If we regress $y$ on $x$, the regression coefficient is $\beta_1=\frac{cov(x,y)}{var(x)}$ and if we regression $x$ on $y$, the coefficient is $\beta_2=\frac{cov(x,y)}{var(y)}$. If we want to draw these two fitted lines on the same $(y-x)$ coordinates we know that the fitted line of regression $x$ on $y$ will be steeper. I have learned a potential way to solve this problem is to run PCR (Principal component regression). But why is it the case? And is that possible to derive that mathematically?
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Of possible interest: https://stats.stackexchange.com/a/136597/930. – chl Nov 04 '20 at 09:02
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To what does "this problem" refer? – whuber Nov 04 '20 at 13:19