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

Question

Quizplus · Accepted Answer

The Answer of (Requires Appendix material) A necessary and sufficient condition to derive the OLS estimator is that the following two conditions hold: $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } X _ { i } = 0$
Show that these conditions imply that $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } \hat { Y } _ { i } = 0$

(Requires Appendix Material) a Necessary and Sufficient Condition to Derive ∑i=1nu^i=0 and ∑i=1nu^iXi=0\sum _ { i = 1 } ^ { n } \hat { u } _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } X _ { i } = 0∑i=1n​u^i​=0 and ∑i=1n​u^i​Xi​=0

(Requires Appendix Material) a Necessary and Sufficient Condition to Derive $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } = 0 \text { and } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } X _ { i } = 0$