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(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

question 38

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


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