Biased estimator with smaller variances than unbiased estimators are easy to find. The MSE estimator has not been as popular as the best unbiased estimator because of the mathematical difficulties in its derivation. Furthermore, when it can be derived its formula often involves unknown coefficients (the value of beta), making its application impossible. Monte Carlo studies have shown that approximating the estimator by using OLS estimates of the unknown parameters can sometimes circumvent this problem (a little confused here, using approximated OLS estimates to substitute the real beta?)
Note: Weighted Square(d) Error Criterion can be a very interested topic to explore!
Peter Kennedy: When the weights are equal, the criterion is the popular mean square error (MSE) criterion. It happens that the expected value of a loss function consisting of the square of the difference between beta and its estimate (i.e. the square of the estimation error) is the same as the sum of the variance and the squared bias.
Please refer to following derivation: