$(Y_i  \hat{Y}_i)(\hat{Y}_i  \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
Asked
Active
Viewed 1,271 times
0
Alex M.
 33,823
 15
 43
 81
Sidney Bookeart
 11
 2

I do agree to the first line. And I cannot comprehend the transforming from third line to the fourth line too. – callculus42 Nov 19 '15 at 18:41
1 Answers
1
By $\hat{Y}_i$ you mean the MMSE estimator (or the posterior mean) ?
If so, remember a very important property of this estimator is that the MMSE estimator of $Y$ is that the error $\hat Y  Y$ is orthogonal to any function of $Y$. Thus $E[(Y_i  \hat{Y}_i)(\hat{Y}_i  \bar{Y})] = 0$. But in your case there are no expectations. Please verify that or give the full context of this derivation
Nir Regev
 304
 1
 6

I am taking my first course in regression and was trying to see how square of correlation of y and y hats is the r squared value. I got the above picture from: https://en.m.wikipedia.org/wiki/Pearson_productmoment_correlation_coefficient#Pearson.E2.80.99s_distance. Thank you so much for your response. – Sidney Bookeart Nov 19 '15 at 19:24

@Bhrigu Aneja: yes so please see what I wrote. It holds also for the sample covariance which is the approximation of the expectation I wrote above. The error is orthogonal to any function of Y – Nir Regev Nov 19 '15 at 19:30