I'm trying to proof some results in Multiple Linear Regression.
In matrix notation, why $Cov(\pmb{y}, \pmb{\hat{y}}) = \pmb{y}^T \pmb{\hat{y}}$?
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$\begingroup$ Do you have a source for the equality? $\endgroup$– user603Commented Jul 3, 2019 at 12:22
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$\begingroup$ In "Solutions Manual to Accompany Introduction to Linear Regression", 5th edition, Exercise 3.33, the author says this in the page 30. $\endgroup$– igorkfCommented Jul 3, 2019 at 12:26
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$\begingroup$ This does not look right. See math.stackexchange.com/questions/2978027/…. $\endgroup$– StubbornAtomCommented Sep 25, 2019 at 15:08
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That is simply not true!
$y^T\hat{y}$ can take values larger than 1, and it's not that hard to come with an example. Try the data
$x=(0,1,2,3)$
$y=(6,8.1,9.9,12)$
Estimate the regression model for $y$ on $x$ and see how $y^T\hat{y}$ will be above 340! Correlation between $y$ and $\hat{y}$ stays at $0.9995$ while their covariance is close to $6.5$
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$\begingroup$ Sorry! I mean the Covariance. I'm editing the post. $\endgroup$– igorkfCommented Jul 3, 2019 at 12:05
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1$\begingroup$ Still not true, Covariance stays at about 6.5 in the example I presented. Indeed, the product $y^t\hat{y}$ always increases with sample size while covariance does not necessarely $\endgroup$– DavidCommented Jul 3, 2019 at 13:10