MSE Bias Variance tradeoff in estimating the variance of noise for MLE linear regression

Question

I am just grasping the bias variance trade-off as it is explained by the MSE heuristic. We have that if $y = f(x) + \epsilon$ for $\epsilon \sim N(0,\sigma^2)$ we can show that

\begin{aligned} MSE(y, \hat y) &= Var(\epsilon) + Bias^2(\hat y) + Var(\hat y) \\ &= Var(\epsilon) + Var(\hat y) \qquad \text{ because we assume OLS is unbiased} \end{aligned}

I am confused because when we solve linear regression via MLE, we assume the same $y = f(x) + \epsilon$ for $\epsilon$ normally distributed with variance

$$Var(\epsilon) = MSE(y, \hat y)$$

What happened to the $Var(\hat y)$ term?

Thank you!

Answer 1

Hi: In your second formula, the bias is assumed to be zero so there is no bias term. The variance term is the MSE