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$\begingroup$ Then, I think that the additive property of the regression ultimately reduces to proving the following, $cov(y_2+y_3,X)=cov(y_2,X)+cov(y_3,X)$, which should hold. Any idea how to prove this? this is, I think, harder than proving it in the matrix form. $\endgroup$
– Kernel
Commented Oct 16, 2023 at 19:03
$\begingroup$ I just found that the previous relationship did hold. See the distributive property of the covariance dlsun.github.io/probability/cov-properties.html $\endgroup$
– Kernel
Commented Oct 18, 2023 at 20:55

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