(Requires Appendix Material) Consider the Following Population Regression Function Model $\widehat { Y } _ { i } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } X _ { 1 i } + \widehat { \beta } _ { 2 } X _ { 2 i }$

question 19

Essay

(Requires Appendix material) Consider the following population regression function model with two explanatory variables: $\widehat { Y } _ { i } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } X _ { 1 i } + \widehat { \beta } _ { 2 } X _ { 2 i }$
It is easy but tedious to
show that $\operatorname { SE } \left( \widehat { \beta _ { 2 } } \right)$ is given by the following formula: $\sigma _ { \widehat { \beta _ { 1 } } } ^ { 2 } = \frac { 1 } { n } \left[ \frac { 1 } { 1 - \rho _ { x _ { 1 } , x _ { 2 } } ^ { 2 } } \right] \frac { \sigma _ { u } ^ { 2 } } { \sigma _ { X _ { 1 } } ^ { 2 } }$ Sketch how
$\operatorname { SE } \left( \widehat { \beta _ { 2 } } \right)$ increases with the correlation between $X _ { 1 i } \text { and } X _ { 2 i } \text {. }$

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Definitions:

Activity Variance

The difference between the expected activity level and the actual activity level, often analyzed in budgeting and variance analysis.

Net Operating Income

A company's revenue minus its operating expenses, not including taxes and interest.

Service Company

A business that provides intangible products or services to customers, as opposed to selling physical goods.

Flexible Budget

An elastic budget that varies according to fluctuations in volume or operational activities.