A Necessary and Sufficient Condition to Derive the OLS Estimator $\sum _ { i = 1 } ^ { n } \hat { u _ { i } }$

Essay

A necessary and sufficient condition to derive the OLS estimator is that the following two conditions hold: $\sum _ { i = 1 } ^ { n } \hat { u _ { i } }$ = 0 and $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } X _ { i }$ = 0. Show that these conditions imply that $\sum _ { i = 1 } ^ { n } \hat { u _ { i } } Y _ { i }$ = 0.