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 basic interface elements of Windows and common applications.

Identify and use basic computer commands to manage files, apps, and system settings.

Recognize and utilize the fundamental features of word processing applications.

Comprehend the role and functionality of an operating system.

---

Definitions:

Statistic

A numerical value derived from a sample that is used to estimate properties of the population from which it was drawn.

Marketing Majors

Students who specialize in the study of marketing strategies, consumer behavior, and market research as part of their academic education.

Parameter

A measurable attribute that defines or limits the characteristics of a population or a statistical model.

Population

The entire group of individuals or instances about whom we seek to make conclusions.