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It seems that minimizing the sum of squared residuals (SSR) in linear regression is equivalent to minimizing MSE (both use true value - prediction) and OLS is the best estimator for minimizing SSR.

I also read that least squares can sometimes produce estimators with large variance under multicollinearity, in which case a biased estimator might produce a better MSE.

I am a bit confused why OLS is the best for SSR but sometimes is not the best on MSE, as these 2 metrics seemingly are proportional to each other.

Thanks.

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This is because OLS only gives the best linear unbiased estimator (BLUE) under the assumptions of the Gauss-Markov theorem. It is only the best among linear, unbiased estimators. It doesn't mean that there can't be biased estimators that perform better than OLS.

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