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So given that the least squares estimator of $\beta$ is:

$$ \hat{\beta} = (\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T \mathbf{Y} $$

What is the covariance matrix?

Here is my attempt, I'm not sure what $E[\epsilon \epsilon^T]$ and $E[\epsilon \beta^T]$ are:

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Does anyone know how to proceed?

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$\beta $ is a constant so $$ E(\epsilon\beta^T) = E(\epsilon)\beta^T .$$

You can compute $E(\epsilon)$ and $E(\epsilon\epsilon^T)$ from your assumptions about the distribution of $\epsilon.$ You didn't specify these here, but usually it is assumed that $\epsilon$ is a vector of independent zero-mean normals all with the same variance $\sigma^2.$ So then $E(\epsilon)$ is the zero vector and $$E(\epsilon\epsilon^T) = E(\epsilon_1^2+\ldots+\epsilon_n^2)$$ where $\epsilon_1\ldots \epsilon_n$ are independent $N(0,\sigma^2).$ (I'm assuming $n$ is the number of data-points.)

I'll leave that for your to compute.

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