# #StackBounty: #regression #multiple-regression #least-squares #mse What's the MSE of \$hat{Y}\$ in ordinary least squares using bias…

### Bounty: 100

Suppose I have the following model: $$Y = mu + epsilon = Xbeta + epsilon,$$ where $$Y$$ is $$n times 1$$, $$X$$ is $$n times p$$, $$beta$$ is $$p times 1$$, and $$epsilon$$ is $$n times 1$$. I assume that $$epsilon$$ are independent with mean 0 and variance $$sigma^2I$$.

In OLS, the fitted values are $$hat{Y} = HY$$, where $$H = X(X^TX)^{-1}X^T$$ is the $$N times N$$ hat matrix. I want to find the MSE of $$hat{Y}$$.

By the bias-variance decomposition, I know that

begin{align} MSE(hat{Y}) &= bias^2(hat{Y}) + var(hat{Y})\ &= (E[HY] – mu)^T(E[HY] – mu) + var(HY)\ &= (Hmu – mu)^T(Hmu – mu) + sigma^2H\ &= 0 + sigma^2H end{align}

I’m confused by the dimension in the last step. The $$bias^2$$ term is a scalar. However, $$var(hat{Y})$$ is an $$N times N$$ matrix. How can one add a scalar to an $$N times N$$ matrix where $$N neq 1$$?

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