Chapter 13.1, Problem 10E

a. Show that ${\displaystyle {\sum }_{i=1}^n{e}_i=0}$ when the e_i’s are the residuals from a simple linear regression.

b. Are the residuals from a simple linear regression independent of one another, positively correlated, or negatively correlated? Explain.

c. Show that ${\displaystyle {\sum }_{i=1}^n{x}_i{e}_i=0}$ for the residuals from a simple linear regression. (This result along with part (a) shows that there are two linear restrictions on the e_i’s, resulting in a loss of 2 df when the squared residuals are used to estimate σ².)

d. Is it true that ${\displaystyle {\sum }_{i=1}^n{e}_i^{*}=0}?$ Give a proof or a counter example.