Best Linear Unbiased Estimator

What is Best Linear Unbiased Estimator (BLUE)?

The OLS estimators are the best linear unbiased estimators (in the sense of having the most negligible variance among all linear unbiased estimators) under certain assumptions, regardless of whether the variables are normally distributed or not (Gauss–Markov theorem). The assumptions underpinning the linear regression must be satisfied for OLS to provide the best linear unbiased estimator (BLUE).

1. Y and X(s) should have a linear connection. 
2. The residuals must be homoscedastic, independent, and have a zero expected value.
3. ε ~ N(i.i.d).

We can relax the normality condition for the residual distribution if there are no outliers and the residuals have an expected value of zero.

Why is it important?

The idea of unbiased estimation is particularly significant in the theory of point estimation because it is crucial to get an unbiased estimator with no systematical errors in many real-world circumstances.