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?

Understand the importance of combining emotional appeals with other types of evidence for effective persuasion.

Recognize ethical guidelines in the manipulation of photographs and videos within arguments.

Identify and avoid logical fallacies such as post hoc reasoning in argument construction.

Distinguish between different types of arguments including restatements, popularity claims, hasty generalizations, false dilemmas, and ad hominem attacks.

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Definitions:

Neuroscience

The scientific study of the nervous system, including its structure, functions, and abnormalities.

Behavioral Economists

Behavioral economists study how psychological, emotional, and cultural factors affect economic decisions of individuals and institutions.

Payroll Deduction

A system where an employer withholds part of an employee's wage or salary for various purposes, such as taxes, benefits, or retirement savings.

Automatic Withdrawals

Electronic funds transfers or payments automatically deducted from a bank account, typically used for recurring expenses.