(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:

IQ

stands for Intelligence Quotient, a measure of a person's intellectual abilities relative to others, based on performance on standardized tests.

Consistent

Showing steady and uniform characteristics or behavior over a period of time, lacking any significant variation.

Reliable

The extent to which an assessment tool produces consistent and stable results over multiple applications or time periods.

Standardized

The process of making something conform to a standard in order to ensure consistency and comparability.