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

question 9

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.

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Definitions:

Nicotine

A stimulant and addictive substance found primarily in cigarettes and other tobacco products.

Amphetamines

Stimulant drugs that increase the activity of the nervous system to enhance alertness and physical performance.

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A persistent pattern of use of a substance characterized by frequent intoxication; impairment of physical, social, or emotional well-being; and possible physical addiction.

Physical Addiction

A physiological dependency on a substance, characterized by the emergence of withdrawal symptoms when the substance is not consumed.