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

question 25

Essay

In the multiple regression model with two explanatory variables $$Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$$ the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ):
$$\begin{array} { c } \hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 } \\\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } } \\\\\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }\end{array}$$ 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 $\left( X _ { 1 i } \right)$ and the saving rate $\left( X _ { 2 i } \right)$ (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: $$\begin{array} { l } \sum _ { i = 1 } ^ { n } Y _ { i } = 33.33 ; \sum _ { i = 1 } ^ { n } X _ { 1 i } = 2.025 ; \sum _ { i = 1 } ^ { n } X _ { 2 i } = 17.313 \\\\\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } = 8.3103 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } = .0122 ; \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } = 0.6422\end{array}$$ $$\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } = - 0.2304 ; \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } = 1.5676 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } = - 0.0520$$ (a)What are your expected signs for the regression coefficient? Calculate the coefficients
and see if their signs correspond to your intuition.

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Definitions:

Fixed Manufacturing Cost

Costs that do not change with the level of production, such as rent, salaries, and insurance.

Fixed Manufacturing Cost

Expenses that do not change with the level of production, such as rent, salaries, and equipment depreciation in a manufacturing setting.

Production Volume

The amount of products a factory or plant has manufactured in a given time frame; it measures the output level of production activities.

Factory Utility Cost

The expense associated with the utilities consumed in the production process, such as electricity, gas, and water.