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 46

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: $$\begin{array} { c } \sum _ { i = 1 } ^ { n } Y _ { i } = 33.33 ; \sum _ { i = 1 } ^ { n } X _ { 1 i } = 2.025 ; \sum _ { i = 1 } ^ { n } X _ { 2 i } = 17.313 \\\\\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } = 8.3103 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } = .0122 ; \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } = 0.6422 \\\\\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } = - 0.2304 ; \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } = 1.5676 ; \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } = - 0.0520\end{array}$$ The heteroskedasticity-robust standard errors of the two slope coefficients are 1.99 (for
population growth)and 0.23 (for the saving rate).Calculate the 95% confidence interval
for both coefficients.How many standard deviations are the coefficients away from zero?

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