Answer the Following Questions $\sum _ { i = 1 } ^ { n }$

Answer the following questions.
a. Show that $\sum _ { i = 1 } ^ { n }$ $e _ { i } = 0$
when the $e _ { i } ^ { \prime } s$
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 $\sum _ { i = 1 } ^ { n } x _ { i } e _ { i } = 0$
for the residuals from a simple linear regression. [This result along with part (a) shows that there are two linear restrictions on the $e _ { i } ^ { \prime } s$
, resulting in a loss of 2 df when the squared residuals are used to estimate $\sigma ^ { 2 }$
]