimator is unbiased cannot be eliminated.
If the error components are uncorrelated, have an identical variance, and have an expected value of zero, the Gauss-Markov theorem predicts that the OLS estimator will have the least sample variance. The variance is lower for the biased estimators. There are various models that use both normal and non-normal distributions to demonstrate this theorem, therefore normality of the model is not a condition. The test statistic's z value can be transformed to satisfy the normality requirements. When random variables are either biased or regularly distributed, the theorem is valid.
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