Correlation between Residuals and Explanatory Variables. Consider a generic sequence of pairs of numbers (left(x_{1}, y_{1} ight),

Correlation between Residuals and Explanatory Variables. Consider a generic sequence of pairs of numbers \(\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)\) with the correlation coefficient computed as \(r(y, x)=\left[(n-1) s_{y} s_{x}\right]^{-1} \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\) \(\left(x_{i}-\bar{x}\right)\).

a. Suppose that either \(\bar{y}=0, \bar{x}=0\) or both \(\bar{x}\) and \(\bar{y}=0\). Then, check that \(r(y, x)=0\) implies \(\sum_{i=1}^{n} y_{i} x_{i}=0\) and vice versa.

b. Show that the correlation between the residuals and the explanatory variables is zero. Do this by using part (a) of Exercise 2.13 to show that \(\sum_{i=1}^{n} x_{i} e_{i}=0\)

c. Show that the correlation between the residuals and fitted values is zero. Do this by showing that \(\sum_{i=1}^{n} \widehat{y}_{i} e_{i}=0\) and then apply part (a).