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:

Postganglionic Neurons

Postganglionic Neurons are the neurons in the autonomic nervous system that connect ganglia with the organs they innervate, transmitting signals from the central nervous system.

Norepinephrine

A neurotransmitter that is also a hormone secreted by the adrenal medulla.

Sympathetic System

A part of the autonomic nervous system responsible for the body's 'fight or flight' response, regulating involuntary body functions like heart rate and blood flow.

Ganglionic Neurons

Neurons that are part of the autonomic nervous system located in ganglia, which act as relay points for transmitting signals from the central nervous system to target organs.