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

question 38

Essay

(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 $\sum _ { i = 1 } ^ { n } \hat { u } _ { i } \hat { Y } _ { i } = 0$

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Definitions:

Community Approach

A method or strategy for addressing issues that involves the collective participation and effort of the entire community.

Psychiatric Hospital

A medical facility specializing in the diagnosis and treatment of mental illness.

Psychotic Symptoms

Manifestations of severe mental disorders characterized by a loss of contact with reality, such as hallucinations, delusions, disorganized thinking, and speech.

Sheltered Workshop

A supervised workplace for people who are not yet ready for competitive jobs.