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Exercise :

For the Simple Linear Model $\mathbb E[y_x] = b_0 + b_1x$, prove that for a newly given $x_0$ and $y_{x_0}$ a new observation while $\hat{y_{x_0}}$ its point estimate, it is : $$\operatorname{cov}(y_{x_0},\hat{y_{x_0}}) = 0$$

Question :

It is :

$$\operatorname{cov}(y_{x_0},\hat{y_{x_0}}) = \mathbb E[(y_{x_0}-\mathbb E[y_{x_0}])(\hat{y_{x_0}}-\mathbb E[\hat{y_{x_0}}])]$$

but how would one continue to prove the expression asked from that point on ?

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  • $\begingroup$ Did you search the site? There are several posts on this topic, one of which likely has the answer. $\endgroup$ – StubbornAtom Nov 1 '18 at 19:50
  • $\begingroup$ @StubbornAtom found none regarding this specific one on either. The fact that covariance operator is not standard in latex makes it hard to find posts. $\endgroup$ – Rebellos Nov 1 '18 at 20:11

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