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.