In the Multiple Regression Model with Two Explanatory Variables $Y _ { i } = \mathcal { P } _ { 0 } + \mathcal { p } _ { 1 } X _ { 1 i } + \mathcal { \rho } _ { 2 } X _ { 2 i } + u _ { i }$

question 27

Essay

In the multiple regression model with two explanatory variables $Y _ { i } = \mathcal { P } _ { 0 } + \mathcal { p } _ { 1 } X _ { 1 i } + \mathcal { \rho } _ { 2 } X _ { 2 i } + u _ { i }$ the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in z_i = Z_i - $\bar { Z }$ ): $$\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 }$$ $$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$ $$\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 23 } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$ You have collected data for 104 countries of the world from the Penn World Tables and want to estimate the effect of the population growth rate ( $X _ { 1 i }$ )and the saving rate ( $X _ { 2 i }$ )(average investment share of GDP from 1980 to 1990)on GDP per worker (relative to the U.S.)in 1990. The various sums needed to calculate the OLS estimates are given below: $\sum _ { i = 1 } ^ { n } Y _ { i }$ = 33.33; $\sum _ { i = 1 } ^ { n } X _ { 1 i }$ = 2.025; $\sum _ { i = 1 } ^ { n } X _ { 2 i }$ = 17.313 $\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 }$ = 8.3103; $\sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 }$ = .0122; $\sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 }$ = 0.6422 $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i }$ = -0.2304; $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i }$ = 1.5676; $\sum _ { i = 1 } ^ { n } x _ { 1 \mathrm { i } } x _ { 2 i }$ = -0.0520
(a)What are your expected signs for the regression coefficient? Calculate the coefficients and see if their signs correspond to your intuition.
(b)Find the regression $R ^ { 2 }$ , and interpret it. What other factors can you think of that might have an influence on productivity?

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Definitions:

Pathos

One of the three modes of persuasion in rhetoric (alongside ethos and logos), appealing to the audience's emotions.

Emotional Intelligence

The ability to recognize, understand, and manage one's own emotions and the emotions of others in a productive and empathetic way.

Self-Management

Relates to the ability to control disruptive emotions and impulses.

Lincoln

The 16th President of the United States, Abraham Lincoln, known for leading the country during the Civil War and emancipating slaves with the Emancipation Proclamation.