(Requires Calculus)For the Simple Linear Regression Model of Chapter 4 $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

question 6

Essay

(Requires Calculus)For the simple linear regression model of Chapter 4, $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ , the OLS estimator for the intercept was $\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X }$ , and $\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } X _ { i } Y _ { i } - n \overline { X Y } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - n \bar { X } ^ { 2 } }$ Intuitively, the OLS estimators for the regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$ might be $\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 } , \hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } \bar { X } _ { 1i } Y _ { i } - n \bar { X } _ { 1 } \bar { Y } } { \sum _ { i = 1 } ^ { n } \bar { X } _ { 1 i } ^ { 2 } - n \bar { X } _ { 1 } ^ { 2 } }$ and $\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } \bar { X } _ { 2 i } Y _ { i } - n \bar { X } _ { 2 } \bar { Y } } { \sum _ { i = 1 } ^ { n } \bar { X } _ { 2 i } ^ { 2 } - n \bar { X } _ { 2 } ^ { 2 } }$ By minimizing the prediction mistakes of the regression model with two explanatory variables, show that this cannot be the case.

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Definitions:

Economic Profit

The variance between gross receipts and all outlays, including both tangible and intangible costs.

Accounting Profit

The total revenue of a business minus the explicit costs of operating, not including opportunity costs.

Implicit Costs

The opportunity costs that are not directly paid out but represent the loss of potential income from using resources in a certain way.

Opportunity Costs

Missing out on possible advantages from other choices by opting for a particular alternative.