(Requires Appedix Material and Calculus) Equation (5 $\widetilde { \beta } _ { 1 }$

question 26

Essay

(Requires Appedix material and Calculus) Equation (5.36) in your textbook derives the conditional variance for any old conditionally unbiased estimator $\widetilde { \beta } _ { 1 }$ to be $\operatorname { var } \left( \widetilde { \beta } _ { 1 } \mid X _ { 1 } , \ldots , X _ { n } \right) = \sigma _ { u } ^ { 2 } \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ (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 $X _ { 1 } , \ldots , X _ { n }$ are nonrandom (fixed over repeated samples.)) 22

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Definitions:

Meaningful Stimulus

An external event or situation that holds significance to an individual, influencing their psychological or emotional state.

Neutral Stimulus

A stimulus that initially produces no specific response except focusing attention, until it is paired with an unconditioned stimulus.

Dynamic-Systems Approach

A view of human development as an ongoing, ever-changing interaction between the physical, cognitive, and psychosocial influences.

Static Process

A process or condition that remains constant or unchanged over time, lacking in dynamic or evolutionary development.