(Requires Appendix Material and Calculus)Equation (5 $\beta$ ₁ to Be Var $\beta$

question 32

Essay

(Requires Appendix material and Calculus)Equation (5.36)in your textbook derives the conditional variance for any old conditionally unbiased estimator $\beta$ ₁ to be var( $\beta$ 1 X₁, ..., X_n)= $\sigma _ { u } ^ { 2 } \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ where the conditions for conditional unbiasedness are $\sum _ { i = 1 } ^ { n } a _ { i }$ = 0 and $\sum _ { i = 1 } ^ { n } a _ { i } X _ { i }$ = 1. As an alternative to the BLUE proof presented in your textbook, you recall from one of your calculus courses that you could minimize the variance subject to the two constraints, thereby making the variance as small as possible while the constraints are holding. Show that in doing so you get the OLS weights $\hat { a } _ { i }$ (You may assume that X₁,..., X_n are nonrandom (fixed over repeated samples).)

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Definitions:

Secondary Sexual Characteristics

The physical traits that emerge during puberty which are not directly involved in reproduction but distinguish the two sexes, such as breast development in females and facial hair in males.

Primary Sexual Characteristics

The biological and physical characteristics present at birth that are directly involved in reproduction, such as the ovaries in females and testes in males.

Tertiary Sexual Characteristics

Traits influenced by hormones that differentiate between sexes but are not directly part of the reproductive system, often developing during puberty.

Object Permanence

The realization by infants that objects still exist even when they are out of sight, sound, or touch.