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I know this is very elementary but I cannot remember how to show $\hat\alpha$ as below.

enter image description here

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    $\begingroup$ Are you asking how to show that that's what the least-squares estimator of $\alpha$ is, or are you asking how to show that that's what its distribution is? $\endgroup$ – Michael Hardy May 27 '13 at 19:30
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You minimize the squared error $$\epsilon^T\epsilon=(y-X\alpha)^T(y-X\alpha)=y^Ty-2\alpha^TX^Ty+\alpha^TX^TX\alpha$$ This expression can be minimized by setting its derivative w.r.t. $\alpha$ equal to zero:

$$-2X^Ty+2X^TX\alpha=0\Rightarrow \alpha=(X^TX)^{-1}X^Ty$$

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