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If I have $n$ variables $x_1,\dots,x_d$ in my dataset, and I regress the first one against all the others i.e.

$x_1=a_{12}x_2 + a_{13}x_3 + \dots + a_{1d} x_d + \epsilon_1$,

how can I calculate $\text{Var}(\epsilon_1)$?

If there are $n$ datapoints for each variable, then it is possible to calculate $\Sigma_{2:d}$ i.e. the covariance matrix for variables $x_2,x_3,\dots,x_d$. I can also easily calculate an unbiased estimate for $\text{Var}(x_1) = \frac{1}{n-1} \displaystyle\sum_{i=1}^n(x_1^i - \bar{x}_1)^2$.

But I don't know what to do with these?

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